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There is a pleasant symmetry in Wagner's work between heaps, semiheaps, and generalised heaps on the one hand, and groups, semigroups, and generalised groups on the other. Essentially, the various types of semiheaps appear whenever we consider binary relations (and partial one-one mappings) between
16491:
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11605:
In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set
16111:
11935:, as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.) With this definition one can for instance define a binary relation over every set and its power set.
4967:
8893:
9494:, which exists when spatial events are "normal" to a time characterized by a velocity. He used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in a
16932:
16844:
18117:
1389:
6940:"; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.
1318:
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11577:
Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of
8807:
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13993:
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1633:
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2800:, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as
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6227:
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1474:
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Ali Jaoua, Rehab
Duwairi, Samir Elloumi, and Sadok Ben Yahia (2009) "Data mining, reasoning and incremental information retrieval through non enlargeable rectangular relation coverage", pages 199 to 210 in
15550:
The corresponding logical matrix of a general binary relation has rows which finish with a sequence of ones. Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix.
16384:
9567:
9150:{\displaystyle R={\begin{pmatrix}0&0&1&0&1&1&1\\1&0&0&1&1&0&0\\1&1&1&1&0&0&1\\1&1&0&0&1&1&1\end{pmatrix}}.}
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indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by
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Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "... a variant of the theory has evolved that treats relations from the very beginning as
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The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".
15325:
15025:
11897:: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple
11377:: a function that is injective. For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function.
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However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "
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or a logical matrix of all ones). This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand,
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13051:. A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
13912:
6376:
15890:
14042:. The necessity of matching target to source of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of
1583:
1197:
16939:
11263:. For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to
7058:
However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation
17082:
16217:
8449:
That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing; see the 1st example. As a set,
21026:
11886:
10753:: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.
10747:: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
10741:: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
1480:
14936:
6169:
16534:
15557:
1436:
15765:
14553:
which counts the number of concepts necessary to cover a relation. Structural analysis of relations with concepts provides an approach for
14420:
or left-total, functional relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order
1395:
19614:
11085:
is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given
10735:: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
15832:
20490:
19698:"Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic"
6641:
66:
18993:
16486:{\displaystyle R^{\textsf {T}}{\bar {R}}\subseteq {\bar {I}}\implies I\subseteq {\overline {R^{\textsf {T}}{\bar {R}}}}=R\backslash R}
12904:. A relation is asymmetric if and only if it is both antisymmetric and irreflexive. For example, > is an asymmetric relation, but
11368:
one image element. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
20473:
14829:
The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of an
9503:
18184:
15040:
9400:
are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.
20003:
16705:
4580:
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4289:
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17:
19771:
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19286:
19253:
19222:
19152:
18933:
18908:
18693:
18523:
G. Schmidt, Claudia
Haltensperger, and Michael Winter (1997) "Heterogeneous relation algebra", chapter 3 (pages 37 to 53) in
18508:
18478:
18448:
18390:
18287:
17810:, heaps, and generalized heaps. The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:
8492:
10254:. For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both
8304:
20715:
20535:
20320:
19246:
17535:
13995:
the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to
10952:. For example, the red and green binary relations in the diagram are total, but the blue one is not (as it does not relate
3926:
are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a
Cartesian product
19168:
East, James; Vernitski, Alexei (February 2018). "Ranks of ideals in inverse semigroups of difunctional binary relations".
17930:
11391:
8270:
11386:: a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not.
10603:. For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates
21043:
19608:
5732:
5398:
15410:
9856:
2908:
20456:
20315:
19793:
19683:
19538:
19270:
19095:
19078:
19028:
18958:
18883:
18841:
18535:
18418:
18362:
9704:
9454:: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of
8549:
19573:
18234:
15089:
has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as a
2770:
20310:
11403:
one preimage element. For example, the green binary relation in the diagram is a bijection, but the red one is not.
17275:
15612:
13996:
21021:
19946:
16663:
16106:{\displaystyle C^{\textsf {T}}{\bar {C}}\subseteq \ni {\bar {C}}\equiv C{\overline {\ni {\bar {C}}}}\subseteq C,}
14739:
14625:
14142:
2545:
may be defined as a binary relation that meets additional constraints. Binary relations are also heavily used in
20901:
18596:
17480:
17393:
17333:
20028:
18321:
16282:
14293:
the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion
13422:
9601:
59:
20795:
20674:
20347:
20267:
19810:
19506:
17313:
16211:
6978:
2492:, are used in many branches of mathematics to model a wide variety of concepts. These include, among others:
17890:
Christopher
Hollings, "Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups"
4962:{\displaystyle S\circ R=\{(x,z)\mid {\text{ there exists }}y\in Y{\text{ such that }}xRy{\text{ and }}ySz\}}
21038:
20132:
20061:
19941:
19351:
Sitzungsberichte der mathematisch-physikalischen Klasse der
Bayerischen Akademie der Wissenschaften MΓΌnchen
17445:
15516:
14328:) is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is
14321:
14261:
9698:
used in these geometrical contexts corresponds to the logical matrix used generally with binary relations.
9377:
11602:, take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory.
21031:
20669:
20632:
20035:
20023:
19986:
19961:
19936:
19890:
19859:
19805:
18830:
17924:
17651:
14836:
14334:
12285:
11890:
8888:{\displaystyle B=\{{\text{NA}},{\text{SA}},{\text{AF}},{\text{EU}},{\text{AS}},{\text{AU}},{\text{AA}}\}}
7947:
6944:
1167:
17693:
14890:
9645:
2859:
20332:
19966:
19956:
19832:
19763:
19399:
18683:
18044:
17913:
15146:
12307:
12198:
20686:
18547:
16927:{\displaystyle \equiv (R\backslash R)(R\backslash R)\subseteq {\overline {R^{\textsf {T}}{\bar {R}}}}}
16839:{\displaystyle \equiv R^{\textsf {T}}{\bar {R}}\subseteq {\overline {(R\backslash R)(R\backslash R)}}}
15283:
14998:
10943:
9298:
9194:
9162:
1551:
21176:
20720:
20612:
20600:
20595:
20305:
19971:
19800:
17902:
17752:
16596:
16323:
14596:
12338:
11745:
8267:
The following example shows that the choice of codomain is important. Suppose there are four objects
6850:
6284:
6140:
5990:
5950:
2746:
52:
19237:
Ali Jaoua, Nadin
Belkhiter, Habib Ounalli, and Theodore Moukam (1997) "Databases", pages 197β210 in
16628:
15456:
11503:
7739:
7666:
20528:
20237:
19864:
16496:
16187:
14926:
14035:
13827:
13817:
13547:
13496:
12297:
12289:
12254:
11564:
is set-like, and every relation on two sets is set-like. The usual ordering < over the class of
11364:): a binary relation that is functional and total. In other words, every element of the domain has
9573:
9451:
7212:
7036:
6834:
5152:
4749:
2762:
2497:
1126:
950:
119:
14296:
9917:
9834:
9772:
7010:
2115:
2090:
1384:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
21140:
21058:
20933:
20885:
20699:
20622:
20485:
20468:
19697:
19454:
19420:
19044:
15504:
In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in
12935:
10384:
9584:
9459:
6947:(not to be confused with being "total") do not carry over to restrictions. For example, over the
2750:
19591:
19140:
18468:
16352:
15971:
14084:
14006:
11996:
11676:
5970:
5930:
3428:
3167:
2656:
2632:
2599:
21092:
20973:
20785:
20605:
20397:
20013:
19693:
17006:
15994:
15120:
14797:
14683:
14413:
14063:
13906:
13277:
13183:
13062:
12820:
12676:
12669:
12581:
11395:: a function that is injective and surjective. In other words, every element of the domain has
11356:
10425:
9330:
9272:
9226:
6818:
6030:
5664:
5330:
5125:
3929:
3398:
3004:
2626:
2542:
2501:
2199:
1313:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
1096:
89:
31:
18978:
18875:
18869:
11824:
10063:
9982:: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black).
3911:, i.e. as relations where the normal case is that they are relations between different sets."
21008:
20978:
20922:
20842:
20822:
20800:
20375:
20210:
20201:
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19951:
19905:
19869:
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18464:
17918:
17034:
15097:
with rectangular blocks of ones on the (asymmetric) main diagonal. More formally, a relation
14443:
14059:
13809:
13097:
12478:
12396:
11900:
10095:
9393:
7863:
7307:
4038:
3340:
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2512:
1651:
indicates that the column's property is always true the row's term (at the very left), while
1106:
99:
19001:. Prague: School of Mathematics β Physics Charles University. p. 1. Archived from
16018:
In terms of the calculus of relations, sufficient conditions for a contact relation include
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13440:
13341:
11451:
11421:
11170:
11144:
11042:
10839:
10813:
10399:
9587:
can be considered a relation between its points and its lines. The relation is expressed as
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1701:
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20147:
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19709:
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19461:
19375:
19368:
19102:
19002:
15233:
15186:
14884:
14830:
14543:
13752:
13446:
12907:
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12553:
12471:
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10948:
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6954:
6854:
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2489:
2048:
1888:
1672:
883:
158:
19346:
19118:
18851:
17243:{\displaystyle \operatorname {fringe} (R)=R\cap {\overline {R{\bar {R}}^{\textsf {T}}R}}.}
17039:
16136:
16116:
13312:
12763:
12533:
12086:
11855:
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11547:
11068:
10998:
7699:
6234:
6094:
6054:
6010:
5594:
5574:
3877:
3775:
2712:. A binary relation is also called a heterogeneous relation when it is not necessary that
1788:
8:
21077:
20988:
20896:
20891:
20705:
20647:
20585:
20521:
20390:
20301:
20247:
20206:
20196:
20085:
20018:
19981:
18608:
18260:
18018:
17942:
17862:
16528:
14733:
13882:
13856:
13745:
13393:
13367:
13270:
13243:
13217:
13149:
13123:
13028:
13002:
12976:
12928:
12881:
12855:
12813:
12737:
12711:
12642:
12616:
12507:
12424:
12172:
11477:
11382:
11196:
11108:
10865:
10540:
10512:
10486:
10460:
10204:
10176:
10150:
10124:
9687:
9495:
8902:
7961:
7512:
7362:
7336:
6826:
6822:
4972:
4090:
4064:
3702:
3600:
3526:
2810:
2797:
2715:
2689:
1815:
A term's definition may require additional properties that are not listed in this table.
1762:
1736:
1695:
1524:
1141:
1131:
542:
124:
41:
19713:
18153:
11266:
10955:
10689:
10646:
10320:
10257:
8802:{\displaystyle A=\{{\text{Indian}},{\text{Arctic}},{\text{Atlantic}},{\text{Pacific}}\}}
8442:{\displaystyle R=\{({\text{ball, John}}),({\text{doll, Mary}}),({\text{car, Venus}})\}.}
5175:(is mother of) yields (is maternal grandparent of), while the composition (is mother of)
2468:
2308:
2245:
21000:
20995:
20780:
20735:
20642:
20502:
20429:
20282:
20191:
20181:
20122:
20040:
19976:
19735:
19195:
19177:
19010:
Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
18975:
Foundations of Logic and
Mathematics: Applications to Computer Science and Cryptography
18351:
18176:
17998:
17978:
17948:
17842:
17822:
17785:
17631:
17611:
17603:
17573:
17512:
17425:
17365:
17255:
17149:
17059:
17016:
16520:
16164:
15738:
15718:
15694:
15670:
15330:
15260:
15213:
15100:
14777:
14715:
14663:
14572:
14525:
14505:
14481:
14423:
14395:
14375:
14233:
14213:
14118:
13778:
13758:
13724:
13704:
13695:
13674:
13654:
13623:
13592:
13572:
13473:
13453:
13176:
13055:
12574:
12389:
12367:
12347:
12231:
12152:
12132:
12112:
12092:
12068:
12048:
12025:
11976:
11956:
11775:
11725:
11656:
11609:
11585:
11445:
11373:
11333:
11313:
11289:
11246:
11226:
11088:
11022:
10978:
10934:
10915:
10895:
10780:
10760:
10712:
10669:
10626:
10606:
10586:
10363:
10343:
10300:
10280:
10237:
10210:
10029:
10009:
9989:
9942:
9893:
9814:
9794:
9752:
9669:
9588:
9465:
9429:
9409:
9381:
9252:
8948:
8928:
8580:
8529:
8472:
8452:
8361:
8341:
8156:
8135:
8034:
8013:
7989:
7929:
7909:
7889:
7869:
7837:
7817:
7793:
7773:
7646:
7618:
7598:
7578:
7558:
7538:
7488:
7468:
7448:
7428:
7408:
7388:
7261:
7241:
7192:
7172:
7152:
7124:
7104:
7084:
6923:
6903:
6883:
6863:
6814:
6810:
6789:
6769:
6749:
6729:
6621:
6601:
6581:
6561:
6541:
6518:
6498:
6478:
6356:
6336:
6316:
6292:
5890:
5867:
5847:
5827:
5712:
5692:
5672:
5652:
5534:
5511:
5491:
5471:
5378:
5358:
5338:
5298:
5278:
5258:
5238:
5218:
5198:
5105:
5085:
5062:
5042:
5022:
5002:
4857:
4837:
4817:
4797:
4777:
4757:
4720:
4700:
4680:
4660:
4560:
4540:
4520:
4500:
4429:
4409:
4389:
4369:
4269:
4249:
4229:
4209:
4176:
4156:
4136:
4116:
4015:
3995:
3975:
3955:
3834:
3814:
3752:
3728:
3682:
3662:
3626:
3580:
3560:
3506:
3486:
3466:
3378:
3312:
3292:
3272:
3244:
3224:
3196:
3147:
3127:
3107:
2984:
2964:
2837:
2742:
2575:
2555:
2448:
2428:
2408:
2388:
2368:
2348:
2328:
2288:
2268:
2225:
2205:
2181:
2157:
2137:
2024:
2000:
1980:
1960:
1940:
1920:
1865:
1845:
1677:
1121:
1101:
1091:
1017:
114:
94:
84:
20342:
19577:
18806:
18238:
20857:
20694:
20657:
20627:
20558:
20439:
20417:
20277:
20262:
20242:
20045:
19767:
19679:
19660:
19656:
19637:
19604:
19560:
19534:
19509:
19480:
19402:
19325:
19282:
19249:
19218:
19148:
19074:
19024:
18954:
18929:
18904:
18879:
18837:
18739:
18689:
18671:
18531:
18504:
18474:
18444:
18414:
18386:
18358:
18327:
18317:
18283:
18180:
17907:
17779:
17688:
15540:
13988:{\displaystyle P\subseteq Q\equiv (P\cap {\bar {Q}}=\varnothing )\equiv (P\cap Q=P),}
13821:
13645:
12293:
11885:
Another solution to this problem is to use a set theory with proper classes, such as
9974:
9487:
6846:
6460:{\displaystyle R_{\vert S}=\{(x,y)\mid xRy{\text{ and }}x\in S{\text{ and }}y\in S\}}
6271:
6135:
5324:
2903:
2834:
2758:
1831:
19503:
Mathematics across the Iron
Curtain: a history of the algebraic theory of semigroups
19199:
19145:
Finite
Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions
18763:
15143:
is difunctional if and only if it can be written as the union of
Cartesian products
13620:
also apply to homogeneous relations. Beyond that, a homogeneous relation over a set
21145:
21135:
21120:
21115:
20983:
20637:
20252:
20105:
19730:
19725:
19717:
19625:
19581:
19274:
19187:
19048:
18847:
18636:
18600:
18575:
18559:
18378:
18242:
18168:
17330:
operator selects a boundary sub-relation described in terms of its logical matrix:
14325:
14047:
14039:
12301:
11873:
11849:
10545:
9695:
7855:
2786:
2546:
816:
749:
18604:
16011:
is a contact relation. The notion of a general contact relation was introduced by
15954:{\displaystyle {\text{for all }}y\in Y,ygZ{\text{ and }}xgY{\text{ implies }}xgZ.}
2741:
Since relations are sets, they can be manipulated using set operations, including
21014:
20952:
20770:
20590:
20434:
20217:
20095:
20090:
20075:
19991:
19900:
19757:
19753:
19552:
19548:
19487:
19458:
19391:
19319:
19315:
19242:
19099:
18438:
18434:
18275:
18085:
17272:
is a partial identity relation, difunctional, or a block diagonal relation, then
15544:
15508:
management." Furthermore, difunctional relations are fundamental in the study of
15082:
14590:
14075:
14043:
13813:
13805:
13610:
13491:
11019:. But it is not a total relation over the positive integers, because there is no
9389:
7993:
7985:
7859:
7693:
6830:
2778:
2754:
2650:
677:
474:
19278:
18640:
3992:, and reserve the term "correspondence" for a binary relation with reference to
21150:
20947:
20928:
20832:
20817:
20774:
20710:
20652:
20352:
20337:
20327:
20186:
20164:
20142:
19633:
19600:
19477:
19438:
19302:
19114:
18528:
18346:
15965:
15536:
15090:
14988:
14205:
14112:
11582:. For example, to model the general concept of "equality" as a binary relation
11565:
10802:
9596:
9373:
8966:
8606:
7955:
7811:
2781:. A deeper analysis of relations involves decomposing them into subsets called
2774:
2534:
2042:
1628:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
345:
19191:
18208:
1242:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
21170:
21155:
20957:
20871:
20866:
20451:
20407:
20385:
20257:
20127:
20115:
19920:
19653:
Monoids, Acts and Categories: with Applications to Wreath Products and Graphs
19587:
19450:
19071:
19040:
18865:
18675:
18653:
18331:
18149:
17936:
17803:
16994:{\displaystyle \equiv (R\backslash R)(R\backslash R)\subseteq R\backslash R.}
14136:
9592:
9357:
to be universal because at least two oceans must be traversed to voyage from
6838:
1116:
1111:
283:
109:
104:
21125:
17131:{\displaystyle \Omega ={\overline {\ni {\bar {\in }}}}=\in \backslash \in .}
16272:{\displaystyle R\backslash R\equiv {\overline {R^{\textsf {T}}{\bar {R}}}}.}
13444:
is a relation that is irreflexive, asymmetric, transitive and connected. An
13438:
is a relation that is reflexive, antisymmetric, transitive and connected. A
21105:
21100:
20918:
20847:
20805:
20664:
20568:
20272:
20154:
20137:
20055:
19895:
19848:
19265:
Gumm, H. P.; Zarrad, M. (2014). "Coalgebraic Simulations and Congruences".
18807:"Generalization of rough sets using relationships between attribute values"
18592:
17954:
17387:
17383:
16012:
15532:
15528:
15509:
15094:
11894:
11626:, that contains all the objects of interest, and work with the restriction
9691:
9369:
2523:
2175:
2085:
1883:
18172:
17927:, discusses several unusual but fundamental properties of binary relations
13450:
is a relation that is reflexive, symmetric, and transitive. For example, "
21130:
20765:
20478:
20171:
20050:
19915:
19063:
19043:& M. Winter (2013) "Decomposition of relations on concept lattices",
18473:(2nd ed.). Springer Science & Business Media. pp. 299β300.
18463:
18406:
18256:
17591:
14554:
13434:
10230:
9979:
9388:
has edges possibly with more than two nodes, and can be illustrated by a
7951:
7004:
6948:
6842:
6048:
3769:
is the union of its domain of definition and its codomain of definition.
1823:
407:
19739:
19052:
16378:
will be of opposite logical values, so the diagonal is all zeros. Then
1513:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
21110:
20881:
20544:
20446:
20380:
20221:
18631:
Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series.
15093:, the columns and rows of a difunctional relation can be arranged as a
14981:{\displaystyle F\subseteq A\times Z{\text{ and }}G\subseteq B\times Z.}
9483:
9402:
9385:
6222:{\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}
2530:
2505:
605:
19212:
18814:
Proceedings of the 2nd Annual Joint Conference on Information Sciences
16586:{\displaystyle (R\backslash R)(R\backslash R)\subseteq R\backslash R.}
15407:
have a non-empty intersection, then these two sets coincide; formally
2077:
belongs to the set of ordered pairs that defines the binary relation.
1469:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
20913:
20876:
20827:
20725:
20497:
20370:
20176:
19721:
17921:, a category having sets as objects and binary relations as morphisms
17010:
15821:{\displaystyle {\text{for all }}x\in A,Y=\{x\}{\text{ implies }}xgY.}
15688:
14000:
12249:
11772:
Also, the "member of" relation needs to be restricted to have domain
9362:
8896:
6972:
6267:
The complement of a reflexive relation is irreflexiveβand vice versa.
2593:
1425:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
11223:
one preimage element. In other words, the codomain of definition of
9572:
As a relation between some temporal events and some spatial events,
20292:
20159:
19910:
19182:
17807:
16182:
15879:{\displaystyle Y\subseteq Z{\text{ and }}xgY{\text{ implies }}xgZ.}
15505:
14314:
14051:
9397:
7958:
corresponds to the universal relation. Homogeneous relations (when
7030:
5195:(is parent of) yields (is grandmother of). For the former case, if
2516:
222:
19557:
Relations and Graphs: Discrete Mathematics for Computer Scientists
19321:
Relations and Graphs: Discrete Mathematics for Computer Scientists
18440:
Relations and Graphs: Discrete Mathematics for Computer Scientists
15327:, a difunctional relation can also be characterized as a relation
13426:
is a relation that is reflexive, antisymmetric, and transitive. A
5651:. A binary relation is equal to its converse if and only if it is
19453:(1953) "The theory of generalised heaps and generalised groups",
18926:
Mathematical Foundations of Computational Engineering: A Handbook
17975:
Authors who deal with binary relations only as a special case of
17859:, while the various types of semigroups appear in the case where
15554:
An algebraic statement required for a Ferrers type relation R is
15034:
In 1950 Riguet showed that such relations satisfy the inclusion:
13432:
is a relation that is irreflexive, asymmetric, and transitive. A
11310:
Uniqueness and totality properties (only definable if the domain
11016:
9591:. Finite and infinite projective and affine planes are included.
6711:{\displaystyle R_{\vert S}=\{(x,y)\mid xRy{\text{ and }}x\in S\}}
4035:
In a binary relation, the order of the elements is important; if
3870:, suggesting that it does not have the square-like symmetry of a
2110:
2081:
12665:. For example, "is a blood relative of" is a symmetric relation.
3851:
are allowed to be different, a binary relation is also called a
20938:
20760:
19817:
19788:
15712:
11286:), nor the black one (as it does not relate any real number to
10892:
one image element. In other words, the domain of definition of
9426:
axes represent time for observers in motion, the corresponding
9358:
18501:
Goguen Categories: A Categorical Approach to L-fuzzy Relations
14062:
are sets, and the relation-morphisms compose as required in a
10975:
to any real number), nor the black one (as it does not relate
20810:
20577:
20513:
11893:, and allow the domain and codomain (and so the graph) to be
9562:{\displaystyle \langle x,z\rangle =x{\bar {z}}+{\bar {x}}z\;}
8810:
7858:(addition corresponds to OR and multiplication to AND) where
13609:" is an equivalence relation on the set of all lines in the
7866:
corresponds to composition of relations (of a relation over
5082:
The identity element is the identity relation. The order of
30:
This article covers advanced notions. For basic topics, see
18991:
18949:
Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),
18923:
18443:. Springer Science & Business Media. Definition 4.1.1.
18111:"MIT 6.042J Math for Computer Science, Lecture 3T, Slide 2"
14230:
is maximal, not contained in any other outer product. Thus
14074:
Binary relations have been described through their induced
9159:
The connectivity of the planet Earth can be viewed through
18992:
FlaΕ‘ka, V.; JeΕΎek, J.; Kepka, T.; Kortelainen, J. (2007).
17945:, a heterogeneous relation between set of points and lines
16751:{\displaystyle R(R\backslash R)(R\backslash R)\subseteq R}
14542:
identity corresponds to difunctional, a generalization of
4646:{\displaystyle R\cap S=\{(x,y)\mid xRy{\text{ and }}xSy\}}
19147:. Springer Science & Business Media. pp. 35β37.
11993:
and itself, i.e. it is a subset of the Cartesian product
4449:
The identity element is the empty relation. For example,
4355:{\displaystyle R\cup S=\{(x,y)\mid xRy{\text{ or }}xSy\}}
3087:{\displaystyle \{(x,y)\mid x\in X{\text{ and }}y\in Y\},}
19651:
Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander (2000).
18154:"A Relational Model of Data for Large Shared Data Banks"
11872:
to be defined over all sets leads to a contradiction in
11399:
one image element and every element of the codomain has
5151:
used here agrees with the standard notational order for
1997:. It encodes the common concept of relation: an element
19593:
The Relational Model for Database Management: Version 2
8519:{\displaystyle A\times \{{\text{John, Mary, Venus}}\},}
19217:. Springer Science & Business Media. p. 200.
18928:. Springer Science & Business Media. p. 496.
17119:
16982:
16967:
16952:
16879:
16864:
16821:
16806:
16733:
16718:
16676:
16609:
16574:
16559:
16544:
16503:
16477:
16224:
16194:
15600:{\displaystyle R{\bar {R}}^{\textsf {T}}R\subseteq R.}
14833:. One way this can be done is with an intervening set
12344:
Some important properties that a homogeneous relation
11568:
is a set-like relation, while its inverse > is not.
8989:
8331:{\displaystyle B=\{{\text{John, Mary, Ian, Venus}}\}.}
6264:
If a relation is symmetric, then so is the complement.
19423:
19324:. Springer Science & Business Media. p. 77.
18316:. Hochschultext (Springer-Verlag). London: Springer.
18047:
18021:
18001:
17981:
17865:
17845:
17825:
17788:
17755:
17696:
17654:
17634:
17614:
17576:
17563:{\displaystyle \operatorname {fringe} (R)=\emptyset }
17538:
17515:
17483:
17448:
17428:
17396:
17368:
17336:
17316:
17278:
17258:
17176:
17152:
17085:
17062:
17042:
17019:
16942:
16854:
16766:
16708:
16666:
16631:
16599:
16537:
16499:
16387:
16355:
16326:
16285:
16220:
16190:
16167:
16139:
16119:
16024:
15997:
15974:
15893:
15835:
15768:
15741:
15721:
15697:
15673:
15615:
15560:
15547:, to extend ordering to binary relations in general.
15459:
15413:
15383:
15353:
15333:
15286:
15263:
15236:
15216:
15189:
15149:
15123:
15103:
15043:
15001:
14939:
14893:
14839:
14800:
14780:
14742:
14718:
14686:
14666:
14628:
14599:
14575:
14528:
14508:
14484:
14446:
14426:
14398:
14378:
14337:
14299:
14264:
14236:
14216:
14145:
14121:
14087:
14009:
13915:
13885:
13859:
13830:
13781:
13761:
13727:
13707:
13677:
13657:
13626:
13595:
13575:
13550:
13524:
13499:
13476:
13456:
13396:
13370:
13344:
13315:
13280:
13246:
13220:
13186:
13152:
13126:
13100:
13065:
13031:
13005:
12979:
12938:
12910:
12884:
12858:
12823:
12795:
12766:
12740:
12714:
12679:
12645:
12619:
12584:
12556:
12536:
12510:
12481:
12453:
12427:
12399:
12370:
12350:
12310:
12257:
12234:
12201:
12175:
12155:
12135:
12115:
12095:
12071:
12051:
12028:
11999:
11979:
11959:
11903:
11858:
11827:
11798:
11778:
11748:
11728:
11699:
11679:
11659:
11632:
11612:
11588:
11550:
11506:
11480:
11454:
11424:
11336:
11316:
11292:
11269:
11249:
11229:
11199:
11173:
11147:
11111:
11091:
11071:
11045:
11025:
11001:
10981:
10958:
10932:. This property, is different from the definition of
10918:
10898:
10868:
10842:
10816:
10783:
10763:
10715:
10692:
10672:
10649:
10629:
10609:
10589:
10559:
10515:
10489:
10463:
10428:
10402:
10366:
10346:
10323:
10303:
10283:
10260:
10240:
10213:
10179:
10153:
10127:
10098:
10066:
10032:
10012:
9992:
9945:
9920:
9896:
9859:
9837:
9817:
9797:
9775:
9755:
9707:
9672:
9648:
9604:
9595:
pioneered the cataloguing of configurations with the
9506:
9468:
9432:
9412:
9333:
9301:
9275:
9255:
9229:
9197:
9165:
8977:
8951:
8931:
8905:
8819:
8757:
8583:
8552:
8532:
8495:
8475:
8455:
8384:
8364:
8344:
8307:
8273:
8159:
8138:
8037:
8016:
7964:
7932:
7912:
7892:
7872:
7840:
7820:
7796:
7776:
7742:
7722:
7702:
7669:
7649:
7621:
7601:
7581:
7561:
7541:
7515:
7491:
7471:
7451:
7431:
7411:
7391:
7365:
7339:
7310:
7284:
7264:
7244:
7215:
7195:
7175:
7155:
7127:
7107:
7087:
7064:
7039:
7013:
6981:
6957:
6926:
6906:
6886:
6866:
6792:
6772:
6752:
6732:
6644:
6624:
6604:
6584:
6564:
6544:
6521:
6501:
6481:
6379:
6359:
6339:
6319:
6295:
6237:
6172:
6143:
6117:
6097:
6077:
6057:
6033:
6013:
5993:
5973:
5953:
5933:
5913:
5893:
5870:
5850:
5830:
5803:
5735:
5715:
5695:
5675:
5637:
5617:
5597:
5577:
5557:
5537:
5514:
5494:
5474:
5401:
5381:
5361:
5341:
5301:
5281:
5261:
5241:
5221:
5201:
5181:
5161:
5128:
5108:
5088:
5065:
5045:
5025:
5005:
4975:
4880:
4860:
4840:
4820:
4800:
4780:
4760:
4723:
4703:
4683:
4663:
4583:
4563:
4543:
4523:
4503:
4475:
4455:
4432:
4412:
4392:
4372:
4292:
4272:
4252:
4232:
4212:
4179:
4159:
4139:
4119:
4093:
4067:
4041:
4018:
3998:
3978:
3958:
3932:
3880:
3837:
3817:
3778:
3755:
3731:
3705:
3685:
3665:
3629:
3603:
3583:
3563:
3529:
3509:
3489:
3469:
3431:
3401:
3381:
3343:
3315:
3295:
3275:
3247:
3227:
3199:
3170:
3150:
3130:
3110:
3033:
3007:
2987:
2967:
2911:
2862:
2840:
2813:
2718:
2692:
2659:
2635:
2602:
2578:
2558:
2471:
2451:
2431:
2411:
2391:
2371:
2351:
2331:
2311:
2291:
2271:
2248:
2228:
2208:
2184:
2160:
2140:
2118:
2093:
2051:
2027:
2003:
1983:
1963:
1943:
1923:
1891:
1868:
1848:
1791:
1765:
1739:
1704:
1680:
1586:
1554:
1527:
1483:
1439:
1398:
1327:
1256:
1200:
1170:
1144:
19702:
Memoirs of the American Academy of Arts and Sciences
19139:
18917:
17806:
used properties of this ternary operation to define
14324:(1937) (that any partial order may be embedded in a
11219:. In other words, every element of the codomain has
10199:. In other words, every element of the codomain has
9978:
Examples of four types of binary relations over the
9372:: For relations on a set (homogeneous relations), a
8597:
contains no information about the ownership by Ian.
19213:Chris Brink; Wolfram Kahl; Gunther Schmidt (1997).
19119:"Quelques proprietes des relations difonctionelles"
18948:
18740:"Functional relation - Encyclopedia of Mathematics"
18457:
11693:needs to be restricted to have domain and codomain
8294:{\displaystyle A=\{{\text{ball, car, doll, cup}}\}}
2807:A binary relation is the most studied special case
2222:. In this relation, for instance, the prime number
19650:
19429:
18832:Set theory: an introduction to independence proofs
18829:
18707:
18705:
18703:
18701:
18669:
18350:
18076:
18033:
18007:
17987:
17933:, a binary relation defined by algebraic equations
17877:
17851:
17831:
17794:
17770:
17741:
17679:
17640:
17620:
17582:
17562:
17521:
17501:
17469:
17434:
17414:
17374:
17354:
17322:
17302:
17264:
17242:
17158:
17130:
17068:
17048:
17025:
16993:
16926:
16838:
16750:
16693:
16649:
16617:
16585:
16511:
16485:
16370:
16341:
16312:
16271:
16202:
16173:
16145:
16125:
16105:
16003:
15991:"is an element of", satisfies these properties so
15983:
15953:
15878:
15820:
15747:
15727:
15703:
15679:
15651:
15599:
15494:
15445:
15399:
15369:
15339:
15319:
15269:
15249:
15222:
15202:
15175:
15135:
15109:
15070:
15019:
14980:
14917:
14875:
14812:
14786:
14766:
14724:
14698:
14672:
14652:
14614:
14581:
14534:
14514:
14490:
14470:
14432:
14404:
14384:
14364:
14305:
14285:
14242:
14222:
14197:
14127:
14099:
14024:
13987:
13897:
13871:
13845:
13787:
13767:
13733:
13713:
13683:
13663:
13632:
13601:
13581:
13561:
13536:
13510:
13482:
13462:
13408:
13382:
13356:
13330:
13301:
13258:
13232:
13207:
13164:
13138:
13112:
13086:
13043:
13017:
12991:
12965:
12916:
12896:
12870:
12844:
12801:
12781:
12752:
12726:
12700:
12657:
12631:
12605:
12562:
12542:
12522:
12496:
12459:
12439:
12414:
12376:
12356:
12329:
12276:
12240:
12220:
12187:
12161:
12141:
12121:
12101:
12077:
12057:
12034:
12022:It is also simply called a (binary) relation over
12014:
11985:
11965:
11927:
11864:
11840:
11813:
11784:
11764:
11734:
11714:
11685:
11665:
11645:
11618:
11594:
11556:
11536:
11492:
11466:
11436:
11342:
11322:
11298:
11278:
11255:
11235:
11211:
11185:
11159:
11123:
11097:
11077:
11057:
11031:
11007:
10987:
10967:
10924:
10904:
10888:. In other words, every element of the domain has
10880:
10854:
10828:
10789:
10769:
10757:Totality properties (only definable if the domain
10721:
10701:
10678:
10658:
10635:
10615:
10595:
10571:
10535:. In other words, every element of the domain has
10527:
10501:
10475:
10449:
10414:
10372:
10352:
10332:
10309:
10289:
10269:
10246:
10219:
10191:
10165:
10139:
10113:
10084:
10038:
10018:
9998:
9951:
9928:
9902:
9882:
9845:
9823:
9803:
9783:
9761:
9741:
9678:
9654:
9634:
9561:
9474:
9438:
9418:
9345:
9319:
9287:
9261:
9241:
9215:
9183:
9149:
8957:
8937:
8917:
8887:
8801:
8599:While the 2nd example relation is surjective (see
8589:
8569:
8538:
8518:
8481:
8461:
8441:
8370:
8350:
8330:
8293:
8165:
8144:
8043:
8022:
7976:
7938:
7918:
7898:
7878:
7846:
7826:
7802:
7782:
7754:
7728:
7708:
7684:
7655:
7627:
7607:
7587:
7567:
7547:
7527:
7497:
7477:
7457:
7437:
7417:
7397:
7377:
7351:
7325:
7296:
7270:
7250:
7230:
7201:
7181:
7161:
7133:
7113:
7093:
7070:
7050:
7021:
6995:
6963:
6932:
6912:
6892:
6872:
6798:
6778:
6758:
6738:
6710:
6630:
6610:
6590:
6570:
6550:
6527:
6507:
6487:
6459:
6365:
6345:
6325:
6301:
6252:
6221:
6158:
6123:
6103:
6083:
6063:
6039:
6019:
5999:
5979:
5959:
5939:
5919:
5899:
5876:
5856:
5836:
5812:
5789:
5721:
5701:
5681:
5643:
5623:
5603:
5583:
5563:
5543:
5520:
5500:
5480:
5452:
5387:
5367:
5347:
5307:
5287:
5267:
5247:
5227:
5207:
5187:
5167:
5143:
5114:
5094:
5071:
5051:
5031:
5011:
4987:
4961:
4866:
4846:
4826:
4806:
4786:
4766:
4729:
4709:
4689:
4669:
4645:
4569:
4549:
4529:
4509:
4481:
4461:
4438:
4418:
4398:
4378:
4354:
4278:
4258:
4238:
4218:
4185:
4165:
4145:
4125:
4105:
4079:
4053:
4024:
4004:
3984:
3964:
3944:
3895:
3843:
3823:
3793:
3761:
3737:
3717:
3691:
3671:
3635:
3615:
3589:
3569:
3541:
3515:
3495:
3475:
3455:
3413:
3387:
3367:
3321:
3301:
3281:
3253:
3233:
3205:
3185:
3156:
3136:
3116:
3086:
3019:
2993:
2973:
2946:
2894:
2846:
2825:
2730:
2704:
2674:
2641:
2617:
2584:
2564:
2477:
2457:
2437:
2417:
2397:
2377:
2357:
2337:
2317:
2297:
2277:
2257:
2234:
2214:
2190:
2166:
2146:
2126:
2101:
2069:
2033:
2009:
1989:
1969:
1949:
1929:
1909:
1874:
1854:
1806:
1777:
1751:
1725:
1686:
1627:
1571:
1539:
1512:
1468:
1424:
1383:
1312:
1241:
1185:
1156:
19092:Relations and Kleene algebras in computer science
17951:, a theory of relations by Charles Sanders Peirce
8577:see the 2nd example. But in that second example,
5790:{\displaystyle {\bar {R}}=\{(x,y)\mid \neg xRy\}}
5453:{\displaystyle R^{\textsf {T}}=\{(y,x)\mid xRy\}}
21168:
19547:
19314:
18433:
15446:{\displaystyle x_{1}\cap x_{2}\neq \varnothing }
13808:have facilitated usage of binary relations. The
12129:is the edge set (there is an edge from a vertex
11742:): the resulting set relation can be denoted by
9883:{\displaystyle I\subseteq V\times \mathbf {B} .}
2947:{\displaystyle X_{1}\times \cdots \times X_{n}.}
1403:
18871:A formalization of set theory without variables
18804:
18698:
18377:
18230:
18228:
15027:involves functional relations, commonly called
9742:{\displaystyle \mathbf {D} =(V,\mathbf {B} ,I)}
9490:changed that when he articulated the notion of
8570:{\displaystyle \{{\text{John, Mary, Venus}}\};}
18498:
17939:, a graphic means to display an order relation
15531:on a set is a homogeneous relation arising in
11407:If relations over proper classes are allowed:
8600:
7950:corresponds to intersection of relations, the
3289:. In order to specify the choices of the sets
20529:
19833:
19301:J. Riguet (1951) "Les relations de Ferrers",
19206:
19167:
18494:
18492:
18490:
14522:total order corresponds to Ferrers type, and
13640:may be subjected to closure operations like:
6260:the complement has the following properties:
5155:. For example, the composition (is parent of)
1668:in the "Antisymmetric" column, respectively.
60:
19673:
19344:
19308:
18898:
18864:
18591:
18252:
18250:
18225:
17957:, investigates properties of order relations
17910:, a many-valued homomorphism between modules
17303:{\displaystyle \operatorname {fringe} (R)=R}
15798:
15792:
15652:{\displaystyle R,{\bar {R}},R^{\textsf {T}}}
15305:
15287:
15071:{\displaystyle RR^{\textsf {T}}R\subseteq R}
14870:
14846:
11531:
11507:
10566:
10560:
10543:element. Such a binary relation is called a
9969:
9519:
9507:
8882:
8826:
8796:
8764:
8561:
8553:
8510:
8502:
8433:
8391:
8322:
8314:
8288:
8280:
6705:
6661:
6650:
6454:
6396:
6385:
5784:
5751:
5447:
5417:
4956:
4893:
4640:
4596:
4349:
4305:
3078:
3034:
19241:, edited by Chris Brink, Wolfram Kahl, and
18725:
18723:
18721:
18719:
18717:
18311:
17648:, the set of binary relations between them
16694:{\displaystyle R(R\backslash R)\subseteq R}
15515:In the context of homogeneous relations, a
14767:{\displaystyle I\subseteq RR^{\textsf {T}}}
14653:{\displaystyle I\subseteq R^{\textsf {T}}R}
14198:{\displaystyle C_{ij}=u_{i}v_{j},\quad u,v}
7954:corresponds to the empty relation, and the
3866:A heterogeneous relation has been called a
20536:
20522:
20491:Positive cone of a partially ordered group
19840:
19826:
19264:
18972:
18953:(6th ed.), Brooks/Cole, p. 160,
18903:. Cambridge University Press. p. 22.
18892:
18597:"quantum mechanics over a commutative rig"
18487:
18144:
18142:
18140:
18138:
18136:
18134:
18132:
18130:
17509:is a sequence of boundary rectangles when
17502:{\displaystyle \operatorname {fringe} (R)}
17415:{\displaystyle \operatorname {fringe} (R)}
17355:{\displaystyle \operatorname {fringe} (R)}
16431:
16427:
15659:is of Ferrers type, then all of them are.
14440:that belongs to the minimal decomposition
14069:
14034:In contrast to homogeneous relations, the
9558:
67:
53:
19729:
19559:. Springer Science & Business Media.
19181:
19133:
18995:Transitive Closures of Binary Relations I
18874:. American Mathematical Society. p.
18247:
17730:
17222:
16900:
16776:
16447:
16394:
16333:
16313:{\displaystyle R^{\textsf {T}}{\bar {R}}}
16292:
16242:
16031:
15643:
15579:
15053:
14909:
14758:
14641:
14356:
13552:
13501:
13490:" is a partial, but not a total order on
10995:to any real number). As another example,
10317:), nor the black one (as it relates both
9986:Some important types of binary relations
9635:{\displaystyle \operatorname {S} (t,k,n)}
7041:
7015:
6989:
6150:
5408:
3871:
2683:
2120:
2095:
1842:. Precisely, a binary relation over sets
27:Relationship between elements of two sets
20474:Positive cone of an ordered vector space
19624:
18714:
18371:
18299:
16214:. In terms of converse and complements,
13799:
12467:is a reflexive relation but > is not.
12195:). The set of all homogeneous relations
9973:
9401:
8605:
8378:is the relation "is owned by", given by
7765:
2765:are available, satisfying the laws of a
2080:An example of a binary relation is the "
1838:with elements of another set called the
19752:
19528:
19505:, page 265, History of Mathematics 41,
19472:C.D. Hollings & M.V. Lawson (2017)
19387:
19385:
19383:
19267:Coalgebraic Methods in Computer Science
18427:
18127:
17323:{\displaystyle \operatorname {fringe} }
17141:
14560:
14502:Particular cases are considered below:
11938:
9686:elements lies in just one block. These
8610:Oceans and continents (islands omitted)
6996:{\displaystyle S\subseteq \mathbb {R} }
14:
21169:
19692:
19676:On the Shape of Mathematical Arguments
19533:. Berlin: Cambridge University Press.
19239:Relational Methods in Computer Science
19215:Relational Methods in Computer Science
19113:
19068:Boolean Matrix Theory and Applications
18924:Peter J. Pahl; Rudolf Damrath (2001).
18625:
18525:Relational Methods in Computer Science
18345:
18339:
17762:
15011:
14606:
13701:the smallest transitive relation over
12087:directed simple graph permitting loops
11673:. Similarly, the "subset of" relation
11572:
11544:, is a set. For example, the relation
9662:and a set of k-element subsets called
9569:where the overbar denotes conjugation.
9396:is integral to relations on a set, so
9308:
9204:
9175:
8489:could have been viewed as a subset of
7996:corresponds to the identity relation.
6210:
6181:
4087:can be true or false independently of
3425:of the binary relation. The statement
2198:, but not to an integer that is not a
20517:
19821:
19247:Springer Science & Business Media
18827:
18263:, pages 269β279, via internet Archive
18108:
17597:
17470:{\displaystyle R\subseteq {\bar {I}}}
17033:can be obtained in this way from the
14286:{\displaystyle R\subseteq X\times Y,}
13999:, provides a calculus to work in the
13651:the smallest reflexive relation over
10946:. Such a binary relation is called a
7992:over the Boolean semiring) where the
19586:
19553:"Chapter 3: Heterogeneous relations"
19474:Wagner's Theory of Generalised Heaps
19380:
18951:A Transition to Advanced Mathematics
18582:, Book 2, page 339, Houghton Mifflin
18580:Modern Algebra: Structure and Method
18405:
18399:
18383:Set Theory and the Continuum Problem
18271:
18269:
18209:"Relation definition β Math Insight"
18148:
15257:likewise a partition of a subset of
14054:of this category are relations. The
10666:), nor the black one (as it relates
9446:axes are their lines of simultaneity
9368:Visualization of relations leans on
8469:does not involve Ian, and therefore
7810:can be represented algebraically by
6857:, then so too are its restrictions.
5927:are each other's complement, as are
2649:), each relation has a place in the
1671:All definitions tacitly require the
17931:Correspondence (algebraic geometry)
17680:{\displaystyle {\mathcal {B}}(A,B)}
17477:) or upper right block triangular.
16133:is the converse of set membership (
15759:if it satisfies three properties:
14876:{\displaystyle Z=\{x,y,z,\ldots \}}
14365:{\displaystyle R=fEg^{\textsf {T}}}
11039:in the positive integers such that
9701:An incidence structure is a triple
9269:, which is the universal relation (
7862:corresponds to union of relations,
6274:is a total preorderβand vice versa.
6166:is the converse of the complement:
2769:, for which there are textbooks by
2757:. Beyond that, operations like the
2625:Since the latter set is ordered by
1186:{\displaystyle S\neq \varnothing :}
24:
20001:Properties & Types (
19424:
18795:Kilp, Knauer, Mikhalev 2000, p. 4.
18729:Kilp, Knauer, Mikhalev 2000, p. 3.
18314:Introduction to Mathematical Logic
18109:Meyer, Albert (17 November 2021).
17742:{\displaystyle =ab^{\textsf {T}}c}
17657:
17557:
17086:
16625:is the largest relation such that
14918:{\displaystyle R=FG^{\textsf {T}}}
13616:All operations defined in section
12313:
12204:
9655:{\displaystyle \operatorname {S} }
9649:
9605:
6900:" to females yields the relation "
5804:
5772:
5611:are each other's converse, as are
2895:{\displaystyle X_{1},\dots ,X_{n}}
25:
21188:
20457:Positive cone of an ordered field
19781:
19271:Lecture Notes in Computer Science
19096:Lecture Notes in Computer Science
18266:
18077:{\displaystyle Rx_{1}\dots x_{n}}
15440:
15176:{\displaystyle A_{i}\times B_{i}}
13952:
12330:{\displaystyle {\mathcal {B}}(X)}
12221:{\displaystyle {\mathcal {B}}(X)}
11722:(the power set of a specific set
5551:is the converse of itself, as is
2522:the "is adjacent to" relation in
2488:Binary relations, and especially
1177:
20567:
20311:Ordered topological vector space
19847:
19787:
19674:Van Gasteren, Antonetta (1990).
19620:from the original on 2022-10-09.
18805:Yao, Y.Y.; Wong, S.K.M. (1995).
18652:
18527:, Advances in Computer Science,
18123:from the original on 2021-11-17.
15320:{\displaystyle \{y\mid xRy\}=xR}
15020:{\displaystyle FG^{\mathsf {T}}}
12550:is an irreflexive relation, but
12292:of mapping of a relation to its
9922:
9873:
9839:
9777:
9726:
9709:
9384:. For heterogeneous relations a
9320:{\displaystyle R^{\mathsf {T}}R}
9216:{\displaystyle R^{\mathsf {T}}R}
9184:{\displaystyle RR^{\mathsf {T}}}
4469:is the union of < and =, and
3863:, "other, another, different").
2753:, and satisfying the laws of an
1656:
1643:
1572:{\displaystyle {\text{not }}aRa}
1077:
1067:
1037:
1027:
1005:
995:
960:
938:
928:
893:
861:
851:
826:
794:
779:
759:
727:
717:
707:
687:
655:
635:
625:
615:
583:
563:
519:
499:
489:
452:
427:
417:
385:
360:
323:
293:
261:
199:
164:
19522:
19495:
19466:
19444:
19411:
19361:
19338:
19295:
19273:. Vol. 8446. p. 118.
19258:
19231:
19161:
19107:
19083:
19057:
19034:
19013:
18985:
18966:
18942:
18858:
18821:
18798:
18789:
18780:
18756:
18732:
18663:
18646:
18585:
18569:
18553:
18540:
18517:
18190:from the original on 2004-09-08
17771:{\displaystyle b^{\mathsf {T}}}
17170:is the sub-relation defined as
16618:{\displaystyle X=R\backslash R}
16342:{\displaystyle R^{\textsf {T}}}
16156:
15522:
15210:are a partition of a subset of
14824:
14615:{\displaystyle R^{\mathsf {T}}}
14549:Relations may be ranked by the
14185:
13617:
11765:{\displaystyle \subseteq _{A}.}
7736:, and equal to the composition
6558:is a binary relation over sets
6159:{\displaystyle R^{\textsf {T}}}
6000:{\displaystyle \not \supseteq }
5960:{\displaystyle \not \subseteq }
5689:is a binary relation over sets
5355:is a binary relation over sets
5295:is the maternal grandparent of
4834:is a binary relation over sets
4774:is a binary relation over sets
4537:are binary relations over sets
4492:
4246:are binary relations over sets
20543:
18901:Relational Knowledge Discovery
18836:. North-Holland. p. 102.
18305:
18293:
18282:. Cambridge University Press,
18239:Algebra und Logic der Relative
18201:
18102:
17969:
17715:
17697:
17674:
17662:
17551:
17545:
17496:
17490:
17461:
17409:
17403:
17349:
17343:
17291:
17285:
17214:
17189:
17183:
17104:
16973:
16961:
16958:
16946:
16912:
16885:
16873:
16870:
16858:
16827:
16815:
16812:
16800:
16788:
16739:
16727:
16724:
16712:
16682:
16670:
16650:{\displaystyle RX\subseteq R.}
16565:
16553:
16550:
16538:
16459:
16428:
16421:
16406:
16362:
16304:
16254:
16082:
16058:
16043:
15628:
15571:
15495:{\displaystyle x_{1}R=x_{2}R.}
14465:
14447:
13979:
13961:
13955:
13943:
13928:
12324:
12318:
12215:
12209:
11922:
11904:
11808:
11802:
11709:
11703:
11537:{\displaystyle \{y\in Y,yRx\}}
10207:element. For such a relation,
9791:are any two disjoint sets and
9736:
9716:
9629:
9611:
9580:) is a heterogeneous relation.
9549:
9534:
8430:
8422:
8416:
8408:
8402:
8394:
7755:{\displaystyle >\circ >}
7685:{\displaystyle R\subsetneq S.}
6943:Also, the various concepts of
6676:
6664:
6411:
6399:
6278:
6202:
5766:
5754:
5742:
5432:
5420:
4908:
4896:
4743:
4611:
4599:
4320:
4308:
3811:). To emphasize the fact that
3801:a binary relation is called a
3444:
3432:
3362:
3344:
3049:
3037:
2682:A binary relation is called a
2504:", and "divides" relations in
2242:is related to numbers such as
2064:
2052:
1904:
1892:
1664:in the "Symmetric" column and
1600:
1345:
1290:
1219:
13:
1:
20268:Series-parallel partial order
19507:American Mathematical Society
18764:"functional relation in nLab"
18633:Handbook of Weighted Automata
18095:
17995:-ary relations for arbitrary
17382:is an upper right triangular
16512:{\displaystyle R\backslash R}
16203:{\displaystyle R\backslash R}
13846:{\displaystyle R\subseteq S,}
13562:{\displaystyle \mathbb {N} ,}
13544:" is a strict total order on
13511:{\displaystyle \mathbb {N} ,}
12809:is an antisymmetric relation.
12277:{\displaystyle 2^{X\times X}}
11015:is a total relation over the
9811:is a binary relation between
9376:illustrates a relation and a
7231:{\displaystyle R\subseteq S,}
7051:{\displaystyle \mathbb {R} .}
5658:
4196:
3872:homogeneous relation on a set
2956:
1665:
1652:
1062:
1057:
1052:
1047:
1022:
990:
985:
980:
975:
970:
955:
923:
918:
913:
908:
903:
888:
876:
871:
846:
841:
836:
821:
809:
804:
789:
774:
769:
754:
742:
737:
702:
697:
682:
670:
665:
650:
645:
610:
598:
593:
578:
573:
558:
553:
548:
534:
529:
514:
509:
484:
479:
467:
462:
447:
442:
437:
412:
400:
395:
380:
375:
370:
355:
350:
338:
333:
318:
313:
308:
303:
288:
276:
271:
256:
251:
246:
241:
236:
231:
214:
209:
194:
189:
184:
179:
174:
19947:Cantor's isomorphism theorem
19551:; StrΓΆhlein, Thomas (2012).
19501:Christopher Hollings (2014)
19417:In this context, the symbol
19318:; StrΓΆhlein, Thomas (2012).
18470:Encyclopedia of Optimization
18467:; Panos M. Pardalos (2008).
18437:; StrΓΆhlein, Thomas (2012).
17232:
17111:
16919:
16831:
16466:
16261:
16089:
15609:If any one of the relations
15517:partial equivalence relation
14887:. The partitioning relation
14322:MacNeille completion theorem
14306:{\displaystyle \sqsubseteq }
11821:to obtain a binary relation
9929:{\displaystyle \mathbf {B} }
9846:{\displaystyle \mathbf {B} }
9784:{\displaystyle \mathbf {B} }
9642:which have an n-element set
7022:{\displaystyle \mathbb {R} }
6187:
4489:is the union of > and =.
3094:and its elements are called
2796:, relations are extended to
2552:A binary relation over sets
2127:{\displaystyle \mathbb {Z} }
2102:{\displaystyle \mathbb {P} }
7:
19987:Szpilrajn extension theorem
19962:Hausdorff maximal principle
19937:Boolean prime ideal theorem
19806:Encyclopedia of Mathematics
19578:Algebra der Logik, Band III
19430:{\displaystyle \backslash }
19279:10.1007/978-3-662-44124-4_7
18977:, Springer-Verlag, p.
18641:10.1007/978-3-642-01492-5_1
18562:& Thomas Bartee (1970)
18503:. Springer. pp. xβxi.
18381:; Fitting, Melvin (2010) .
17925:Confluence (term rewriting)
17895:
16320:, the corresponding row of
15539:adopted the ordering of an
12966:{\displaystyle x,y,z\in X,}
9690:have been generalized with
7999:
7770:Binary relations over sets
6951:a property of the relation
5318:
2902:, which is a subset of the
2365:, just as the prime number
2154:is related to each integer
2084:" relation over the set of
1830:associates elements of one
10:
21193:
21027:von NeumannβBernaysβGΓΆdel
20333:Topological vector lattice
19764:Cambridge University Press
19400:Cambridge University Press
18684:Cambridge University Press
18548:Basic Algebra II (2nd ed.)
18261:A Survey of Symbolic Logic
17914:Allegory (category theory)
17601:
16371:{\displaystyle {\bar {R}}}
15984:{\displaystyle \epsilon =}
15662:
14107:satisfies two properties:
14100:{\displaystyle C\subset R}
14025:{\displaystyle A\times B.}
12015:{\displaystyle X\times X.}
11973:is a binary relation over
11942:
11686:{\displaystyle \subseteq }
9666:, such that a subset with
9482:determines a simultaneous
7033:(also called supremum) in
6282:
5980:{\displaystyle \supseteq }
5940:{\displaystyle \subseteq }
5662:
5328:
5322:
4747:
3859:is from the Greek αΌΟΞ΅ΟΞΏΟ (
3456:{\displaystyle (x,y)\in R}
3186:{\displaystyle X\times Y.}
2675:{\displaystyle X\times Y.}
2642:{\displaystyle \subseteq }
2618:{\displaystyle X\times Y.}
29:
21091:
21054:
20966:
20856:
20828:One-to-one correspondence
20744:
20685:
20576:
20565:
20551:
20363:
20291:
20230:
20000:
19929:
19878:
19855:
19529:Schmidt, Gunther (2010).
19192:10.1007/s00233-017-9846-9
18973:Nievergelt, Yves (2002),
18868:; Givant, Steven (1987).
18711:Van Gasteren 1990, p. 45.
18546:Jacobson, Nathan (2009),
18161:Communications of the ACM
17903:Abstract rewriting system
16004:{\displaystyle \epsilon }
15136:{\displaystyle X\times Y}
14813:{\displaystyle n\times n}
14699:{\displaystyle m\times m}
14252:non-enlargeable rectangle
13302:{\displaystyle x,y\in X,}
13208:{\displaystyle x,y\in X,}
13087:{\displaystyle x,y\in X,}
12845:{\displaystyle x,y\in X,}
12701:{\displaystyle x,y\in X,}
12606:{\displaystyle x,y\in X,}
12339:semigroup with involution
12085:may be identified with a
10450:{\displaystyle y,z\in Y,}
9970:Types of binary relations
9346:{\displaystyle B\times B}
9288:{\displaystyle A\times A}
9242:{\displaystyle 4\times 4}
8722:
8696:
8670:
8644:
8639:
8636:
8633:
8630:
8627:
8624:
8621:
8244:
8227:
8208:
8189:
8184:
8181:
8178:
8175:
8105:
8086:
8067:
8062:
8059:
8056:
8053:
7078:to the rational numbers.
6722:left-restriction relation
6285:Restriction (mathematics)
6040:{\displaystyle \not \in }
5144:{\displaystyle S\circ R,}
3945:{\displaystyle X\times Y}
3414:{\displaystyle X\times Y}
3020:{\displaystyle X\times Y}
19942:CantorβBernstein theorem
19023:, Academic Press, 1982,
18465:Christodoulos A. Floudas
17962:
17594:, linear, strict order.
17362:is the side diagonal if
16279:Forming the diagonal of
14927:composition of relations
14036:composition of relations
13909:of relations. But since
13818:composition of relations
12298:composition of relations
11852:has shown that assuming
11841:{\displaystyle \in _{A}}
10085:{\displaystyle x,y\in X}
9574:hyperbolic orthogonality
9452:Hyperbolic orthogonality
8615:Ocean borders continent
5153:composition of functions
4916: there exists
4750:Composition of relations
4201:
3329:, some authors define a
3001:, the Cartesian product
2785:, and placing them in a
2763:composition of relations
20486:Partially ordered group
20306:Specialization preorder
19731:2027/hvd.32044019561034
19694:Peirce, Charles Sanders
19455:Matematicheskii Sbornik
19367:Anne K. Steiner (1970)
19045:Fundamenta Informaticae
18828:Kunen, Kenneth (1980).
18499:Michael Winter (2007).
17687:can be equipped with a
17422:is the block fringe if
17009:relation Ω on the
14622:is its transpose, then
14471:{\displaystyle (f,g,E)}
14070:Induced concept lattice
13113:{\displaystyle x\neq y}
12497:{\displaystyle x\in X,}
12415:{\displaystyle x\in X,}
12045:A homogeneous relation
11928:{\displaystyle (X,Y,G)}
11891:MorseβKelley set theory
10553:. For such a relation,
10114:{\displaystyle z\in Y,}
10049:Uniqueness properties:
9585:geometric configuration
9460:absolute time and space
8338:A possible relation on
7326:{\displaystyle y\in Y,}
6920:is mother of the woman
4054:{\displaystyle x\neq y}
3368:{\displaystyle (X,Y,G)}
20786:Constructible universe
20613:Constructibility (V=L)
19972:Kruskal's tree theorem
19967:KnasterβTarski theorem
19957:DushnikβMiller theorem
19759:Relational Mathematics
19630:Elements of Set Theory
19531:Relational Mathematics
19431:
19396:Relational Mathematics
19019:Joseph G. Rosenstein,
18744:encyclopediaofmath.org
18566:, page 35, McGraw-Hill
18564:Modern Applied Algebra
18280:Relational Mathematics
18078:
18035:
18009:
17989:
17893:
17879:
17853:
17833:
17796:
17772:
17743:
17681:
17642:
17622:
17584:
17564:
17523:
17503:
17471:
17436:
17416:
17376:
17356:
17324:
17304:
17266:
17244:
17160:
17132:
17070:
17050:
17027:
16995:
16928:
16840:
16752:
16695:
16651:
16619:
16587:
16513:
16487:
16372:
16343:
16314:
16273:
16204:
16175:
16147:
16127:
16107:
16005:
15985:
15955:
15880:
15822:
15749:
15729:
15705:
15681:
15653:
15601:
15496:
15447:
15401:
15400:{\displaystyle x_{2}R}
15371:
15370:{\displaystyle x_{1}R}
15341:
15321:
15271:
15251:
15224:
15204:
15177:
15137:
15111:
15072:
15021:
14995:since the composition
14991:named these relations
14982:
14919:
14877:
14814:
14788:
14768:
14726:
14700:
14674:
14654:
14616:
14583:
14536:
14516:
14492:
14472:
14434:
14406:
14386:
14366:
14307:
14287:
14244:
14224:
14199:
14129:
14101:
14026:
13989:
13905:, sets the scene in a
13899:
13873:
13847:
13789:
13769:
13735:
13715:
13685:
13665:
13634:
13603:
13583:
13563:
13538:
13537:{\displaystyle x<y}
13512:
13484:
13464:
13410:
13384:
13358:
13357:{\displaystyle z\in X}
13332:
13303:
13260:
13234:
13209:
13166:
13140:
13114:
13088:
13045:
13019:
12993:
12967:
12918:
12898:
12872:
12846:
12803:
12783:
12754:
12728:
12702:
12659:
12633:
12607:
12564:
12544:
12524:
12498:
12461:
12441:
12416:
12378:
12358:
12331:
12278:
12242:
12222:
12189:
12163:
12143:
12123:
12109:is the vertex set and
12103:
12079:
12059:
12036:
12016:
11987:
11967:
11929:
11866:
11842:
11815:
11786:
11766:
11736:
11716:
11687:
11667:
11647:
11620:
11596:
11558:
11538:
11494:
11468:
11467:{\displaystyle y\in Y}
11438:
11437:{\displaystyle x\in X}
11344:
11324:
11300:
11280:
11257:
11237:
11213:
11187:
11186:{\displaystyle x\in X}
11161:
11160:{\displaystyle y\in Y}
11125:
11099:
11079:
11059:
11058:{\displaystyle 1>y}
11033:
11009:
10989:
10969:
10926:
10906:
10882:
10856:
10855:{\displaystyle y\in Y}
10830:
10829:{\displaystyle x\in X}
10791:
10771:
10723:
10703:
10680:
10660:
10637:
10617:
10597:
10573:
10529:
10503:
10477:
10451:
10416:
10415:{\displaystyle x\in X}
10374:
10354:
10334:
10311:
10291:
10271:
10248:
10221:
10193:
10167:
10141:
10115:
10086:
10040:
10020:
10000:
9983:
9953:
9930:
9904:
9884:
9847:
9825:
9805:
9785:
9763:
9743:
9680:
9656:
9636:
9563:
9476:
9447:
9440:
9420:
9347:
9321:
9289:
9263:
9243:
9217:
9185:
9151:
8969:for this relation is:
8959:
8939:
8919:
8889:
8803:
8611:
8591:
8571:
8540:
8520:
8483:
8463:
8443:
8372:
8352:
8332:
8319:John, Mary, Ian, Venus
8295:
8167:
8146:
8045:
8024:
7978:
7940:
7920:
7900:
7880:
7848:
7828:
7804:
7784:
7756:
7730:
7710:
7686:
7657:
7629:
7609:
7589:
7569:
7549:
7529:
7499:
7479:
7459:
7439:
7419:
7399:
7379:
7353:
7327:
7298:
7297:{\displaystyle x\in X}
7272:
7252:
7232:
7203:
7183:
7163:
7135:
7115:
7095:
7072:
7052:
7023:
6997:
6965:
6934:
6914:
6894:
6874:
6800:
6780:
6760:
6740:
6712:
6632:
6612:
6592:
6572:
6552:
6529:
6509:
6489:
6461:
6367:
6347:
6327:
6303:
6254:
6223:
6160:
6134:The complement of the
6125:
6105:
6085:
6065:
6041:
6021:
6001:
5981:
5961:
5941:
5921:
5901:
5878:
5858:
5838:
5822:complementary relation
5814:
5813:{\displaystyle \neg R}
5791:
5723:
5703:
5683:
5665:Complementary relation
5645:
5625:
5605:
5585:
5565:
5545:
5522:
5502:
5482:
5454:
5389:
5369:
5349:
5331:Duality (order theory)
5309:
5289:
5269:
5249:
5229:
5209:
5189:
5188:{\displaystyle \circ }
5169:
5168:{\displaystyle \circ }
5145:
5116:
5096:
5073:
5053:
5033:
5013:
4989:
4963:
4868:
4848:
4828:
4808:
4788:
4768:
4731:
4711:
4691:
4671:
4647:
4571:
4551:
4531:
4511:
4483:
4463:
4440:
4420:
4400:
4380:
4356:
4280:
4260:
4240:
4220:
4187:
4167:
4147:
4127:
4107:
4081:
4055:
4026:
4006:
3986:
3966:
3946:
3897:
3853:heterogeneous relation
3845:
3825:
3795:
3763:
3739:
3719:
3693:
3673:
3645:codomain of definition
3637:
3617:
3591:
3571:
3543:
3517:
3497:
3477:
3457:
3415:
3389:
3369:
3323:
3303:
3283:
3255:
3235:
3207:
3187:
3158:
3138:
3118:
3088:
3021:
2995:
2975:
2948:
2896:
2848:
2827:
2761:of a relation and the
2732:
2706:
2676:
2643:
2619:
2586:
2566:
2479:
2459:
2439:
2419:
2399:
2379:
2359:
2339:
2319:
2299:
2279:
2259:
2236:
2216:
2192:
2168:
2148:
2134:, in which each prime
2128:
2103:
2071:
2035:
2011:
1991:
1971:
1951:
1931:
1911:
1876:
1856:
1808:
1779:
1753:
1727:
1726:{\displaystyle a,b,c,}
1688:
1629:
1573:
1541:
1514:
1470:
1426:
1385:
1314:
1243:
1187:
1158:
32:Relation (mathematics)
18:Heterogeneous relation
21009:Principia Mathematica
20843:Transfinite induction
20702:(i.e. set difference)
19432:
19345:Georg Aumann (1971).
18899:M. E. MΓΌller (2012).
18658:Relative simultaneity
18173:10.1145/362384.362685
18079:
18041:as a special case of
18036:
18010:
17990:
17919:Category of relations
17880:
17854:
17834:
17812:
17797:
17773:
17744:
17682:
17643:
17623:
17585:
17565:
17524:
17504:
17472:
17437:
17417:
17377:
17357:
17325:
17305:
17267:
17245:
17161:
17133:
17071:
17051:
17028:
16996:
16929:
16841:
16753:
16696:
16652:
16620:
16588:
16514:
16488:
16373:
16344:
16315:
16274:
16205:
16176:
16148:
16128:
16108:
16006:
15986:
15956:
15881:
15823:
15750:
15730:
15706:
15682:
15654:
15602:
15497:
15448:
15402:
15372:
15342:
15322:
15272:
15252:
15250:{\displaystyle B_{i}}
15225:
15205:
15203:{\displaystyle A_{i}}
15178:
15138:
15112:
15073:
15022:
14983:
14920:
14878:
14815:
14789:
14769:
14727:
14701:
14675:
14655:
14617:
14584:
14537:
14517:
14493:
14473:
14435:
14407:
14387:
14367:
14308:
14288:
14258:For a given relation
14245:
14225:
14200:
14130:
14102:
14027:
13990:
13900:
13874:
13848:
13810:calculus of relations
13800:Calculus of relations
13790:
13770:
13736:
13716:
13686:
13666:
13635:
13604:
13584:
13564:
13539:
13513:
13485:
13465:
13411:
13385:
13359:
13333:
13304:
13261:
13235:
13210:
13167:
13141:
13115:
13089:
13046:
13020:
12994:
12968:
12919:
12917:{\displaystyle \geq }
12899:
12873:
12847:
12804:
12802:{\displaystyle \geq }
12784:
12755:
12729:
12703:
12660:
12634:
12608:
12565:
12563:{\displaystyle \geq }
12545:
12525:
12499:
12462:
12460:{\displaystyle \geq }
12442:
12417:
12379:
12359:
12332:
12279:
12243:
12223:
12190:
12164:
12144:
12124:
12104:
12080:
12060:
12037:
12017:
11988:
11968:
11930:
11867:
11843:
11816:
11787:
11767:
11737:
11717:
11688:
11668:
11648:
11646:{\displaystyle =_{A}}
11621:
11597:
11559:
11539:
11495:
11469:
11439:
11345:
11325:
11301:
11281:
11258:
11238:
11214:
11188:
11162:
11126:
11100:
11080:
11060:
11034:
11010:
10990:
10970:
10927:
10907:
10883:
10857:
10831:
10792:
10772:
10724:
10704:
10681:
10661:
10638:
10618:
10598:
10574:
10572:{\displaystyle \{X\}}
10530:
10504:
10478:
10452:
10417:
10375:
10355:
10335:
10312:
10292:
10272:
10249:
10222:
10194:
10168:
10142:
10116:
10087:
10041:
10021:
10001:
9977:
9954:
9931:
9905:
9885:
9848:
9826:
9806:
9786:
9764:
9744:
9681:
9657:
9637:
9578:split-complex numbers
9564:
9492:relative simultaneity
9477:
9441:
9421:
9405:
9348:
9322:
9290:
9264:
9244:
9223:, the former being a
9218:
9186:
9152:
8960:
8940:
8925:represent that ocean
8920:
8890:
8804:
8609:
8592:
8572:
8541:
8526:i.e. a relation over
8521:
8484:
8464:
8444:
8373:
8353:
8333:
8296:
8168:
8147:
8127:1st example relation
8046:
8025:
8005:2nd example relation
7979:
7941:
7921:
7901:
7881:
7864:matrix multiplication
7849:
7829:
7805:
7785:
7766:Matrix representation
7757:
7731:
7729:{\displaystyle \geq }
7711:
7687:
7658:
7630:
7610:
7590:
7570:
7550:
7530:
7500:
7480:
7460:
7440:
7420:
7400:
7380:
7354:
7328:
7299:
7273:
7253:
7233:
7204:
7184:
7164:
7136:
7116:
7096:
7073:
7071:{\displaystyle \leq }
7053:
7024:
6998:
6966:
6964:{\displaystyle \leq }
6935:
6915:
6895:
6875:
6801:
6781:
6761:
6741:
6713:
6633:
6613:
6593:
6573:
6553:
6530:
6510:
6490:
6462:
6368:
6348:
6328:
6304:
6255:
6224:
6161:
6126:
6124:{\displaystyle \leq }
6106:
6086:
6084:{\displaystyle \geq }
6066:
6042:
6022:
6002:
5982:
5962:
5942:
5922:
5920:{\displaystyle \neq }
5902:
5879:
5859:
5839:
5815:
5792:
5724:
5704:
5684:
5646:
5644:{\displaystyle \geq }
5626:
5624:{\displaystyle \leq }
5606:
5586:
5566:
5564:{\displaystyle \neq }
5546:
5523:
5503:
5483:
5455:
5390:
5370:
5350:
5310:
5290:
5270:
5250:
5230:
5210:
5190:
5170:
5146:
5117:
5097:
5074:
5054:
5034:
5014:
4990:
4964:
4930: such that
4869:
4849:
4829:
4809:
4789:
4769:
4732:
4712:
4692:
4672:
4655:intersection relation
4648:
4572:
4552:
4532:
4512:
4484:
4482:{\displaystyle \geq }
4464:
4462:{\displaystyle \leq }
4441:
4421:
4401:
4381:
4357:
4281:
4261:
4241:
4221:
4188:
4168:
4148:
4128:
4108:
4082:
4056:
4027:
4007:
3987:
3967:
3952:without reference to
3947:
3898:
3846:
3826:
3796:
3764:
3740:
3720:
3694:
3674:
3638:
3618:
3592:
3572:
3544:
3518:
3498:
3478:
3458:
3416:
3390:
3370:
3337:as an ordered triple
3324:
3304:
3284:
3256:
3236:
3208:
3188:
3159:
3139:
3119:
3089:
3022:
2996:
2976:
2949:
2897:
2849:
2828:
2767:calculus of relations
2733:
2707:
2677:
2644:
2620:
2592:is an element of the
2587:
2567:
2490:homogeneous relations
2480:
2460:
2440:
2420:
2400:
2380:
2360:
2340:
2320:
2300:
2280:
2260:
2237:
2217:
2193:
2169:
2149:
2129:
2104:
2072:
2070:{\displaystyle (x,y)}
2036:
2012:
1992:
1972:
1952:
1932:
1912:
1910:{\displaystyle (x,y)}
1877:
1857:
1809:
1780:
1754:
1728:
1689:
1630:
1574:
1542:
1515:
1471:
1427:
1386:
1315:
1244:
1188:
1159:
1138:Definitions, for all
21083:Burali-Forti paradox
20838:Set-builder notation
20791:Continuum hypothesis
20731:Symmetric difference
20464:Ordered vector space
19796:at Wikimedia Commons
19678:. Berlin: Springer.
19421:
19376:Mathematical Reviews
19347:"Kontakt-Relationen"
19141:Julius Richard BΓΌchi
18786:Schmidt 2010, p. 49.
18605:sci.physics.research
18379:Smullyan, Raymond M.
18353:Axiomatic Set Theory
18312:Hans Hermes (1973).
18045:
18019:
17999:
17979:
17863:
17843:
17823:
17786:
17753:
17694:
17652:
17632:
17612:
17574:
17536:
17529:is of Ferrers type.
17513:
17481:
17446:
17426:
17394:
17366:
17334:
17314:
17276:
17256:
17174:
17150:
17142:Fringe of a relation
17083:
17060:
17049:{\displaystyle \in }
17040:
17017:
16940:
16852:
16764:
16706:
16664:
16629:
16597:
16535:
16531:, one requires that
16497:
16385:
16353:
16324:
16283:
16218:
16188:
16165:
16146:{\displaystyle \in }
16137:
16126:{\displaystyle \ni }
16117:
16022:
15995:
15972:
15891:
15833:
15766:
15739:
15719:
15695:
15671:
15613:
15558:
15457:
15411:
15381:
15351:
15331:
15284:
15261:
15234:
15214:
15187:
15147:
15121:
15101:
15087:rectangular relation
15041:
14999:
14937:
14891:
14837:
14831:equivalence relation
14798:
14778:
14740:
14716:
14684:
14664:
14626:
14597:
14573:
14561:Particular relations
14544:equivalence relation
14526:
14506:
14482:
14444:
14424:
14396:
14376:
14335:
14297:
14262:
14234:
14214:
14143:
14119:
14085:
14038:operation is only a
14007:
13913:
13883:
13857:
13828:
13779:
13759:
13753:equivalence relation
13725:
13705:
13675:
13655:
13624:
13593:
13573:
13548:
13522:
13497:
13474:
13454:
13447:equivalence relation
13429:strict partial order
13394:
13368:
13342:
13331:{\displaystyle xRy,}
13313:
13278:
13244:
13218:
13184:
13150:
13124:
13098:
13063:
13029:
13003:
12977:
12936:
12908:
12882:
12856:
12821:
12793:
12782:{\displaystyle x=y.}
12764:
12738:
12712:
12677:
12643:
12617:
12582:
12554:
12543:{\displaystyle >}
12534:
12508:
12479:
12451:
12425:
12397:
12368:
12348:
12308:
12255:
12232:
12199:
12173:
12153:
12133:
12113:
12093:
12069:
12049:
12026:
11997:
11977:
11957:
11951:homogeneous relation
11945:Homogeneous relation
11939:Homogeneous relation
11901:
11865:{\displaystyle \in }
11856:
11825:
11814:{\displaystyle P(A)}
11796:
11776:
11746:
11726:
11715:{\displaystyle P(A)}
11697:
11677:
11657:
11630:
11610:
11586:
11580:axiomatic set theory
11557:{\displaystyle \in }
11548:
11504:
11478:
11452:
11422:
11334:
11314:
11290:
11267:
11247:
11227:
11197:
11171:
11145:
11109:
11089:
11078:{\displaystyle <}
11069:
11043:
11023:
11008:{\displaystyle >}
10999:
10979:
10956:
10949:multivalued function
10942:by some authors) in
10916:
10896:
10866:
10840:
10814:
10781:
10761:
10713:
10690:
10670:
10647:
10627:
10607:
10587:
10557:
10513:
10487:
10461:
10426:
10400:
10364:
10344:
10321:
10301:
10281:
10258:
10238:
10211:
10177:
10151:
10125:
10096:
10064:
10030:
10010:
9990:
9943:
9918:
9894:
9857:
9835:
9815:
9795:
9773:
9753:
9705:
9688:incidence structures
9670:
9646:
9602:
9504:
9466:
9430:
9410:
9331:
9299:
9273:
9253:
9227:
9195:
9163:
8975:
8949:
8929:
8903:
8817:
8755:
8581:
8550:
8530:
8493:
8473:
8453:
8382:
8362:
8342:
8305:
8285:ball, car, doll, cup
8271:
8157:
8136:
8035:
8014:
7962:
7930:
7910:
7906:and a relation over
7890:
7870:
7854:with entries in the
7838:
7818:
7794:
7774:
7740:
7720:
7709:{\displaystyle >}
7700:
7692:For example, on the
7667:
7647:
7619:
7599:
7595:is not contained in
7579:
7559:
7539:
7513:
7489:
7469:
7449:
7429:
7409:
7389:
7363:
7337:
7308:
7282:
7262:
7242:
7213:
7193:
7173:
7153:
7125:
7105:
7085:
7062:
7037:
7011:
6979:
6955:
6924:
6904:
6884:
6864:
6855:equivalence relation
6853:(weak order), or an
6790:
6770:
6750:
6730:
6642:
6622:
6602:
6582:
6562:
6542:
6519:
6499:
6479:
6471:restriction relation
6377:
6357:
6337:
6317:
6311:homogeneous relation
6293:
6270:The complement of a
6253:{\displaystyle X=Y,}
6235:
6170:
6141:
6115:
6104:{\displaystyle >}
6095:
6075:
6064:{\displaystyle <}
6055:
6031:
6020:{\displaystyle \in }
6011:
5991:
5971:
5951:
5931:
5911:
5891:
5868:
5848:
5828:
5801:
5733:
5713:
5693:
5673:
5635:
5615:
5604:{\displaystyle >}
5595:
5584:{\displaystyle <}
5575:
5555:
5535:
5512:
5492:
5472:
5399:
5379:
5359:
5339:
5299:
5279:
5259:
5239:
5219:
5199:
5179:
5159:
5126:
5106:
5086:
5063:
5043:
5023:
5003:
4997:composition relation
4973:
4878:
4858:
4838:
4818:
4798:
4778:
4758:
4721:
4701:
4681:
4661:
4581:
4561:
4541:
4521:
4501:
4473:
4453:
4430:
4410:
4390:
4370:
4290:
4270:
4250:
4230:
4210:
4177:
4157:
4137:
4117:
4091:
4065:
4039:
4016:
3996:
3976:
3956:
3930:
3896:{\displaystyle A=B.}
3878:
3868:rectangular relation
3835:
3815:
3804:homogeneous relation
3794:{\displaystyle X=Y,}
3776:
3753:
3729:
3703:
3683:
3663:
3627:
3601:
3581:
3561:
3551:domain of definition
3527:
3523:" and is denoted by
3507:
3487:
3467:
3429:
3399:
3379:
3341:
3313:
3293:
3273:
3245:
3225:
3197:
3168:
3148:
3128:
3108:
3031:
3005:
2985:
2965:
2909:
2860:
2838:
2811:
2794:axiomatic set theory
2716:
2690:
2684:homogeneous relation
2657:
2633:
2600:
2576:
2556:
2469:
2449:
2429:
2409:
2389:
2369:
2349:
2329:
2309:
2289:
2269:
2246:
2226:
2206:
2182:
2158:
2138:
2116:
2091:
2049:
2025:
2001:
1981:
1961:
1941:
1921:
1889:
1866:
1846:
1807:{\displaystyle aRc.}
1789:
1763:
1737:
1702:
1678:
1673:homogeneous relation
1584:
1552:
1525:
1481:
1437:
1396:
1325:
1254:
1198:
1168:
1142:
884:Strict partial order
159:Equivalence relation
21044:TarskiβGrothendieck
20302:Alexandrov topology
20248:Lexicographic order
20207:Well-quasi-ordering
19714:1873MAAAS...9..317P
19457:32(74): 545 to 632
19053:10.3233/FI-2013-871
18034:{\displaystyle Rxy}
17943:Incidence structure
17878:{\displaystyle A=B}
17532:On the other hand,
17035:membership relation
15936: implies
15861: implies
15803: implies
15347:such that wherever
15280:Using the notation
14734:surjective relation
14139:of logical vectors
13898:{\displaystyle aSb}
13872:{\displaystyle aRb}
13746:Equivalence closure
13409:{\displaystyle zRy}
13383:{\displaystyle xRz}
13259:{\displaystyle yRx}
13233:{\displaystyle xRy}
13165:{\displaystyle yRx}
13139:{\displaystyle xRy}
13044:{\displaystyle xRz}
13018:{\displaystyle yRz}
12992:{\displaystyle xRy}
12897:{\displaystyle yRx}
12871:{\displaystyle xRy}
12753:{\displaystyle yRx}
12727:{\displaystyle xRy}
12658:{\displaystyle yRx}
12632:{\displaystyle xRy}
12523:{\displaystyle xRx}
12440:{\displaystyle xRx}
12288:augmented with the
12188:{\displaystyle xRy}
11573:Sets versus classes
11493:{\displaystyle yRx}
11212:{\displaystyle xRy}
11124:{\displaystyle y=x}
10881:{\displaystyle xRy}
10528:{\displaystyle y=z}
10502:{\displaystyle xRz}
10476:{\displaystyle xRy}
10192:{\displaystyle x=y}
10166:{\displaystyle yRz}
10140:{\displaystyle xRz}
9496:composition algebra
9486:in that cosmology.
9456:simultaneous events
8918:{\displaystyle aRb}
8616:
8603:), the 1st is not.
8128:
8006:
7977:{\displaystyle X=Y}
7528:{\displaystyle R=S}
7378:{\displaystyle xSy}
7352:{\displaystyle xRy}
7278:, that is, for all
4988:{\displaystyle R;S}
4106:{\displaystyle xRy}
4080:{\displaystyle yRx}
3718:{\displaystyle xRy}
3616:{\displaystyle xRy}
3542:{\displaystyle xRy}
2826:{\displaystyle n=2}
2792:In some systems of
2731:{\displaystyle X=Y}
2705:{\displaystyle X=Y}
1778:{\displaystyle bRc}
1752:{\displaystyle aRb}
1540:{\displaystyle aRa}
1157:{\displaystyle a,b}
543:Well-quasi-ordering
20633:Limitation of size
20283:Transitive closure
20243:Converse/Transpose
19952:Dilworth's theorem
19427:
19371:Kontakt-Relationen
18672:Jungnickel, Dieter
18074:
18031:
18005:
17985:
17949:Logic of relatives
17875:
17849:
17829:
17792:
17768:
17739:
17677:
17638:
17618:
17604:Heap (mathematics)
17598:Mathematical heaps
17580:
17560:
17519:
17499:
17467:
17432:
17412:
17372:
17352:
17320:
17300:
17262:
17240:
17156:
17128:
17066:
17046:
17023:
16991:
16924:
16836:
16748:
16691:
16647:
16615:
16583:
16521:reflexive relation
16509:
16483:
16368:
16339:
16310:
16269:
16200:
16171:
16143:
16123:
16103:
16050:⊆ ∋
16001:
15981:
15951:
15876:
15818:
15745:
15735:. Then a relation
15725:
15701:
15677:
15649:
15597:
15492:
15443:
15397:
15367:
15337:
15317:
15267:
15247:
15220:
15200:
15173:
15133:
15107:
15068:
15017:
14978:
14915:
14873:
14820:identity relation.
14810:
14784:
14764:
14722:
14706:identity relation.
14696:
14670:
14650:
14612:
14579:
14532:
14512:
14488:
14468:
14430:
14402:
14382:
14362:
14303:
14283:
14250:is described as a
14240:
14220:
14195:
14125:
14097:
14050:, except that the
14022:
13985:
13895:
13869:
13843:
13822:converse relations
13785:
13765:
13731:
13711:
13696:Transitive closure
13681:
13661:
13630:
13599:
13579:
13559:
13534:
13508:
13480:
13460:
13441:strict total order
13406:
13380:
13354:
13328:
13299:
13256:
13230:
13205:
13177:Strongly connected
13162:
13136:
13110:
13084:
13041:
13015:
12989:
12963:
12914:
12894:
12868:
12842:
12799:
12779:
12750:
12724:
12698:
12655:
12629:
12603:
12560:
12540:
12520:
12494:
12457:
12437:
12412:
12374:
12354:
12327:
12274:
12238:
12218:
12185:
12159:
12139:
12119:
12099:
12075:
12055:
12032:
12012:
11983:
11963:
11925:
11862:
11838:
11811:
11782:
11762:
11732:
11712:
11683:
11663:
11643:
11616:
11592:
11554:
11534:
11490:
11464:
11434:
11340:
11320:
11296:
11279:{\displaystyle -1}
11276:
11253:
11233:
11209:
11183:
11167:, there exists an
11157:
11121:
11095:
11075:
11055:
11029:
11005:
10985:
10968:{\displaystyle -1}
10965:
10922:
10902:
10878:
10852:
10826:
10787:
10767:
10719:
10702:{\displaystyle -1}
10699:
10676:
10659:{\displaystyle -1}
10656:
10633:
10613:
10593:
10569:
10525:
10499:
10473:
10447:
10412:
10370:
10350:
10333:{\displaystyle -1}
10330:
10307:
10287:
10270:{\displaystyle -1}
10267:
10244:
10217:
10189:
10163:
10137:
10111:
10082:
10046:are listed below.
10036:
10016:
9996:
9984:
9949:
9926:
9900:
9880:
9843:
9821:
9801:
9781:
9759:
9739:
9676:
9652:
9632:
9559:
9472:
9448:
9436:
9416:
9382:symmetric relation
9343:
9317:
9285:
9259:
9239:
9213:
9181:
9147:
9138:
8955:
8945:borders continent
8935:
8915:
8885:
8813:of the globe, and
8799:
8614:
8612:
8587:
8567:
8536:
8516:
8479:
8459:
8439:
8368:
8348:
8328:
8291:
8163:
8142:
8126:
8041:
8020:
8004:
7990:matrix semialgebra
7974:
7936:
7916:
7896:
7876:
7844:
7824:
7800:
7780:
7752:
7726:
7706:
7682:
7653:
7625:
7605:
7585:
7565:
7545:
7525:
7495:
7475:
7455:
7435:
7415:
7395:
7375:
7349:
7323:
7294:
7268:
7248:
7228:
7199:
7179:
7159:
7131:
7111:
7091:
7081:A binary relation
7068:
7048:
7019:
6993:
6961:
6930:
6910:
6890:
6870:
6796:
6776:
6756:
6736:
6708:
6628:
6608:
6588:
6568:
6548:
6525:
6505:
6485:
6457:
6363:
6343:
6323:
6299:
6250:
6219:
6156:
6121:
6101:
6081:
6061:
6037:
6017:
5997:
5977:
5957:
5937:
5917:
5897:
5874:
5854:
5834:
5810:
5787:
5719:
5699:
5679:
5641:
5621:
5601:
5581:
5561:
5541:
5518:
5498:
5478:
5450:
5385:
5365:
5345:
5305:
5285:
5265:
5245:
5225:
5205:
5185:
5165:
5141:
5112:
5092:
5069:
5049:
5029:
5009:
4985:
4959:
4864:
4844:
4824:
4804:
4784:
4764:
4727:
4707:
4687:
4667:
4643:
4567:
4547:
4527:
4507:
4479:
4459:
4436:
4416:
4396:
4376:
4352:
4276:
4256:
4236:
4216:
4183:
4163:
4143:
4123:
4103:
4077:
4051:
4022:
4002:
3982:
3962:
3942:
3924:two-place relation
3893:
3841:
3821:
3791:
3759:
3735:
3715:
3689:
3679:is the set of all
3669:
3633:
3613:
3587:
3577:is the set of all
3567:
3539:
3513:
3493:
3473:
3453:
3411:
3385:
3365:
3319:
3299:
3279:
3267:set of destination
3251:
3231:
3203:
3183:
3154:
3134:
3114:
3084:
3017:
2991:
2971:
2944:
2892:
2844:
2823:
2728:
2702:
2672:
2639:
2615:
2582:
2562:
2478:{\displaystyle 13}
2475:
2455:
2435:
2415:
2395:
2375:
2355:
2335:
2318:{\displaystyle 10}
2315:
2295:
2275:
2258:{\displaystyle -4}
2255:
2232:
2212:
2188:
2164:
2144:
2124:
2099:
2067:
2031:
2007:
1987:
1967:
1947:
1927:
1907:
1872:
1852:
1804:
1775:
1749:
1723:
1684:
1625:
1623:
1569:
1537:
1510:
1508:
1466:
1464:
1422:
1420:
1381:
1379:
1310:
1308:
1239:
1237:
1183:
1154:
1018:Strict total order
21164:
21163:
21073:Russell's paradox
21022:ZermeloβFraenkel
20923:Dedekind-infinite
20796:Diagonal argument
20695:Cartesian product
20559:Set (mathematics)
20511:
20510:
20469:Partially ordered
20278:Symmetric closure
20263:Reflexive closure
20006:
19801:"Binary relation"
19792:Media related to
19773:978-0-521-76268-7
19666:978-3-11-015248-7
19643:978-0-12-238440-0
19626:Enderton, Herbert
19588:Codd, Edgar Frank
19566:978-3-642-77968-8
19514:978-1-4704-1493-1
19485:978-3-319-63620-7
19407:978-0-521-76268-7
19331:978-3-642-77968-8
19288:978-3-662-44123-7
19254:978-3-211-82971-4
19224:978-3-211-82971-4
19154:978-1-4613-8853-1
18935:978-3-540-67995-0
18910:978-0-521-19021-3
18694:978-0-521-44432-3
18688:. 2nd ed. (1999)
18510:978-1-4020-6164-6
18480:978-0-387-74758-3
18450:978-3-642-77968-8
18392:978-0-486-47484-7
18288:978-0-521-76268-7
18150:Codd, Edgar Frank
18008:{\displaystyle n}
17988:{\displaystyle n}
17908:Additive relation
17852:{\displaystyle B}
17832:{\displaystyle A}
17795:{\displaystyle b}
17780:converse relation
17732:
17689:ternary operation
17641:{\displaystyle B}
17621:{\displaystyle A}
17583:{\displaystyle R}
17522:{\displaystyle R}
17464:
17435:{\displaystyle R}
17375:{\displaystyle R}
17265:{\displaystyle R}
17235:
17224:
17217:
17159:{\displaystyle R}
17146:Given a relation
17114:
17107:
17069:{\displaystyle U}
17026:{\displaystyle U}
16934:(complementation)
16922:
16915:
16902:
16846:(SchrΓΆder's rule)
16834:
16791:
16778:
16469:
16462:
16449:
16424:
16409:
16396:
16365:
16335:
16307:
16294:
16264:
16257:
16244:
16174:{\displaystyle R}
16092:
16085:
16061:
16046:
16033:
15937:
15923:
15897:
15862:
15848:
15804:
15772:
15748:{\displaystyle g}
15728:{\displaystyle A}
15711:, the set of all
15704:{\displaystyle A}
15680:{\displaystyle B}
15645:
15631:
15581:
15574:
15541:integer partition
15519:is difunctional.
15340:{\displaystyle R}
15270:{\displaystyle Y}
15223:{\displaystyle X}
15110:{\displaystyle R}
15055:
15029:partial functions
14958:
14911:
14787:{\displaystyle I}
14760:
14725:{\displaystyle R}
14673:{\displaystyle I}
14643:
14582:{\displaystyle R}
14535:{\displaystyle E}
14515:{\displaystyle E}
14491:{\displaystyle R}
14433:{\displaystyle E}
14405:{\displaystyle g}
14385:{\displaystyle f}
14358:
14243:{\displaystyle C}
14223:{\displaystyle C}
14128:{\displaystyle C}
13946:
13788:{\displaystyle R}
13768:{\displaystyle X}
13734:{\displaystyle R}
13714:{\displaystyle X}
13684:{\displaystyle R}
13664:{\displaystyle X}
13646:Reflexive closure
13633:{\displaystyle X}
13618:Β§ Operations
13602:{\displaystyle y}
13582:{\displaystyle x}
13483:{\displaystyle y}
13463:{\displaystyle x}
13364:exists such that
12377:{\displaystyle X}
12357:{\displaystyle R}
12294:converse relation
12241:{\displaystyle X}
12162:{\displaystyle y}
12142:{\displaystyle x}
12122:{\displaystyle R}
12102:{\displaystyle X}
12078:{\displaystyle X}
12058:{\displaystyle R}
12035:{\displaystyle X}
11986:{\displaystyle X}
11966:{\displaystyle X}
11879:Russell's paradox
11785:{\displaystyle A}
11735:{\displaystyle A}
11666:{\displaystyle =}
11619:{\displaystyle A}
11595:{\displaystyle =}
11343:{\displaystyle Y}
11323:{\displaystyle X}
11299:{\displaystyle 2}
11256:{\displaystyle Y}
11236:{\displaystyle R}
11098:{\displaystyle x}
11032:{\displaystyle y}
10988:{\displaystyle 2}
10925:{\displaystyle X}
10905:{\displaystyle R}
10790:{\displaystyle Y}
10770:{\displaystyle X}
10722:{\displaystyle 1}
10679:{\displaystyle 0}
10636:{\displaystyle 1}
10616:{\displaystyle 1}
10596:{\displaystyle R}
10373:{\displaystyle 0}
10353:{\displaystyle 1}
10310:{\displaystyle 1}
10290:{\displaystyle 1}
10247:{\displaystyle R}
10220:{\displaystyle Y}
10039:{\displaystyle Y}
10019:{\displaystyle X}
9999:{\displaystyle R}
9952:{\displaystyle I}
9903:{\displaystyle V}
9824:{\displaystyle V}
9804:{\displaystyle I}
9762:{\displaystyle V}
9679:{\displaystyle t}
9552:
9537:
9488:Hermann Minkowski
9475:{\displaystyle t}
9439:{\displaystyle x}
9419:{\displaystyle t}
9327:is a relation on
9262:{\displaystyle A}
8958:{\displaystyle b}
8938:{\displaystyle a}
8880:
8872:
8864:
8856:
8848:
8840:
8832:
8794:
8786:
8778:
8770:
8748:
8747:
8590:{\displaystyle R}
8559:
8558:John, Mary, Venus
8539:{\displaystyle A}
8508:
8507:John, Mary, Venus
8482:{\displaystyle R}
8462:{\displaystyle R}
8428:
8414:
8400:
8371:{\displaystyle B}
8351:{\displaystyle A}
8320:
8286:
8263:
8262:
8166:{\displaystyle B}
8145:{\displaystyle A}
8124:
8123:
8044:{\displaystyle B}
8023:{\displaystyle A}
7939:{\displaystyle Z}
7919:{\displaystyle Y}
7899:{\displaystyle Y}
7879:{\displaystyle X}
7847:{\displaystyle Y}
7827:{\displaystyle X}
7803:{\displaystyle Y}
7783:{\displaystyle X}
7656:{\displaystyle S}
7628:{\displaystyle R}
7608:{\displaystyle R}
7588:{\displaystyle S}
7568:{\displaystyle S}
7548:{\displaystyle R}
7498:{\displaystyle S}
7478:{\displaystyle R}
7458:{\displaystyle R}
7438:{\displaystyle S}
7418:{\displaystyle S}
7398:{\displaystyle R}
7271:{\displaystyle S}
7251:{\displaystyle R}
7202:{\displaystyle Y}
7182:{\displaystyle X}
7162:{\displaystyle S}
7134:{\displaystyle Y}
7114:{\displaystyle X}
7094:{\displaystyle R}
7031:least upper bound
6933:{\displaystyle y}
6913:{\displaystyle x}
6893:{\displaystyle y}
6873:{\displaystyle x}
6847:strict weak order
6809:If a relation is
6799:{\displaystyle Y}
6779:{\displaystyle X}
6759:{\displaystyle S}
6739:{\displaystyle R}
6694:
6631:{\displaystyle X}
6611:{\displaystyle S}
6591:{\displaystyle Y}
6571:{\displaystyle X}
6551:{\displaystyle R}
6528:{\displaystyle X}
6508:{\displaystyle S}
6488:{\displaystyle R}
6443:
6429:
6366:{\displaystyle X}
6346:{\displaystyle S}
6326:{\displaystyle X}
6302:{\displaystyle R}
6272:strict weak order
6205:
6190:
6152:
6136:converse relation
5900:{\displaystyle =}
5877:{\displaystyle Y}
5857:{\displaystyle X}
5837:{\displaystyle R}
5797:(also denoted by
5745:
5722:{\displaystyle Y}
5702:{\displaystyle X}
5682:{\displaystyle R}
5544:{\displaystyle =}
5521:{\displaystyle X}
5501:{\displaystyle Y}
5481:{\displaystyle R}
5462:converse relation
5410:
5388:{\displaystyle Y}
5368:{\displaystyle X}
5348:{\displaystyle R}
5325:Converse relation
5308:{\displaystyle z}
5288:{\displaystyle x}
5268:{\displaystyle z}
5255:is the mother of
5248:{\displaystyle y}
5228:{\displaystyle y}
5215:is the parent of
5208:{\displaystyle x}
5115:{\displaystyle S}
5095:{\displaystyle R}
5072:{\displaystyle Z}
5052:{\displaystyle X}
5032:{\displaystyle S}
5012:{\displaystyle R}
4969:(also denoted by
4945:
4931:
4917:
4867:{\displaystyle Z}
4847:{\displaystyle Y}
4827:{\displaystyle S}
4807:{\displaystyle Y}
4787:{\displaystyle X}
4767:{\displaystyle R}
4730:{\displaystyle Y}
4710:{\displaystyle X}
4690:{\displaystyle S}
4670:{\displaystyle R}
4629:
4570:{\displaystyle Y}
4550:{\displaystyle X}
4530:{\displaystyle S}
4510:{\displaystyle R}
4439:{\displaystyle Y}
4419:{\displaystyle X}
4399:{\displaystyle S}
4379:{\displaystyle R}
4338:
4279:{\displaystyle Y}
4259:{\displaystyle X}
4239:{\displaystyle S}
4219:{\displaystyle R}
4186:{\displaystyle 3}
4166:{\displaystyle 9}
4146:{\displaystyle 9}
4126:{\displaystyle 3}
4025:{\displaystyle Y}
4005:{\displaystyle X}
3985:{\displaystyle Y}
3965:{\displaystyle X}
3844:{\displaystyle Y}
3824:{\displaystyle X}
3762:{\displaystyle R}
3738:{\displaystyle x}
3725:for at least one
3692:{\displaystyle y}
3672:{\displaystyle R}
3636:{\displaystyle y}
3623:for at least one
3590:{\displaystyle x}
3570:{\displaystyle R}
3516:{\displaystyle y}
3496:{\displaystyle R}
3476:{\displaystyle x}
3388:{\displaystyle G}
3322:{\displaystyle Y}
3302:{\displaystyle X}
3282:{\displaystyle R}
3254:{\displaystyle Y}
3234:{\displaystyle R}
3206:{\displaystyle X}
3157:{\displaystyle Y}
3137:{\displaystyle X}
3117:{\displaystyle R}
3067:
2994:{\displaystyle Y}
2974:{\displaystyle X}
2904:Cartesian product
2847:{\displaystyle n}
2802:Russell's paradox
2585:{\displaystyle Y}
2565:{\displaystyle X}
2458:{\displaystyle 4}
2438:{\displaystyle 9}
2418:{\displaystyle 6}
2398:{\displaystyle 0}
2378:{\displaystyle 3}
2358:{\displaystyle 9}
2338:{\displaystyle 1}
2298:{\displaystyle 6}
2278:{\displaystyle 0}
2235:{\displaystyle 2}
2215:{\displaystyle p}
2191:{\displaystyle p}
2167:{\displaystyle z}
2147:{\displaystyle p}
2034:{\displaystyle y}
2010:{\displaystyle x}
1990:{\displaystyle Y}
1970:{\displaystyle y}
1950:{\displaystyle X}
1930:{\displaystyle x}
1875:{\displaystyle Y}
1855:{\displaystyle X}
1820:
1819:
1687:{\displaystyle R}
1638:
1637:
1610:
1558:
1504:
1460:
1416:
1364:
1273:
951:Strict weak order
137:Total, Semiconnex
16:(Redirected from
21184:
21177:Binary relations
21146:Bertrand Russell
21136:John von Neumann
21121:Abraham Fraenkel
21116:Richard Dedekind
21078:Suslin's problem
20989:Cantor's theorem
20706:De Morgan's laws
20571:
20538:
20531:
20524:
20515:
20514:
20253:Linear extension
20002:
19982:Mirsky's theorem
19842:
19835:
19828:
19819:
19818:
19814:
19794:Binary relations
19791:
19777:
19754:Schmidt, Gunther
19749:
19747:
19746:
19733:
19722:10.2307/25058006
19689:
19670:
19647:
19621:
19619:
19598:
19582:Internet Archive
19570:
19549:Schmidt, Gunther
19544:
19516:
19499:
19493:
19470:
19464:
19448:
19442:
19436:
19434:
19433:
19428:
19415:
19409:
19398:, pages 211β15,
19389:
19378:
19365:
19359:
19358:
19342:
19336:
19335:
19316:Schmidt, Gunther
19312:
19306:
19299:
19293:
19292:
19262:
19256:
19235:
19229:
19228:
19210:
19204:
19203:
19185:
19165:
19159:
19158:
19137:
19131:
19130:
19117:(January 1950).
19111:
19105:
19087:
19081:
19061:
19055:
19038:
19032:
19021:Linear orderings
19017:
19011:
19009:
19007:
19000:
18989:
18983:
18981:
18970:
18964:
18963:
18946:
18940:
18939:
18921:
18915:
18914:
18896:
18890:
18889:
18862:
18856:
18855:
18835:
18825:
18819:
18817:
18811:
18802:
18796:
18793:
18787:
18784:
18778:
18777:
18775:
18774:
18760:
18754:
18753:
18751:
18750:
18736:
18730:
18727:
18712:
18709:
18696:
18687:
18667:
18661:
18656:
18650:
18644:
18629:
18623:
18622:
18620:
18618:
18589:
18583:
18576:Mary P. Dolciani
18573:
18567:
18560:Garrett Birkhoff
18557:
18551:
18544:
18538:
18521:
18515:
18514:
18496:
18485:
18484:
18461:
18455:
18454:
18435:Schmidt, Gunther
18431:
18425:
18424:
18411:Basic Set Theory
18403:
18397:
18396:
18375:
18369:
18368:
18356:
18343:
18337:
18335:
18309:
18303:
18297:
18291:
18273:
18264:
18254:
18245:
18243:Internet Archive
18232:
18223:
18222:
18220:
18219:
18205:
18199:
18198:
18196:
18195:
18189:
18158:
18146:
18125:
18124:
18122:
18115:
18106:
18089:
18083:
18081:
18080:
18075:
18073:
18072:
18060:
18059:
18040:
18038:
18037:
18032:
18014:
18012:
18011:
18006:
17994:
17992:
17991:
17986:
17973:
17891:
17884:
17882:
17881:
17876:
17858:
17856:
17855:
17850:
17838:
17836:
17835:
17830:
17801:
17799:
17798:
17793:
17777:
17775:
17774:
17769:
17767:
17766:
17765:
17748:
17746:
17745:
17740:
17735:
17734:
17733:
17686:
17684:
17683:
17678:
17661:
17660:
17647:
17645:
17644:
17639:
17627:
17625:
17624:
17619:
17589:
17587:
17586:
17581:
17569:
17567:
17566:
17561:
17528:
17526:
17525:
17520:
17508:
17506:
17505:
17500:
17476:
17474:
17473:
17468:
17466:
17465:
17457:
17442:is irreflexive (
17441:
17439:
17438:
17433:
17421:
17419:
17418:
17413:
17381:
17379:
17378:
17373:
17361:
17359:
17358:
17353:
17329:
17327:
17326:
17321:
17310:. Otherwise the
17309:
17307:
17306:
17301:
17271:
17269:
17268:
17263:
17249:
17247:
17246:
17241:
17236:
17231:
17227:
17226:
17225:
17219:
17218:
17210:
17202:
17165:
17163:
17162:
17157:
17137:
17135:
17134:
17129:
17115:
17110:
17109:
17108:
17100:
17093:
17075:
17073:
17072:
17067:
17055:
17053:
17052:
17047:
17032:
17030:
17029:
17024:
17000:
16998:
16997:
16992:
16933:
16931:
16930:
16925:
16923:
16918:
16917:
16916:
16908:
16905:
16904:
16903:
16892:
16845:
16843:
16842:
16837:
16835:
16830:
16798:
16793:
16792:
16784:
16781:
16780:
16779:
16757:
16755:
16754:
16749:
16700:
16698:
16697:
16692:
16656:
16654:
16653:
16648:
16624:
16622:
16621:
16616:
16592:
16590:
16589:
16584:
16518:
16516:
16515:
16510:
16492:
16490:
16489:
16484:
16470:
16465:
16464:
16463:
16455:
16452:
16451:
16450:
16439:
16426:
16425:
16417:
16411:
16410:
16402:
16399:
16398:
16397:
16377:
16375:
16374:
16369:
16367:
16366:
16358:
16348:
16346:
16345:
16340:
16338:
16337:
16336:
16319:
16317:
16316:
16311:
16309:
16308:
16300:
16297:
16296:
16295:
16278:
16276:
16275:
16270:
16265:
16260:
16259:
16258:
16250:
16247:
16246:
16245:
16234:
16209:
16207:
16206:
16201:
16180:
16178:
16177:
16172:
16152:
16150:
16149:
16144:
16132:
16130:
16129:
16124:
16112:
16110:
16109:
16104:
16093:
16088:
16087:
16086:
16078:
16071:
16063:
16062:
16054:
16048:
16047:
16039:
16036:
16035:
16034:
16010:
16008:
16007:
16002:
15990:
15988:
15987:
15982:
15960:
15958:
15957:
15952:
15938:
15935:
15924:
15921:
15898:
15895:
15885:
15883:
15882:
15877:
15863:
15860:
15849:
15846:
15827:
15825:
15824:
15819:
15805:
15802:
15773:
15770:
15757:contact relation
15754:
15752:
15751:
15746:
15734:
15732:
15731:
15726:
15710:
15708:
15707:
15702:
15686:
15684:
15683:
15678:
15658:
15656:
15655:
15650:
15648:
15647:
15646:
15633:
15632:
15624:
15606:
15604:
15603:
15598:
15584:
15583:
15582:
15576:
15575:
15567:
15501:
15499:
15498:
15493:
15485:
15484:
15469:
15468:
15452:
15450:
15449:
15444:
15436:
15435:
15423:
15422:
15406:
15404:
15403:
15398:
15393:
15392:
15376:
15374:
15373:
15368:
15363:
15362:
15346:
15344:
15343:
15338:
15326:
15324:
15323:
15318:
15276:
15274:
15273:
15268:
15256:
15254:
15253:
15248:
15246:
15245:
15229:
15227:
15226:
15221:
15209:
15207:
15206:
15201:
15199:
15198:
15182:
15180:
15179:
15174:
15172:
15171:
15159:
15158:
15142:
15140:
15139:
15134:
15116:
15114:
15113:
15108:
15077:
15075:
15074:
15069:
15058:
15057:
15056:
15026:
15024:
15023:
15018:
15016:
15015:
15014:
14987:
14985:
14984:
14979:
14959:
14956:
14924:
14922:
14921:
14916:
14914:
14913:
14912:
14882:
14880:
14879:
14874:
14819:
14817:
14816:
14811:
14793:
14791:
14790:
14785:
14773:
14771:
14770:
14765:
14763:
14762:
14761:
14731:
14729:
14728:
14723:
14705:
14703:
14702:
14697:
14679:
14677:
14676:
14671:
14659:
14657:
14656:
14651:
14646:
14645:
14644:
14621:
14619:
14618:
14613:
14611:
14610:
14609:
14588:
14586:
14585:
14580:
14541:
14539:
14538:
14533:
14521:
14519:
14518:
14513:
14497:
14495:
14494:
14489:
14478:of the relation
14477:
14475:
14474:
14469:
14439:
14437:
14436:
14431:
14411:
14409:
14408:
14403:
14391:
14389:
14388:
14383:
14371:
14369:
14368:
14363:
14361:
14360:
14359:
14326:complete lattice
14312:
14310:
14309:
14304:
14292:
14290:
14289:
14284:
14249:
14247:
14246:
14241:
14229:
14227:
14226:
14221:
14204:
14202:
14201:
14196:
14181:
14180:
14171:
14170:
14158:
14157:
14134:
14132:
14131:
14126:
14106:
14104:
14103:
14098:
14076:concept lattices
14058:of the category
14048:category of sets
14040:partial function
14031:
14029:
14028:
14023:
13994:
13992:
13991:
13986:
13948:
13947:
13939:
13904:
13902:
13901:
13896:
13878:
13876:
13875:
13870:
13852:
13850:
13849:
13844:
13824:. The inclusion
13804:Developments in
13794:
13792:
13791:
13786:
13774:
13772:
13771:
13766:
13740:
13738:
13737:
13732:
13720:
13718:
13717:
13712:
13690:
13688:
13687:
13682:
13670:
13668:
13667:
13662:
13639:
13637:
13636:
13631:
13608:
13606:
13605:
13600:
13588:
13586:
13585:
13580:
13568:
13566:
13565:
13560:
13555:
13543:
13541:
13540:
13535:
13517:
13515:
13514:
13509:
13504:
13489:
13487:
13486:
13481:
13469:
13467:
13466:
13461:
13415:
13413:
13412:
13407:
13389:
13387:
13386:
13381:
13363:
13361:
13360:
13355:
13337:
13335:
13334:
13329:
13308:
13306:
13305:
13300:
13265:
13263:
13262:
13257:
13239:
13237:
13236:
13231:
13214:
13212:
13211:
13206:
13171:
13169:
13168:
13163:
13145:
13143:
13142:
13137:
13119:
13117:
13116:
13111:
13093:
13091:
13090:
13085:
13050:
13048:
13047:
13042:
13024:
13022:
13021:
13016:
12998:
12996:
12995:
12990:
12972:
12970:
12969:
12964:
12923:
12921:
12920:
12915:
12903:
12901:
12900:
12895:
12877:
12875:
12874:
12869:
12851:
12849:
12848:
12843:
12808:
12806:
12805:
12800:
12788:
12786:
12785:
12780:
12759:
12757:
12756:
12751:
12733:
12731:
12730:
12725:
12707:
12705:
12704:
12699:
12664:
12662:
12661:
12656:
12638:
12636:
12635:
12630:
12612:
12610:
12609:
12604:
12569:
12567:
12566:
12561:
12549:
12547:
12546:
12541:
12529:
12527:
12526:
12521:
12503:
12501:
12500:
12495:
12466:
12464:
12463:
12458:
12446:
12444:
12443:
12438:
12421:
12419:
12418:
12413:
12383:
12381:
12380:
12375:
12363:
12361:
12360:
12355:
12336:
12334:
12333:
12328:
12317:
12316:
12302:binary operation
12283:
12281:
12280:
12275:
12273:
12272:
12247:
12245:
12244:
12239:
12227:
12225:
12224:
12219:
12208:
12207:
12194:
12192:
12191:
12186:
12168:
12166:
12165:
12160:
12148:
12146:
12145:
12140:
12128:
12126:
12125:
12120:
12108:
12106:
12105:
12100:
12084:
12082:
12081:
12076:
12064:
12062:
12061:
12056:
12041:
12039:
12038:
12033:
12021:
12019:
12018:
12013:
11992:
11990:
11989:
11984:
11972:
11970:
11969:
11964:
11934:
11932:
11931:
11926:
11874:naive set theory
11871:
11869:
11868:
11863:
11850:Bertrand Russell
11847:
11845:
11844:
11839:
11837:
11836:
11820:
11818:
11817:
11812:
11791:
11789:
11788:
11783:
11771:
11769:
11768:
11763:
11758:
11757:
11741:
11739:
11738:
11733:
11721:
11719:
11718:
11713:
11692:
11690:
11689:
11684:
11672:
11670:
11669:
11664:
11652:
11650:
11649:
11644:
11642:
11641:
11625:
11623:
11622:
11617:
11601:
11599:
11598:
11593:
11563:
11561:
11560:
11555:
11543:
11541:
11540:
11535:
11499:
11497:
11496:
11491:
11473:
11471:
11470:
11465:
11443:
11441:
11440:
11435:
11350:are specified):
11349:
11347:
11346:
11341:
11329:
11327:
11326:
11321:
11305:
11303:
11302:
11297:
11285:
11283:
11282:
11277:
11262:
11260:
11259:
11254:
11242:
11240:
11239:
11234:
11218:
11216:
11215:
11210:
11192:
11190:
11189:
11184:
11166:
11164:
11163:
11158:
11130:
11128:
11127:
11122:
11104:
11102:
11101:
11096:
11084:
11082:
11081:
11076:
11064:
11062:
11061:
11056:
11038:
11036:
11035:
11030:
11014:
11012:
11011:
11006:
10994:
10992:
10991:
10986:
10974:
10972:
10971:
10966:
10931:
10929:
10928:
10923:
10911:
10909:
10908:
10903:
10887:
10885:
10884:
10879:
10861:
10859:
10858:
10853:
10835:
10833:
10832:
10827:
10797:are specified):
10796:
10794:
10793:
10788:
10776:
10774:
10773:
10768:
10728:
10726:
10725:
10720:
10708:
10706:
10705:
10700:
10685:
10683:
10682:
10677:
10665:
10663:
10662:
10657:
10642:
10640:
10639:
10634:
10622:
10620:
10619:
10614:
10602:
10600:
10599:
10594:
10578:
10576:
10575:
10570:
10546:partial function
10534:
10532:
10531:
10526:
10508:
10506:
10505:
10500:
10482:
10480:
10479:
10474:
10456:
10454:
10453:
10448:
10421:
10419:
10418:
10413:
10379:
10377:
10376:
10371:
10359:
10357:
10356:
10351:
10339:
10337:
10336:
10331:
10316:
10314:
10313:
10308:
10296:
10294:
10293:
10288:
10276:
10274:
10273:
10268:
10253:
10251:
10250:
10245:
10226:
10224:
10223:
10218:
10198:
10196:
10195:
10190:
10172:
10170:
10169:
10164:
10146:
10144:
10143:
10138:
10120:
10118:
10117:
10112:
10091:
10089:
10088:
10083:
10045:
10043:
10042:
10037:
10025:
10023:
10022:
10017:
10005:
10003:
10002:
9997:
9958:
9956:
9955:
9950:
9935:
9933:
9932:
9927:
9925:
9909:
9907:
9906:
9901:
9890:The elements of
9889:
9887:
9886:
9881:
9876:
9852:
9850:
9849:
9844:
9842:
9830:
9828:
9827:
9822:
9810:
9808:
9807:
9802:
9790:
9788:
9787:
9782:
9780:
9768:
9766:
9765:
9760:
9748:
9746:
9745:
9740:
9729:
9712:
9696:incidence matrix
9685:
9683:
9682:
9677:
9661:
9659:
9658:
9653:
9641:
9639:
9638:
9633:
9568:
9566:
9565:
9560:
9554:
9553:
9545:
9539:
9538:
9530:
9481:
9479:
9478:
9473:
9462:since each time
9445:
9443:
9442:
9437:
9425:
9423:
9422:
9417:
9352:
9350:
9349:
9344:
9326:
9324:
9323:
9318:
9313:
9312:
9311:
9294:
9292:
9291:
9286:
9268:
9266:
9265:
9260:
9248:
9246:
9245:
9240:
9222:
9220:
9219:
9214:
9209:
9208:
9207:
9190:
9188:
9187:
9182:
9180:
9179:
9178:
9156:
9154:
9153:
9148:
9143:
9142:
8964:
8962:
8961:
8956:
8944:
8942:
8941:
8936:
8924:
8922:
8921:
8916:
8894:
8892:
8891:
8886:
8881:
8878:
8873:
8870:
8865:
8862:
8857:
8854:
8849:
8846:
8841:
8838:
8833:
8830:
8808:
8806:
8805:
8800:
8795:
8792:
8787:
8784:
8779:
8776:
8771:
8768:
8617:
8613:
8596:
8594:
8593:
8588:
8576:
8574:
8573:
8568:
8560:
8557:
8545:
8543:
8542:
8537:
8525:
8523:
8522:
8517:
8509:
8506:
8488:
8486:
8485:
8480:
8468:
8466:
8465:
8460:
8448:
8446:
8445:
8440:
8429:
8426:
8415:
8412:
8401:
8398:
8377:
8375:
8374:
8369:
8357:
8355:
8354:
8349:
8337:
8335:
8334:
8329:
8321:
8318:
8301:and four people
8300:
8298:
8297:
8292:
8287:
8284:
8172:
8170:
8169:
8164:
8151:
8149:
8148:
8143:
8129:
8125:
8050:
8048:
8047:
8042:
8029:
8027:
8026:
8021:
8007:
8003:
7983:
7981:
7980:
7975:
7948:Hadamard product
7945:
7943:
7942:
7937:
7925:
7923:
7922:
7917:
7905:
7903:
7902:
7897:
7885:
7883:
7882:
7877:
7856:Boolean semiring
7853:
7851:
7850:
7845:
7833:
7831:
7830:
7825:
7812:logical matrices
7809:
7807:
7806:
7801:
7789:
7787:
7786:
7781:
7761:
7759:
7758:
7753:
7735:
7733:
7732:
7727:
7716:is smaller than
7715:
7713:
7712:
7707:
7694:rational numbers
7691:
7689:
7688:
7683:
7662:
7660:
7659:
7654:
7641:
7640:
7634:
7632:
7631:
7626:
7614:
7612:
7611:
7606:
7594:
7592:
7591:
7586:
7574:
7572:
7571:
7566:
7555:is contained in
7554:
7552:
7551:
7546:
7534:
7532:
7531:
7526:
7504:
7502:
7501:
7496:
7484:
7482:
7481:
7476:
7464:
7462:
7461:
7456:
7445:is contained in
7444:
7442:
7441:
7436:
7424:
7422:
7421:
7416:
7405:is contained in
7404:
7402:
7401:
7396:
7384:
7382:
7381:
7376:
7358:
7356:
7355:
7350:
7332:
7330:
7329:
7324:
7303:
7301:
7300:
7295:
7277:
7275:
7274:
7269:
7257:
7255:
7254:
7249:
7237:
7235:
7234:
7229:
7208:
7206:
7205:
7200:
7188:
7186:
7185:
7180:
7168:
7166:
7165:
7160:
7147:
7146:
7140:
7138:
7137:
7132:
7120:
7118:
7117:
7112:
7100:
7098:
7097:
7092:
7077:
7075:
7074:
7069:
7057:
7055:
7054:
7049:
7044:
7028:
7026:
7025:
7020:
7018:
7002:
7000:
6999:
6994:
6992:
6970:
6968:
6967:
6962:
6939:
6937:
6936:
6931:
6919:
6917:
6916:
6911:
6899:
6897:
6896:
6891:
6879:
6877:
6876:
6871:
6805:
6803:
6802:
6797:
6785:
6783:
6782:
6777:
6765:
6763:
6762:
6757:
6745:
6743:
6742:
6737:
6724:
6723:
6717:
6715:
6714:
6709:
6695:
6692:
6657:
6656:
6637:
6635:
6634:
6629:
6617:
6615:
6614:
6609:
6597:
6595:
6594:
6589:
6577:
6575:
6574:
6569:
6557:
6555:
6554:
6549:
6534:
6532:
6531:
6526:
6514:
6512:
6511:
6506:
6494:
6492:
6491:
6486:
6473:
6472:
6466:
6464:
6463:
6458:
6444:
6441:
6430:
6427:
6392:
6391:
6372:
6370:
6369:
6364:
6352:
6350:
6349:
6344:
6332:
6330:
6329:
6324:
6308:
6306:
6305:
6300:
6259:
6257:
6256:
6251:
6228:
6226:
6225:
6220:
6215:
6214:
6213:
6207:
6206:
6198:
6191:
6186:
6185:
6184:
6174:
6165:
6163:
6162:
6157:
6155:
6154:
6153:
6130:
6128:
6127:
6122:
6110:
6108:
6107:
6102:
6090:
6088:
6087:
6082:
6070:
6068:
6067:
6062:
6046:
6044:
6043:
6038:
6026:
6024:
6023:
6018:
6006:
6004:
6003:
5998:
5986:
5984:
5983:
5978:
5966:
5964:
5963:
5958:
5946:
5944:
5943:
5938:
5926:
5924:
5923:
5918:
5906:
5904:
5903:
5898:
5883:
5881:
5880:
5875:
5863:
5861:
5860:
5855:
5843:
5841:
5840:
5835:
5819:
5817:
5816:
5811:
5796:
5794:
5793:
5788:
5747:
5746:
5738:
5728:
5726:
5725:
5720:
5708:
5706:
5705:
5700:
5688:
5686:
5685:
5680:
5650:
5648:
5647:
5642:
5630:
5628:
5627:
5622:
5610:
5608:
5607:
5602:
5590:
5588:
5587:
5582:
5570:
5568:
5567:
5562:
5550:
5548:
5547:
5542:
5527:
5525:
5524:
5519:
5507:
5505:
5504:
5499:
5487:
5485:
5484:
5479:
5466:inverse relation
5459:
5457:
5456:
5451:
5413:
5412:
5411:
5394:
5392:
5391:
5386:
5374:
5372:
5371:
5366:
5354:
5352:
5351:
5346:
5314:
5312:
5311:
5306:
5294:
5292:
5291:
5286:
5274:
5272:
5271:
5266:
5254:
5252:
5251:
5246:
5234:
5232:
5231:
5226:
5214:
5212:
5211:
5206:
5194:
5192:
5191:
5186:
5174:
5172:
5171:
5166:
5150:
5148:
5147:
5142:
5122:in the notation
5121:
5119:
5118:
5113:
5101:
5099:
5098:
5093:
5078:
5076:
5075:
5070:
5058:
5056:
5055:
5050:
5038:
5036:
5035:
5030:
5018:
5016:
5015:
5010:
4994:
4992:
4991:
4986:
4968:
4966:
4965:
4960:
4946:
4943:
4932:
4929:
4918:
4915:
4873:
4871:
4870:
4865:
4853:
4851:
4850:
4845:
4833:
4831:
4830:
4825:
4813:
4811:
4810:
4805:
4793:
4791:
4790:
4785:
4773:
4771:
4770:
4765:
4736:
4734:
4733:
4728:
4716:
4714:
4713:
4708:
4696:
4694:
4693:
4688:
4676:
4674:
4673:
4668:
4652:
4650:
4649:
4644:
4630:
4627:
4576:
4574:
4573:
4568:
4556:
4554:
4553:
4548:
4536:
4534:
4533:
4528:
4516:
4514:
4513:
4508:
4488:
4486:
4485:
4480:
4468:
4466:
4465:
4460:
4445:
4443:
4442:
4437:
4425:
4423:
4422:
4417:
4405:
4403:
4402:
4397:
4385:
4383:
4382:
4377:
4361:
4359:
4358:
4353:
4339:
4336:
4285:
4283:
4282:
4277:
4265:
4263:
4262:
4257:
4245:
4243:
4242:
4237:
4225:
4223:
4222:
4217:
4192:
4190:
4189:
4184:
4173:does not divide
4172:
4170:
4169:
4164:
4152:
4150:
4149:
4144:
4132:
4130:
4129:
4124:
4112:
4110:
4109:
4104:
4086:
4084:
4083:
4078:
4060:
4058:
4057:
4052:
4031:
4029:
4028:
4023:
4011:
4009:
4008:
4003:
3991:
3989:
3988:
3983:
3971:
3969:
3968:
3963:
3951:
3949:
3948:
3943:
3902:
3900:
3899:
3894:
3850:
3848:
3847:
3842:
3830:
3828:
3827:
3822:
3800:
3798:
3797:
3792:
3768:
3766:
3765:
3760:
3744:
3742:
3741:
3736:
3724:
3722:
3721:
3716:
3698:
3696:
3695:
3690:
3678:
3676:
3675:
3670:
3642:
3640:
3639:
3634:
3622:
3620:
3619:
3614:
3596:
3594:
3593:
3588:
3576:
3574:
3573:
3568:
3548:
3546:
3545:
3540:
3522:
3520:
3519:
3514:
3502:
3500:
3499:
3494:
3482:
3480:
3479:
3474:
3462:
3460:
3459:
3454:
3420:
3418:
3417:
3412:
3394:
3392:
3391:
3386:
3374:
3372:
3371:
3366:
3328:
3326:
3325:
3320:
3308:
3306:
3305:
3300:
3288:
3286:
3285:
3280:
3260:
3258:
3257:
3252:
3240:
3238:
3237:
3232:
3219:set of departure
3212:
3210:
3209:
3204:
3192:
3190:
3189:
3184:
3163:
3161:
3160:
3155:
3143:
3141:
3140:
3135:
3123:
3121:
3120:
3115:
3093:
3091:
3090:
3085:
3068:
3065:
3026:
3024:
3023:
3018:
3000:
2998:
2997:
2992:
2980:
2978:
2977:
2972:
2953:
2951:
2950:
2945:
2940:
2939:
2921:
2920:
2901:
2899:
2898:
2893:
2891:
2890:
2872:
2871:
2853:
2851:
2850:
2845:
2832:
2830:
2829:
2824:
2787:complete lattice
2737:
2735:
2734:
2729:
2711:
2709:
2708:
2703:
2681:
2679:
2678:
2673:
2648:
2646:
2645:
2640:
2624:
2622:
2621:
2616:
2591:
2589:
2588:
2583:
2571:
2569:
2568:
2563:
2547:computer science
2533:to" relation in
2484:
2482:
2481:
2476:
2464:
2462:
2461:
2456:
2444:
2442:
2441:
2436:
2424:
2422:
2421:
2416:
2404:
2402:
2401:
2396:
2384:
2382:
2381:
2376:
2364:
2362:
2361:
2356:
2344:
2342:
2341:
2336:
2324:
2322:
2321:
2316:
2304:
2302:
2301:
2296:
2284:
2282:
2281:
2276:
2264:
2262:
2261:
2256:
2241:
2239:
2238:
2233:
2221:
2219:
2218:
2213:
2197:
2195:
2194:
2189:
2173:
2171:
2170:
2165:
2153:
2151:
2150:
2145:
2133:
2131:
2130:
2125:
2123:
2108:
2106:
2105:
2100:
2098:
2076:
2074:
2073:
2068:
2040:
2038:
2037:
2032:
2016:
2014:
2013:
2008:
1996:
1994:
1993:
1988:
1976:
1974:
1973:
1968:
1956:
1954:
1953:
1948:
1936:
1934:
1933:
1928:
1916:
1914:
1913:
1908:
1881:
1879:
1878:
1873:
1861:
1859:
1858:
1853:
1813:
1811:
1810:
1805:
1784:
1782:
1781:
1776:
1758:
1756:
1755:
1750:
1732:
1730:
1729:
1724:
1693:
1691:
1690:
1685:
1667:
1663:
1660:
1659:
1654:
1650:
1647:
1646:
1634:
1632:
1631:
1626:
1624:
1611:
1608:
1578:
1576:
1575:
1570:
1559:
1556:
1546:
1544:
1543:
1538:
1519:
1517:
1516:
1511:
1509:
1505:
1502:
1475:
1473:
1472:
1467:
1465:
1461:
1458:
1431:
1429:
1428:
1423:
1421:
1417:
1414:
1390:
1388:
1387:
1382:
1380:
1365:
1362:
1339:
1319:
1317:
1316:
1311:
1309:
1300:
1274:
1271:
1248:
1246:
1245:
1240:
1238:
1223:
1204:
1192:
1190:
1189:
1184:
1163:
1161:
1160:
1155:
1084:
1081:
1080:
1074:
1071:
1070:
1064:
1059:
1054:
1049:
1044:
1041:
1040:
1034:
1031:
1030:
1024:
1012:
1009:
1008:
1002:
999:
998:
992:
987:
982:
977:
972:
967:
964:
963:
957:
945:
942:
941:
935:
932:
931:
925:
920:
915:
910:
905:
900:
897:
896:
890:
878:
873:
868:
865:
864:
858:
855:
854:
848:
843:
838:
833:
830:
829:
823:
817:Meet-semilattice
811:
806:
801:
798:
797:
791:
786:
783:
782:
776:
771:
766:
763:
762:
756:
750:Join-semilattice
744:
739:
734:
731:
730:
724:
721:
720:
714:
711:
710:
704:
699:
694:
691:
690:
684:
672:
667:
662:
659:
658:
652:
647:
642:
639:
638:
632:
629:
628:
622:
619:
618:
612:
600:
595:
590:
587:
586:
580:
575:
570:
567:
566:
560:
555:
550:
545:
536:
531:
526:
523:
522:
516:
511:
506:
503:
502:
496:
493:
492:
486:
481:
469:
464:
459:
456:
455:
449:
444:
439:
434:
431:
430:
424:
421:
420:
414:
402:
397:
392:
389:
388:
382:
377:
372:
367:
364:
363:
357:
352:
340:
335:
330:
327:
326:
320:
315:
310:
305:
300:
297:
296:
290:
278:
273:
268:
265:
264:
258:
253:
248:
243:
238:
233:
228:
226:
216:
211:
206:
203:
202:
196:
191:
186:
181:
176:
171:
168:
167:
161:
79:
78:
69:
62:
55:
48:
46:binary relations
37:
36:
21:
21192:
21191:
21187:
21186:
21185:
21183:
21182:
21181:
21167:
21166:
21165:
21160:
21087:
21066:
21050:
21015:New Foundations
20962:
20852:
20771:Cardinal number
20754:
20740:
20681:
20572:
20563:
20547:
20542:
20512:
20507:
20503:Young's lattice
20359:
20287:
20226:
20076:Heyting algebra
20024:Boolean algebra
19996:
19977:Laver's theorem
19925:
19891:Boolean algebra
19886:Binary relation
19874:
19851:
19846:
19799:
19784:
19774:
19744:
19742:
19686:
19667:
19644:
19617:
19611:
19596:
19567:
19541:
19525:
19520:
19519:
19500:
19496:
19471:
19467:
19449:
19445:
19437:does not mean "
19422:
19419:
19418:
19416:
19412:
19392:Gunther Schmidt
19390:
19381:
19366:
19362:
19343:
19339:
19332:
19313:
19309:
19300:
19296:
19289:
19263:
19259:
19243:Gunther Schmidt
19236:
19232:
19225:
19211:
19207:
19170:Semigroup Forum
19166:
19162:
19155:
19138:
19134:
19115:Riguet, Jacques
19112:
19108:
19098:5827, Springer
19088:
19084:
19062:
19058:
19039:
19035:
19018:
19014:
19005:
18998:
18990:
18986:
18971:
18967:
18961:
18947:
18943:
18936:
18922:
18918:
18911:
18897:
18893:
18886:
18863:
18859:
18844:
18826:
18822:
18809:
18803:
18799:
18794:
18790:
18785:
18781:
18772:
18770:
18762:
18761:
18757:
18748:
18746:
18738:
18737:
18733:
18728:
18715:
18710:
18699:
18668:
18664:
18651:
18647:
18630:
18626:
18616:
18614:
18590:
18586:
18574:
18570:
18558:
18554:
18545:
18541:
18522:
18518:
18511:
18497:
18488:
18481:
18462:
18458:
18451:
18432:
18428:
18421:
18404:
18400:
18393:
18376:
18372:
18365:
18347:Suppes, Patrick
18344:
18340:
18324:
18310:
18306:
18298:
18294:
18276:Gunther Schmidt
18274:
18267:
18255:
18248:
18233:
18226:
18217:
18215:
18213:mathinsight.org
18207:
18206:
18202:
18193:
18191:
18187:
18156:
18147:
18128:
18120:
18113:
18107:
18103:
18098:
18093:
18092:
18086:prefix notation
18068:
18064:
18055:
18051:
18046:
18043:
18042:
18020:
18017:
18016:
18000:
17997:
17996:
17980:
17977:
17976:
17974:
17970:
17965:
17960:
17898:
17892:
17889:
17864:
17861:
17860:
17844:
17841:
17840:
17824:
17821:
17820:
17787:
17784:
17783:
17761:
17760:
17756:
17754:
17751:
17750:
17729:
17728:
17724:
17695:
17692:
17691:
17656:
17655:
17653:
17650:
17649:
17633:
17630:
17629:
17613:
17610:
17609:
17608:Given two sets
17606:
17600:
17575:
17572:
17571:
17537:
17534:
17533:
17514:
17511:
17510:
17482:
17479:
17478:
17456:
17455:
17447:
17444:
17443:
17427:
17424:
17423:
17395:
17392:
17391:
17367:
17364:
17363:
17335:
17332:
17331:
17315:
17312:
17311:
17277:
17274:
17273:
17257:
17254:
17253:
17221:
17220:
17209:
17208:
17207:
17203:
17201:
17175:
17172:
17171:
17151:
17148:
17147:
17144:
17099:
17098:
17094:
17092:
17084:
17081:
17080:
17061:
17058:
17057:
17041:
17038:
17037:
17018:
17015:
17014:
16941:
16938:
16937:
16907:
16906:
16899:
16898:
16894:
16893:
16891:
16853:
16850:
16849:
16799:
16797:
16783:
16782:
16775:
16774:
16770:
16765:
16762:
16761:
16707:
16704:
16703:
16665:
16662:
16661:
16630:
16627:
16626:
16598:
16595:
16594:
16536:
16533:
16532:
16498:
16495:
16494:
16454:
16453:
16446:
16445:
16441:
16440:
16438:
16416:
16415:
16401:
16400:
16393:
16392:
16388:
16386:
16383:
16382:
16357:
16356:
16354:
16351:
16350:
16332:
16331:
16327:
16325:
16322:
16321:
16299:
16298:
16291:
16290:
16286:
16284:
16281:
16280:
16249:
16248:
16241:
16240:
16236:
16235:
16233:
16219:
16216:
16215:
16189:
16186:
16185:
16166:
16163:
16162:
16161:Every relation
16159:
16138:
16135:
16134:
16118:
16115:
16114:
16077:
16076:
16072:
16070:
16053:
16052:
16038:
16037:
16030:
16029:
16025:
16023:
16020:
16019:
15996:
15993:
15992:
15973:
15970:
15969:
15934:
15922: and
15920:
15894:
15892:
15889:
15888:
15859:
15847: and
15845:
15834:
15831:
15830:
15801:
15769:
15767:
15764:
15763:
15740:
15737:
15736:
15720:
15717:
15716:
15696:
15693:
15692:
15672:
15669:
15668:
15665:
15642:
15641:
15637:
15623:
15622:
15614:
15611:
15610:
15578:
15577:
15566:
15565:
15564:
15559:
15556:
15555:
15545:Ferrers diagram
15525:
15480:
15476:
15464:
15460:
15458:
15455:
15454:
15431:
15427:
15418:
15414:
15412:
15409:
15408:
15388:
15384:
15382:
15379:
15378:
15358:
15354:
15352:
15349:
15348:
15332:
15329:
15328:
15285:
15282:
15281:
15262:
15259:
15258:
15241:
15237:
15235:
15232:
15231:
15215:
15212:
15211:
15194:
15190:
15188:
15185:
15184:
15167:
15163:
15154:
15150:
15148:
15145:
15144:
15122:
15119:
15118:
15102:
15099:
15098:
15083:automata theory
15052:
15051:
15047:
15042:
15039:
15038:
15010:
15009:
15005:
15000:
14997:
14996:
14957: and
14955:
14938:
14935:
14934:
14908:
14907:
14903:
14892:
14889:
14888:
14838:
14835:
14834:
14827:
14799:
14796:
14795:
14779:
14776:
14775:
14757:
14756:
14752:
14741:
14738:
14737:
14717:
14714:
14713:
14685:
14682:
14681:
14665:
14662:
14661:
14640:
14639:
14635:
14627:
14624:
14623:
14605:
14604:
14600:
14598:
14595:
14594:
14591:serial relation
14574:
14571:
14570:
14563:
14527:
14524:
14523:
14507:
14504:
14503:
14483:
14480:
14479:
14445:
14442:
14441:
14425:
14422:
14421:
14397:
14394:
14393:
14377:
14374:
14373:
14355:
14354:
14350:
14336:
14333:
14332:
14298:
14295:
14294:
14263:
14260:
14259:
14235:
14232:
14231:
14215:
14212:
14211:
14206:logical vectors
14176:
14172:
14166:
14162:
14150:
14146:
14144:
14141:
14140:
14120:
14117:
14116:
14086:
14083:
14082:
14072:
14044:category theory
14008:
14005:
14004:
13938:
13937:
13914:
13911:
13910:
13884:
13881:
13880:
13858:
13855:
13854:
13829:
13826:
13825:
13820:and the use of
13814:algebra of sets
13806:algebraic logic
13802:
13780:
13777:
13776:
13760:
13757:
13756:
13726:
13723:
13722:
13706:
13703:
13702:
13676:
13673:
13672:
13656:
13653:
13652:
13625:
13622:
13621:
13611:Euclidean plane
13594:
13591:
13590:
13589:is parallel to
13574:
13571:
13570:
13551:
13549:
13546:
13545:
13523:
13520:
13519:
13500:
13498:
13495:
13494:
13492:natural numbers
13475:
13472:
13471:
13455:
13452:
13451:
13395:
13392:
13391:
13369:
13366:
13365:
13343:
13340:
13339:
13314:
13311:
13310:
13279:
13276:
13275:
13245:
13242:
13241:
13219:
13216:
13215:
13185:
13182:
13181:
13151:
13148:
13147:
13125:
13122:
13121:
13099:
13096:
13095:
13064:
13061:
13060:
13030:
13027:
13026:
13004:
13001:
13000:
12978:
12975:
12974:
12937:
12934:
12933:
12909:
12906:
12905:
12883:
12880:
12879:
12857:
12854:
12853:
12822:
12819:
12818:
12794:
12791:
12790:
12765:
12762:
12761:
12739:
12736:
12735:
12713:
12710:
12709:
12678:
12675:
12674:
12644:
12641:
12640:
12618:
12615:
12614:
12583:
12580:
12579:
12555:
12552:
12551:
12535:
12532:
12531:
12530:. For example,
12509:
12506:
12505:
12480:
12477:
12476:
12452:
12449:
12448:
12447:. For example,
12426:
12423:
12422:
12398:
12395:
12394:
12369:
12366:
12365:
12349:
12346:
12345:
12312:
12311:
12309:
12306:
12305:
12286:Boolean algebra
12262:
12258:
12256:
12253:
12252:
12233:
12230:
12229:
12203:
12202:
12200:
12197:
12196:
12174:
12171:
12170:
12169:if and only if
12154:
12151:
12150:
12134:
12131:
12130:
12114:
12111:
12110:
12094:
12091:
12090:
12070:
12067:
12066:
12050:
12047:
12046:
12027:
12024:
12023:
11998:
11995:
11994:
11978:
11975:
11974:
11958:
11955:
11954:
11947:
11941:
11902:
11899:
11898:
11857:
11854:
11853:
11848:that is a set.
11832:
11828:
11826:
11823:
11822:
11797:
11794:
11793:
11777:
11774:
11773:
11753:
11749:
11747:
11744:
11743:
11727:
11724:
11723:
11698:
11695:
11694:
11678:
11675:
11674:
11658:
11655:
11654:
11637:
11633:
11631:
11628:
11627:
11611:
11608:
11607:
11587:
11584:
11583:
11575:
11566:ordinal numbers
11549:
11546:
11545:
11505:
11502:
11501:
11479:
11476:
11475:
11453:
11450:
11449:
11423:
11420:
11419:
11335:
11332:
11331:
11315:
11312:
11311:
11291:
11288:
11287:
11268:
11265:
11264:
11248:
11245:
11244:
11228:
11225:
11224:
11198:
11195:
11194:
11172:
11169:
11168:
11146:
11143:
11142:
11110:
11107:
11106:
11090:
11087:
11086:
11070:
11067:
11066:
11044:
11041:
11040:
11024:
11021:
11020:
11000:
10997:
10996:
10980:
10977:
10976:
10957:
10954:
10953:
10917:
10914:
10913:
10897:
10894:
10893:
10867:
10864:
10863:
10841:
10838:
10837:
10836:there exists a
10815:
10812:
10811:
10782:
10779:
10778:
10762:
10759:
10758:
10714:
10711:
10710:
10691:
10688:
10687:
10671:
10668:
10667:
10648:
10645:
10644:
10628:
10625:
10624:
10608:
10605:
10604:
10588:
10585:
10584:
10558:
10555:
10554:
10551:partial mapping
10514:
10511:
10510:
10488:
10485:
10484:
10462:
10459:
10458:
10427:
10424:
10423:
10401:
10398:
10397:
10365:
10362:
10361:
10345:
10342:
10341:
10322:
10319:
10318:
10302:
10299:
10298:
10282:
10279:
10278:
10259:
10256:
10255:
10239:
10236:
10235:
10212:
10209:
10208:
10178:
10175:
10174:
10152:
10149:
10148:
10126:
10123:
10122:
10097:
10094:
10093:
10065:
10062:
10061:
10031:
10028:
10027:
10011:
10008:
10007:
9991:
9988:
9987:
9972:
9967:
9944:
9941:
9940:
9939:, and those of
9921:
9919:
9916:
9915:
9910:will be called
9895:
9892:
9891:
9872:
9858:
9855:
9854:
9838:
9836:
9833:
9832:
9816:
9813:
9812:
9796:
9793:
9792:
9776:
9774:
9771:
9770:
9754:
9751:
9750:
9725:
9708:
9706:
9703:
9702:
9671:
9668:
9667:
9647:
9644:
9643:
9603:
9600:
9599:
9597:Steiner systems
9544:
9543:
9529:
9528:
9505:
9502:
9501:
9467:
9464:
9463:
9431:
9428:
9427:
9411:
9408:
9407:
9390:bipartite graph
9332:
9329:
9328:
9307:
9306:
9302:
9300:
9297:
9296:
9274:
9271:
9270:
9254:
9251:
9250:
9228:
9225:
9224:
9203:
9202:
9198:
9196:
9193:
9192:
9174:
9173:
9169:
9164:
9161:
9160:
9137:
9136:
9131:
9126:
9121:
9116:
9111:
9106:
9100:
9099:
9094:
9089:
9084:
9079:
9074:
9069:
9063:
9062:
9057:
9052:
9047:
9042:
9037:
9032:
9026:
9025:
9020:
9015:
9010:
9005:
9000:
8995:
8985:
8984:
8976:
8973:
8972:
8950:
8947:
8946:
8930:
8927:
8926:
8904:
8901:
8900:
8877:
8869:
8861:
8853:
8845:
8837:
8829:
8818:
8815:
8814:
8791:
8783:
8775:
8767:
8756:
8753:
8752:
8582:
8579:
8578:
8556:
8551:
8548:
8547:
8531:
8528:
8527:
8505:
8494:
8491:
8490:
8474:
8471:
8470:
8454:
8451:
8450:
8425:
8411:
8397:
8383:
8380:
8379:
8363:
8360:
8359:
8343:
8340:
8339:
8317:
8306:
8303:
8302:
8283:
8272:
8269:
8268:
8173:
8158:
8155:
8154:
8152:
8137:
8134:
8133:
8051:
8036:
8033:
8032:
8030:
8015:
8012:
8011:
8002:
7994:identity matrix
7986:matrix semiring
7963:
7960:
7959:
7931:
7928:
7927:
7911:
7908:
7907:
7891:
7888:
7887:
7871:
7868:
7867:
7860:matrix addition
7839:
7836:
7835:
7819:
7816:
7815:
7795:
7792:
7791:
7775:
7772:
7771:
7768:
7741:
7738:
7737:
7721:
7718:
7717:
7701:
7698:
7697:
7696:, the relation
7668:
7665:
7664:
7648:
7645:
7644:
7638:
7637:
7620:
7617:
7616:
7600:
7597:
7596:
7580:
7577:
7576:
7560:
7557:
7556:
7540:
7537:
7536:
7514:
7511:
7510:
7490:
7487:
7486:
7470:
7467:
7466:
7450:
7447:
7446:
7430:
7427:
7426:
7410:
7407:
7406:
7390:
7387:
7386:
7364:
7361:
7360:
7338:
7335:
7334:
7309:
7306:
7305:
7283:
7280:
7279:
7263:
7260:
7259:
7258:is a subset of
7243:
7240:
7239:
7214:
7211:
7210:
7194:
7191:
7190:
7174:
7171:
7170:
7154:
7151:
7150:
7144:
7143:
7126:
7123:
7122:
7106:
7103:
7102:
7086:
7083:
7082:
7063:
7060:
7059:
7040:
7038:
7035:
7034:
7014:
7012:
7009:
7008:
6988:
6980:
6977:
6976:
6956:
6953:
6952:
6925:
6922:
6921:
6905:
6902:
6901:
6885:
6882:
6881:
6865:
6862:
6861:
6813:, irreflexive,
6791:
6788:
6787:
6771:
6768:
6767:
6751:
6748:
6747:
6731:
6728:
6727:
6721:
6720:
6693: and
6691:
6649:
6645:
6643:
6640:
6639:
6623:
6620:
6619:
6618:is a subset of
6603:
6600:
6599:
6583:
6580:
6579:
6563:
6560:
6559:
6543:
6540:
6539:
6520:
6517:
6516:
6500:
6497:
6496:
6480:
6477:
6476:
6470:
6469:
6442: and
6440:
6428: and
6426:
6384:
6380:
6378:
6375:
6374:
6358:
6355:
6354:
6353:is a subset of
6338:
6335:
6334:
6318:
6315:
6314:
6294:
6291:
6290:
6287:
6281:
6236:
6233:
6232:
6209:
6208:
6197:
6196:
6195:
6180:
6179:
6175:
6173:
6171:
6168:
6167:
6149:
6148:
6144:
6142:
6139:
6138:
6116:
6113:
6112:
6096:
6093:
6092:
6076:
6073:
6072:
6056:
6053:
6052:
6032:
6029:
6028:
6012:
6009:
6008:
5992:
5989:
5988:
5972:
5969:
5968:
5952:
5949:
5948:
5932:
5929:
5928:
5912:
5909:
5908:
5892:
5889:
5888:
5869:
5866:
5865:
5849:
5846:
5845:
5829:
5826:
5825:
5802:
5799:
5798:
5737:
5736:
5734:
5731:
5730:
5714:
5711:
5710:
5694:
5691:
5690:
5674:
5671:
5670:
5667:
5661:
5636:
5633:
5632:
5616:
5613:
5612:
5596:
5593:
5592:
5576:
5573:
5572:
5556:
5553:
5552:
5536:
5533:
5532:
5513:
5510:
5509:
5493:
5490:
5489:
5473:
5470:
5469:
5407:
5406:
5402:
5400:
5397:
5396:
5380:
5377:
5376:
5360:
5357:
5356:
5340:
5337:
5336:
5333:
5327:
5321:
5300:
5297:
5296:
5280:
5277:
5276:
5260:
5257:
5256:
5240:
5237:
5236:
5220:
5217:
5216:
5200:
5197:
5196:
5180:
5177:
5176:
5160:
5157:
5156:
5127:
5124:
5123:
5107:
5104:
5103:
5087:
5084:
5083:
5064:
5061:
5060:
5044:
5041:
5040:
5024:
5021:
5020:
5004:
5001:
5000:
4974:
4971:
4970:
4944: and
4942:
4928:
4914:
4879:
4876:
4875:
4859:
4856:
4855:
4839:
4836:
4835:
4819:
4816:
4815:
4799:
4796:
4795:
4779:
4776:
4775:
4759:
4756:
4755:
4752:
4746:
4722:
4719:
4718:
4702:
4699:
4698:
4682:
4679:
4678:
4662:
4659:
4658:
4628: and
4626:
4582:
4579:
4578:
4562:
4559:
4558:
4542:
4539:
4538:
4522:
4519:
4518:
4502:
4499:
4498:
4495:
4474:
4471:
4470:
4454:
4451:
4450:
4431:
4428:
4427:
4411:
4408:
4407:
4391:
4388:
4387:
4371:
4368:
4367:
4335:
4291:
4288:
4287:
4271:
4268:
4267:
4251:
4248:
4247:
4231:
4228:
4227:
4211:
4208:
4207:
4204:
4199:
4178:
4175:
4174:
4158:
4155:
4154:
4138:
4135:
4134:
4118:
4115:
4114:
4113:. For example,
4092:
4089:
4088:
4066:
4063:
4062:
4040:
4037:
4036:
4017:
4014:
4013:
3997:
3994:
3993:
3977:
3974:
3973:
3957:
3954:
3953:
3931:
3928:
3927:
3920:dyadic relation
3879:
3876:
3875:
3836:
3833:
3832:
3816:
3813:
3812:
3777:
3774:
3773:
3754:
3751:
3750:
3730:
3727:
3726:
3704:
3701:
3700:
3684:
3681:
3680:
3664:
3661:
3660:
3649:active codomain
3628:
3625:
3624:
3602:
3599:
3598:
3582:
3579:
3578:
3562:
3559:
3558:
3528:
3525:
3524:
3508:
3505:
3504:
3488:
3485:
3484:
3468:
3465:
3464:
3430:
3427:
3426:
3400:
3397:
3396:
3395:is a subset of
3380:
3377:
3376:
3342:
3339:
3338:
3331:binary relation
3314:
3311:
3310:
3294:
3291:
3290:
3274:
3271:
3270:
3246:
3243:
3242:
3226:
3223:
3222:
3198:
3195:
3194:
3169:
3166:
3165:
3164:is a subset of
3149:
3146:
3145:
3129:
3126:
3125:
3109:
3106:
3105:
3103:binary relation
3066: and
3064:
3032:
3029:
3028:
3006:
3003:
3002:
2986:
2983:
2982:
2966:
2963:
2962:
2959:
2935:
2931:
2916:
2912:
2910:
2907:
2906:
2886:
2882:
2867:
2863:
2861:
2858:
2857:
2839:
2836:
2835:
2812:
2809:
2808:
2779:Gunther Schmidt
2755:algebra of sets
2751:complementation
2717:
2714:
2713:
2691:
2688:
2687:
2658:
2655:
2654:
2634:
2631:
2630:
2601:
2598:
2597:
2577:
2574:
2573:
2557:
2554:
2553:
2513:is congruent to
2498:is greater than
2470:
2467:
2466:
2450:
2447:
2446:
2430:
2427:
2426:
2410:
2407:
2406:
2390:
2387:
2386:
2370:
2367:
2366:
2350:
2347:
2346:
2330:
2327:
2326:
2310:
2307:
2306:
2290:
2287:
2286:
2270:
2267:
2266:
2247:
2244:
2243:
2227:
2224:
2223:
2207:
2204:
2203:
2183:
2180:
2179:
2159:
2156:
2155:
2139:
2136:
2135:
2119:
2117:
2114:
2113:
2109:and the set of
2094:
2092:
2089:
2088:
2050:
2047:
2046:
2026:
2023:
2022:
2002:
1999:
1998:
1982:
1979:
1978:
1962:
1959:
1958:
1942:
1939:
1938:
1922:
1919:
1918:
1890:
1887:
1886:
1867:
1864:
1863:
1847:
1844:
1843:
1828:binary relation
1814:
1790:
1787:
1786:
1764:
1761:
1760:
1738:
1735:
1734:
1703:
1700:
1699:
1679:
1676:
1675:
1669:
1661:
1657:
1648:
1644:
1622:
1621:
1607:
1604:
1603:
1587:
1585:
1582:
1581:
1555:
1553:
1550:
1549:
1526:
1523:
1522:
1507:
1506:
1501:
1498:
1497:
1484:
1482:
1479:
1478:
1463:
1462:
1457:
1454:
1453:
1440:
1438:
1435:
1434:
1419:
1418:
1413:
1410:
1409:
1399:
1397:
1394:
1393:
1378:
1377:
1366:
1361:
1349:
1348:
1340:
1338:
1328:
1326:
1323:
1322:
1307:
1306:
1301:
1299:
1287:
1286:
1275:
1272: and
1270:
1257:
1255:
1252:
1251:
1236:
1235:
1224:
1222:
1216:
1215:
1201:
1199:
1196:
1195:
1169:
1166:
1165:
1143:
1140:
1139:
1082:
1078:
1072:
1068:
1042:
1038:
1032:
1028:
1010:
1006:
1000:
996:
965:
961:
943:
939:
933:
929:
898:
894:
866:
862:
856:
852:
831:
827:
799:
795:
784:
780:
764:
760:
732:
728:
722:
718:
712:
708:
692:
688:
660:
656:
640:
636:
630:
626:
620:
616:
588:
584:
568:
564:
541:
524:
520:
504:
500:
494:
490:
475:Prewellordering
457:
453:
432:
428:
422:
418:
390:
386:
365:
361:
328:
324:
298:
294:
266:
262:
224:
221:
204:
200:
169:
165:
157:
149:
73:
40:
35:
28:
23:
22:
15:
12:
11:
5:
21190:
21180:
21179:
21162:
21161:
21159:
21158:
21153:
21151:Thoralf Skolem
21148:
21143:
21138:
21133:
21128:
21123:
21118:
21113:
21108:
21103:
21097:
21095:
21089:
21088:
21086:
21085:
21080:
21075:
21069:
21067:
21065:
21064:
21061:
21055:
21052:
21051:
21049:
21048:
21047:
21046:
21041:
21036:
21035:
21034:
21019:
21018:
21017:
21005:
21004:
21003:
20992:
20991:
20986:
20981:
20976:
20970:
20968:
20964:
20963:
20961:
20960:
20955:
20950:
20945:
20936:
20931:
20926:
20916:
20911:
20910:
20909:
20904:
20899:
20889:
20879:
20874:
20869:
20863:
20861:
20854:
20853:
20851:
20850:
20845:
20840:
20835:
20833:Ordinal number
20830:
20825:
20820:
20815:
20814:
20813:
20808:
20798:
20793:
20788:
20783:
20778:
20768:
20763:
20757:
20755:
20753:
20752:
20749:
20745:
20742:
20741:
20739:
20738:
20733:
20728:
20723:
20718:
20713:
20711:Disjoint union
20708:
20703:
20697:
20691:
20689:
20683:
20682:
20680:
20679:
20678:
20677:
20672:
20661:
20660:
20658:Martin's axiom
20655:
20650:
20645:
20640:
20635:
20630:
20625:
20623:Extensionality
20620:
20615:
20610:
20609:
20608:
20603:
20598:
20588:
20582:
20580:
20574:
20573:
20566:
20564:
20562:
20561:
20555:
20553:
20549:
20548:
20541:
20540:
20533:
20526:
20518:
20509:
20508:
20506:
20505:
20500:
20495:
20494:
20493:
20483:
20482:
20481:
20476:
20471:
20461:
20460:
20459:
20449:
20444:
20443:
20442:
20437:
20430:Order morphism
20427:
20426:
20425:
20415:
20410:
20405:
20400:
20395:
20394:
20393:
20383:
20378:
20373:
20367:
20365:
20361:
20360:
20358:
20357:
20356:
20355:
20350:
20348:Locally convex
20345:
20340:
20330:
20328:Order topology
20325:
20324:
20323:
20321:Order topology
20318:
20308:
20298:
20296:
20289:
20288:
20286:
20285:
20280:
20275:
20270:
20265:
20260:
20255:
20250:
20245:
20240:
20234:
20232:
20228:
20227:
20225:
20224:
20214:
20204:
20199:
20194:
20189:
20184:
20179:
20174:
20169:
20168:
20167:
20157:
20152:
20151:
20150:
20145:
20140:
20135:
20133:Chain-complete
20125:
20120:
20119:
20118:
20113:
20108:
20103:
20098:
20088:
20083:
20078:
20073:
20068:
20058:
20053:
20048:
20043:
20038:
20033:
20032:
20031:
20021:
20016:
20010:
20008:
19998:
19997:
19995:
19994:
19989:
19984:
19979:
19974:
19969:
19964:
19959:
19954:
19949:
19944:
19939:
19933:
19931:
19927:
19926:
19924:
19923:
19918:
19913:
19908:
19903:
19898:
19893:
19888:
19882:
19880:
19876:
19875:
19873:
19872:
19867:
19862:
19856:
19853:
19852:
19845:
19844:
19837:
19830:
19822:
19816:
19815:
19797:
19783:
19782:External links
19780:
19779:
19778:
19772:
19750:
19708:(2): 317β178.
19690:
19684:
19671:
19665:
19648:
19642:
19634:Academic Press
19622:
19610:978-0201141924
19609:
19601:Addison-Wesley
19584:
19574:Ernst SchrΓΆder
19571:
19565:
19545:
19539:
19524:
19521:
19518:
19517:
19494:
19478:Springer books
19465:
19443:
19439:set difference
19426:
19410:
19379:
19360:
19337:
19330:
19307:
19303:Comptes Rendus
19294:
19287:
19257:
19230:
19223:
19205:
19160:
19153:
19132:
19123:Comptes rendus
19106:
19082:
19056:
19047:126(1): 37β82
19033:
19012:
19008:on 2013-11-02.
18984:
18965:
18959:
18941:
18934:
18916:
18909:
18891:
18884:
18866:Tarski, Alfred
18857:
18842:
18820:
18797:
18788:
18779:
18755:
18731:
18713:
18697:
18676:Lenz, Hanfried
18670:Beth, Thomas;
18662:
18645:
18624:
18595:(6 Nov 2001).
18584:
18568:
18552:
18539:
18529:Springer books
18516:
18509:
18486:
18479:
18456:
18449:
18426:
18419:
18398:
18391:
18370:
18363:
18338:
18336:Sect.II.Β§1.1.4
18322:
18304:
18302:, Ch 3. pg. 40
18292:
18265:
18246:
18235:Ernst SchrΓΆder
18224:
18200:
18167:(6): 377β387.
18126:
18100:
18099:
18097:
18094:
18091:
18090:
18071:
18067:
18063:
18058:
18054:
18050:
18030:
18027:
18024:
18015:usually write
18004:
17984:
17967:
17966:
17964:
17961:
17959:
17958:
17952:
17946:
17940:
17934:
17928:
17922:
17916:
17911:
17905:
17899:
17897:
17894:
17887:
17874:
17871:
17868:
17848:
17828:
17791:
17764:
17759:
17738:
17727:
17723:
17720:
17717:
17714:
17711:
17708:
17705:
17702:
17699:
17676:
17673:
17670:
17667:
17664:
17659:
17637:
17617:
17602:Main article:
17599:
17596:
17579:
17559:
17556:
17553:
17550:
17547:
17544:
17541:
17518:
17498:
17495:
17492:
17489:
17486:
17463:
17460:
17454:
17451:
17431:
17411:
17408:
17405:
17402:
17399:
17371:
17351:
17348:
17345:
17342:
17339:
17319:
17299:
17296:
17293:
17290:
17287:
17284:
17281:
17261:
17239:
17234:
17230:
17216:
17213:
17206:
17200:
17197:
17194:
17191:
17188:
17185:
17182:
17179:
17155:
17143:
17140:
17139:
17138:
17127:
17124:
17121:
17118:
17113:
17106:
17103:
17097:
17091:
17088:
17065:
17056:on subsets of
17045:
17022:
17003:
17002:
16990:
16987:
16984:
16981:
16978:
16975:
16972:
16969:
16966:
16963:
16960:
16957:
16954:
16951:
16948:
16945:
16935:
16921:
16914:
16911:
16897:
16890:
16887:
16884:
16881:
16878:
16875:
16872:
16869:
16866:
16863:
16860:
16857:
16847:
16833:
16829:
16826:
16823:
16820:
16817:
16814:
16811:
16808:
16805:
16802:
16796:
16790:
16787:
16773:
16769:
16759:
16747:
16744:
16741:
16738:
16735:
16732:
16729:
16726:
16723:
16720:
16717:
16714:
16711:
16701:
16690:
16687:
16684:
16681:
16678:
16675:
16672:
16669:
16646:
16643:
16640:
16637:
16634:
16614:
16611:
16608:
16605:
16602:
16582:
16579:
16576:
16573:
16570:
16567:
16564:
16561:
16558:
16555:
16552:
16549:
16546:
16543:
16540:
16525:
16524:
16508:
16505:
16502:
16482:
16479:
16476:
16473:
16468:
16461:
16458:
16444:
16437:
16434:
16430:
16423:
16420:
16414:
16408:
16405:
16391:
16364:
16361:
16349:and column of
16330:
16306:
16303:
16289:
16268:
16263:
16256:
16253:
16239:
16232:
16229:
16226:
16223:
16199:
16196:
16193:
16170:
16158:
16155:
16142:
16122:
16102:
16099:
16096:
16091:
16084:
16081:
16075:
16069:
16066:
16060:
16057:
16051:
16045:
16042:
16028:
16000:
15980:
15977:
15966:set membership
15962:
15961:
15950:
15947:
15944:
15941:
15933:
15930:
15927:
15919:
15916:
15913:
15910:
15907:
15904:
15901:
15886:
15875:
15872:
15869:
15866:
15858:
15855:
15852:
15844:
15841:
15838:
15828:
15817:
15814:
15811:
15808:
15800:
15797:
15794:
15791:
15788:
15785:
15782:
15779:
15776:
15744:
15724:
15700:
15676:
15664:
15661:
15640:
15636:
15630:
15627:
15621:
15618:
15596:
15593:
15590:
15587:
15573:
15570:
15563:
15537:Jacques Riguet
15524:
15521:
15491:
15488:
15483:
15479:
15475:
15472:
15467:
15463:
15442:
15439:
15434:
15430:
15426:
15421:
15417:
15396:
15391:
15387:
15366:
15361:
15357:
15336:
15316:
15313:
15310:
15307:
15304:
15301:
15298:
15295:
15292:
15289:
15266:
15244:
15240:
15219:
15197:
15193:
15170:
15166:
15162:
15157:
15153:
15132:
15129:
15126:
15106:
15091:logical matrix
15079:
15078:
15067:
15064:
15061:
15050:
15046:
15013:
15008:
15004:
14989:Jacques Riguet
14977:
14974:
14971:
14968:
14965:
14962:
14954:
14951:
14948:
14945:
14942:
14932:
14906:
14902:
14899:
14896:
14872:
14869:
14866:
14863:
14860:
14857:
14854:
14851:
14848:
14845:
14842:
14826:
14823:
14822:
14821:
14809:
14806:
14803:
14783:
14755:
14751:
14748:
14745:
14721:
14707:
14695:
14692:
14689:
14669:
14649:
14638:
14634:
14631:
14608:
14603:
14578:
14562:
14559:
14531:
14511:
14500:
14499:
14487:
14467:
14464:
14461:
14458:
14455:
14452:
14449:
14429:
14419:
14401:
14381:
14353:
14349:
14346:
14343:
14340:
14302:
14282:
14279:
14276:
14273:
14270:
14267:
14256:
14255:
14239:
14219:
14209:
14194:
14191:
14188:
14184:
14179:
14175:
14169:
14165:
14161:
14156:
14153:
14149:
14124:
14113:logical matrix
14096:
14093:
14090:
14071:
14068:
14057:
14021:
14018:
14015:
14012:
13997:SchrΓΆder rules
13984:
13981:
13978:
13975:
13972:
13969:
13966:
13963:
13960:
13957:
13954:
13951:
13945:
13942:
13936:
13933:
13930:
13927:
13924:
13921:
13918:
13894:
13891:
13888:
13868:
13865:
13862:
13842:
13839:
13836:
13833:
13816:, extended by
13801:
13798:
13797:
13796:
13784:
13764:
13749:
13748:
13742:
13730:
13710:
13699:
13698:
13692:
13680:
13660:
13649:
13648:
13629:
13598:
13578:
13558:
13554:
13533:
13530:
13527:
13507:
13503:
13479:
13459:
13449:
13443:
13437:
13431:
13425:
13418:
13417:
13405:
13402:
13399:
13379:
13376:
13373:
13353:
13350:
13347:
13327:
13324:
13321:
13318:
13298:
13295:
13292:
13289:
13286:
13283:
13273:
13267:
13255:
13252:
13249:
13229:
13226:
13223:
13204:
13201:
13198:
13195:
13192:
13189:
13179:
13173:
13161:
13158:
13155:
13135:
13132:
13129:
13109:
13106:
13103:
13083:
13080:
13077:
13074:
13071:
13068:
13058:
13052:
13040:
13037:
13034:
13014:
13011:
13008:
12988:
12985:
12982:
12962:
12959:
12956:
12953:
12950:
12947:
12944:
12941:
12931:
12925:
12913:
12893:
12890:
12887:
12867:
12864:
12861:
12841:
12838:
12835:
12832:
12829:
12826:
12816:
12810:
12798:
12778:
12775:
12772:
12769:
12749:
12746:
12743:
12723:
12720:
12717:
12697:
12694:
12691:
12688:
12685:
12682:
12672:
12666:
12654:
12651:
12648:
12628:
12625:
12622:
12602:
12599:
12596:
12593:
12590:
12587:
12577:
12571:
12559:
12539:
12519:
12516:
12513:
12493:
12490:
12487:
12484:
12474:
12468:
12456:
12436:
12433:
12430:
12411:
12408:
12405:
12402:
12392:
12384:may have are:
12373:
12353:
12326:
12323:
12320:
12315:
12296:. Considering
12271:
12268:
12265:
12261:
12237:
12217:
12214:
12211:
12206:
12184:
12181:
12178:
12158:
12138:
12118:
12098:
12074:
12054:
12031:
12011:
12008:
12005:
12002:
11982:
11962:
11943:Main article:
11940:
11937:
11924:
11921:
11918:
11915:
11912:
11909:
11906:
11895:proper classes
11861:
11835:
11831:
11810:
11807:
11804:
11801:
11781:
11761:
11756:
11752:
11731:
11711:
11708:
11705:
11702:
11682:
11662:
11640:
11636:
11615:
11591:
11574:
11571:
11570:
11569:
11553:
11533:
11530:
11527:
11524:
11521:
11518:
11515:
11512:
11509:
11489:
11486:
11483:
11463:
11460:
11457:
11433:
11430:
11427:
11405:
11404:
11387:
11378:
11369:
11339:
11319:
11308:
11307:
11295:
11275:
11272:
11252:
11232:
11208:
11205:
11202:
11182:
11179:
11176:
11156:
11153:
11150:
11132:
11120:
11117:
11114:
11094:
11074:
11054:
11051:
11048:
11028:
11004:
10984:
10964:
10961:
10951:
10941:
10937:
10921:
10901:
10877:
10874:
10871:
10851:
10848:
10845:
10825:
10822:
10819:
10786:
10766:
10755:
10754:
10748:
10742:
10736:
10730:
10718:
10698:
10695:
10675:
10655:
10652:
10632:
10612:
10592:
10582:
10568:
10565:
10562:
10552:
10548:
10524:
10521:
10518:
10498:
10495:
10492:
10472:
10469:
10466:
10446:
10443:
10440:
10437:
10434:
10431:
10411:
10408:
10405:
10381:
10369:
10349:
10329:
10326:
10306:
10286:
10266:
10263:
10243:
10216:
10188:
10185:
10182:
10162:
10159:
10156:
10136:
10133:
10130:
10110:
10107:
10104:
10101:
10081:
10078:
10075:
10072:
10069:
10035:
10015:
9995:
9971:
9968:
9966:
9965:
9964:
9963:
9961:
9948:
9938:
9924:
9913:
9899:
9879:
9875:
9871:
9868:
9865:
9862:
9841:
9820:
9800:
9779:
9758:
9738:
9735:
9732:
9728:
9724:
9721:
9718:
9715:
9711:
9675:
9651:
9631:
9628:
9625:
9622:
9619:
9616:
9613:
9610:
9607:
9581:
9571:
9570:
9557:
9551:
9548:
9542:
9536:
9533:
9527:
9524:
9521:
9518:
9515:
9512:
9509:
9493:
9471:
9457:
9449:
9435:
9415:
9392:. Just as the
9374:directed graph
9366:
9342:
9339:
9336:
9316:
9310:
9305:
9284:
9281:
9278:
9258:
9238:
9235:
9232:
9212:
9206:
9201:
9177:
9172:
9168:
9158:
9157:
9146:
9141:
9135:
9132:
9130:
9127:
9125:
9122:
9120:
9117:
9115:
9112:
9110:
9107:
9105:
9102:
9101:
9098:
9095:
9093:
9090:
9088:
9085:
9083:
9080:
9078:
9075:
9073:
9070:
9068:
9065:
9064:
9061:
9058:
9056:
9053:
9051:
9048:
9046:
9043:
9041:
9038:
9036:
9033:
9031:
9028:
9027:
9024:
9021:
9019:
9016:
9014:
9011:
9009:
9006:
9004:
9001:
8999:
8996:
8994:
8991:
8990:
8988:
8983:
8980:
8967:logical matrix
8954:
8934:
8914:
8911:
8908:
8884:
8876:
8868:
8860:
8852:
8844:
8836:
8828:
8825:
8822:
8798:
8790:
8782:
8774:
8766:
8763:
8760:
8749:
8746:
8745:
8742:
8739:
8736:
8733:
8730:
8727:
8724:
8720:
8719:
8716:
8713:
8710:
8707:
8704:
8701:
8698:
8694:
8693:
8690:
8687:
8684:
8681:
8678:
8675:
8672:
8668:
8667:
8664:
8661:
8658:
8655:
8652:
8649:
8646:
8642:
8641:
8638:
8635:
8632:
8629:
8626:
8623:
8620:
8586:
8566:
8563:
8555:
8535:
8515:
8512:
8504:
8501:
8498:
8478:
8458:
8438:
8435:
8432:
8424:
8421:
8418:
8410:
8407:
8404:
8396:
8393:
8390:
8387:
8367:
8347:
8327:
8324:
8316:
8313:
8310:
8290:
8282:
8279:
8276:
8264:
8261:
8260:
8257:
8254:
8249:
8246:
8242:
8241:
8238:
8235:
8232:
8229:
8225:
8224:
8221:
8216:
8213:
8210:
8206:
8205:
8202:
8199:
8196:
8191:
8187:
8186:
8183:
8180:
8177:
8174:
8162:
8153:
8141:
8132:
8122:
8121:
8118:
8115:
8110:
8107:
8103:
8102:
8099:
8094:
8091:
8088:
8084:
8083:
8080:
8077:
8074:
8069:
8065:
8064:
8061:
8058:
8055:
8052:
8040:
8031:
8019:
8010:
8001:
7998:
7973:
7970:
7967:
7956:matrix of ones
7935:
7915:
7895:
7875:
7843:
7823:
7799:
7779:
7767:
7764:
7751:
7748:
7745:
7725:
7705:
7681:
7678:
7675:
7672:
7652:
7642:
7635:is said to be
7624:
7604:
7584:
7564:
7544:
7524:
7521:
7518:
7508:
7494:
7474:
7454:
7434:
7414:
7394:
7374:
7371:
7368:
7348:
7345:
7342:
7322:
7319:
7316:
7313:
7293:
7290:
7287:
7267:
7247:
7227:
7224:
7221:
7218:
7198:
7178:
7158:
7148:
7141:is said to be
7130:
7110:
7090:
7067:
7047:
7043:
7017:
6991:
6987:
6984:
6971:is that every
6960:
6929:
6909:
6889:
6869:
6851:total preorder
6795:
6775:
6755:
6735:
6725:
6707:
6704:
6701:
6698:
6690:
6687:
6684:
6681:
6678:
6675:
6672:
6669:
6666:
6663:
6660:
6655:
6652:
6648:
6627:
6607:
6587:
6567:
6547:
6524:
6504:
6484:
6474:
6456:
6453:
6450:
6447:
6439:
6436:
6433:
6425:
6422:
6419:
6416:
6413:
6410:
6407:
6404:
6401:
6398:
6395:
6390:
6387:
6383:
6362:
6342:
6322:
6298:
6283:Main article:
6280:
6277:
6276:
6275:
6268:
6265:
6249:
6246:
6243:
6240:
6218:
6212:
6204:
6201:
6194:
6189:
6183:
6178:
6147:
6120:
6100:
6080:
6060:
6036:
6016:
5996:
5976:
5956:
5936:
5916:
5896:
5873:
5853:
5833:
5823:
5809:
5806:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5744:
5741:
5718:
5698:
5678:
5663:Main article:
5660:
5657:
5640:
5620:
5600:
5580:
5560:
5540:
5517:
5497:
5477:
5467:
5464:, also called
5463:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5425:
5422:
5419:
5416:
5405:
5384:
5364:
5344:
5323:Main article:
5320:
5317:
5304:
5284:
5264:
5244:
5224:
5204:
5184:
5164:
5140:
5137:
5134:
5131:
5111:
5091:
5068:
5048:
5028:
5008:
4998:
4984:
4981:
4978:
4958:
4955:
4952:
4949:
4941:
4938:
4935:
4927:
4924:
4921:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4883:
4863:
4843:
4823:
4803:
4783:
4763:
4748:Main article:
4745:
4742:
4726:
4706:
4686:
4666:
4656:
4642:
4639:
4636:
4633:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4566:
4546:
4526:
4506:
4494:
4491:
4478:
4458:
4435:
4415:
4395:
4375:
4365:
4364:union relation
4351:
4348:
4345:
4342:
4337: or
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4295:
4275:
4255:
4235:
4215:
4203:
4200:
4198:
4195:
4182:
4162:
4142:
4122:
4102:
4099:
4096:
4076:
4073:
4070:
4050:
4047:
4044:
4021:
4001:
3981:
3961:
3941:
3938:
3935:
3916:correspondence
3910:
3906:
3892:
3889:
3886:
3883:
3840:
3820:
3810:
3806:
3790:
3787:
3784:
3781:
3758:
3748:
3734:
3714:
3711:
3708:
3688:
3668:
3658:
3654:
3650:
3632:
3612:
3609:
3606:
3586:
3566:
3556:
3552:
3538:
3535:
3532:
3512:
3492:
3472:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3424:
3410:
3407:
3404:
3384:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3336:
3335:correspondence
3332:
3318:
3298:
3278:
3268:
3264:
3250:
3241:, and the set
3230:
3220:
3216:
3213:is called the
3202:
3182:
3179:
3176:
3173:
3153:
3133:
3113:
3104:
3083:
3080:
3077:
3074:
3071:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3027:is defined as
3016:
3013:
3010:
2990:
2970:
2958:
2955:
2943:
2938:
2934:
2930:
2927:
2924:
2919:
2915:
2889:
2885:
2881:
2878:
2875:
2870:
2866:
2843:
2822:
2819:
2816:
2775:Clarence Lewis
2771:Ernst SchrΓΆder
2727:
2724:
2721:
2701:
2698:
2695:
2671:
2668:
2665:
2662:
2653:of subsets of
2638:
2614:
2611:
2608:
2605:
2581:
2561:
2539:
2538:
2535:linear algebra
2527:
2520:
2515:" relation in
2509:
2474:
2454:
2434:
2414:
2394:
2385:is related to
2374:
2354:
2334:
2314:
2294:
2274:
2254:
2251:
2231:
2211:
2187:
2163:
2143:
2122:
2097:
2066:
2063:
2060:
2057:
2054:
2043:if and only if
2030:
2021:to an element
2006:
1986:
1966:
1946:
1926:
1906:
1903:
1900:
1897:
1894:
1871:
1851:
1818:
1817:
1803:
1800:
1797:
1794:
1774:
1771:
1768:
1748:
1745:
1742:
1722:
1719:
1716:
1713:
1710:
1707:
1683:
1640:
1639:
1636:
1635:
1620:
1617:
1614:
1606:
1605:
1602:
1599:
1596:
1593:
1590:
1589:
1579:
1568:
1565:
1562:
1547:
1536:
1533:
1530:
1520:
1500:
1499:
1496:
1493:
1490:
1487:
1486:
1476:
1456:
1455:
1452:
1449:
1446:
1443:
1442:
1432:
1412:
1411:
1408:
1405:
1402:
1401:
1391:
1376:
1373:
1370:
1367:
1363: or
1360:
1357:
1354:
1351:
1350:
1347:
1344:
1341:
1337:
1334:
1331:
1330:
1320:
1305:
1302:
1298:
1295:
1292:
1289:
1288:
1285:
1282:
1279:
1276:
1269:
1266:
1263:
1260:
1259:
1249:
1234:
1231:
1228:
1225:
1221:
1218:
1217:
1214:
1211:
1208:
1205:
1203:
1193:
1182:
1179:
1176:
1173:
1153:
1150:
1147:
1135:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1086:
1085:
1075:
1065:
1060:
1055:
1050:
1045:
1035:
1025:
1020:
1014:
1013:
1003:
993:
988:
983:
978:
973:
968:
958:
953:
947:
946:
936:
926:
921:
916:
911:
906:
901:
891:
886:
880:
879:
874:
869:
859:
849:
844:
839:
834:
824:
819:
813:
812:
807:
802:
792:
787:
777:
772:
767:
757:
752:
746:
745:
740:
735:
725:
715:
705:
700:
695:
685:
680:
674:
673:
668:
663:
653:
648:
643:
633:
623:
613:
608:
602:
601:
596:
591:
581:
576:
571:
561:
556:
551:
546:
538:
537:
532:
527:
517:
512:
507:
497:
487:
482:
477:
471:
470:
465:
460:
450:
445:
440:
435:
425:
415:
410:
404:
403:
398:
393:
383:
378:
373:
368:
358:
353:
348:
346:Total preorder
342:
341:
336:
331:
321:
316:
311:
306:
301:
291:
286:
280:
279:
274:
269:
259:
254:
249:
244:
239:
234:
229:
218:
217:
212:
207:
197:
192:
187:
182:
177:
172:
162:
154:
153:
151:
146:
144:
142:
140:
138:
135:
133:
131:
128:
127:
122:
117:
112:
107:
102:
97:
92:
87:
82:
75:
74:
72:
71:
64:
57:
49:
26:
9:
6:
4:
3:
2:
21189:
21178:
21175:
21174:
21172:
21157:
21156:Ernst Zermelo
21154:
21152:
21149:
21147:
21144:
21142:
21141:Willard Quine
21139:
21137:
21134:
21132:
21129:
21127:
21124:
21122:
21119:
21117:
21114:
21112:
21109:
21107:
21104:
21102:
21099:
21098:
21096:
21094:
21093:Set theorists
21090:
21084:
21081:
21079:
21076:
21074:
21071:
21070:
21068:
21062:
21060:
21057:
21056:
21053:
21045:
21042:
21040:
21039:KripkeβPlatek
21037:
21033:
21030:
21029:
21028:
21025:
21024:
21023:
21020:
21016:
21013:
21012:
21011:
21010:
21006:
21002:
20999:
20998:
20997:
20994:
20993:
20990:
20987:
20985:
20982:
20980:
20977:
20975:
20972:
20971:
20969:
20965:
20959:
20956:
20954:
20951:
20949:
20946:
20944:
20942:
20937:
20935:
20932:
20930:
20927:
20924:
20920:
20917:
20915:
20912:
20908:
20905:
20903:
20900:
20898:
20895:
20894:
20893:
20890:
20887:
20883:
20880:
20878:
20875:
20873:
20870:
20868:
20865:
20864:
20862:
20859:
20855:
20849:
20846:
20844:
20841:
20839:
20836:
20834:
20831:
20829:
20826:
20824:
20821:
20819:
20816:
20812:
20809:
20807:
20804:
20803:
20802:
20799:
20797:
20794:
20792:
20789:
20787:
20784:
20782:
20779:
20776:
20772:
20769:
20767:
20764:
20762:
20759:
20758:
20756:
20750:
20747:
20746:
20743:
20737:
20734:
20732:
20729:
20727:
20724:
20722:
20719:
20717:
20714:
20712:
20709:
20707:
20704:
20701:
20698:
20696:
20693:
20692:
20690:
20688:
20684:
20676:
20675:specification
20673:
20671:
20668:
20667:
20666:
20663:
20662:
20659:
20656:
20654:
20651:
20649:
20646:
20644:
20641:
20639:
20636:
20634:
20631:
20629:
20626:
20624:
20621:
20619:
20616:
20614:
20611:
20607:
20604:
20602:
20599:
20597:
20594:
20593:
20592:
20589:
20587:
20584:
20583:
20581:
20579:
20575:
20570:
20560:
20557:
20556:
20554:
20550:
20546:
20539:
20534:
20532:
20527:
20525:
20520:
20519:
20516:
20504:
20501:
20499:
20496:
20492:
20489:
20488:
20487:
20484:
20480:
20477:
20475:
20472:
20470:
20467:
20466:
20465:
20462:
20458:
20455:
20454:
20453:
20452:Ordered field
20450:
20448:
20445:
20441:
20438:
20436:
20433:
20432:
20431:
20428:
20424:
20421:
20420:
20419:
20416:
20414:
20411:
20409:
20408:Hasse diagram
20406:
20404:
20401:
20399:
20396:
20392:
20389:
20388:
20387:
20386:Comparability
20384:
20382:
20379:
20377:
20374:
20372:
20369:
20368:
20366:
20362:
20354:
20351:
20349:
20346:
20344:
20341:
20339:
20336:
20335:
20334:
20331:
20329:
20326:
20322:
20319:
20317:
20314:
20313:
20312:
20309:
20307:
20303:
20300:
20299:
20297:
20294:
20290:
20284:
20281:
20279:
20276:
20274:
20271:
20269:
20266:
20264:
20261:
20259:
20258:Product order
20256:
20254:
20251:
20249:
20246:
20244:
20241:
20239:
20236:
20235:
20233:
20231:Constructions
20229:
20223:
20219:
20215:
20212:
20208:
20205:
20203:
20200:
20198:
20195:
20193:
20190:
20188:
20185:
20183:
20180:
20178:
20175:
20173:
20170:
20166:
20163:
20162:
20161:
20158:
20156:
20153:
20149:
20146:
20144:
20141:
20139:
20136:
20134:
20131:
20130:
20129:
20128:Partial order
20126:
20124:
20121:
20117:
20116:Join and meet
20114:
20112:
20109:
20107:
20104:
20102:
20099:
20097:
20094:
20093:
20092:
20089:
20087:
20084:
20082:
20079:
20077:
20074:
20072:
20069:
20067:
20063:
20059:
20057:
20054:
20052:
20049:
20047:
20044:
20042:
20039:
20037:
20034:
20030:
20027:
20026:
20025:
20022:
20020:
20017:
20015:
20014:Antisymmetric
20012:
20011:
20009:
20005:
19999:
19993:
19990:
19988:
19985:
19983:
19980:
19978:
19975:
19973:
19970:
19968:
19965:
19963:
19960:
19958:
19955:
19953:
19950:
19948:
19945:
19943:
19940:
19938:
19935:
19934:
19932:
19928:
19922:
19921:Weak ordering
19919:
19917:
19914:
19912:
19909:
19907:
19906:Partial order
19904:
19902:
19899:
19897:
19894:
19892:
19889:
19887:
19884:
19883:
19881:
19877:
19871:
19868:
19866:
19863:
19861:
19858:
19857:
19854:
19850:
19843:
19838:
19836:
19831:
19829:
19824:
19823:
19820:
19812:
19808:
19807:
19802:
19798:
19795:
19790:
19786:
19785:
19775:
19769:
19765:
19762:. Cambridge:
19761:
19760:
19755:
19751:
19741:
19737:
19732:
19727:
19723:
19719:
19715:
19711:
19707:
19703:
19699:
19695:
19691:
19687:
19685:9783540528494
19681:
19677:
19672:
19668:
19662:
19658:
19654:
19649:
19645:
19639:
19635:
19631:
19627:
19623:
19616:
19612:
19606:
19602:
19595:
19594:
19589:
19585:
19583:
19579:
19575:
19572:
19568:
19562:
19558:
19554:
19550:
19546:
19542:
19540:9780511778810
19536:
19532:
19527:
19526:
19515:
19511:
19508:
19504:
19498:
19492:
19489:
19486:
19482:
19479:
19475:
19469:
19463:
19460:
19456:
19452:
19451:Viktor Wagner
19447:
19440:
19414:
19408:
19404:
19401:
19397:
19393:
19388:
19386:
19384:
19377:
19373:
19372:
19364:
19356:
19352:
19348:
19341:
19333:
19327:
19323:
19322:
19317:
19311:
19304:
19298:
19290:
19284:
19280:
19276:
19272:
19268:
19261:
19255:
19251:
19248:
19244:
19240:
19234:
19226:
19220:
19216:
19209:
19201:
19197:
19193:
19189:
19184:
19179:
19175:
19171:
19164:
19156:
19150:
19146:
19142:
19136:
19128:
19125:(in French).
19124:
19120:
19116:
19110:
19104:
19101:
19097:
19093:
19086:
19080:
19079:0-8247-1788-0
19076:
19073:
19072:Marcel Dekker
19069:
19065:
19060:
19054:
19050:
19046:
19042:
19041:R. Berghammer
19037:
19030:
19029:0-12-597680-1
19026:
19022:
19016:
19004:
18997:
18996:
18988:
18980:
18976:
18969:
18962:
18960:0-534-39900-2
18956:
18952:
18945:
18937:
18931:
18927:
18920:
18912:
18906:
18902:
18895:
18887:
18885:0-8218-1041-3
18881:
18877:
18873:
18872:
18867:
18861:
18853:
18849:
18845:
18843:0-444-85401-0
18839:
18834:
18833:
18824:
18815:
18808:
18801:
18792:
18783:
18769:
18765:
18759:
18745:
18741:
18735:
18726:
18724:
18722:
18720:
18718:
18708:
18706:
18704:
18702:
18695:
18691:
18686:. p. 15.
18685:
18681:
18680:Design Theory
18677:
18673:
18666:
18659:
18655:
18649:
18642:
18638:
18634:
18628:
18613:
18610:
18606:
18602:
18598:
18594:
18588:
18581:
18577:
18572:
18565:
18561:
18556:
18549:
18543:
18537:
18536:3-211-82971-7
18533:
18530:
18526:
18520:
18512:
18506:
18502:
18495:
18493:
18491:
18482:
18476:
18472:
18471:
18466:
18460:
18452:
18446:
18442:
18441:
18436:
18430:
18422:
18420:0-486-42079-5
18416:
18412:
18408:
18402:
18394:
18388:
18384:
18380:
18374:
18366:
18364:0-486-61630-4
18360:
18355:
18354:
18348:
18342:
18333:
18329:
18325:
18319:
18315:
18308:
18301:
18300:Enderton 1977
18296:
18289:
18285:
18281:
18277:
18272:
18270:
18262:
18258:
18253:
18251:
18244:
18240:
18236:
18231:
18229:
18214:
18210:
18204:
18186:
18182:
18178:
18174:
18170:
18166:
18162:
18155:
18152:(June 1970).
18151:
18145:
18143:
18141:
18139:
18137:
18135:
18133:
18131:
18119:
18112:
18105:
18101:
18087:
18069:
18065:
18061:
18056:
18052:
18048:
18028:
18025:
18022:
18002:
17982:
17972:
17968:
17956:
17953:
17950:
17947:
17944:
17941:
17938:
17937:Hasse diagram
17935:
17932:
17929:
17926:
17923:
17920:
17917:
17915:
17912:
17909:
17906:
17904:
17901:
17900:
17886:
17872:
17869:
17866:
17846:
17826:
17818:
17811:
17809:
17805:
17804:Viktor Wagner
17789:
17781:
17757:
17736:
17725:
17721:
17718:
17712:
17709:
17706:
17703:
17700:
17690:
17671:
17668:
17665:
17635:
17615:
17605:
17595:
17593:
17577:
17554:
17548:
17542:
17539:
17530:
17516:
17493:
17487:
17484:
17458:
17452:
17449:
17429:
17406:
17400:
17397:
17389:
17385:
17369:
17346:
17340:
17337:
17317:
17297:
17294:
17288:
17282:
17279:
17259:
17250:
17237:
17228:
17211:
17204:
17198:
17195:
17192:
17186:
17180:
17177:
17169:
17153:
17125:
17122:
17116:
17101:
17095:
17089:
17079:
17078:
17077:
17063:
17043:
17036:
17020:
17012:
17008:
16988:
16985:
16979:
16976:
16970:
16964:
16955:
16949:
16943:
16936:
16909:
16895:
16888:
16882:
16876:
16867:
16861:
16855:
16848:
16824:
16818:
16809:
16803:
16794:
16785:
16771:
16767:
16760:
16745:
16742:
16736:
16730:
16721:
16715:
16709:
16702:
16688:
16685:
16679:
16673:
16667:
16660:
16659:
16658:
16644:
16641:
16638:
16635:
16632:
16612:
16606:
16603:
16600:
16580:
16577:
16571:
16568:
16562:
16556:
16547:
16541:
16530:
16522:
16506:
16500:
16480:
16474:
16471:
16456:
16442:
16435:
16432:
16418:
16412:
16403:
16389:
16381:
16380:
16379:
16359:
16328:
16301:
16287:
16266:
16251:
16237:
16230:
16227:
16221:
16213:
16212:left residual
16210:which is the
16197:
16191:
16184:
16168:
16154:
16140:
16120:
16100:
16097:
16094:
16079:
16073:
16067:
16064:
16055:
16049:
16040:
16026:
16016:
16014:
15998:
15978:
15975:
15967:
15948:
15945:
15942:
15939:
15931:
15928:
15925:
15917:
15914:
15911:
15908:
15905:
15902:
15899:
15896:for all
15887:
15873:
15870:
15867:
15864:
15856:
15853:
15850:
15842:
15839:
15836:
15829:
15815:
15812:
15809:
15806:
15795:
15789:
15786:
15783:
15780:
15777:
15774:
15771:for all
15762:
15761:
15760:
15758:
15742:
15722:
15714:
15698:
15690:
15674:
15660:
15638:
15634:
15625:
15619:
15616:
15607:
15594:
15591:
15588:
15585:
15568:
15561:
15552:
15548:
15546:
15542:
15538:
15534:
15530:
15520:
15518:
15513:
15511:
15510:bisimulations
15507:
15502:
15489:
15486:
15481:
15477:
15473:
15470:
15465:
15461:
15437:
15432:
15428:
15424:
15419:
15415:
15394:
15389:
15385:
15364:
15359:
15355:
15334:
15314:
15311:
15308:
15302:
15299:
15296:
15293:
15290:
15278:
15264:
15242:
15238:
15217:
15195:
15191:
15168:
15164:
15160:
15155:
15151:
15130:
15127:
15124:
15104:
15096:
15092:
15088:
15084:
15065:
15062:
15059:
15048:
15044:
15037:
15036:
15035:
15032:
15030:
15006:
15002:
14994:
14990:
14975:
14972:
14969:
14966:
14963:
14960:
14952:
14949:
14946:
14943:
14940:
14930:
14928:
14904:
14900:
14897:
14894:
14886:
14867:
14864:
14861:
14858:
14855:
14852:
14849:
14843:
14840:
14832:
14807:
14804:
14801:
14781:
14753:
14749:
14746:
14743:
14735:
14719:
14711:
14708:
14693:
14690:
14687:
14667:
14647:
14636:
14632:
14629:
14601:
14592:
14576:
14568:
14565:
14564:
14558:
14556:
14552:
14547:
14545:
14529:
14509:
14485:
14462:
14459:
14456:
14453:
14450:
14427:
14417:
14415:
14399:
14379:
14351:
14347:
14344:
14341:
14338:
14331:
14330:
14329:
14327:
14323:
14318:
14316:
14300:
14280:
14277:
14274:
14271:
14268:
14265:
14253:
14237:
14217:
14210:
14207:
14192:
14189:
14186:
14182:
14177:
14173:
14167:
14163:
14159:
14154:
14151:
14147:
14138:
14137:outer product
14122:
14114:
14110:
14109:
14108:
14094:
14091:
14088:
14081:
14077:
14067:
14065:
14061:
14055:
14053:
14049:
14045:
14041:
14037:
14032:
14019:
14016:
14013:
14010:
14002:
13998:
13982:
13976:
13973:
13970:
13967:
13964:
13958:
13949:
13940:
13934:
13931:
13925:
13922:
13919:
13916:
13908:
13892:
13889:
13886:
13866:
13863:
13860:
13853:meaning that
13840:
13837:
13834:
13831:
13823:
13819:
13815:
13812:includes the
13811:
13807:
13782:
13762:
13754:
13751:the smallest
13750:
13747:
13744:
13743:
13728:
13708:
13700:
13697:
13694:
13693:
13678:
13658:
13650:
13647:
13644:
13643:
13642:
13641:
13627:
13619:
13614:
13612:
13596:
13576:
13556:
13531:
13528:
13525:
13505:
13493:
13477:
13457:
13448:
13445:
13442:
13439:
13436:
13433:
13430:
13427:
13424:
13423:partial order
13421:
13403:
13400:
13397:
13377:
13374:
13371:
13351:
13348:
13345:
13325:
13322:
13319:
13316:
13296:
13293:
13290:
13287:
13284:
13281:
13272:
13269:
13268:
13253:
13250:
13247:
13227:
13224:
13221:
13202:
13199:
13196:
13193:
13190:
13187:
13178:
13175:
13174:
13159:
13156:
13153:
13133:
13130:
13127:
13107:
13104:
13101:
13081:
13078:
13075:
13072:
13069:
13066:
13057:
13054:
13053:
13038:
13035:
13032:
13012:
13009:
13006:
12986:
12983:
12980:
12960:
12957:
12954:
12951:
12948:
12945:
12942:
12939:
12930:
12927:
12926:
12911:
12891:
12888:
12885:
12865:
12862:
12859:
12839:
12836:
12833:
12830:
12827:
12824:
12815:
12812:
12811:
12796:
12789:For example,
12776:
12773:
12770:
12767:
12747:
12744:
12741:
12721:
12718:
12715:
12695:
12692:
12689:
12686:
12683:
12680:
12671:
12670:Antisymmetric
12668:
12667:
12652:
12649:
12646:
12626:
12623:
12620:
12600:
12597:
12594:
12591:
12588:
12585:
12576:
12573:
12572:
12557:
12537:
12517:
12514:
12511:
12491:
12488:
12485:
12482:
12473:
12470:
12469:
12454:
12434:
12431:
12428:
12409:
12406:
12403:
12400:
12391:
12388:
12387:
12386:
12385:
12371:
12351:
12342:
12340:
12337:, it forms a
12321:
12303:
12299:
12295:
12291:
12287:
12269:
12266:
12263:
12259:
12251:
12235:
12212:
12182:
12179:
12176:
12156:
12136:
12116:
12096:
12088:
12072:
12052:
12043:
12029:
12009:
12006:
12003:
12000:
11980:
11960:
11952:
11946:
11936:
11919:
11916:
11913:
11910:
11907:
11896:
11892:
11888:
11883:
11881:
11880:
11875:
11859:
11851:
11833:
11829:
11805:
11799:
11792:and codomain
11779:
11759:
11754:
11750:
11729:
11706:
11700:
11680:
11660:
11638:
11634:
11613:
11603:
11589:
11581:
11567:
11551:
11528:
11525:
11522:
11519:
11516:
11513:
11510:
11487:
11484:
11481:
11461:
11458:
11455:
11447:
11431:
11428:
11425:
11417:
11414:(also called
11413:
11410:
11409:
11408:
11402:
11398:
11394:
11393:
11388:
11385:
11384:
11379:
11376:
11375:
11370:
11367:
11363:
11360:(also called
11359:
11358:
11353:
11352:
11351:
11337:
11330:and codomain
11317:
11293:
11273:
11270:
11250:
11230:
11222:
11206:
11203:
11200:
11180:
11177:
11174:
11154:
11151:
11148:
11140:
11137:(also called
11136:
11133:
11118:
11115:
11112:
11092:
11072:
11052:
11049:
11046:
11026:
11018:
11002:
10982:
10962:
10959:
10950:
10947:
10945:
10939:
10938:(also called
10936:
10933:
10919:
10899:
10891:
10875:
10872:
10869:
10849:
10846:
10843:
10823:
10820:
10817:
10809:
10806:(also called
10805:
10804:
10800:
10799:
10798:
10784:
10777:and codomain
10764:
10752:
10749:
10746:
10743:
10740:
10737:
10734:
10731:
10716:
10696:
10693:
10673:
10653:
10650:
10630:
10610:
10590:
10581:a primary key
10580:
10563:
10550:
10547:
10544:
10542:
10538:
10522:
10519:
10516:
10496:
10493:
10490:
10470:
10467:
10464:
10444:
10441:
10438:
10435:
10432:
10429:
10409:
10406:
10403:
10395:
10391:
10388:(also called
10387:
10386:
10382:
10367:
10347:
10327:
10324:
10304:
10284:
10264:
10261:
10241:
10233:
10232:
10214:
10206:
10202:
10186:
10183:
10180:
10160:
10157:
10154:
10134:
10131:
10128:
10108:
10105:
10102:
10099:
10079:
10076:
10073:
10070:
10067:
10059:
10056:(also called
10055:
10052:
10051:
10050:
10047:
10033:
10013:
9993:
9981:
9976:
9959:
9946:
9936:
9911:
9897:
9877:
9869:
9866:
9863:
9860:
9818:
9798:
9756:
9733:
9730:
9722:
9719:
9713:
9700:
9699:
9697:
9693:
9692:block designs
9689:
9673:
9665:
9626:
9623:
9620:
9617:
9614:
9608:
9598:
9594:
9593:Jakob Steiner
9590:
9586:
9582:
9579:
9576:(as found in
9575:
9555:
9546:
9540:
9531:
9525:
9522:
9516:
9513:
9510:
9500:
9499:
9497:
9491:
9489:
9485:
9469:
9461:
9458:is simple in
9455:
9453:
9450:
9433:
9413:
9404:
9399:
9395:
9391:
9387:
9383:
9379:
9375:
9371:
9367:
9364:
9360:
9356:
9340:
9337:
9334:
9314:
9303:
9282:
9279:
9276:
9256:
9236:
9233:
9230:
9210:
9199:
9170:
9166:
9144:
9139:
9133:
9128:
9123:
9118:
9113:
9108:
9103:
9096:
9091:
9086:
9081:
9076:
9071:
9066:
9059:
9054:
9049:
9044:
9039:
9034:
9029:
9022:
9017:
9012:
9007:
9002:
8997:
8992:
8986:
8981:
8978:
8971:
8970:
8968:
8952:
8932:
8912:
8909:
8906:
8898:
8874:
8866:
8858:
8850:
8842:
8834:
8823:
8820:
8812:
8788:
8780:
8772:
8761:
8758:
8750:
8743:
8740:
8737:
8734:
8731:
8728:
8725:
8721:
8717:
8714:
8711:
8708:
8705:
8702:
8699:
8695:
8691:
8688:
8685:
8682:
8679:
8676:
8673:
8669:
8665:
8662:
8659:
8656:
8653:
8650:
8647:
8643:
8619:
8618:
8608:
8604:
8602:
8584:
8564:
8533:
8513:
8499:
8496:
8476:
8456:
8436:
8419:
8405:
8388:
8385:
8365:
8345:
8325:
8311:
8308:
8277:
8274:
8266:
8265:
8258:
8255:
8253:
8250:
8247:
8243:
8239:
8236:
8233:
8230:
8226:
8222:
8220:
8217:
8214:
8211:
8207:
8203:
8200:
8197:
8195:
8192:
8188:
8160:
8139:
8131:
8130:
8119:
8116:
8114:
8111:
8108:
8104:
8100:
8098:
8095:
8092:
8089:
8085:
8081:
8078:
8075:
8073:
8070:
8066:
8038:
8017:
8009:
8008:
7997:
7995:
7991:
7987:
7971:
7968:
7965:
7957:
7953:
7949:
7933:
7913:
7893:
7873:
7865:
7861:
7857:
7841:
7821:
7813:
7797:
7777:
7763:
7749:
7746:
7743:
7723:
7703:
7695:
7679:
7676:
7673:
7670:
7650:
7636:
7622:
7602:
7582:
7562:
7542:
7522:
7519:
7516:
7506:
7492:
7472:
7452:
7432:
7412:
7392:
7372:
7369:
7366:
7346:
7343:
7340:
7320:
7317:
7314:
7311:
7291:
7288:
7285:
7265:
7245:
7225:
7222:
7219:
7216:
7196:
7176:
7156:
7142:
7128:
7108:
7088:
7079:
7065:
7045:
7032:
7006:
6985:
6982:
6974:
6958:
6950:
6946:
6941:
6927:
6907:
6887:
6880:is parent of
6867:
6858:
6856:
6852:
6848:
6844:
6840:
6839:partial order
6836:
6832:
6828:
6824:
6820:
6819:antisymmetric
6816:
6812:
6807:
6793:
6773:
6753:
6733:
6719:
6702:
6699:
6696:
6688:
6685:
6682:
6679:
6673:
6670:
6667:
6658:
6653:
6646:
6625:
6605:
6585:
6565:
6545:
6536:
6522:
6502:
6482:
6468:
6451:
6448:
6445:
6437:
6434:
6431:
6423:
6420:
6417:
6414:
6408:
6405:
6402:
6393:
6388:
6381:
6360:
6340:
6320:
6312:
6296:
6286:
6273:
6269:
6266:
6263:
6262:
6261:
6247:
6244:
6241:
6238:
6229:
6216:
6199:
6192:
6176:
6145:
6137:
6132:
6118:
6098:
6078:
6058:
6050:
6034:
6014:
5994:
5974:
5954:
5934:
5914:
5894:
5887:For example,
5885:
5871:
5851:
5831:
5821:
5807:
5781:
5778:
5775:
5769:
5763:
5760:
5757:
5748:
5739:
5716:
5696:
5676:
5666:
5656:
5654:
5638:
5618:
5598:
5578:
5558:
5538:
5531:For example,
5529:
5515:
5495:
5475:
5465:
5461:
5444:
5441:
5438:
5435:
5429:
5426:
5423:
5414:
5403:
5382:
5362:
5342:
5332:
5326:
5316:
5302:
5282:
5262:
5242:
5222:
5202:
5182:
5162:
5154:
5138:
5135:
5132:
5129:
5109:
5089:
5080:
5066:
5046:
5026:
5006:
4996:
4982:
4979:
4976:
4953:
4950:
4947:
4939:
4936:
4933:
4925:
4922:
4919:
4911:
4905:
4902:
4899:
4890:
4887:
4884:
4881:
4861:
4841:
4821:
4801:
4781:
4761:
4751:
4741:
4738:
4724:
4704:
4684:
4664:
4654:
4637:
4634:
4631:
4623:
4620:
4617:
4614:
4608:
4605:
4602:
4593:
4590:
4587:
4584:
4564:
4544:
4524:
4504:
4490:
4476:
4456:
4447:
4433:
4413:
4393:
4373:
4363:
4346:
4343:
4340:
4332:
4329:
4326:
4323:
4317:
4314:
4311:
4302:
4299:
4296:
4293:
4273:
4253:
4233:
4213:
4194:
4180:
4160:
4140:
4120:
4100:
4097:
4094:
4074:
4071:
4068:
4048:
4045:
4042:
4033:
4019:
3999:
3979:
3959:
3939:
3936:
3933:
3925:
3921:
3917:
3912:
3908:
3905:heterogeneous
3904:
3890:
3887:
3884:
3881:
3873:
3869:
3864:
3862:
3858:
3855:. The prefix
3854:
3838:
3818:
3808:
3805:
3802:
3788:
3785:
3782:
3779:
3770:
3756:
3746:
3732:
3712:
3709:
3706:
3686:
3666:
3656:
3652:
3648:
3646:
3630:
3610:
3607:
3604:
3584:
3564:
3555:active domain
3554:
3550:
3536:
3533:
3530:
3510:
3490:
3470:
3450:
3447:
3441:
3438:
3435:
3422:
3408:
3405:
3402:
3382:
3359:
3356:
3353:
3350:
3347:
3334:
3330:
3316:
3296:
3276:
3266:
3262:
3248:
3228:
3218:
3214:
3200:
3180:
3177:
3174:
3171:
3151:
3131:
3111:
3102:
3099:
3097:
3096:ordered pairs
3081:
3075:
3072:
3069:
3061:
3058:
3055:
3052:
3046:
3043:
3040:
3014:
3011:
3008:
2988:
2968:
2954:
2941:
2936:
2932:
2928:
2925:
2922:
2917:
2913:
2905:
2887:
2883:
2879:
2876:
2873:
2868:
2864:
2855:
2854:-ary relation
2841:
2820:
2817:
2814:
2805:
2803:
2799:
2795:
2790:
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2756:
2752:
2748:
2744:
2739:
2725:
2722:
2719:
2699:
2696:
2693:
2685:
2669:
2666:
2663:
2660:
2652:
2636:
2628:
2612:
2609:
2606:
2603:
2595:
2579:
2559:
2550:
2548:
2544:
2536:
2532:
2528:
2525:
2521:
2518:
2514:
2510:
2507:
2503:
2499:
2495:
2494:
2493:
2491:
2486:
2472:
2452:
2445:, but not to
2432:
2412:
2392:
2372:
2352:
2332:
2325:, but not to
2312:
2292:
2272:
2252:
2249:
2229:
2209:
2201:
2185:
2177:
2161:
2141:
2112:
2087:
2086:prime numbers
2083:
2078:
2061:
2058:
2055:
2044:
2028:
2020:
2004:
1984:
1964:
1944:
1924:
1901:
1898:
1895:
1885:
1884:ordered pairs
1869:
1849:
1841:
1837:
1833:
1829:
1825:
1816:
1801:
1798:
1795:
1792:
1772:
1769:
1766:
1746:
1743:
1740:
1720:
1717:
1714:
1711:
1708:
1705:
1697:
1681:
1674:
1642:
1641:
1618:
1615:
1612:
1597:
1594:
1591:
1580:
1566:
1563:
1560:
1548:
1534:
1531:
1528:
1521:
1494:
1491:
1488:
1477:
1450:
1447:
1444:
1433:
1406:
1392:
1374:
1371:
1368:
1358:
1355:
1352:
1342:
1335:
1332:
1321:
1303:
1296:
1293:
1283:
1280:
1277:
1267:
1264:
1261:
1250:
1232:
1229:
1226:
1212:
1209:
1206:
1194:
1180:
1174:
1171:
1151:
1148:
1145:
1137:
1136:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1097:Antisymmetric
1095:
1093:
1090:
1088:
1087:
1076:
1066:
1061:
1056:
1051:
1046:
1036:
1026:
1021:
1019:
1016:
1015:
1004:
994:
989:
984:
979:
974:
969:
959:
954:
952:
949:
948:
937:
927:
922:
917:
912:
907:
902:
892:
887:
885:
882:
881:
875:
870:
860:
850:
845:
840:
835:
825:
820:
818:
815:
814:
808:
803:
793:
788:
778:
773:
768:
758:
753:
751:
748:
747:
741:
736:
726:
716:
706:
701:
696:
686:
681:
679:
676:
675:
669:
664:
654:
649:
644:
634:
624:
614:
609:
607:
606:Well-ordering
604:
603:
597:
592:
582:
577:
572:
562:
557:
552:
547:
544:
540:
539:
533:
528:
518:
513:
508:
498:
488:
483:
478:
476:
473:
472:
466:
461:
451:
446:
441:
436:
426:
416:
411:
409:
406:
405:
399:
394:
384:
379:
374:
369:
359:
354:
349:
347:
344:
343:
337:
332:
322:
317:
312:
307:
302:
292:
287:
285:
284:Partial order
282:
281:
275:
270:
260:
255:
250:
245:
240:
235:
230:
227:
220:
219:
213:
208:
198:
193:
188:
183:
178:
173:
163:
160:
156:
155:
152:
147:
145:
143:
141:
139:
136:
134:
132:
130:
129:
126:
123:
121:
118:
116:
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
90:Antisymmetric
88:
86:
83:
81:
80:
77:
76:
70:
65:
63:
58:
56:
51:
50:
47:
43:
39:
38:
33:
19:
21106:Georg Cantor
21101:Paul Bernays
21032:MorseβKelley
21007:
20940:
20939:Subset
20886:hereditarily
20848:Venn diagram
20806:ordered pair
20721:Intersection
20665:Axiom schema
20295:& Orders
20273:Star product
20202:Well-founded
20155:Prefix order
20111:Distributive
20101:Complemented
20071:Foundational
20036:Completeness
19992:Zorn's lemma
19896:Cyclic order
19885:
19879:Key concepts
19849:Order theory
19804:
19758:
19743:. Retrieved
19705:
19701:
19675:
19652:
19629:
19592:
19556:
19530:
19523:Bibliography
19502:
19497:
19473:
19468:
19446:
19413:
19395:
19370:
19363:
19357:(II): 67β77.
19354:
19350:
19340:
19320:
19310:
19305:232: 1729,30
19297:
19266:
19260:
19238:
19233:
19214:
19208:
19176:(1): 21β30.
19173:
19169:
19163:
19144:
19135:
19129:: 1999β2000.
19126:
19122:
19109:
19091:
19085:
19067:
19059:
19036:
19020:
19015:
19003:the original
18994:
18987:
18974:
18968:
18950:
18944:
18925:
18919:
18900:
18894:
18870:
18860:
18831:
18823:
18813:
18800:
18791:
18782:
18771:. Retrieved
18767:
18758:
18747:. Retrieved
18743:
18734:
18679:
18665:
18660:at Wikibooks
18648:
18632:
18627:
18617:November 25,
18615:. Retrieved
18593:John C. Baez
18587:
18579:
18571:
18563:
18555:
18542:
18524:
18519:
18500:
18469:
18459:
18439:
18429:
18410:
18407:Levy, Azriel
18401:
18382:
18373:
18352:
18341:
18313:
18307:
18295:
18279:
18216:. Retrieved
18212:
18203:
18192:. Retrieved
18164:
18160:
18104:
17971:
17955:Order theory
17816:
17813:
17778:denotes the
17607:
17531:
17388:strict order
17384:linear order
17251:
17167:
17145:
17004:
17001:(definition)
16593:Recall that
16529:transitivity
16526:
16181:generates a
16160:
16157:Preorder R\R
16017:
16013:Georg Aumann
15963:
15756:
15666:
15608:
15553:
15549:
15533:order theory
15529:strict order
15526:
15523:Ferrers type
15514:
15503:
15279:
15183:, where the
15095:block matrix
15086:
15080:
15033:
15028:
14993:difunctional
14992:
14828:
14825:Difunctional
14709:
14566:
14550:
14548:
14501:
14319:
14257:
14251:
14079:
14073:
14033:
13803:
13615:
13419:
12343:
12149:to a vertex
12044:
11950:
11948:
11884:
11877:
11604:
11576:
11415:
11411:
11406:
11400:
11396:
11390:
11381:
11372:
11365:
11361:
11355:
11309:
11243:is equal to
11220:
11138:
11134:
10912:is equal to
10889:
10807:
10801:
10756:
10751:Many-to-many
10750:
10744:
10738:
10732:
10536:
10393:
10390:right-unique
10389:
10383:
10228:
10200:
10057:
10053:
10048:
9985:
9980:real numbers
9663:
9498:is given by
9406:The various
9370:graph theory
9354:
9249:relation on
8598:
8251:
8218:
8193:
8112:
8096:
8071:
7769:
7145:contained in
7080:
6949:real numbers
6945:completeness
6942:
6859:
6835:trichotomous
6808:
6537:
6309:is a binary
6288:
6230:
6133:
6049:total orders
5886:
5668:
5530:
5334:
5081:
4753:
4739:
4496:
4493:Intersection
4448:
4205:
4034:
3923:
3919:
3915:
3913:
3867:
3865:
3860:
3856:
3852:
3809:endorelation
3771:
3644:
3503:-related to
3100:
3095:
2960:
2806:
2791:
2782:
2747:intersection
2740:
2551:
2540:
2524:graph theory
2487:
2079:
2018:
1882:is a set of
1839:
1835:
1827:
1821:
1670:
1107:Well-founded
225:(Quasiorder)
100:Well-founded
45:
21131:Thomas Jech
20974:Alternative
20953:Uncountable
20907:Ultrafilter
20766:Cardinality
20670:replacement
20618:Determinacy
20479:Riesz space
20440:Isomorphism
20316:Normal cone
20238:Composition
20172:Semilattice
20081:Homogeneous
20066:Equivalence
19916:Total order
19070:, page 37,
19064:Ki-Hang Kim
19031:, p. 4
18768:ncatlab.org
18550:Β§ 2.1.
18257:C. I. Lewis
15543:, called a
15085:, the term
14710:Proposition
14567:Proposition
14555:data mining
14551:Schein rank
13775:containing
13721:containing
13671:containing
13435:total order
12472:Irreflexive
12364:over a set
12284:which is a
12228:over a set
12065:over a set
11953:over a set
11653:instead of
11418:): for all
11141:): for all
11139:right-total
11065:. However,
10810:): for all
10745:Many-to-one
10739:One-to-many
10396:): for all
10231:primary key
10060:): for all
10058:left-unique
9914:, those of
8965:. Then the
7988:(indeed, a
7952:zero matrix
7814:indexed by
7505:are called
7149:a relation
7005:upper bound
6843:total order
6313:over a set
6279:Restriction
4744:Composition
3909:rectangular
3421:called the
2961:Given sets
2502:is equal to
1834:called the
1824:mathematics
1127:Irreflexive
408:Total order
120:Irreflexive
21126:Kurt GΓΆdel
21111:Paul Cohen
20948:Transitive
20716:Identities
20700:Complement
20687:Operations
20648:Regularity
20586:Adjunction
20545:Set theory
20447:Order type
20381:Cofinality
20222:Well-order
20197:Transitive
20086:Idempotent
20019:Asymmetric
19745:2020-05-05
19657:De Gruyter
19655:. Berlin:
19632:. Boston:
19599:. Boston:
19183:1612.04935
18852:0443.03021
18773:2024-06-13
18749:2024-06-13
18643:, pp. 7-10
18323:3540058192
18290:, Chapt. 5
18218:2019-12-11
18194:2020-04-29
18096:References
17802:. In 1953
17117:=∈
16493:, so that
15968:relation,
15535:. In 1951
14933:relations
14931:functional
14885:indicators
14546:on a set.
14313:forming a
14046:as in the
13338:then some
13274:: for all
13180:: for all
13059:: for all
12932:: for all
12929:Transitive
12817:: for all
12814:Asymmetric
12673:: for all
12578:: for all
12475:: for all
12393:: for all
12290:involution
11474:such that
11383:surjection
11193:such that
11135:Surjective
10944:Properties
10862:such that
10808:left-total
10733:One-to-one
10579:is called
10385:Functional
10227:is called
10006:over sets
9484:hyperplane
9386:hypergraph
8897:continents
8427:car, Venus
8413:doll, Mary
8399:ball, John
7663:, written
7209:, written
7101:over sets
6827:transitive
6823:asymmetric
6047:, and for
5659:Complement
5329:See also:
4197:Operations
3914:The terms
3699:such that
3597:such that
3124:over sets
2957:Definition
2856:over sets
2531:orthogonal
2506:arithmetic
2174:that is a
1698:: for all
1696:transitive
1132:Asymmetric
125:Asymmetric
42:Transitive
21059:Paradoxes
20979:Axiomatic
20958:Universal
20934:Singleton
20929:Recursive
20872:Countable
20867:Amorphous
20726:Power set
20643:Power set
20601:dependent
20596:countable
20498:Upper set
20435:Embedding
20371:Antichain
20192:Tolerance
20182:Symmetric
20177:Semiorder
20123:Reflexive
20041:Connected
19811:EMS Press
19425:∖
18601:Newsgroup
18413:. Dover.
18409:(2002) .
18385:. Dover.
18357:. Dover.
18349:(1972) .
18332:1431-4657
18181:207549016
18062:…
17817:different
17808:semiheaps
17558:∅
17543:
17488:
17462:¯
17453:⊆
17401:
17341:
17283:
17233:¯
17215:¯
17199:∩
17181:
17123:∈
17120:∖
17112:¯
17105:¯
17102:∈
17096:∋
17087:Ω
17044:∈
17011:power set
17007:inclusion
16983:∖
16977:⊆
16968:∖
16953:∖
16944:≡
16920:¯
16913:¯
16889:⊆
16880:∖
16865:∖
16856:≡
16832:¯
16822:∖
16807:∖
16795:⊆
16789:¯
16768:≡
16743:⊆
16734:∖
16719:∖
16686:⊆
16677:∖
16639:⊆
16610:∖
16575:∖
16569:⊆
16560:∖
16545:∖
16504:∖
16478:∖
16467:¯
16460:¯
16436:⊆
16429:⟹
16422:¯
16413:⊆
16407:¯
16363:¯
16305:¯
16262:¯
16255:¯
16231:≡
16225:∖
16195:∖
16141:∈
16121:∋
16095:⊆
16090:¯
16083:¯
16074:∋
16065:≡
16059:¯
16044:¯
16015:in 1970.
15999:ϵ
15976:ϵ
15903:∈
15840:⊆
15778:∈
15689:power set
15629:¯
15589:⊆
15572:¯
15441:∅
15438:≠
15425:∩
15294:∣
15161:×
15128:×
15063:⊆
14970:×
14964:⊆
14950:×
14944:⊆
14868:…
14805:×
14747:⊆
14691:×
14633:⊆
14416:, called
14414:functions
14301:⊑
14275:×
14269:⊆
14092:⊂
14052:morphisms
14014:×
14001:power set
13968:∩
13959:≡
13953:∅
13944:¯
13935:∩
13926:≡
13920:⊆
13835:⊆
13349:∈
13291:∈
13197:∈
13105:≠
13076:∈
13056:Connected
12955:∈
12912:≥
12878:then not
12834:∈
12797:≥
12690:∈
12595:∈
12575:Symmetric
12558:≥
12486:∈
12455:≥
12404:∈
12390:Reflexive
12267:×
12250:power set
12004:×
11860:∈
11830:∈
11751:⊆
11681:⊆
11552:∈
11514:∈
11459:∈
11429:∈
11392:bijection
11374:injection
11271:−
11178:∈
11152:∈
11105:, choose
10960:−
10935:connected
10847:∈
10821:∈
10694:−
10651:−
10439:∈
10407:∈
10394:univalent
10325:−
10262:−
10103:∈
10077:∈
10054:Injective
9870:×
9864:⊆
9609:
9589:incidence
9550:¯
9535:¯
9520:⟩
9508:⟨
9398:bicliques
9363:Australia
9338:×
9280:×
9234:×
8697:Atlantic
8500:×
7984:) form a
7747:∘
7724:≥
7674:⊊
7315:∈
7289:∈
7220:⊆
7066:≤
6986:⊆
6973:non-empty
6959:≤
6815:symmetric
6811:reflexive
6700:∈
6680:∣
6449:∈
6435:∈
6415:∣
6203:¯
6188:¯
6119:≤
6079:≥
6015:∈
5975:⊇
5935:⊆
5915:≠
5820:) is the
5805:¬
5773:¬
5770:∣
5743:¯
5653:symmetric
5639:≥
5619:≤
5559:≠
5436:∣
5183:∘
5163:∘
5133:∘
4995:) is the
4923:∈
4912:∣
4885:∘
4615:∣
4588:∩
4477:≥
4457:≤
4324:∣
4297:∪
4046:≠
3937:×
3448:∈
3406:×
3175:×
3073:∈
3059:∈
3053:∣
3012:×
2929:×
2926:⋯
2923:×
2877:…
2664:×
2637:⊆
2627:inclusion
2607:×
2594:power set
2250:−
2045:the pair
1609:not
1601:⇒
1557:not
1492:∧
1448:∨
1346:⇒
1336:≠
1291:⇒
1220:⇒
1178:∅
1175:≠
1122:Reflexive
1117:Has meets
1112:Has joins
1102:Connected
1092:Symmetric
223:Preorder
150:reflexive
115:Reflexive
110:Has meets
105:Has joins
95:Connected
85:Symmetric
21171:Category
21063:Problems
20967:Theories
20943:Superset
20919:Infinite
20748:Concepts
20628:Infinity
20552:Overview
20293:Topology
20160:Preorder
20143:Eulerian
20106:Complete
20056:Directed
20046:Covering
19911:Preorder
19870:Category
19865:Glossary
19756:(2010).
19740:25058006
19696:(1873).
19628:(1977).
19615:Archived
19590:(1990).
19200:54527913
19143:(1989).
18816:: 30β33.
18678:(1986).
18635:, 3β28.
18278:, 2010.
18185:Archived
18118:Archived
17896:See also
17888:β
16758:(repeat)
16527:To show
16183:preorder
15667:Suppose
15506:database
15453:implies
15230:and the
14418:mappings
14372:, where
14315:preorder
14064:category
13879:implies
13470:divides
12089:, where
11412:Set-like
11357:function
11221:at least
11017:integers
10890:at least
10686:to both
10623:to both
10422:and all
10205:preimage
10092:and all
8785:Atlantic
8723:Pacific
8000:Examples
7509:written
7003:with an
6035:∉
5995:⊉
5955:⊈
5319:Converse
4133:divides
3375:, where
3263:codomain
3193:The set
2783:concepts
2759:converse
2543:function
2529:the "is
2517:geometry
2200:multiple
2176:multiple
2111:integers
1840:codomain
1666:✗
1653:✗
1063:✗
1058:✗
1053:✗
1048:✗
1023:✗
991:✗
986:✗
981:✗
976:✗
971:✗
956:✗
924:✗
919:✗
914:✗
909:✗
904:✗
889:✗
877:✗
872:✗
847:✗
842:✗
837:✗
822:✗
810:✗
805:✗
790:✗
775:✗
770:✗
755:✗
743:✗
738:✗
703:✗
698:✗
683:✗
671:✗
666:✗
651:✗
646:✗
611:✗
599:✗
594:✗
579:✗
574:✗
559:✗
554:✗
549:✗
535:✗
530:✗
515:✗
510:✗
485:✗
480:✗
468:✗
463:✗
448:✗
443:✗
438:✗
413:✗
401:✗
396:✗
381:✗
376:✗
371:✗
356:✗
351:✗
339:✗
334:✗
319:✗
314:✗
309:✗
304:✗
289:✗
277:✗
272:✗
257:✗
252:✗
247:✗
242:✗
237:✗
232:✗
215:✗
210:✗
195:✗
190:✗
185:✗
180:✗
175:✗
21001:General
20996:Zermelo
20902:subbase
20884: (
20823:Forcing
20801:Element
20773: (
20751:Methods
20638:Pairing
20398:Duality
20376:Cofinal
20364:Related
20343:FrΓ©chet
20220:)
20096:Bounded
20091:Lattice
20064:)
20062:Partial
19930:Results
19901:Lattice
19813:, 2001
19710:Bibcode
19576:(1895)
19491:3729305
19462:0059267
19394:(2011)
19369:Review:
19103:2781235
19066:(1982)
18609:Usenet:
18603::
18578:(1962)
18259:(1918)
18237:(1895)
15713:subsets
15687:is the
15663:Contact
14794:is the
14736:, then
14680:is the
14135:is the
14080:concept
14056:objects
13907:lattice
12924:is not.
12570:is not.
12248:is the
11500:, i.e.
11448:of all
11401:exactly
11397:exactly
11366:exactly
11362:mapping
10537:at most
10201:at most
9853:, i.e.
8793:Pacific
8671:Arctic
8645:Indian
7946:), the
7639:smaller
7615:, then
7465:, then
7359:, then
6975:subset
6718:is the
6598:and if
6467:is the
5460:is the
5275:, then
4653:is the
4362:is the
3861:heteros
3463:reads "
2798:classes
2651:lattice
2082:divides
2019:related
678:Lattice
20892:Filter
20882:Finite
20818:Family
20761:Almost
20606:global
20591:Choice
20578:Axioms
20423:Subnet
20403:Filter
20353:Normed
20338:Banach
20304:&
20211:Better
20148:Strict
20138:Graded
20029:topics
19860:Topics
19770:
19738:
19682:
19663:
19640:
19607:
19580:, via
19563:
19537:
19512:
19483:
19405:
19328:
19285:
19252:
19221:
19198:
19151:
19077:
19027:
18957:
18932:
18907:
18882:
18850:
18840:
18692:
18611:
18534:
18507:
18477:
18447:
18417:
18389:
18361:
18330:
18320:
18286:
18241:, via
18179:
17749:where
17540:fringe
17485:fringe
17398:fringe
17338:fringe
17318:fringe
17280:fringe
17178:fringe
17168:fringe
17166:, its
16657:Then
16113:where
14929:using
14774:where
14660:where
11876:, see
11444:, the
9937:blocks
9912:points
9749:where
9694:. The
9664:blocks
9394:clique
9359:Europe
9353:which
8899:. Let
8895:, the
8811:oceans
8809:, the
8777:Arctic
8769:Indian
8245:Venus
8106:Venus
7029:has a
6091:, and
4814:, and
4153:, but
3874:where
3857:hetero
3745:. The
3643:. The
3549:. The
3215:domain
2833:of an
2777:, and
2749:, and
2425:, and
1977:is in
1937:is in
1917:where
1836:domain
1503:exists
1459:exists
1415:exists
44:
20984:Naive
20914:Fuzzy
20877:Empty
20860:types
20811:tuple
20781:Class
20775:large
20736:Union
20653:Union
20413:Ideal
20391:Graph
20187:Total
20165:Total
20051:Dense
19736:JSTOR
19618:(PDF)
19597:(PDF)
19374:from
19196:S2CID
19178:arXiv
19006:(PDF)
18999:(PDF)
18810:(PDF)
18188:(PDF)
18177:S2CID
18157:(PDF)
18121:(PDF)
18114:(PDF)
17963:Notes
17819:sets
17592:dense
17590:is a
17570:when
17252:When
16519:is a
15755:is a
14925:is a
14732:is a
14712:: If
14589:is a
14569:: If
13755:over
13569:and "
13271:Dense
13120:then
13025:then
12760:then
12639:then
12300:as a
11446:class
11416:local
10940:total
10803:Total
10541:image
10509:then
10173:then
9960:flags
9378:graph
9355:fails
8601:below
8209:Mary
8190:John
8182:doll
8176:ball
8087:Mary
8068:John
8060:doll
8054:ball
7643:than
7535:. If
7507:equal
7385:. If
7169:over
6831:total
6766:over
6638:then
6515:over
6373:then
6051:also
5844:over
5729:then
5488:over
5468:, of
5395:then
5039:over
4874:then
4697:over
4577:then
4406:over
4286:then
4202:Union
4061:then
3772:When
3747:field
3657:range
3653:image
3423:graph
2743:union
2686:when
2511:the "
2496:the "
1785:then
148:Anti-
20897:base
20004:list
19768:ISBN
19680:ISBN
19661:ISBN
19638:ISBN
19605:ISBN
19561:ISBN
19535:ISBN
19510:ISBN
19481:ISBN
19403:ISBN
19355:1970
19326:ISBN
19283:ISBN
19250:ISBN
19219:ISBN
19149:ISBN
19075:ISBN
19025:ISBN
18955:ISBN
18930:ISBN
18905:ISBN
18880:ISBN
18838:ISBN
18690:ISBN
18619:2018
18532:ISBN
18505:ISBN
18475:ISBN
18445:ISBN
18415:ISBN
18387:ISBN
18359:ISBN
18328:ISSN
18318:ISBN
18284:ISBN
17839:and
17628:and
17005:The
15964:The
15377:and
14593:and
14412:are
14392:and
14320:The
14111:The
14078:: A
13529:<
13390:and
12999:and
12734:and
12538:>
12504:not
11073:<
11050:>
11003:>
10709:and
10643:and
10539:one
10483:and
10340:and
10277:and
10203:one
10147:and
10026:and
9831:and
9769:and
9191:and
8751:Let
8546:and
8358:and
8228:Ian
8185:cup
8179:car
8063:cup
8057:car
7926:and
7886:and
7834:and
7790:and
7750:>
7744:>
7704:>
7575:but
7485:and
7425:and
7304:and
7189:and
7121:and
6837:, a
6786:and
6578:and
6333:and
6111:and
6099:>
6071:and
6059:<
6027:and
5987:and
5947:and
5907:and
5864:and
5709:and
5631:and
5599:>
5591:and
5579:<
5571:and
5508:and
5375:and
5235:and
5102:and
5059:and
5019:and
4854:and
4794:and
4717:and
4677:and
4557:and
4517:and
4426:and
4386:and
4266:and
4226:and
4012:and
3972:and
3922:and
3831:and
3807:(or
3309:and
3261:the
3144:and
2981:and
2572:and
2500:", "
1957:and
1862:and
1826:, a
1759:and
1164:and
20858:Set
20418:Net
20218:Pre
19726:hdl
19718:doi
19275:doi
19188:doi
19127:230
19049:doi
18979:158
18848:Zbl
18637:doi
18169:doi
17782:of
17386:or
17013:of
16153:).
15715:of
15691:of
15117:on
15081:In
14883:of
14115:of
14060:Rel
14003:of
13309:if
13240:or
13146:or
13094:if
12973:if
12852:if
12708:if
12613:if
12304:on
11889:or
11887:NBG
11371:An
10583:of
10549:or
10457:if
10392:or
10360:to
10297:to
10234:of
10121:if
9361:to
8640:AA
8637:AU
8634:AS
8631:EU
8628:AF
8625:SA
8622:NA
7333:if
7238:if
7007:in
6746:to
6726:of
6538:If
6495:to
6475:of
6289:If
6231:If
5824:of
5669:If
5335:If
4999:of
4754:If
4657:of
4497:If
4366:of
4206:If
3907:or
3749:of
3659:of
3655:or
3557:of
3553:or
3483:is
3333:or
3269:of
3265:or
3221:of
3217:or
2596:of
2465:or
2345:or
2202:of
2178:of
2017:is
1832:set
1822:In
1733:if
1694:be
1404:min
21173::
19809:,
19803:,
19766:.
19734:.
19724:.
19716:.
19704:.
19700:.
19659:.
19636:.
19613:.
19603:.
19555:.
19488:MR
19476:,
19459:MR
19441:".
19382:^
19353:.
19349:.
19281:.
19269:.
19245:,
19194:.
19186:.
19174:96
19172:.
19121:.
19100:MR
19094:,
18878:.
18846:.
18812:.
18766:.
18742:.
18716:^
18700:^
18682:.
18674:;
18607:.
18599:.
18489:^
18326:.
18268:^
18249:^
18227:^
18211:.
18183:.
18175:.
18165:13
18163:.
18159:.
18129:^
18116:.
18088:).
17390:.
17076::
15527:A
15512:.
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15031:.
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14498:."
14317:.
14066:.
13613:.
13420:A
12341:.
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11949:A
11882:.
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11306:).
10729:).
10380:).
10229:a
9583:A
9380:a
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8871:AU
8863:AS
8855:EU
8847:AF
8839:SA
8831:NA
8744:1
8718:1
8692:0
8666:1
8259:β
8240:β
8223:β
8204:β
8120:β
8101:β
8082:β
7762:.
6849:,
6845:,
6841:,
6833:,
6829:,
6825:,
6821:,
6817:,
6806:.
6535:.
6131:.
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5967:,
5884:.
5655:.
5528:.
5315:.
5079:.
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4032:.
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3651:,
3647:,
3101:A
3098:.
2804:.
2789:.
2773:,
2745:,
2738:.
2549:.
2541:A
2485:.
2473:13
2405:,
2313:10
2305:,
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2265:,
2041:,
20941:Β·
20925:)
20921:(
20888:)
20777:)
20537:e
20530:t
20523:v
20216:(
20213:)
20209:(
20060:(
20007:)
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19834:t
19827:v
19776:.
19748:.
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19669:.
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18367:.
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18171::
18084:(
18070:n
18066:x
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18029:y
18026:x
18023:R
18003:n
17983:n
17885:.
17873:B
17870:=
17867:A
17847:B
17827:A
17790:b
17763:T
17758:b
17737:c
17731:T
17726:b
17722:a
17719:=
17716:]
17713:c
17710:,
17707:b
17704:,
17701:a
17698:[
17675:)
17672:B
17669:,
17666:A
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17658:B
17636:B
17616:A
17578:R
17555:=
17552:)
17549:R
17546:(
17517:R
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17459:I
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17090:=
17064:U
17021:U
16989:.
16986:R
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16962:(
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16950:R
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16910:R
16901:T
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16868:R
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16828:)
16825:R
16819:R
16816:(
16813:)
16810:R
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16801:(
16786:R
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16728:(
16725:)
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16710:R
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16680:R
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16668:R
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16636:X
16633:R
16613:R
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16604:=
16601:X
16581:.
16578:R
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16566:)
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16548:R
16542:R
16539:(
16523:.
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16475:R
16472:=
16457:R
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16419:I
16404:R
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16329:R
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16098:C
16080:C
16068:C
16056:C
16041:C
16032:T
16027:C
15979:=
15949:.
15946:Z
15943:g
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15868:g
15865:x
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15816:.
15813:Y
15810:g
15807:x
15799:}
15796:x
15793:{
15790:=
15787:Y
15784:,
15781:A
15775:x
15743:g
15723:A
15699:A
15675:B
15644:T
15639:R
15635:,
15626:R
15620:,
15617:R
15595:.
15592:R
15586:R
15580:T
15569:R
15562:R
15490:.
15487:R
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15478:x
15474:=
15471:R
15466:1
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15429:x
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15315:R
15312:x
15309:=
15306:}
15303:y
15300:R
15297:x
15291:y
15288:{
15265:Y
15243:i
15239:B
15218:X
15196:i
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15131:Y
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15066:R
15060:R
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15045:R
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15007:G
15003:F
14976:.
14973:Z
14967:B
14961:G
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14947:A
14941:F
14910:T
14905:G
14901:F
14898:=
14895:R
14871:}
14865:,
14862:z
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14850:x
14847:{
14844:=
14841:Z
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13977:P
13974:=
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13950:=
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13923:Q
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13134:y
13131:R
13128:x
13108:y
13102:x
13082:,
13079:X
13073:y
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13067:x
13039:z
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13033:x
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13010:R
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12987:y
12984:R
12981:x
12961:,
12958:X
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12949:,
12946:y
12943:,
12940:x
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12860:x
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12837:X
12831:y
12828:,
12825:x
12777:.
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12771:=
12768:x
12748:x
12745:R
12742:y
12722:y
12719:R
12716:x
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12325:)
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12319:(
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12260:2
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11981:X
11961:X
11923:)
11920:G
11917:,
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11911:,
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11905:(
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11780:A
11760:.
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11661:=
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11635:=
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11590:=
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11529:x
11526:R
11523:y
11520:,
11517:Y
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11508:{
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11119:x
11116:=
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10567:}
10564:X
10561:{
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10520:=
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10184:=
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9962:.
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9737:)
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9630:)
9627:n
9624:,
9621:k
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9612:(
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9556:z
9547:x
9541:+
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9523:=
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9414:t
9365:.
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9315:R
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8883:}
8875:,
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8859:,
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8835:,
8827:{
8824:=
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8789:,
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8773:,
8765:{
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8437:.
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8423:(
8420:,
8417:)
8409:(
8406:,
8403:)
8395:(
8392:{
8389:=
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8326:.
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8315:{
8312:=
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8289:}
8281:{
8278:=
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8234:β
8231:β
8219:+
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8212:β
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8109:β
8097:+
8093:β
8090:β
8079:β
8076:β
8072:+
8039:B
8018:A
7972:Y
7969:=
7966:X
7934:Z
7914:Y
7894:Y
7874:X
7842:Y
7822:X
7798:Y
7778:X
7680:.
7677:S
7671:R
7651:S
7623:R
7603:R
7583:S
7563:S
7543:R
7523:S
7520:=
7517:R
7493:S
7473:R
7453:R
7433:S
7413:S
7393:R
7373:y
7370:S
7367:x
7347:y
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7109:X
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7046:.
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6990:R
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6928:y
6908:x
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6754:S
6734:R
6706:}
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6689:y
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6683:x
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6665:(
6662:{
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6483:R
6455:}
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6432:x
6424:y
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6412:)
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6400:(
6397:{
6394:=
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6341:S
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6248:,
6245:Y
6242:=
6239:X
6217:.
6211:T
6200:R
6193:=
6182:T
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6151:T
6146:R
5895:=
5872:Y
5852:X
5832:R
5808:R
5785:}
5782:y
5779:R
5776:x
5767:)
5764:y
5761:,
5758:x
5755:(
5752:{
5749:=
5740:R
5717:Y
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5539:=
5516:X
5496:Y
5476:R
5448:}
5445:y
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5433:)
5430:x
5427:,
5424:y
5421:(
5418:{
5415:=
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5243:y
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5203:x
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4980:;
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4957:}
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4951:S
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4920:y
4909:)
4906:z
4903:,
4900:x
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4894:{
4891:=
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4842:Y
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4725:Y
4705:X
4685:S
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4641:}
4638:y
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4624:y
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4612:)
4609:y
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4600:(
4597:{
4594:=
4591:S
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4350:}
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4341:x
4333:y
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4327:x
4321:)
4318:y
4315:,
4312:x
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4306:{
4303:=
4300:S
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4181:3
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4141:9
4121:3
4101:y
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4020:Y
4000:X
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3891:.
3888:B
3885:=
3882:A
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3783:=
3780:X
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3608:R
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3317:Y
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3172:X
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3082:,
3079:}
3076:Y
3070:y
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3047:y
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3038:(
3035:{
3015:Y
3009:X
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2942:.
2937:n
2933:X
2918:1
2914:X
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2884:X
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2869:1
2865:X
2842:n
2821:2
2818:=
2815:n
2726:Y
2723:=
2720:X
2700:Y
2697:=
2694:X
2670:.
2667:Y
2661:X
2629:(
2613:.
2610:Y
2604:X
2580:Y
2560:X
2537:.
2526:;
2519:;
2508:;
2453:4
2433:9
2413:6
2393:0
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2333:1
2293:6
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2253:4
2230:2
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2162:z
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2029:y
2005:x
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1945:X
1925:x
1905:)
1902:y
1899:,
1896:x
1893:(
1870:Y
1850:X
1802:.
1799:c
1796:R
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1767:b
1747:b
1744:R
1741:a
1721:,
1718:c
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1706:a
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1662:Y
1649:Y
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1613:b
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1561:a
1535:a
1532:R
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1451:b
1445:a
1407:S
1375:a
1372:R
1369:b
1359:b
1356:R
1353:a
1343:b
1333:a
1304:b
1297:=
1294:a
1284:a
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1268:b
1265:R
1262:a
1233:a
1230:R
1227:b
1213:b
1210:R
1207:a
1181::
1172:S
1152:b
1149:,
1146:a
1083:Y
1073:Y
1043:Y
1033:Y
1011:Y
1001:Y
966:Y
944:Y
934:Y
899:Y
867:Y
857:Y
832:Y
800:Y
785:Y
765:Y
733:Y
723:Y
713:Y
693:Y
661:Y
641:Y
631:Y
621:Y
589:Y
569:Y
525:Y
505:Y
495:Y
458:Y
433:Y
423:Y
391:Y
366:Y
329:Y
299:Y
267:Y
205:Y
170:Y
68:e
61:t
54:v
34:.
20:)
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