6429:
5113:
5025:
7228:
2823:
1820:
7129:
4497:
5069:
1653:
1640:
1074:
1064:
1034:
1024:
1002:
992:
957:
935:
925:
890:
858:
848:
823:
791:
776:
756:
724:
714:
704:
684:
652:
632:
622:
612:
580:
560:
516:
496:
486:
449:
424:
414:
382:
357:
320:
290:
258:
196:
161:
7284:
4487:
of this DAG is then the Hasse diagram. Similarly this process can be reversed to construct strict partial orders from certain DAGs. In contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted
7213:
which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. If the number 0 is included, this will be the greatest element, since this is a multiple of every integer (see Fig. 6).
2806:
Irreflexivity and transitivity together imply asymmetry. Also, asymmetry implies irreflexivity. In other words, a transitive relation is asymmetric if and only if it is irreflexive. So the definition is the same if it omits either irreflexivity or asymmetry (but not both).
10541:
does not hold, all these intervals are empty. Every interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig. 7b), the set {1, 2, 4, 5, 8} is convex, but not an interval.
1384:
4765:
1313:
2196:
partial orders. However some authors use the term for the other common type of partial order relations, the irreflexive partial order relations, also called strict partial orders. Strict and non-strict partial orders can be put into a
9612:
is an extension that is also a linear (that is, total) order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Every partial order can be extended to a total order
1628:
1242:
11177:
1513:
1469:
1425:
6885:
If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements. In our running example,
4368:
4503:
Graph of the divisibility of numbers from 1 to 4. This set is partially, but not totally, ordered because there is a relationship from 1 to every other number, but there is no relationship from 2 to 3 or 3 to
3262:
8221:
8114:
3343:
2998:
1650:
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
5580:
3523:. The dual of a non-strict partial order is a non-strict partial order, and the dual of a strict partial order is a strict partial order. The dual of a dual of a relation is the original relation.
3181:
4250:
1583:
1480:
1436:
1395:
1324:
1253:
1197:
5152:
10672:
5105:
5061:
3061:
10112:
10060:
7793:
5968:
2658:
9851:
8819:
6039:
2007:
is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize
7162:
is not maximal. If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any
1186:
9968:
4833:
10177:
8285:
7123:
1572:
7557:
4805:
3495:
6280:
3946:
3113:
7211:
3438:
3391:
9606:
8307:
4573:
4543:
4001:
2876:
9281:
9079:
8861:
2613:
2307:
2074:
11200:
10004:
6418:
5625:
1897:
1873:
10737:
9541:
9442:
8939:
7753:
7628:
7400:
6922:
6691:
5824:
3876:
3091:
1319:
10579:
10206:
9570:
8044:
8018:
7992:
7960:
7894:
7675:
7589:
7516:
7484:
7349:
6377:
6151:
6090:
5720:
5011:
descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.
4684:
4437:
4021:
3908:
2949:
9407:
9248:
9105:
8170:
7425:
7072:
6883:
6802:
6653:
6624:
6568:
6539:
6229:
4918:
2508:
2482:
2456:
2398:
2372:
2340:
1929:
8223:
that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it, for instance, does not map any number to the set
7842:
6946:
6854:
6828:
6773:
6747:
6715:
6594:
6509:
5507:
are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is the
3636:
3592:
2800:
2774:
2748:
2716:
2690:
1726:
1248:
9486:
9358:
9308:
9222:
9135:
9026:
8959:
8888:
8247:
7274:
6342:
6116:
5876:
5850:
5786:
5685:
4191:
4168:
4128:
4108:
4044:
3981:
3800:
3780:
3760:
3676:
3614:
3569:
3282:
3201:
2920:
2138:
1981:
1955:
10644:
if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product
4481:
4148:
4088:
4068:
3840:
3820:
3716:
3656:
1807:
7819:
7150:
all other elements; on the other hand this poset does not have a greatest element. This partially ordered set does not even have any maximal elements, since any
3521:
2424:
1778:
1752:
1540:
1157:
10135:
9746:
9699:
6476:
10980:
9914:
9723:
9676:
9506:
9462:
9328:
9195:
9175:
9155:
9046:
8999:
8979:
8912:
4461:
3736:
3696:
3549:
3462:
3411:
3137:
2900:
2560:
2254:
2178:
2158:
2118:
2094:
1687:
3963:
can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than
3210:
1578:
1192:
3291:
1475:
11657:
11319:. CALCULEMUS-2003 β 11th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning. Roma, Italy: Aracne. pp. 89β100.
1431:
1390:
4309:
12384:
61:
11024:
8751:
8336:
12367:
10599:. Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let
8179:
8072:
11897:
5364:
11733:
11135:
So we can think of every partial order as really being a pair, consisting of a weak partial order and an associated strict one.
10741:
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the
8313:. The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called
3846:
refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation,
2954:
11625:
11604:
11555:
11482:
11438:
11034:
11615:
6948:
are the maximal and minimal elements. Removing these, there are 3 maximal elements and 3 minimal elements (see Fig. 5).
5522:
12214:
8319:
3142:
10950:
8731:
8250:
4212:
11545:
11398:
6435:
The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are all
5124:
12350:
12209:
11647:
11390:
11351:
11096:
11063:
10647:
5080:
5036:
7292:
Order isomorphism between the divisors of 120 (partially ordered by divisibility) and the divisor-closed subsets of
2832:
about the connections between strict/non-strict relations and their duals, via the operations of reflexive closure (
12204:
3003:
10065:
10013:
7758:
5282:
11840:
5511:
of two partially ordered sets, with no order relation between elements of one set and elements of the other set.
11120:
7166:
is a minimal element for it. In this poset, 60 is an upper bound (though not a least upper bound) of the subset
5907:
11922:
10827:
54:
2624:
2201:, so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.
12241:
12161:
9785:
8765:
5504:
5173:
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the
17:
11301:
A comparison between two elements s, t in S returns one of three distinct values, namely sβ€t, s>t or s|t.
5985:
12026:
11955:
11835:
11929:
11917:
11880:
11855:
11830:
11784:
11753:
10882:
2015:
1162:
12226:
11860:
11850:
11726:
9923:
7359:
4810:
10139:
8256:
7081:
6428:
1546:
12422:
12199:
11865:
11331:
9867:
9863:
9614:
8120:
of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its
7524:
4770:
4606:
3467:
3139:
may be converted to a non-strict partial order by adjoining all relationships of that form; that is,
2198:
47:
7227:
6250:
3913:
3096:
2160:. When the meaning is clear from context and there is no ambiguity about the partial order, the set
12131:
11758:
11706:
sequence A000112 (Number of partially ordered sets ("posets") with n unlabeled elements.)
10993:
7169:
5892:
5112:
3416:
3369:
1121:
945:
114:
11430:
10758:, a formalization of orderings on a set that allows more general families of orderings than posets
9575:
8290:
8172:) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of its
5024:
4556:
4526:
3986:
2855:
1379:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
12379:
12362:
9891:
9253:
9051:
8828:
5980:
5979:
is a subset of a poset in which no two distinct elements are comparable. For example, the set of
4760:{\displaystyle \left(a_{n}\right)_{n\in \mathbb {N} }\leq \left(b_{n}\right)_{n\in \mathbb {N} }}
2577:
2271:
2041:
11119:
Avigad, Jeremy; Lewis, Robert Y.; van Doorn, Floris (29 March 2021). "13.2. More on
Orderings".
9980:
9878:. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is
6382:
5589:
1882:
1837:
12291:
11907:
11599:. Encyclopedia of Mathematics and its Applications. Vol. 132. Cambridge University Press.
10773:
10689:
10305:
10278:
10216:
9655:
9629:
9511:
9420:
8917:
7714:
7598:
7370:
6951:
6889:
6658:
5791:
4628:
4440:
4397:
4047:
3959:, as long as it is clear from the context that no other kind of order is meant. In particular,
3849:
3353:
3264:
Conversely, if < is a strict partial order, then the corresponding non-strict partial order
3070:
2345:
2261:
2023:
1308:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
1091:
84:
11051:
10552:
10182:
9546:
8023:
7997:
7965:
7933:
7867:
7651:
7562:
7489:
7457:
7322:
6347:
6121:
6066:
5690:
4410:
4006:
3881:
2925:
12417:
12269:
12104:
12095:
11964:
11845:
11763:
11719:
11408:
11343:
11336:
11106:
9363:
9227:
9084:
8143:
7404:
7045:
6859:
6778:
6629:
6603:
6544:
6518:
6208:
5182:
4894:
2487:
2461:
2435:
2377:
2351:
2319:
2003:
is an arrangement such that, for certain pairs of elements, one precedes the other. The word
1902:
1646:
indicates that the column's property is always true the row's term (at the very left), while
1101:
94:
7824:
6927:
6833:
6807:
6752:
6726:
6696:
6573:
6488:
3618:
3574:
2779:
2753:
2727:
2695:
2669:
1696:
12357:
12316:
12306:
12296:
12041:
12004:
11994:
11974:
11959:
11148:
10894:
10888:
10852:
10809:
10641:
10324:
10320:
9971:
9471:
9336:
9286:
9200:
9113:
9004:
8944:
8866:
8399:
8389:
8314:
8226:
7519:
7259:
6321:
6095:
5900:
5855:
5829:
5765:
5664:
5353:
4484:
4389:
4378:
4370:
that returns one of four codes when given two elements. This definition is equivalent to a
4176:
4153:
4113:
4093:
4029:
3966:
3785:
3765:
3745:
3661:
3596:
3554:
3267:
3186:
2905:
2618:
2563:
2230:
2123:
1960:
1934:
1667:
878:
153:
11423:
4466:
4133:
4073:
4053:
3825:
3805:
3701:
3641:
1783:
8:
12284:
12195:
12141:
12100:
12090:
11979:
11912:
11875:
10767:
10007:
9633:
8364:
7798:
7135:
5883:
5357:
4923:
4836:
4393:
3500:
2829:
2721:
2663:
2571:
2567:
2429:
2403:
2265:
2027:
1810:
A term's definition may require additional properties that are not listed in this table.
1757:
1731:
1690:
1519:
1136:
1126:
537:
119:
36:
10319:. Every nonempty convex sublattice can be uniquely represented as the intersection of a
10117:
9728:
9681:
6458:
2192:
usually refers to the reflexive partial order relations, referred to in this article as
12396:
12323:
12176:
12085:
12075:
12016:
11934:
11870:
11642:. Cambridge Studies in Advanced Mathematics. Vol. 49. Cambridge University Press.
11635:
11474:
11093:
Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
10965:
9899:
9879:
9708:
9661:
9491:
9447:
9313:
9180:
9160:
9140:
9031:
8984:
8964:
8897:
8374:
8369:
5658:
4952:
4948:
4446:
4261:
3739:
3721:
3681:
3534:
3447:
3396:
3204:
3203:
is a non-strict partial order, then the corresponding strict partial order < is the
3122:
2885:
2545:
2313:
2257:
2239:
2163:
2143:
2103:
2079:
2019:
1672:
1116:
1096:
1086:
1012:
109:
89:
79:
12236:
11583:
11566:
11516:
11499:
12333:
12311:
12171:
12156:
12136:
11939:
11643:
11621:
11600:
11551:
11478:
11434:
11386:
11347:
11059:
11030:
10821:
10797:
9917:
8047:
7902:
6655:
A poset can only have one greatest or least element. In our running example, the set
6161:
5278:
5174:
5007:
One familiar example of a partially ordered set is a collection of people ordered by
4579:
3441:
3285:
3064:
2922:
may be converted to a strict partial order by removing all relationships of the form
2234:
2000:
11314:
5349:
All three can similarly be defined for the
Cartesian product of more than two sets.
12146:
11999:
11578:
11521:
11511:
10864:
9855:
9645:
9621:
9609:
7353:
6482:
4202:
811:
744:
2817:
12328:
12111:
11989:
11984:
11969:
11885:
11794:
11779:
11592:
10928:
10803:
9875:
9624:, algorithms for finding linear extensions of partial orders (represented as the
8359:
7683:
6720:
4639:
4170:
3116:
2822:
2811:
1819:
672:
469:
40:
4926:, a partially ordered set defined by an alternating sequence of order relations
12246:
12231:
12221:
12080:
12058:
12036:
11669:
Kalmbach, G. (1976). "Extension of
Homology Theory to Partially Ordered Sets".
11381:
Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and
Lexicographic Order",
10858:
10832:
10742:
9871:
7128:
5508:
4851:
4654:
4617:
2845:
1623:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
340:
11525:
1237:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
12411:
12345:
12301:
12279:
12151:
12021:
12009:
11814:
11617:
Ordered Sets: An
Introduction with Connections from Combinatorics to Topology
11056:
Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics
10843:
10837:
9975:
8121:
7927:
5734:
5239:
4404:
4206:
3571:, we may uniquely extend our notation to define four partial order relations
2849:
1876:
1827:
1111:
1106:
278:
104:
99:
5014:
4549:
relation < is a strict partial order. The same is also true of the usual
12166:
12048:
12031:
11949:
11789:
11742:
11283:
10788:
10782:
9625:
8737:
The number of strict partial orders is the same as that of partial orders.
7163:
5498:
4643:
4632:
4550:
4400:. This includes both reflexive and irreflexive partial orders as subtypes.
1992:
10824: β Associative algebra used in combinatorics, a branch of mathematics
10800: β Associative algebra used in combinatorics, a branch of mathematics
5891:
is a partial order under which every pair of elements is comparable, i.e.
12372:
12065:
11944:
11809:
11466:
10922:
10870:
10755:
10282:
8394:
8173:
4602:
4513:
3960:
2008:
1988:
402:
1508:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
12340:
12274:
12115:
11091:(1). Prague: School of Mathematics β Physics Charles University: 55β69.
10761:
10286:
9916:
is a partially ordered set that has also been given the structure of a
6455:
There are several notions of "greatest" and "least" element in a poset
4978:
3910:, or, in rare instances, the non-strict and strict relations together,
600:
10633:, so it cannot be written in interval notation using elements of
4496:
4488:
by a Hasse diagram, actually the corresponding strict order is shown.
1464:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
12391:
12264:
12070:
10876:
10770: β Graph linking pairs of comparable elements in a partial order
9779:
8961:. The latter condition is equivalent to the requirement that for any
8117:
7897:
7708:
5975:
5583:
5068:
5008:
4610:
4590:
4546:
3551:
and a partial order relation, typically the non-strict partial order
1831:
1420:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
7681:
is both order-preserving and order-reflecting, then it is called an
4363:{\displaystyle {\text{compare}}:P\times P\to \{<,>,=,\vert \}}
2660:, i.e. no element is related to itself (also called anti-reflexive).
12186:
12053:
11804:
10626:
10290:
9702:
9651:
8379:
8140:. However, it is neither injective (since it maps both 12 and 6 to
5904:
is a subset of a poset that is a totally ordered set. For example,
4658:
4598:
2515:
217:
11176:
Kwong, Harris (25 April 2018). "7.4: Partial and Total
Ordering".
10931: β Mathematical proposition equivalent to the axiom of choice
10006:
Under this assumption partial order relations are well behaved at
7930:(see Fig. 7a). It can be shown that if order-preserving maps
5895:
holds. For example, the natural numbers with their standard order.
11702:
11690:
10622:
8744:
isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... (sequence
8116:
from the set of natural numbers (ordered by divisibility) to the
7147:
7143:
4621:
4516:, or in general any totally ordered set, ordered by the standard
11049:
10861:, a kind of topological space that can be defined from any poset
3257:{\displaystyle a<b{\text{ if }}a\leq b{\text{ and }}a\neq b.}
11711:
11447:
8891:
4594:
4586:
4373:
2818:
Correspondence of strict and non-strict partial order relations
7283:
11080:
8741:
2514:
A non-strict partial order is also known as an antisymmetric
8216:{\displaystyle g:\mathbb {N} \to \mathbb {P} (\mathbb {N} )}
8109:{\displaystyle f:\mathbb {N} \to \mathbb {P} (\mathbb {N} )}
4508:
Standard examples of posets arising in mathematics include:
4252:
as defined previously, it can be observed that two elements
11705:
11693:
11078:
10791: β partially ordered set equipped with a rank function
8746:
8707:
8702:
8697:
8692:
8687:
8682:
8677:
8672:
8667:
8662:
5360:, the result is in each case also an ordered vector space.
3338:{\displaystyle a\leq b{\text{ if }}a<b{\text{ or }}a=b.}
11079:
FlaΕ‘ka, V.; JeΕΎek, J.; Kepka, T.; Kortelainen, J. (2007).
7217:
11365:
A partially ordered set is conveniently represented by a
5015:
Orders on the
Cartesian product of partially ordered sets
2574:; that is, it satisfies the following conditions for all
10342:
is a subset that can be defined with interval notation:
9870:. In a poset, the smallest element, if it exists, is an
2993:{\displaystyle <\;:=\ \leq \ \setminus \ \Delta _{P}}
11316:
Making proofs in a hierarchy of mathematical structures
11313:
Prevosto, Virgile; Jaume, Mathieu (11 September 2003).
10885:β every partial order is contained in some total order.
10848:
Pages displaying short descriptions of redirect targets
9885:
7146:, ordered by divisibility: 1 is a least element, as it
5365:
orders on the
Cartesian product of totally ordered sets
4407:. Specifically, taking a strict partial order relation
11550:(2nd ed.). New York: Cambridge University Press.
11029:. Springer Science & Business Media. p. 100.
10867:β continuity of a function between two partial orders.
5575:{\displaystyle ({\mathcal {P}}(\{x,y,z\}),\subseteq )}
2878:
so row 3, column 4 of the bottom left matrix is empty.
2810:
A strict partial order is also known as an asymmetric
10968:
10692:
10650:
10555:
10185:
10142:
10120:
10068:
10016:
9983:
9926:
9902:
9788:
9731:
9711:
9684:
9664:
9578:
9549:
9514:
9494:
9474:
9450:
9423:
9366:
9339:
9316:
9289:
9256:
9230:
9203:
9183:
9163:
9143:
9116:
9087:
9054:
9034:
9007:
8987:
8967:
8947:
8920:
8900:
8869:
8831:
8768:
8332:
8293:
8259:
8229:
8182:
8146:
8075:
8026:
8000:
7968:
7936:
7870:
7827:
7801:
7761:
7717:
7654:
7601:
7565:
7527:
7492:
7460:
7407:
7373:
7325:
7262:
7172:
7084:
7048:
6930:
6892:
6862:
6836:
6810:
6781:
6755:
6729:
6699:
6661:
6632:
6606:
6576:
6547:
6521:
6491:
6461:
6385:
6350:
6324:
6253:
6211:
6124:
6098:
6069:
5988:
5910:
5858:
5832:
5794:
5768:
5693:
5667:
5592:
5525:
5177:
of two partially ordered sets are (see Fig. 4):
5127:
5083:
5039:
4897:
4813:
4773:
4687:
4559:
4529:
4469:
4449:
4413:
4312:
4215:
4179:
4156:
4136:
4116:
4096:
4076:
4056:
4032:
4009:
3989:
3969:
3916:
3884:
3852:
3828:
3808:
3788:
3768:
3748:
3724:
3704:
3684:
3664:
3644:
3621:
3599:
3577:
3557:
3537:
3503:
3470:
3450:
3419:
3399:
3372:
3294:
3270:
3213:
3189:
3176:{\displaystyle \leq \;:=\;\Delta _{P}\;\cup \;<\;}
3145:
3125:
3099:
3073:
3006:
2957:
2928:
2908:
2888:
2858:
2782:
2756:
2730:
2698:
2672:
2627:
2580:
2548:
2490:
2464:
2438:
2406:
2380:
2354:
2322:
2274:
2242:
2166:
2146:
2126:
2106:
2082:
2044:
1963:
1937:
1905:
1885:
1840:
1786:
1760:
1734:
1699:
1675:
1581:
1549:
1522:
1478:
1434:
1393:
1322:
1251:
1195:
1165:
1139:
11118:
11085:
10897:β strict partial order "<" in which the relation
10814:
Pages displaying wikidata descriptions as a fallback
10793:
Pages displaying wikidata descriptions as a fallback
10778:
Pages displaying wikidata descriptions as a fallback
10418:
Using the corresponding strict relation "<", the
5375:
Another way to combine two (disjoint) posets is the
5121:
Reflexive closure of strict direct product order on
10485:on the integers is empty since there is no integer
4443:(DAG) may be constructed by taking each element of
4245:{\displaystyle \leq ,<,\geq ,{\text{ and }}>}
3718:is the associated strict partial order relation on
2426:, i.e. no two distinct elements precede each other.
11694:sequence A001035 (Number of posets with
11422:
11335:
10974:
10840: β Algebraic object with an ordered structure
10817:Pages displaying short descriptions with no spaces
10731:
10666:
10573:
10200:
10171:
10129:
10106:
10054:
9998:
9962:
9908:
9845:
9740:
9717:
9693:
9670:
9600:
9564:
9535:
9500:
9480:
9456:
9436:
9401:
9352:
9322:
9302:
9275:
9242:
9216:
9189:
9169:
9149:
9129:
9099:
9073:
9040:
9020:
8993:
8973:
8953:
8933:
8906:
8882:
8855:
8813:
8301:
8279:
8241:
8215:
8164:
8108:
8038:
8012:
7986:
7954:
7888:
7836:
7813:
7787:
7747:
7669:
7622:
7583:
7551:
7510:
7478:
7419:
7394:
7343:
7268:
7236:Order-preserving, but not order-reflecting (since
7205:
7117:
7066:
6940:
6916:
6877:
6848:
6822:
6796:
6767:
6741:
6709:
6685:
6647:
6618:
6588:
6562:
6533:
6503:
6470:
6412:
6371:
6336:
6274:
6223:
6145:
6110:
6084:
6033:
5962:
5870:
5844:
5818:
5780:
5714:
5679:
5619:
5574:
5146:
5099:
5055:
4912:
4827:
4799:
4759:
4567:
4537:
4475:
4455:
4431:
4362:
4244:
4201:Another way of defining a partial order, found in
4185:
4162:
4142:
4122:
4102:
4082:
4062:
4038:
4015:
3995:
3975:
3940:
3902:
3870:
3834:
3814:
3794:
3774:
3754:
3730:
3710:
3690:
3670:
3650:
3630:
3608:
3586:
3563:
3543:
3515:
3489:
3456:
3432:
3405:
3385:
3337:
3276:
3256:
3195:
3175:
3131:
3107:
3085:
3055:
2992:
2943:
2914:
2894:
2870:
2794:
2768:
2742:
2710:
2684:
2652:
2607:
2554:
2502:
2476:
2450:
2418:
2392:
2366:
2334:
2301:
2248:
2172:
2152:
2132:
2112:
2088:
2068:
1975:
1949:
1923:
1891:
1867:
1801:
1772:
1746:
1720:
1681:
1622:
1566:
1534:
1507:
1463:
1419:
1378:
1307:
1236:
1180:
1151:
11329:
10879: β Numerical ordering with a margin of error
5370:
4023:to distinguish partial orders from total orders.
12409:
11571:Proceedings of the American Mathematical Society
11504:Proceedings of the American Mathematical Society
11050:Simovici, Dan A. & Djeraba, Chabane (2008).
10676:(1, 2) β€ (1, 3) β€ (1, 4) β€ (1, 5) β€ ... β€ (2, 1)
10277:. This definition generalizes the definition of
10070:
10018:
7038:itself, and a least element is a lower bound of
5147:{\displaystyle \mathbb {N} \times \mathbb {N} .}
4110:, which is always a subset of the complement of
2902:are closely related. A non-strict partial order
1398:
10925: β Order whose elements are all comparable
10785: β Mathematical ordering with upper bounds
10667:{\displaystyle \mathbb {N} \times \mathbb {N} }
8339:gives the number of partial orders on a set of
5166:are highlighted in green and red, respectively.
5100:{\displaystyle \mathbb {N} \times \mathbb {N} }
5056:{\displaystyle \mathbb {N} \times \mathbb {N} }
11613:
11543:
11453:
11404:
11205:Sage 9.2.beta2 Reference Manual: Combinatorics
11102:
10891: β Partial order between random variables
10806: β Set whose pairs have minima and maxima
10776: β term used in mathematical order theory
10617:as a subposet of the real numbers. The subset
9874:, and the largest element, if it exists, is a
7926:. Isomorphic orders have structurally similar
5397:, defined on the union of the underlying sets
2882:Strict and non-strict partial orders on a set
11727:
11312:
10846: β Group with a compatible partial order
10678:. Using the interval notation, the property "
8351:-element binary relations of different types
4403:A finite poset can be visualized through its
4090:is the converse of the irreflexive kernel of
3119:. Conversely, a strict partial order < on
3056:{\displaystyle \Delta _{P}:=\{(p,p):p\in P\}}
2951:that is, the strict partial order is the set
1663:in the "Antisymmetric" column, respectively.
55:
11420:
11342:. New York: John Wiley & Sons. pp.
10723:
10711:
10107:{\displaystyle \lim _{i\to \infty }b_{i}=b,}
10055:{\displaystyle \lim _{i\to \infty }a_{i}=a,}
9957:
9927:
8236:
8230:
8159:
8147:
7788:{\displaystyle x\leq y{\text{ and }}y\leq x}
7197:
7173:
7109:
7106:
7100:
7094:
7088:
7085:
7061:
7049:
6935:
6931:
6911:
6893:
6830:is a minimal element if there is no element
6749:is a maximal element if there is no element
6704:
6700:
6680:
6662:
6404:
6386:
6363:
6351:
6331:
6325:
6137:
6125:
6105:
6099:
6025:
6022:
6016:
6010:
6004:
5998:
5992:
5989:
5957:
5954:
5936:
5930:
5924:
5918:
5914:
5911:
5865:
5859:
5839:
5833:
5813:
5795:
5775:
5769:
5706:
5694:
5674:
5668:
5627:ordered by set inclusion (see Fig. 1).
5611:
5593:
5557:
5539:
5285:of the corresponding strict orders:
4357:
4354:
4333:
3050:
3020:
1970:
1964:
1944:
1938:
1918:
1906:
1859:
1841:
11081:"Transitive Closures of Binary Relations I"
10764:, a poset-based approach to quantum gravity
8326:
12385:Positive cone of a partially ordered group
11734:
11720:
11380:
11281:
11179:A Spiral Workbook for Discrete Mathematics
11072:
11026:A Beginner's Guide to Discrete Mathematics
7454:is also a partially ordered set, and both
7142:As another example, consider the positive
6282:Using the strict order <, the relation
5963:{\displaystyle \{\{\,\},\{x\},\{x,y,z\}\}}
4384:Wallis defines a more general notion of a
4377:, where equality is taken to be a defined
4196:
3678:is a non-strict partial order relation on
3172:
3168:
3164:
3153:
3149:
3104:
3100:
2961:
2342:, i.e. every element is related to itself.
2183:
62:
48:
11656:
11582:
11564:
11515:
11491:
11282:Chen, Peter; Ding, Guoli; Seiden, Steve.
10962:which always exists and is unique, since
10855: β Vector space with a partial order
10660:
10652:
10471:). An open interval may be empty even if
8295:
8267:
8206:
8198:
8190:
8099:
8091:
8083:
7844:If an order-embedding between two posets
6934:
6703:
5917:
5137:
5129:
5093:
5085:
5049:
5041:
4821:
4751:
4715:
4597:(see Fig. 1). Similarly, the set of
4561:
4531:
1879:. Sets connected by an upward path, like
12368:Positive cone of an ordered vector space
11668:
11018:
11016:
11014:
10285:. When there is possible confusion with
7127:
6427:
4495:
4306:. This can be represented by a function
3183:is a non-strict partial order. Thus, if
2821:
2653:{\displaystyle \neg \left(a<a\right)}
2542:is a homogeneous relation < on a set
2521:
1818:
11634:
11591:
11544:Davey, B. A.; Priestley, H. A. (2002).
9866:to one another if and only if they are
9846:{\displaystyle (y,z)\circ (x,y)=(x,z).}
8814:{\displaystyle P^{*}=(X^{*},\leq ^{*})}
7218:Mappings between partially ordered sets
2852:is depicted in the center. For example
14:
12410:
11500:"Partially Ordered Topological Spaces"
11146:
11022:
9920:, then it is customary to assume that
6034:{\displaystyle \{\{x\},\{y\},\{z\}\}.}
5155:
11715:
11175:
11011:
10994:General relativity Β§ Time travel
10621:is a bounded interval, but it has no
10407:). It contains at least the elements
9853:Such categories are sometimes called
9639:
7559:is order-preserving, too. A function
6439:elements, and the bottom row are all
6197:fits between them; formally: if both
3955:is sometimes used as a shorthand for
2011:, in which every pair is comparable.
11497:
11465:
11385:, World Scientific, pp. 62β63,
9886:Partial orders in topological spaces
7296:(partially ordered by set inclusion)
4264:relationships to each other: either
2844:). Each relation is depicted by its
2180:itself is sometimes called a poset.
1666:All definitions tacitly require the
11147:Rounds, William C. (7 March 2002).
10686:" can be rephrased equivalently as
10549:is bounded if there exist elements
9412:
8732:Stirling numbers of the second kind
5657:, because the relation need not be
4677:precedes the corresponding item in
4624:. (see Fig. 3 and Fig. 6)
2225:, commonly referred to simply as a
1181:{\displaystyle S\neq \varnothing :}
24:
11895:Properties & Types (
11567:"On Continuity of a Partial Order"
11547:Introduction to Lattices and Order
10481:. For example, the open interval
10080:
10028:
6290:can be equivalently rephrased as "
5531:
5514:
4026:When referring to partial orders,
3155:
3008:
2981:
2628:
1886:
25:
12434:
12351:Positive cone of an ordered field
11683:
11662:Foundations of Algebraic Topology
11584:10.1090/S0002-9939-1968-0236071-7
11517:10.1090/S0002-9939-1954-0063016-5
9963:{\displaystyle \{(a,b):a\leq b\}}
8320:Birkhoff's representation theorem
7821:according to the antisymmetry of
7303:Given two partially ordered sets
6723:and minimal elements: An element
4828:{\displaystyle n\in \mathbb {N} }
4463:to be a node and each element of
2974:
2204:
1172:
27:Mathematical set with an ordering
12205:Ordered topological vector space
11741:
11338:Topological Methods in Chemistry
11270:are not comparable, return None.
10873: β Partial order with joins
10237:with the property that, for any
10172:{\displaystyle a_{i}\leq b_{i},}
8280:{\displaystyle g(\mathbb {N} ).}
7282:
7226:
7158:, which is distinct from it, so
7118:{\displaystyle \{\{x\},\{y\}\}.}
5501:, then so is their ordinal sum.
5111:
5067:
5023:
1651:
1638:
1567:{\displaystyle {\text{not }}aRa}
1072:
1062:
1032:
1022:
1000:
990:
955:
933:
923:
888:
856:
846:
821:
789:
774:
754:
722:
712:
702:
682:
650:
630:
620:
610:
578:
558:
514:
494:
484:
447:
422:
412:
380:
355:
318:
288:
256:
194:
159:
11459:
11414:
11374:
11323:
11306:
11275:
11023:Wallis, W. D. (14 March 2013).
10985:
10956:
9508:provided that for all elements
8309:and its isomorphic image under
8287:Fig. 7b shows a subset of
7552:{\displaystyle g\circ f:S\to U}
7078:for the collection of elements
4800:{\displaystyle a_{n}\leq b_{n}}
4520:relation β€, is a partial order.
3490:{\displaystyle xR^{\text{op}}y}
2014:Formally, a partial order is a
11614:Bernd SchrΓΆder (11 May 2016).
11193:
11169:
11156:EECS 203: DISCRETE MATHEMATICS
11140:
11112:
11043:
10943:
10705:
10693:
10077:
10025:
9942:
9930:
9837:
9825:
9819:
9807:
9801:
9789:
9396:
9383:
8850:
8838:
8808:
8782:
8271:
8263:
8210:
8202:
8194:
8103:
8095:
8087:
7978:
7946:
7880:
7742:
7736:
7727:
7721:
7575:
7543:
7502:
7470:
7335:
6485:and least element: An element
6275:{\displaystyle a\neq c\neq b.}
5569:
5560:
5536:
5526:
5505:Series-parallel partial orders
5371:Sums of partially ordered sets
4854:containing all functions from
4620:equipped with the relation of
4426:
4414:
4330:
4150:is equal to the complement of
3941:{\displaystyle (P,\leq ,<)}
3935:
3917:
3897:
3885:
3865:
3853:
3842:. Strictly speaking, the term
3347:
3108:{\displaystyle \;\setminus \;}
3035:
3023:
2063:
2051:
2038:for short) is an ordered pair
1659:in the "Symmetric" column and
1595:
1340:
1285:
1214:
13:
1:
12162:Series-parallel partial order
11664:. Princeton University Press.
11565:Deshpande, Jayant V. (1968).
11536:
11291:(Technical report). p. 2
10674:is not locally finite, since
10315:that is also a convex set of
8249:), but it can be made one by
8132:, then each prime divisor of
8124:. It is order-preserving: if
7206:{\displaystyle \{2,3,5,10\},}
6693:is the greatest element, and
3433:{\displaystyle R^{\text{op}}}
3386:{\displaystyle R^{\text{op}}}
1931:, are comparable, while e.g.
1660:
1647:
1057:
1052:
1047:
1042:
1017:
985:
980:
975:
970:
965:
950:
918:
913:
908:
903:
898:
883:
871:
866:
841:
836:
831:
816:
804:
799:
784:
769:
764:
749:
737:
732:
697:
692:
677:
665:
660:
645:
640:
605:
593:
588:
573:
568:
553:
548:
543:
529:
524:
509:
504:
479:
474:
462:
457:
442:
437:
432:
407:
395:
390:
375:
370:
365:
350:
345:
333:
328:
313:
308:
303:
298:
283:
271:
266:
251:
246:
241:
236:
231:
226:
209:
204:
189:
184:
179:
174:
169:
11841:Cantor's isomorphism theorem
11454:Davey & Priestley (2002)
11405:Davey & Priestley (2002)
11103:Davey & Priestley (2002)
11004:
10210:
9601:{\displaystyle x\leq ^{*}y.}
8757:
8302:{\displaystyle \mathbb {N} }
7518:are order-preserving, their
4989:can be causally affected by
4846:and a partially ordered set
4649:For a partially ordered set
4568:{\displaystyle \mathbb {R} }
4538:{\displaystyle \mathbb {R} }
3996:{\displaystyle \sqsubseteq }
3393:of a partial order relation
2871:{\displaystyle 3\not \leq 4}
7:
11881:Szpilrajn extension theorem
11856:Hausdorff maximal principle
11831:Boolean prime ideal theorem
11640:Enumerative Combinatorics 1
10883:Szpilrajn extension theorem
10748:
9276:{\displaystyle x\leq ^{*}y}
9074:{\displaystyle x\leq ^{*}y}
8856:{\displaystyle P=(X,\leq )}
8136:is also a prime divisor of
5649:. This does not imply that
5519:The examples use the poset
4491:
4046:should not be taken as the
3526:
2608:{\displaystyle a,b,c\in P:}
2302:{\displaystyle a,b,c\in P,}
2069:{\displaystyle P=(X,\leq )}
2016:homogeneous binary relation
10:
12439:
12227:Topological vector lattice
10214:
9999:{\displaystyle P\times P.}
9974:subset of the topological
9889:
9643:
8740:If the count is made only
7042:. In our example, the set
6443:elements, but there is no
6423:
6413:{\displaystyle \{x,y,z\}.}
5620:{\displaystyle \{x,y,z\},}
4582:is a strict partial order.
4381:rather than set equality.
3351:
2840:), and converse relation (
1892:{\displaystyle \emptyset }
1868:{\displaystyle \{x,y,z\},}
12257:
12185:
12124:
11894:
11823:
11772:
11749:
11125:(Release 3.18.4 ed.)
10822:MΓΆbius function on posets
10732:{\displaystyle =\{a,b\}.}
9654:) may be considered as a
9615:order-extension principle
9572:it is also the case that
9536:{\displaystyle x,y\in X,}
9468:of another partial order
9437:{\displaystyle \leq ^{*}}
9157:and furthermore, for all
8934:{\displaystyle \leq ^{*}}
7906:, and the partial orders
7748:{\displaystyle f(x)=f(y)}
7623:{\displaystyle x,y\in S,}
7395:{\displaystyle x,y\in S,}
7138:, ordered by divisibility
6917:{\displaystyle \{x,y,z\}}
6686:{\displaystyle \{x,y,z\}}
5819:{\displaystyle \{x,y,z\}}
4260:may stand in any of four
3871:{\displaystyle (P,\leq )}
3086:{\displaystyle P\times P}
2199:one-to-one correspondence
11836:CantorβBernstein theorem
11330:Merrifield, Richard E.;
11052:"Partially Ordered Sets"
10936:
10574:{\displaystyle a,b\in P}
10201:{\displaystyle a\leq b.}
9565:{\displaystyle x\leq y,}
8327:Number of partial orders
8251:restricting its codomain
8039:{\displaystyle f\circ g}
8013:{\displaystyle g\circ f}
7987:{\displaystyle g:T\to U}
7955:{\displaystyle f:S\to T}
7889:{\displaystyle f:S\to T}
7864:. If an order-embedding
7670:{\displaystyle x\leq y.}
7584:{\displaystyle f:S\to T}
7511:{\displaystyle g:T\to U}
7479:{\displaystyle f:S\to T}
7344:{\displaystyle f:S\to T}
7030:. A greatest element of
7002:. Similarly, an element
6998:to be an upper bound of
6372:{\displaystyle \{x,z\},}
6146:{\displaystyle \{x,y\}.}
6085:{\displaystyle a\neq b.}
5715:{\displaystyle \{x,y\},}
4432:{\displaystyle (P,<)}
4016:{\displaystyle \preceq }
3903:{\displaystyle (P,<)}
2944:{\displaystyle a\leq a;}
2221:non-strict partial order
12380:Partially ordered group
12200:Specialization preorder
11498:Ward, L. E. Jr (1954).
10982:is assumed to be finite
10433:is the set of elements
10369:is the set of elements
9892:Partially ordered space
9650:Every poset (and every
9630:directed acyclic graphs
9402:{\displaystyle P^{*}=P}
9243:{\displaystyle x\leq y}
9100:{\displaystyle x\leq y}
8176:divisors defines a map
8165:{\displaystyle \{2,3\}}
8069:For example, a mapping
7420:{\displaystyle x\leq y}
7067:{\displaystyle \{x,y\}}
6878:{\displaystyle a<m.}
6797:{\displaystyle a>g.}
6648:{\displaystyle a\in P.}
6619:{\displaystyle m\leq a}
6563:{\displaystyle a\in P.}
6534:{\displaystyle a\leq g}
6224:{\displaystyle a\neq b}
5586:of a three-element set
5033:Lexicographic order on
4955:, where for two events
4913:{\displaystyle x\in X.}
4197:Alternative definitions
2836:), irreflexive kernel (
2503:{\displaystyle a\leq c}
2477:{\displaystyle b\leq c}
2451:{\displaystyle a\leq b}
2393:{\displaystyle b\leq a}
2367:{\displaystyle a\leq b}
2335:{\displaystyle a\leq a}
2184:Partial order relations
1924:{\displaystyle \{x,y\}}
1834:of a three-element set
11866:Kruskal's tree theorem
11861:KnasterβTarski theorem
11851:DushnikβMiller theorem
11597:Relational Mathematics
10976:
10774:Complete partial order
10733:
10668:
10575:
10527:are defined similarly.
10217:Interval (mathematics)
10202:
10173:
10131:
10108:
10056:
10000:
9964:
9910:
9847:
9742:
9719:
9695:
9672:
9602:
9566:
9537:
9502:
9482:
9458:
9438:
9403:
9354:
9324:
9304:
9277:
9244:
9218:
9191:
9171:
9151:
9131:
9101:
9075:
9042:
9022:
8995:
8975:
8955:
8935:
8908:
8884:
8857:
8815:
8303:
8281:
8243:
8217:
8166:
8110:
8066:are order-isomorphic.
8040:
8014:
7988:
7956:
7890:
7852:exists, one says that
7838:
7837:{\displaystyle \leq .}
7815:
7789:
7749:
7703:. In the latter case,
7671:
7624:
7585:
7553:
7512:
7480:
7421:
7396:
7345:
7270:
7207:
7154:divides for instance 2
7139:
7119:
7068:
6952:Upper and lower bounds
6942:
6941:{\displaystyle \{\,\}}
6918:
6879:
6850:
6849:{\displaystyle a\in P}
6824:
6823:{\displaystyle m\in P}
6804:Similarly, an element
6798:
6769:
6768:{\displaystyle a\in P}
6743:
6742:{\displaystyle g\in P}
6711:
6710:{\displaystyle \{\,\}}
6687:
6649:
6620:
6590:
6589:{\displaystyle m\in P}
6564:
6535:
6505:
6504:{\displaystyle g\in P}
6472:
6452:
6414:
6379:but is not covered by
6373:
6338:
6276:
6225:
6189:is strictly less than
6147:
6118:is strictly less than
6112:
6086:
6035:
5964:
5872:
5846:
5826:are comparable, while
5820:
5782:
5716:
5681:
5621:
5576:
5148:
5101:
5057:
4914:
4829:
4801:
4761:
4629:directed acyclic graph
4569:
4539:
4505:
4477:
4457:
4441:directed acyclic graph
4433:
4386:partial order relation
4364:
4246:
4209:. Specifically, given
4187:
4164:
4144:
4124:
4104:
4084:
4064:
4040:
4017:
3997:
3977:
3942:
3904:
3872:
3836:
3816:
3796:
3776:
3756:
3732:
3712:
3692:
3672:
3652:
3632:
3631:{\displaystyle \geq ,}
3610:
3588:
3587:{\displaystyle \leq ,}
3565:
3545:
3517:
3491:
3458:
3434:
3413:is defined by letting
3407:
3387:
3354:Duality (order theory)
3339:
3278:
3258:
3197:
3177:
3133:
3109:
3087:
3057:
2994:
2945:
2916:
2896:
2879:
2872:
2796:
2795:{\displaystyle a<c}
2770:
2769:{\displaystyle b<c}
2744:
2743:{\displaystyle a<b}
2712:
2711:{\displaystyle b<a}
2686:
2685:{\displaystyle a<b}
2654:
2609:
2556:
2504:
2478:
2452:
2420:
2394:
2368:
2336:
2303:
2250:
2174:
2154:
2134:
2120:) and a partial order
2114:
2090:
2070:
1984:
1977:
1951:
1925:
1893:
1869:
1803:
1774:
1748:
1722:
1721:{\displaystyle a,b,c,}
1683:
1624:
1568:
1536:
1509:
1465:
1421:
1380:
1309:
1238:
1182:
1153:
11421:P. R. Halmos (1974).
10977:
10949:A proof can be found
10828:Nested set collection
10734:
10669:
10576:
10297:instead of "convex".
10203:
10174:
10132:
10109:
10057:
10010:in the sense that if
10001:
9965:
9911:
9848:
9748:More explicitly, let
9743:
9720:
9701:there is at most one
9696:
9673:
9603:
9567:
9538:
9503:
9483:
9481:{\displaystyle \leq }
9459:
9439:
9404:
9355:
9353:{\displaystyle X^{*}}
9325:
9305:
9303:{\displaystyle P^{*}}
9278:
9245:
9219:
9217:{\displaystyle X^{*}}
9192:
9172:
9152:
9132:
9130:{\displaystyle P^{*}}
9102:
9076:
9043:
9023:
9021:{\displaystyle X^{*}}
8996:
8976:
8956:
8954:{\displaystyle \leq }
8936:
8909:
8885:
8883:{\displaystyle X^{*}}
8858:
8816:
8315:distributive lattices
8304:
8282:
8244:
8242:{\displaystyle \{4\}}
8218:
8167:
8111:
8058:, respectively, then
8041:
8015:
7989:
7957:
7891:
7839:
7816:
7790:
7750:
7672:
7625:
7586:
7554:
7513:
7481:
7422:
7397:
7346:
7271:
7269:{\displaystyle \leq }
7208:
7131:
7120:
7069:
7034:is an upper bound of
6970:is an upper bound of
6943:
6919:
6880:
6851:
6825:
6799:
6770:
6744:
6712:
6688:
6650:
6621:
6591:
6565:
6536:
6506:
6473:
6431:
6415:
6374:
6339:
6337:{\displaystyle \{x\}}
6277:
6226:
6193:and no third element
6148:
6113:
6111:{\displaystyle \{x\}}
6087:
6036:
5965:
5873:
5871:{\displaystyle \{y\}}
5847:
5845:{\displaystyle \{x\}}
5821:
5783:
5781:{\displaystyle \{x\}}
5758:. Otherwise they are
5717:
5682:
5680:{\displaystyle \{x\}}
5622:
5577:
5354:ordered vector spaces
5183:lexicographical order
5149:
5102:
5058:
4947:The set of events in
4915:
4830:
4802:
4762:
4646:ordered by inclusion.
4578:By definition, every
4570:
4540:
4499:
4478:
4458:
4434:
4365:
4247:
4205:, is via a notion of
4188:
4186:{\displaystyle \leq }
4165:
4163:{\displaystyle \leq }
4145:
4125:
4123:{\displaystyle \leq }
4105:
4103:{\displaystyle \leq }
4085:
4065:
4041:
4039:{\displaystyle \leq }
4018:
3998:
3978:
3976:{\displaystyle \leq }
3957:partially ordered set
3943:
3905:
3873:
3844:partially ordered set
3837:
3817:
3797:
3795:{\displaystyle \leq }
3777:
3775:{\displaystyle \geq }
3757:
3755:{\displaystyle \leq }
3733:
3713:
3693:
3673:
3671:{\displaystyle \leq }
3653:
3633:
3611:
3609:{\displaystyle <,}
3589:
3566:
3564:{\displaystyle \leq }
3546:
3518:
3492:
3459:
3435:
3408:
3388:
3340:
3279:
3277:{\displaystyle \leq }
3259:
3198:
3196:{\displaystyle \leq }
3178:
3134:
3110:
3088:
3058:
2995:
2946:
2917:
2915:{\displaystyle \leq }
2897:
2873:
2825:
2797:
2771:
2745:
2713:
2687:
2655:
2610:
2557:
2522:Strict partial orders
2505:
2479:
2453:
2421:
2395:
2369:
2337:
2304:
2251:
2175:
2155:
2135:
2133:{\displaystyle \leq }
2115:
2091:
2071:
2032:partially ordered set
1978:
1976:{\displaystyle \{y\}}
1952:
1950:{\displaystyle \{x\}}
1926:
1894:
1870:
1822:
1804:
1775:
1749:
1723:
1684:
1625:
1569:
1537:
1510:
1466:
1422:
1381:
1310:
1239:
1183:
1154:
1133:Definitions, for all
12358:Ordered vector space
11671:J. Reine Angew. Math
11429:. Springer. p.
10966:
10895:Strict weak ordering
10889:Stochastic dominance
10853:Ordered vector space
10810:Locally finite poset
10690:
10648:
10553:
10183:
10140:
10118:
10066:
10014:
9981:
9924:
9900:
9786:
9729:
9709:
9682:
9662:
9576:
9547:
9512:
9492:
9472:
9448:
9421:
9364:
9337:
9314:
9287:
9254:
9228:
9201:
9181:
9161:
9141:
9114:
9085:
9052:
9032:
9005:
8985:
8965:
8945:
8918:
8898:
8867:
8829:
8766:
8400:Equivalence relation
8291:
8257:
8227:
8180:
8144:
8073:
8024:
7998:
7966:
7934:
7868:
7825:
7799:
7759:
7715:
7652:
7599:
7563:
7525:
7490:
7458:
7405:
7371:
7323:
7260:
7170:
7136:Nonnegative integers
7082:
7046:
7010:is a lower bound of
6928:
6890:
6860:
6834:
6808:
6779:
6753:
6727:
6697:
6659:
6630:
6604:
6574:
6545:
6519:
6489:
6459:
6383:
6348:
6322:
6251:
6209:
6122:
6096:
6067:
5986:
5908:
5856:
5830:
5792:
5766:
5722:but not the reverse.
5691:
5665:
5590:
5523:
5125:
5081:
5037:
4951:and, in most cases,
4895:
4811:
4771:
4685:
4627:The vertex set of a
4589:of a given set (its
4557:
4527:
4523:On the real numbers
4485:transitive reduction
4476:{\displaystyle <}
4467:
4447:
4411:
4390:homogeneous relation
4379:equivalence relation
4310:
4213:
4177:
4154:
4143:{\displaystyle >}
4134:
4114:
4094:
4083:{\displaystyle >}
4074:
4063:{\displaystyle >}
4054:
4030:
4007:
3987:
3967:
3961:totally ordered sets
3914:
3882:
3850:
3835:{\displaystyle <}
3826:
3815:{\displaystyle >}
3806:
3786:
3766:
3746:
3722:
3711:{\displaystyle <}
3702:
3682:
3662:
3651:{\displaystyle >}
3642:
3619:
3597:
3575:
3555:
3535:
3501:
3468:
3448:
3417:
3397:
3370:
3292:
3268:
3211:
3187:
3143:
3123:
3097:
3071:
3004:
2955:
2926:
2906:
2886:
2856:
2848:for the poset whose
2780:
2754:
2728:
2696:
2670:
2625:
2578:
2546:
2538:strict partial order
2488:
2462:
2436:
2404:
2378:
2352:
2320:
2272:
2240:
2231:homogeneous relation
2164:
2144:
2124:
2104:
2080:
2076:consisting of a set
2042:
1961:
1935:
1903:
1883:
1838:
1802:{\displaystyle aRc.}
1784:
1758:
1732:
1697:
1673:
1668:homogeneous relation
1579:
1547:
1520:
1476:
1432:
1391:
1320:
1249:
1193:
1163:
1137:
879:Strict partial order
154:Equivalence relation
12196:Alexandrov topology
12142:Lexicographic order
12101:Well-quasi-ordering
11636:Stanley, Richard P.
11471:The Axiom of Choice
10812: β Mathematics
10768:Comparability graph
10502:half-open intervals
10311:is a sublattice of
9778:(and otherwise the
9658:where, for objects
9634:topological sorting
8352:
7814:{\displaystyle x=y}
7022:, for each element
6982:, for each element
6165:by another element
5653:is also related to
4837:componentwise order
4483:to be an edge. The
4372:partial order on a
3516:{\displaystyle yRx}
2830:Commutative diagram
2419:{\displaystyle a=b}
2268:. That is, for all
1773:{\displaystyle bRc}
1747:{\displaystyle aRb}
1535:{\displaystyle aRa}
1152:{\displaystyle a,b}
538:Well-quasi-ordering
12177:Transitive closure
12137:Converse/Transpose
11846:Dilworth's theorem
11526:10338.dmlcz/101379
11475:Dover Publications
11332:Simmons, Howard E.
10972:
10729:
10664:
10640:A poset is called
10571:
10198:
10169:
10130:{\displaystyle i,}
10127:
10104:
10084:
10052:
10032:
9996:
9960:
9906:
9880:isomorphism-closed
9843:
9741:{\displaystyle y.}
9738:
9715:
9694:{\displaystyle y,}
9691:
9668:
9640:In category theory
9598:
9562:
9533:
9498:
9478:
9454:
9434:
9399:
9350:
9320:
9300:
9273:
9240:
9214:
9187:
9167:
9147:
9127:
9097:
9071:
9038:
9028:(and thus also in
9018:
8991:
8971:
8951:
8931:
8904:
8880:
8853:
8811:
8346:
8343:labeled elements:
8299:
8277:
8239:
8213:
8162:
8106:
8036:
8010:
7984:
7952:
7900:, it is called an
7886:
7834:
7811:
7785:
7745:
7667:
7620:
7581:
7549:
7508:
7476:
7417:
7392:
7341:
7266:
7203:
7140:
7115:
7064:
6938:
6914:
6875:
6846:
6820:
6794:
6765:
6739:
6707:
6683:
6645:
6626:for every element
6616:
6586:
6560:
6541:for every element
6531:
6501:
6471:{\displaystyle P,}
6468:
6453:
6410:
6369:
6334:
6272:
6243:is false for each
6221:
6143:
6108:
6082:
6049:strictly less than
6031:
5960:
5868:
5842:
5816:
5778:
5712:
5677:
5617:
5584:set of all subsets
5582:consisting of the
5572:
5497:If two posets are
5144:
5097:
5053:
4953:general relativity
4949:special relativity
4910:
4825:
4797:
4757:
4669:precedes sequence
4565:
4535:
4518:less-than-or-equal
4506:
4473:
4453:
4429:
4360:
4262:mutually exclusive
4242:
4193:is a total order.
4183:
4160:
4140:
4120:
4100:
4080:
4060:
4036:
4013:
3993:
3973:
3938:
3900:
3868:
3832:
3812:
3792:
3772:
3752:
3740:irreflexive kernel
3728:
3708:
3688:
3668:
3648:
3628:
3606:
3584:
3561:
3541:
3513:
3487:
3454:
3430:
3403:
3383:
3335:
3274:
3254:
3205:irreflexive kernel
3193:
3173:
3129:
3105:
3083:
3053:
2990:
2941:
2912:
2892:
2880:
2868:
2792:
2766:
2740:
2708:
2682:
2650:
2605:
2552:
2500:
2474:
2448:
2416:
2390:
2364:
2332:
2299:
2246:
2170:
2150:
2130:
2110:
2086:
2066:
1985:
1973:
1947:
1921:
1889:
1865:
1832:set of all subsets
1799:
1770:
1744:
1718:
1679:
1620:
1618:
1564:
1532:
1505:
1503:
1461:
1459:
1417:
1415:
1376:
1374:
1305:
1303:
1234:
1232:
1178:
1149:
1013:Strict total order
12405:
12404:
12363:Partially ordered
12172:Symmetric closure
12157:Reflexive closure
11900:
11698:labeled elements)
11627:978-3-319-29788-0
11606:978-0-521-76268-7
11557:978-0-521-78451-1
11484:978-0-486-46624-8
11456:, pp. 23β24.
11440:978-1-4757-1645-0
11232:in the poset. If
11216:compare_elements(
11149:"Lectures slides"
11036:978-1-4757-3826-1
10975:{\displaystyle P}
10798:Incidence algebra
10302:convex sublattice
10069:
10017:
9918:topological space
9909:{\displaystyle P}
9718:{\displaystyle x}
9671:{\displaystyle x}
9501:{\displaystyle X}
9457:{\displaystyle X}
9323:{\displaystyle P}
9190:{\displaystyle y}
9170:{\displaystyle x}
9150:{\displaystyle P}
9137:is a subposet of
9041:{\displaystyle X}
8994:{\displaystyle y}
8974:{\displaystyle x}
8907:{\displaystyle X}
8825:of another poset
8713:
8712:
8048:identity function
7903:order isomorphism
7774:
6990:. In particular,
5279:reflexive closure
5175:Cartesian product
5077:Product order on
4977:is in the future
4673:if every item in
4665:, where sequence
4661:of elements from
4605:, and the set of
4580:strict weak order
4553:relation > on
4456:{\displaystyle P}
4316:
4237:
3731:{\displaystyle P}
3691:{\displaystyle P}
3544:{\displaystyle P}
3481:
3457:{\displaystyle R}
3442:converse relation
3427:
3406:{\displaystyle R}
3380:
3321:
3307:
3286:reflexive closure
3240:
3226:
3132:{\displaystyle P}
3065:identity relation
2979:
2973:
2967:
2895:{\displaystyle P}
2555:{\displaystyle P}
2309:it must satisfy:
2249:{\displaystyle P}
2173:{\displaystyle X}
2153:{\displaystyle X}
2113:{\displaystyle P}
2089:{\displaystyle X}
1815:
1814:
1682:{\displaystyle R}
1633:
1632:
1605:
1553:
1499:
1455:
1411:
1359:
1268:
946:Strict weak order
132:Total, Semiconnex
16:(Redirected from
12430:
12423:Binary relations
12147:Linear extension
11896:
11876:Mirsky's theorem
11736:
11729:
11722:
11713:
11712:
11704:
11692:
11678:
11665:
11653:
11631:
11610:
11593:Schmidt, Gunther
11588:
11586:
11561:
11530:
11529:
11519:
11495:
11489:
11488:
11463:
11457:
11451:
11445:
11444:
11428:
11425:Naive Set Theory
11418:
11412:
11402:
11396:
11395:
11378:
11372:
11371:
11362:
11360:
11341:
11327:
11321:
11320:
11310:
11304:
11303:
11298:
11296:
11290:
11285:On Poset Merging
11279:
11273:
11272:
11261:
11251:
11242:, return β1. If
11241:
11213:
11211:
11197:
11191:
11190:
11188:
11186:
11173:
11167:
11166:
11164:
11162:
11153:
11144:
11138:
11137:
11132:
11130:
11116:
11110:
11100:
11094:
11092:
11076:
11070:
11069:
11047:
11041:
11040:
11020:
10998:
10989:
10983:
10981:
10979:
10978:
10973:
10960:
10954:
10947:
10918:
10907:
10865:Scott continuity
10849:
10818:
10815:
10794:
10779:
10738:
10736:
10735:
10730:
10677:
10673:
10671:
10670:
10665:
10663:
10655:
10620:
10616:
10615:
10611:
10607:
10598:
10597:
10580:
10578:
10577:
10572:
10548:
10540:
10526:
10514:
10496:
10488:
10484:
10480:
10470:
10460:
10450:
10432:
10406:
10396:
10386:
10368:
10276:
10248:
10232:
10207:
10205:
10204:
10199:
10178:
10176:
10175:
10170:
10165:
10164:
10152:
10151:
10136:
10134:
10133:
10128:
10113:
10111:
10110:
10105:
10094:
10093:
10083:
10061:
10059:
10058:
10053:
10042:
10041:
10031:
10005:
10003:
10002:
9997:
9969:
9967:
9966:
9961:
9915:
9913:
9912:
9907:
9852:
9850:
9849:
9844:
9777:
9767:
9747:
9745:
9744:
9739:
9724:
9722:
9721:
9716:
9700:
9698:
9697:
9692:
9677:
9675:
9674:
9669:
9646:Posetal category
9622:computer science
9610:linear extension
9607:
9605:
9604:
9599:
9591:
9590:
9571:
9569:
9568:
9563:
9542:
9540:
9539:
9534:
9507:
9505:
9504:
9499:
9487:
9485:
9484:
9479:
9463:
9461:
9460:
9455:
9443:
9441:
9440:
9435:
9433:
9432:
9417:A partial order
9413:Linear extension
9408:
9406:
9405:
9400:
9395:
9394:
9376:
9375:
9359:
9357:
9356:
9351:
9349:
9348:
9329:
9327:
9326:
9321:
9310:the subposet of
9309:
9307:
9306:
9301:
9299:
9298:
9282:
9280:
9279:
9274:
9269:
9268:
9249:
9247:
9246:
9241:
9223:
9221:
9220:
9215:
9213:
9212:
9196:
9194:
9193:
9188:
9176:
9174:
9173:
9168:
9156:
9154:
9153:
9148:
9136:
9134:
9133:
9128:
9126:
9125:
9106:
9104:
9103:
9098:
9080:
9078:
9077:
9072:
9067:
9066:
9047:
9045:
9044:
9039:
9027:
9025:
9024:
9019:
9017:
9016:
9000:
8998:
8997:
8992:
8980:
8978:
8977:
8972:
8960:
8958:
8957:
8952:
8940:
8938:
8937:
8932:
8930:
8929:
8913:
8911:
8910:
8905:
8889:
8887:
8886:
8881:
8879:
8878:
8862:
8860:
8859:
8854:
8820:
8818:
8817:
8812:
8807:
8806:
8794:
8793:
8778:
8777:
8749:
8729:
8657:
8644:
8643:
8623:
8606:
8605:
8554:
8549:
8544:
8539:
8353:
8345:
8312:
8308:
8306:
8305:
8300:
8298:
8286:
8284:
8283:
8278:
8270:
8248:
8246:
8245:
8240:
8222:
8220:
8219:
8214:
8209:
8201:
8193:
8171:
8169:
8168:
8163:
8139:
8135:
8131:
8127:
8115:
8113:
8112:
8107:
8102:
8094:
8086:
8045:
8043:
8042:
8037:
8019:
8017:
8016:
8011:
7994:exist such that
7993:
7991:
7990:
7985:
7961:
7959:
7958:
7953:
7921:
7913:
7895:
7893:
7892:
7887:
7843:
7841:
7840:
7835:
7820:
7818:
7817:
7812:
7794:
7792:
7791:
7786:
7775:
7772:
7754:
7752:
7751:
7746:
7706:
7702:
7694:
7680:
7676:
7674:
7673:
7668:
7647:
7629:
7627:
7626:
7621:
7593:order-reflecting
7590:
7588:
7587:
7582:
7558:
7556:
7555:
7550:
7517:
7515:
7514:
7509:
7485:
7483:
7482:
7477:
7453:
7445:
7426:
7424:
7423:
7418:
7401:
7399:
7398:
7393:
7354:order-preserving
7350:
7348:
7347:
7342:
7318:
7310:
7295:
7286:
7276:
7275:
7273:
7272:
7267:
7254:
7230:
7212:
7210:
7209:
7204:
7124:
7122:
7121:
7116:
7073:
7071:
7070:
7065:
6947:
6945:
6944:
6939:
6923:
6921:
6920:
6915:
6884:
6882:
6881:
6876:
6855:
6853:
6852:
6847:
6829:
6827:
6826:
6821:
6803:
6801:
6800:
6795:
6774:
6772:
6771:
6766:
6748:
6746:
6745:
6740:
6721:Maximal elements
6716:
6714:
6713:
6708:
6692:
6690:
6689:
6684:
6654:
6652:
6651:
6646:
6625:
6623:
6622:
6617:
6595:
6593:
6592:
6587:
6569:
6567:
6566:
6561:
6540:
6538:
6537:
6532:
6513:greatest element
6510:
6508:
6507:
6502:
6483:Greatest element
6477:
6475:
6474:
6469:
6419:
6417:
6416:
6411:
6378:
6376:
6375:
6370:
6343:
6341:
6340:
6335:
6318:". For example,
6313:
6299:
6281:
6279:
6278:
6273:
6230:
6228:
6227:
6222:
6152:
6150:
6149:
6144:
6117:
6115:
6114:
6109:
6091:
6089:
6088:
6083:
6040:
6038:
6037:
6032:
5969:
5967:
5966:
5961:
5877:
5875:
5874:
5869:
5851:
5849:
5848:
5843:
5825:
5823:
5822:
5817:
5787:
5785:
5784:
5779:
5757:
5747:
5721:
5719:
5718:
5713:
5686:
5684:
5683:
5678:
5626:
5624:
5623:
5618:
5581:
5579:
5578:
5573:
5535:
5534:
5420:if and only if:
5419:
5396:
5344:
5334:
5324:
5314:
5304:
5234:
5224:
5214:
5204:
5165:
5161:
5153:
5151:
5150:
5145:
5140:
5132:
5115:
5106:
5104:
5103:
5098:
5096:
5088:
5071:
5062:
5060:
5059:
5054:
5052:
5044:
5027:
5002:
4972:
4944:
4919:
4917:
4916:
4911:
4890:
4871:
4834:
4832:
4831:
4826:
4824:
4806:
4804:
4803:
4798:
4796:
4795:
4783:
4782:
4766:
4764:
4763:
4758:
4756:
4755:
4754:
4742:
4738:
4737:
4720:
4719:
4718:
4706:
4702:
4701:
4574:
4572:
4571:
4566:
4564:
4544:
4542:
4541:
4536:
4534:
4482:
4480:
4479:
4474:
4462:
4460:
4459:
4454:
4438:
4436:
4435:
4430:
4369:
4367:
4366:
4361:
4317:
4314:
4293:
4283:
4273:
4251:
4249:
4248:
4243:
4238:
4235:
4203:computer science
4192:
4190:
4189:
4184:
4169:
4167:
4166:
4161:
4149:
4147:
4146:
4141:
4129:
4127:
4126:
4121:
4109:
4107:
4106:
4101:
4089:
4087:
4086:
4081:
4069:
4067:
4066:
4061:
4045:
4043:
4042:
4037:
4022:
4020:
4019:
4014:
4002:
4000:
3999:
3994:
3982:
3980:
3979:
3974:
3947:
3945:
3944:
3939:
3909:
3907:
3906:
3901:
3877:
3875:
3874:
3869:
3841:
3839:
3838:
3833:
3821:
3819:
3818:
3813:
3801:
3799:
3798:
3793:
3781:
3779:
3778:
3773:
3761:
3759:
3758:
3753:
3737:
3735:
3734:
3729:
3717:
3715:
3714:
3709:
3697:
3695:
3694:
3689:
3677:
3675:
3674:
3669:
3657:
3655:
3654:
3649:
3637:
3635:
3634:
3629:
3615:
3613:
3612:
3607:
3593:
3591:
3590:
3585:
3570:
3568:
3567:
3562:
3550:
3548:
3547:
3542:
3522:
3520:
3519:
3514:
3496:
3494:
3493:
3488:
3483:
3482:
3479:
3463:
3461:
3460:
3455:
3439:
3437:
3436:
3431:
3429:
3428:
3425:
3412:
3410:
3409:
3404:
3392:
3390:
3389:
3384:
3382:
3381:
3378:
3344:
3342:
3341:
3336:
3322:
3319:
3308:
3305:
3283:
3281:
3280:
3275:
3263:
3261:
3260:
3255:
3241:
3238:
3227:
3224:
3202:
3200:
3199:
3194:
3182:
3180:
3179:
3174:
3163:
3162:
3138:
3136:
3135:
3130:
3114:
3112:
3111:
3106:
3092:
3090:
3089:
3084:
3062:
3060:
3059:
3054:
3016:
3015:
2999:
2997:
2996:
2991:
2989:
2988:
2977:
2971:
2965:
2950:
2948:
2947:
2942:
2921:
2919:
2918:
2913:
2901:
2899:
2898:
2893:
2877:
2875:
2874:
2869:
2801:
2799:
2798:
2793:
2775:
2773:
2772:
2767:
2749:
2747:
2746:
2741:
2717:
2715:
2714:
2709:
2691:
2689:
2688:
2683:
2659:
2657:
2656:
2651:
2649:
2645:
2614:
2612:
2611:
2606:
2561:
2559:
2558:
2553:
2540:
2539:
2509:
2507:
2506:
2501:
2483:
2481:
2480:
2475:
2457:
2455:
2454:
2449:
2425:
2423:
2422:
2417:
2399:
2397:
2396:
2391:
2373:
2371:
2370:
2365:
2341:
2339:
2338:
2333:
2308:
2306:
2305:
2300:
2255:
2253:
2252:
2247:
2223:
2222:
2179:
2177:
2176:
2171:
2159:
2157:
2156:
2151:
2139:
2137:
2136:
2131:
2119:
2117:
2116:
2111:
2095:
2093:
2092:
2087:
2075:
2073:
2072:
2067:
1982:
1980:
1979:
1974:
1956:
1954:
1953:
1948:
1930:
1928:
1927:
1922:
1898:
1896:
1895:
1890:
1874:
1872:
1871:
1866:
1808:
1806:
1805:
1800:
1779:
1777:
1776:
1771:
1753:
1751:
1750:
1745:
1727:
1725:
1724:
1719:
1688:
1686:
1685:
1680:
1662:
1658:
1655:
1654:
1649:
1645:
1642:
1641:
1629:
1627:
1626:
1621:
1619:
1606:
1603:
1573:
1571:
1570:
1565:
1554:
1551:
1541:
1539:
1538:
1533:
1514:
1512:
1511:
1506:
1504:
1500:
1497:
1470:
1468:
1467:
1462:
1460:
1456:
1453:
1426:
1424:
1423:
1418:
1416:
1412:
1409:
1385:
1383:
1382:
1377:
1375:
1360:
1357:
1334:
1314:
1312:
1311:
1306:
1304:
1295:
1269:
1266:
1243:
1241:
1240:
1235:
1233:
1218:
1199:
1187:
1185:
1184:
1179:
1158:
1156:
1155:
1150:
1079:
1076:
1075:
1069:
1066:
1065:
1059:
1054:
1049:
1044:
1039:
1036:
1035:
1029:
1026:
1025:
1019:
1007:
1004:
1003:
997:
994:
993:
987:
982:
977:
972:
967:
962:
959:
958:
952:
940:
937:
936:
930:
927:
926:
920:
915:
910:
905:
900:
895:
892:
891:
885:
873:
868:
863:
860:
859:
853:
850:
849:
843:
838:
833:
828:
825:
824:
818:
812:Meet-semilattice
806:
801:
796:
793:
792:
786:
781:
778:
777:
771:
766:
761:
758:
757:
751:
745:Join-semilattice
739:
734:
729:
726:
725:
719:
716:
715:
709:
706:
705:
699:
694:
689:
686:
685:
679:
667:
662:
657:
654:
653:
647:
642:
637:
634:
633:
627:
624:
623:
617:
614:
613:
607:
595:
590:
585:
582:
581:
575:
570:
565:
562:
561:
555:
550:
545:
540:
531:
526:
521:
518:
517:
511:
506:
501:
498:
497:
491:
488:
487:
481:
476:
464:
459:
454:
451:
450:
444:
439:
434:
429:
426:
425:
419:
416:
415:
409:
397:
392:
387:
384:
383:
377:
372:
367:
362:
359:
358:
352:
347:
335:
330:
325:
322:
321:
315:
310:
305:
300:
295:
292:
291:
285:
273:
268:
263:
260:
259:
253:
248:
243:
238:
233:
228:
223:
221:
211:
206:
201:
198:
197:
191:
186:
181:
176:
171:
166:
163:
162:
156:
74:
73:
64:
57:
50:
43:
41:binary relations
32:
31:
21:
12438:
12437:
12433:
12432:
12431:
12429:
12428:
12427:
12408:
12407:
12406:
12401:
12397:Young's lattice
12253:
12181:
12120:
11970:Heyting algebra
11918:Boolean algebra
11890:
11871:Laver's theorem
11819:
11785:Boolean algebra
11780:Binary relation
11768:
11745:
11740:
11710:
11686:
11681:
11650:
11628:
11607:
11558:
11539:
11534:
11533:
11496:
11492:
11485:
11464:
11460:
11452:
11448:
11441:
11419:
11415:
11403:
11399:
11393:
11379:
11375:
11358:
11356:
11354:
11328:
11324:
11311:
11307:
11294:
11292:
11288:
11280:
11276:
11262:, return 1. If
11253:
11252:, return 0. If
11243:
11233:
11209:
11207:
11201:"Finite posets"
11199:
11198:
11194:
11184:
11182:
11174:
11170:
11160:
11158:
11151:
11145:
11141:
11128:
11126:
11122:Logic and Proof
11117:
11113:
11101:
11097:
11077:
11073:
11066:
11048:
11044:
11037:
11021:
11012:
11007:
11002:
11001:
10990:
10986:
10967:
10964:
10963:
10961:
10957:
10948:
10944:
10939:
10934:
10908:
10898:
10847:
10816:
10813:
10792:
10777:
10751:
10743:interval orders
10691:
10688:
10687:
10675:
10659:
10651:
10649:
10646:
10645:
10618:
10613:
10609:
10605:
10600:
10587:
10585:
10582:
10554:
10551:
10550:
10546:
10532:
10516:
10504:
10494:
10490:
10486:
10482:
10472:
10462:
10452:
10438:
10422:
10398:
10388:
10374:
10358:
10356:closed interval
10274:
10246:
10230:
10219:
10213:
10184:
10181:
10180:
10160:
10156:
10147:
10143:
10141:
10138:
10137:
10119:
10116:
10115:
10089:
10085:
10073:
10067:
10064:
10063:
10037:
10033:
10021:
10015:
10012:
10011:
9982:
9979:
9978:
9925:
9922:
9921:
9901:
9898:
9897:
9894:
9888:
9876:terminal object
9787:
9784:
9783:
9769:
9749:
9730:
9727:
9726:
9710:
9707:
9706:
9683:
9680:
9679:
9663:
9660:
9659:
9648:
9642:
9586:
9582:
9577:
9574:
9573:
9548:
9545:
9544:
9513:
9510:
9509:
9493:
9490:
9489:
9473:
9470:
9469:
9449:
9446:
9445:
9428:
9424:
9422:
9419:
9418:
9415:
9390:
9386:
9371:
9367:
9365:
9362:
9361:
9344:
9340:
9338:
9335:
9334:
9315:
9312:
9311:
9294:
9290:
9288:
9285:
9284:
9283:, then we call
9264:
9260:
9255:
9252:
9251:
9229:
9226:
9225:
9208:
9204:
9202:
9199:
9198:
9182:
9179:
9178:
9162:
9159:
9158:
9142:
9139:
9138:
9121:
9117:
9115:
9112:
9111:
9086:
9083:
9082:
9062:
9058:
9053:
9050:
9049:
9033:
9030:
9029:
9012:
9008:
9006:
9003:
9002:
8986:
8983:
8982:
8966:
8963:
8962:
8946:
8943:
8942:
8941:is a subset of
8925:
8921:
8919:
8916:
8915:
8899:
8896:
8895:
8874:
8870:
8868:
8865:
8864:
8830:
8827:
8826:
8802:
8798:
8789:
8785:
8773:
8769:
8767:
8764:
8763:
8760:
8754:) is obtained.
8745:
8716:
8642:
8636:
8635:
8634:
8632:
8604:
8598:
8597:
8596:
8594:
8552:
8547:
8542:
8537:
8356:Elements
8329:
8310:
8294:
8292:
8289:
8288:
8266:
8258:
8255:
8254:
8228:
8225:
8224:
8205:
8197:
8189:
8181:
8178:
8177:
8145:
8142:
8141:
8137:
8133:
8129:
8125:
8098:
8090:
8082:
8074:
8071:
8070:
8025:
8022:
8021:
7999:
7996:
7995:
7967:
7964:
7963:
7935:
7932:
7931:
7922:are said to be
7915:
7907:
7869:
7866:
7865:
7826:
7823:
7822:
7800:
7797:
7796:
7773: and
7771:
7760:
7757:
7756:
7716:
7713:
7712:
7707:is necessarily
7704:
7696:
7688:
7684:order-embedding
7678:
7653:
7650:
7649:
7630:
7600:
7597:
7596:
7564:
7561:
7560:
7526:
7523:
7522:
7491:
7488:
7487:
7459:
7456:
7455:
7447:
7428:
7406:
7403:
7402:
7372:
7369:
7368:
7324:
7321:
7320:
7312:
7304:
7301:
7300:
7299:
7298:
7297:
7294:{2, 3, 4, 5, 8}
7293:
7287:
7279:
7278:
7261:
7258:
7257:
7256:
7237:
7231:
7220:
7171:
7168:
7167:
7083:
7080:
7079:
7047:
7044:
7043:
6994:need not be in
6954:: For a subset
6929:
6926:
6925:
6891:
6888:
6887:
6861:
6858:
6857:
6835:
6832:
6831:
6809:
6806:
6805:
6780:
6777:
6776:
6754:
6751:
6750:
6728:
6725:
6724:
6698:
6695:
6694:
6660:
6657:
6656:
6631:
6628:
6627:
6605:
6602:
6601:
6575:
6572:
6571:
6546:
6543:
6542:
6520:
6517:
6516:
6490:
6487:
6486:
6460:
6457:
6456:
6426:
6384:
6381:
6380:
6349:
6346:
6345:
6323:
6320:
6319:
6301:
6291:
6252:
6249:
6248:
6210:
6207:
6206:
6123:
6120:
6119:
6097:
6094:
6093:
6068:
6065:
6064:
5987:
5984:
5983:
5909:
5906:
5905:
5857:
5854:
5853:
5831:
5828:
5827:
5793:
5790:
5789:
5767:
5764:
5763:
5762:. For example,
5749:
5739:
5692:
5689:
5688:
5666:
5663:
5662:
5661:. For example,
5591:
5588:
5587:
5530:
5529:
5524:
5521:
5520:
5517:
5515:Derived notions
5471:
5444:
5415:
5406:
5384:
5373:
5336:
5326:
5316:
5306:
5286:
5226:
5216:
5206:
5186:
5171:
5170:
5169:
5168:
5167:
5163:
5159:
5136:
5128:
5126:
5123:
5122:
5116:
5108:
5107:
5092:
5084:
5082:
5079:
5078:
5072:
5064:
5063:
5048:
5040:
5038:
5035:
5034:
5028:
5017:
4994:
4973:if and only if
4964:
4927:
4896:
4893:
4892:
4873:
4872:if and only if
4863:
4820:
4812:
4809:
4808:
4791:
4787:
4778:
4774:
4772:
4769:
4768:
4767:if and only if
4750:
4743:
4733:
4729:
4725:
4724:
4714:
4707:
4697:
4693:
4689:
4688:
4686:
4683:
4682:
4657:containing all
4618:natural numbers
4560:
4558:
4555:
4554:
4530:
4528:
4525:
4524:
4494:
4468:
4465:
4464:
4448:
4445:
4444:
4412:
4409:
4408:
4313:
4311:
4308:
4307:
4285:
4275:
4265:
4236: and
4234:
4214:
4211:
4210:
4199:
4178:
4175:
4174:
4171:if, and only if
4155:
4152:
4151:
4135:
4132:
4131:
4115:
4112:
4111:
4095:
4092:
4091:
4075:
4072:
4071:
4070:. The relation
4055:
4052:
4051:
4031:
4028:
4027:
4008:
4005:
4004:
3988:
3985:
3984:
3968:
3965:
3964:
3915:
3912:
3911:
3883:
3880:
3879:
3851:
3848:
3847:
3827:
3824:
3823:
3822:is the dual of
3807:
3804:
3803:
3787:
3784:
3783:
3782:is the dual of
3767:
3764:
3763:
3747:
3744:
3743:
3723:
3720:
3719:
3703:
3700:
3699:
3683:
3680:
3679:
3663:
3660:
3659:
3643:
3640:
3639:
3620:
3617:
3616:
3598:
3595:
3594:
3576:
3573:
3572:
3556:
3553:
3552:
3536:
3533:
3532:
3529:
3502:
3499:
3498:
3497:if and only if
3478:
3474:
3469:
3466:
3465:
3449:
3446:
3445:
3424:
3420:
3418:
3415:
3414:
3398:
3395:
3394:
3377:
3373:
3371:
3368:
3367:
3356:
3350:
3318:
3304:
3293:
3290:
3289:
3269:
3266:
3265:
3239: and
3237:
3223:
3212:
3209:
3208:
3188:
3185:
3184:
3158:
3154:
3144:
3141:
3140:
3124:
3121:
3120:
3117:set subtraction
3098:
3095:
3094:
3072:
3069:
3068:
3011:
3007:
3005:
3002:
3001:
2984:
2980:
2956:
2953:
2952:
2927:
2924:
2923:
2907:
2904:
2903:
2887:
2884:
2883:
2857:
2854:
2853:
2820:
2812:strict preorder
2781:
2778:
2777:
2755:
2752:
2751:
2729:
2726:
2725:
2697:
2694:
2693:
2671:
2668:
2667:
2635:
2631:
2626:
2623:
2622:
2579:
2576:
2575:
2547:
2544:
2543:
2537:
2536:
2524:
2489:
2486:
2485:
2463:
2460:
2459:
2437:
2434:
2433:
2405:
2402:
2401:
2379:
2376:
2375:
2353:
2350:
2349:
2321:
2318:
2317:
2273:
2270:
2269:
2241:
2238:
2237:
2220:
2219:
2207:
2186:
2165:
2162:
2161:
2145:
2142:
2141:
2125:
2122:
2121:
2105:
2102:
2101:
2081:
2078:
2077:
2043:
2040:
2039:
1962:
1959:
1958:
1936:
1933:
1932:
1904:
1901:
1900:
1884:
1881:
1880:
1839:
1836:
1835:
1817:
1816:
1809:
1785:
1782:
1781:
1759:
1756:
1755:
1733:
1730:
1729:
1698:
1695:
1694:
1674:
1671:
1670:
1664:
1656:
1652:
1643:
1639:
1617:
1616:
1602:
1599:
1598:
1582:
1580:
1577:
1576:
1550:
1548:
1545:
1544:
1521:
1518:
1517:
1502:
1501:
1496:
1493:
1492:
1479:
1477:
1474:
1473:
1458:
1457:
1452:
1449:
1448:
1435:
1433:
1430:
1429:
1414:
1413:
1408:
1405:
1404:
1394:
1392:
1389:
1388:
1373:
1372:
1361:
1356:
1344:
1343:
1335:
1333:
1323:
1321:
1318:
1317:
1302:
1301:
1296:
1294:
1282:
1281:
1270:
1267: and
1265:
1252:
1250:
1247:
1246:
1231:
1230:
1219:
1217:
1211:
1210:
1196:
1194:
1191:
1190:
1164:
1161:
1160:
1138:
1135:
1134:
1077:
1073:
1067:
1063:
1037:
1033:
1027:
1023:
1005:
1001:
995:
991:
960:
956:
938:
934:
928:
924:
893:
889:
861:
857:
851:
847:
826:
822:
794:
790:
779:
775:
759:
755:
727:
723:
717:
713:
707:
703:
687:
683:
655:
651:
635:
631:
625:
621:
615:
611:
583:
579:
563:
559:
536:
519:
515:
499:
495:
489:
485:
470:Prewellordering
452:
448:
427:
423:
417:
413:
385:
381:
360:
356:
323:
319:
293:
289:
261:
257:
219:
216:
199:
195:
164:
160:
152:
144:
68:
35:
28:
23:
22:
15:
12:
11:
5:
12436:
12426:
12425:
12420:
12403:
12402:
12400:
12399:
12394:
12389:
12388:
12387:
12377:
12376:
12375:
12370:
12365:
12355:
12354:
12353:
12343:
12338:
12337:
12336:
12331:
12324:Order morphism
12321:
12320:
12319:
12309:
12304:
12299:
12294:
12289:
12288:
12287:
12277:
12272:
12267:
12261:
12259:
12255:
12254:
12252:
12251:
12250:
12249:
12244:
12242:Locally convex
12239:
12234:
12224:
12222:Order topology
12219:
12218:
12217:
12215:Order topology
12212:
12202:
12192:
12190:
12183:
12182:
12180:
12179:
12174:
12169:
12164:
12159:
12154:
12149:
12144:
12139:
12134:
12128:
12126:
12122:
12121:
12119:
12118:
12108:
12098:
12093:
12088:
12083:
12078:
12073:
12068:
12063:
12062:
12061:
12051:
12046:
12045:
12044:
12039:
12034:
12029:
12027:Chain-complete
12019:
12014:
12013:
12012:
12007:
12002:
11997:
11992:
11982:
11977:
11972:
11967:
11962:
11952:
11947:
11942:
11937:
11932:
11927:
11926:
11925:
11915:
11910:
11904:
11902:
11892:
11891:
11889:
11888:
11883:
11878:
11873:
11868:
11863:
11858:
11853:
11848:
11843:
11838:
11833:
11827:
11825:
11821:
11820:
11818:
11817:
11812:
11807:
11802:
11797:
11792:
11787:
11782:
11776:
11774:
11770:
11769:
11767:
11766:
11761:
11756:
11750:
11747:
11746:
11739:
11738:
11731:
11724:
11716:
11709:
11708:
11700:
11687:
11685:
11684:External links
11682:
11680:
11679:
11666:
11654:
11648:
11632:
11626:
11620:. BirkhΓ€user.
11611:
11605:
11589:
11577:(2): 383β386.
11562:
11556:
11540:
11538:
11535:
11532:
11531:
11510:(1): 144β161.
11490:
11483:
11458:
11446:
11439:
11413:
11397:
11391:
11373:
11352:
11322:
11305:
11274:
11192:
11168:
11139:
11111:
11095:
11071:
11064:
11042:
11035:
11009:
11008:
11006:
11003:
11000:
10999:
10984:
10971:
10955:
10941:
10940:
10938:
10935:
10933:
10932:
10926:
10920:
10919:is transitive.
10892:
10886:
10880:
10874:
10868:
10862:
10859:Poset topology
10856:
10850:
10841:
10835:
10833:Order polytope
10830:
10825:
10819:
10807:
10801:
10795:
10786:
10780:
10771:
10765:
10759:
10752:
10750:
10747:
10728:
10725:
10722:
10719:
10716:
10713:
10710:
10707:
10704:
10701:
10698:
10695:
10682:is covered by
10662:
10658:
10654:
10642:locally finite
10583:
10570:
10567:
10564:
10561:
10558:
10529:
10528:
10498:
10492:
10416:
10212:
10209:
10197:
10194:
10191:
10188:
10168:
10163:
10159:
10155:
10150:
10146:
10126:
10123:
10103:
10100:
10097:
10092:
10088:
10082:
10079:
10076:
10072:
10051:
10048:
10045:
10040:
10036:
10030:
10027:
10024:
10020:
9995:
9992:
9989:
9986:
9959:
9956:
9953:
9950:
9947:
9944:
9941:
9938:
9935:
9932:
9929:
9905:
9890:Main article:
9887:
9884:
9872:initial object
9842:
9839:
9836:
9833:
9830:
9827:
9824:
9821:
9818:
9815:
9812:
9809:
9806:
9803:
9800:
9797:
9794:
9791:
9737:
9734:
9714:
9690:
9687:
9667:
9652:preordered set
9644:Main article:
9641:
9638:
9597:
9594:
9589:
9585:
9581:
9561:
9558:
9555:
9552:
9532:
9529:
9526:
9523:
9520:
9517:
9497:
9477:
9453:
9431:
9427:
9414:
9411:
9398:
9393:
9389:
9385:
9382:
9379:
9374:
9370:
9347:
9343:
9319:
9297:
9293:
9272:
9267:
9263:
9259:
9239:
9236:
9233:
9211:
9207:
9186:
9166:
9146:
9124:
9120:
9096:
9093:
9090:
9070:
9065:
9061:
9057:
9037:
9015:
9011:
8990:
8970:
8950:
8928:
8924:
8903:
8877:
8873:
8863:provided that
8852:
8849:
8846:
8843:
8840:
8837:
8834:
8810:
8805:
8801:
8797:
8792:
8788:
8784:
8781:
8776:
8772:
8759:
8756:
8711:
8710:
8705:
8700:
8695:
8690:
8685:
8680:
8675:
8670:
8665:
8659:
8658:
8637:
8630:
8624:
8599:
8592:
8590:
8588:
8585:
8582:
8580:
8577:
8571:
8570:
8567:
8564:
8561:
8558:
8555:
8550:
8545:
8540:
8535:
8531:
8530:
8527:
8524:
8521:
8518:
8515:
8512:
8509:
8506:
8503:
8499:
8498:
8495:
8492:
8489:
8486:
8483:
8480:
8477:
8474:
8471:
8467:
8466:
8463:
8460:
8457:
8454:
8451:
8448:
8445:
8442:
8439:
8435:
8434:
8431:
8428:
8425:
8422:
8419:
8416:
8413:
8410:
8407:
8403:
8402:
8397:
8392:
8390:Total preorder
8387:
8382:
8377:
8372:
8367:
8362:
8357:
8328:
8325:
8297:
8276:
8273:
8269:
8265:
8262:
8238:
8235:
8232:
8212:
8208:
8204:
8200:
8196:
8192:
8188:
8185:
8161:
8158:
8155:
8152:
8149:
8122:prime divisors
8105:
8101:
8097:
8093:
8089:
8085:
8081:
8078:
8035:
8032:
8029:
8009:
8006:
8003:
7983:
7980:
7977:
7974:
7971:
7951:
7948:
7945:
7942:
7939:
7928:Hasse diagrams
7885:
7882:
7879:
7876:
7873:
7833:
7830:
7810:
7807:
7804:
7784:
7781:
7778:
7770:
7767:
7764:
7744:
7741:
7738:
7735:
7732:
7729:
7726:
7723:
7720:
7666:
7663:
7660:
7657:
7619:
7616:
7613:
7610:
7607:
7604:
7580:
7577:
7574:
7571:
7568:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7507:
7504:
7501:
7498:
7495:
7475:
7472:
7469:
7466:
7463:
7416:
7413:
7410:
7391:
7388:
7385:
7382:
7379:
7376:
7340:
7337:
7334:
7331:
7328:
7288:
7281:
7280:
7265:
7232:
7225:
7224:
7223:
7222:
7221:
7219:
7216:
7202:
7199:
7196:
7193:
7190:
7187:
7184:
7181:
7178:
7175:
7126:
7125:
7114:
7111:
7108:
7105:
7102:
7099:
7096:
7093:
7090:
7087:
7077:
7063:
7060:
7057:
7054:
7051:
6949:
6937:
6933:
6913:
6910:
6907:
6904:
6901:
6898:
6895:
6874:
6871:
6868:
6865:
6845:
6842:
6839:
6819:
6816:
6813:
6793:
6790:
6787:
6784:
6764:
6761:
6758:
6738:
6735:
6732:
6718:
6706:
6702:
6682:
6679:
6676:
6673:
6670:
6667:
6664:
6644:
6641:
6638:
6635:
6615:
6612:
6609:
6599:
6585:
6582:
6579:
6559:
6556:
6553:
6550:
6530:
6527:
6524:
6514:
6500:
6497:
6494:
6467:
6464:
6450:
6446:
6442:
6438:
6425:
6422:
6421:
6420:
6409:
6406:
6403:
6400:
6397:
6394:
6391:
6388:
6368:
6365:
6362:
6359:
6356:
6353:
6344:is covered by
6333:
6330:
6327:
6271:
6268:
6265:
6262:
6259:
6256:
6231:are true, and
6220:
6217:
6214:
6159:is said to be
6153:
6142:
6139:
6136:
6133:
6130:
6127:
6107:
6104:
6101:
6081:
6078:
6075:
6072:
6047:is said to be
6041:
6030:
6027:
6024:
6021:
6018:
6015:
6012:
6009:
6006:
6003:
6000:
5997:
5994:
5991:
5971:
5959:
5956:
5953:
5950:
5947:
5944:
5941:
5938:
5935:
5932:
5929:
5926:
5923:
5920:
5916:
5913:
5896:
5879:
5867:
5864:
5861:
5841:
5838:
5835:
5815:
5812:
5809:
5806:
5803:
5800:
5797:
5777:
5774:
5771:
5723:
5711:
5708:
5705:
5702:
5699:
5696:
5687:is related to
5676:
5673:
5670:
5616:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5571:
5568:
5565:
5562:
5559:
5556:
5553:
5550:
5547:
5544:
5541:
5538:
5533:
5528:
5516:
5513:
5509:disjoint union
5495:
5494:
5476:
5467:
5449:
5440:
5411:
5372:
5369:
5356:over the same
5347:
5346:
5283:direct product
5275:
5236:
5143:
5139:
5135:
5131:
5117:
5110:
5109:
5095:
5091:
5087:
5073:
5066:
5065:
5051:
5047:
5043:
5029:
5022:
5021:
5020:
5019:
5018:
5016:
5013:
5005:
5004:
4945:
4920:
4909:
4906:
4903:
4900:
4852:function space
4840:
4823:
4819:
4816:
4794:
4790:
4786:
4781:
4777:
4753:
4749:
4746:
4741:
4736:
4732:
4728:
4723:
4717:
4713:
4710:
4705:
4700:
4696:
4692:
4655:sequence space
4647:
4636:
4625:
4614:
4583:
4576:
4563:
4533:
4521:
4493:
4490:
4472:
4452:
4428:
4425:
4422:
4419:
4416:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4241:
4233:
4230:
4227:
4224:
4221:
4218:
4198:
4195:
4182:
4159:
4139:
4119:
4099:
4079:
4059:
4035:
4012:
3992:
3972:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3899:
3896:
3893:
3890:
3887:
3867:
3864:
3861:
3858:
3855:
3831:
3811:
3791:
3771:
3751:
3727:
3707:
3687:
3667:
3647:
3627:
3624:
3605:
3602:
3583:
3580:
3560:
3540:
3528:
3525:
3512:
3509:
3506:
3486:
3477:
3473:
3453:
3423:
3402:
3376:
3352:Main article:
3349:
3346:
3334:
3331:
3328:
3325:
3320: or
3317:
3314:
3311:
3306: if
3303:
3300:
3297:
3273:
3253:
3250:
3247:
3244:
3236:
3233:
3230:
3225: if
3222:
3219:
3216:
3192:
3171:
3167:
3161:
3157:
3152:
3148:
3128:
3103:
3082:
3079:
3076:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3014:
3010:
2987:
2983:
2976:
2970:
2964:
2960:
2940:
2937:
2934:
2931:
2911:
2891:
2867:
2864:
2861:
2846:logical matrix
2819:
2816:
2804:
2803:
2791:
2788:
2785:
2765:
2762:
2759:
2739:
2736:
2733:
2719:
2707:
2704:
2701:
2681:
2678:
2675:
2661:
2648:
2644:
2641:
2638:
2634:
2630:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2551:
2523:
2520:
2512:
2511:
2499:
2496:
2493:
2473:
2470:
2467:
2447:
2444:
2441:
2427:
2415:
2412:
2409:
2389:
2386:
2383:
2363:
2360:
2357:
2343:
2331:
2328:
2325:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2245:
2206:
2205:Partial orders
2203:
2185:
2182:
2169:
2149:
2129:
2109:
2085:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
1972:
1969:
1966:
1946:
1943:
1940:
1920:
1917:
1914:
1911:
1908:
1888:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1813:
1812:
1798:
1795:
1792:
1789:
1769:
1766:
1763:
1743:
1740:
1737:
1717:
1714:
1711:
1708:
1705:
1702:
1678:
1635:
1634:
1631:
1630:
1615:
1612:
1609:
1601:
1600:
1597:
1594:
1591:
1588:
1585:
1584:
1574:
1563:
1560:
1557:
1542:
1531:
1528:
1525:
1515:
1495:
1494:
1491:
1488:
1485:
1482:
1481:
1471:
1451:
1450:
1447:
1444:
1441:
1438:
1437:
1427:
1407:
1406:
1403:
1400:
1397:
1396:
1386:
1371:
1368:
1365:
1362:
1358: or
1355:
1352:
1349:
1346:
1345:
1342:
1339:
1336:
1332:
1329:
1326:
1325:
1315:
1300:
1297:
1293:
1290:
1287:
1284:
1283:
1280:
1277:
1274:
1271:
1264:
1261:
1258:
1255:
1254:
1244:
1229:
1226:
1223:
1220:
1216:
1213:
1212:
1209:
1206:
1203:
1200:
1198:
1188:
1177:
1174:
1171:
1168:
1148:
1145:
1142:
1130:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1081:
1080:
1070:
1060:
1055:
1050:
1045:
1040:
1030:
1020:
1015:
1009:
1008:
998:
988:
983:
978:
973:
968:
963:
953:
948:
942:
941:
931:
921:
916:
911:
906:
901:
896:
886:
881:
875:
874:
869:
864:
854:
844:
839:
834:
829:
819:
814:
808:
807:
802:
797:
787:
782:
772:
767:
762:
752:
747:
741:
740:
735:
730:
720:
710:
700:
695:
690:
680:
675:
669:
668:
663:
658:
648:
643:
638:
628:
618:
608:
603:
597:
596:
591:
586:
576:
571:
566:
556:
551:
546:
541:
533:
532:
527:
522:
512:
507:
502:
492:
482:
477:
472:
466:
465:
460:
455:
445:
440:
435:
430:
420:
410:
405:
399:
398:
393:
388:
378:
373:
368:
363:
353:
348:
343:
341:Total preorder
337:
336:
331:
326:
316:
311:
306:
301:
296:
286:
281:
275:
274:
269:
264:
254:
249:
244:
239:
234:
229:
224:
213:
212:
207:
202:
192:
187:
182:
177:
172:
167:
157:
149:
148:
146:
141:
139:
137:
135:
133:
130:
128:
126:
123:
122:
117:
112:
107:
102:
97:
92:
87:
82:
77:
70:
69:
67:
66:
59:
52:
44:
30:
29:
26:
9:
6:
4:
3:
2:
12435:
12424:
12421:
12419:
12416:
12415:
12413:
12398:
12395:
12393:
12390:
12386:
12383:
12382:
12381:
12378:
12374:
12371:
12369:
12366:
12364:
12361:
12360:
12359:
12356:
12352:
12349:
12348:
12347:
12346:Ordered field
12344:
12342:
12339:
12335:
12332:
12330:
12327:
12326:
12325:
12322:
12318:
12315:
12314:
12313:
12310:
12308:
12305:
12303:
12302:Hasse diagram
12300:
12298:
12295:
12293:
12290:
12286:
12283:
12282:
12281:
12280:Comparability
12278:
12276:
12273:
12271:
12268:
12266:
12263:
12262:
12260:
12256:
12248:
12245:
12243:
12240:
12238:
12235:
12233:
12230:
12229:
12228:
12225:
12223:
12220:
12216:
12213:
12211:
12208:
12207:
12206:
12203:
12201:
12197:
12194:
12193:
12191:
12188:
12184:
12178:
12175:
12173:
12170:
12168:
12165:
12163:
12160:
12158:
12155:
12153:
12152:Product order
12150:
12148:
12145:
12143:
12140:
12138:
12135:
12133:
12130:
12129:
12127:
12125:Constructions
12123:
12117:
12113:
12109:
12106:
12102:
12099:
12097:
12094:
12092:
12089:
12087:
12084:
12082:
12079:
12077:
12074:
12072:
12069:
12067:
12064:
12060:
12057:
12056:
12055:
12052:
12050:
12047:
12043:
12040:
12038:
12035:
12033:
12030:
12028:
12025:
12024:
12023:
12022:Partial order
12020:
12018:
12015:
12011:
12010:Join and meet
12008:
12006:
12003:
12001:
11998:
11996:
11993:
11991:
11988:
11987:
11986:
11983:
11981:
11978:
11976:
11973:
11971:
11968:
11966:
11963:
11961:
11957:
11953:
11951:
11948:
11946:
11943:
11941:
11938:
11936:
11933:
11931:
11928:
11924:
11921:
11920:
11919:
11916:
11914:
11911:
11909:
11908:Antisymmetric
11906:
11905:
11903:
11899:
11893:
11887:
11884:
11882:
11879:
11877:
11874:
11872:
11869:
11867:
11864:
11862:
11859:
11857:
11854:
11852:
11849:
11847:
11844:
11842:
11839:
11837:
11834:
11832:
11829:
11828:
11826:
11822:
11816:
11815:Weak ordering
11813:
11811:
11808:
11806:
11803:
11801:
11800:Partial order
11798:
11796:
11793:
11791:
11788:
11786:
11783:
11781:
11778:
11777:
11775:
11771:
11765:
11762:
11760:
11757:
11755:
11752:
11751:
11748:
11744:
11737:
11732:
11730:
11725:
11723:
11718:
11717:
11714:
11707:
11701:
11699:
11697:
11689:
11688:
11676:
11672:
11667:
11663:
11659:
11658:Eilenberg, S.
11655:
11651:
11649:0-521-66351-2
11645:
11641:
11637:
11633:
11629:
11623:
11619:
11618:
11612:
11608:
11602:
11598:
11594:
11590:
11585:
11580:
11576:
11572:
11568:
11563:
11559:
11553:
11549:
11548:
11542:
11541:
11527:
11523:
11518:
11513:
11509:
11505:
11501:
11494:
11486:
11480:
11476:
11472:
11468:
11462:
11455:
11450:
11442:
11436:
11432:
11427:
11426:
11417:
11410:
11406:
11401:
11394:
11392:9789810235895
11388:
11384:
11377:
11370:
11368:
11367:Hasse diagram
11355:
11353:0-471-83817-9
11349:
11345:
11340:
11339:
11333:
11326:
11318:
11317:
11309:
11302:
11287:
11286:
11278:
11271:
11269:
11265:
11260:
11256:
11250:
11246:
11240:
11236:
11231:
11227:
11223:
11219:
11206:
11202:
11196:
11181:
11180:
11172:
11157:
11150:
11143:
11136:
11124:
11123:
11115:
11108:
11104:
11099:
11090:
11086:
11082:
11075:
11067:
11065:9781848002012
11061:
11057:
11053:
11046:
11038:
11032:
11028:
11027:
11019:
11017:
11015:
11010:
10996:
10995:
10988:
10969:
10959:
10952:
10946:
10942:
10930:
10927:
10924:
10921:
10916:
10912:
10906:
10902:
10896:
10893:
10890:
10887:
10884:
10881:
10878:
10875:
10872:
10869:
10866:
10863:
10860:
10857:
10854:
10851:
10845:
10844:Ordered group
10842:
10839:
10838:Ordered field
10836:
10834:
10831:
10829:
10826:
10823:
10820:
10811:
10808:
10805:
10802:
10799:
10796:
10790:
10787:
10784:
10781:
10775:
10772:
10769:
10766:
10763:
10760:
10757:
10754:
10753:
10746:
10744:
10739:
10726:
10720:
10717:
10714:
10708:
10702:
10699:
10696:
10685:
10681:
10656:
10643:
10638:
10636:
10632:
10628:
10624:
10603:
10595:
10591:
10568:
10565:
10562:
10559:
10556:
10543:
10539:
10535:
10524:
10520:
10512:
10508:
10503:
10499:
10479:
10475:
10469:
10465:
10459:
10455:
10449:
10445:
10441:
10436:
10430:
10426:
10421:
10420:open interval
10417:
10414:
10410:
10405:
10401:
10395:
10391:
10385:
10381:
10377:
10372:
10366:
10362:
10357:
10353:
10349:
10345:
10344:
10343:
10341:
10337:
10332:
10330:
10326:
10322:
10318:
10314:
10310:
10307:
10303:
10298:
10296:
10292:
10288:
10284:
10280:
10272:
10268:
10264:
10260:
10256:
10252:
10244:
10240:
10236:
10228:
10224:
10218:
10208:
10195:
10192:
10189:
10186:
10166:
10161:
10157:
10153:
10148:
10144:
10124:
10121:
10101:
10098:
10095:
10090:
10086:
10074:
10049:
10046:
10043:
10038:
10034:
10022:
10009:
9993:
9990:
9987:
9984:
9977:
9976:product space
9973:
9954:
9951:
9948:
9945:
9939:
9936:
9933:
9919:
9903:
9893:
9883:
9881:
9877:
9873:
9869:
9865:
9860:
9858:
9857:
9840:
9834:
9831:
9828:
9822:
9816:
9813:
9810:
9804:
9798:
9795:
9792:
9781:
9776:
9772:
9765:
9761:
9757:
9753:
9735:
9732:
9712:
9704:
9688:
9685:
9665:
9657:
9653:
9647:
9637:
9635:
9632:) are called
9631:
9627:
9623:
9618:
9616:
9611:
9595:
9592:
9587:
9583:
9579:
9559:
9556:
9553:
9550:
9530:
9527:
9524:
9521:
9518:
9515:
9495:
9475:
9467:
9464:is called an
9451:
9429:
9425:
9410:
9391:
9387:
9380:
9377:
9372:
9368:
9345:
9341:
9332:
9317:
9295:
9291:
9270:
9265:
9261:
9257:
9250:we also have
9237:
9234:
9231:
9209:
9205:
9184:
9164:
9144:
9122:
9118:
9108:
9094:
9091:
9088:
9068:
9063:
9059:
9055:
9035:
9013:
9009:
8988:
8968:
8948:
8926:
8922:
8901:
8893:
8875:
8871:
8847:
8844:
8841:
8835:
8832:
8824:
8803:
8799:
8795:
8790:
8786:
8779:
8774:
8770:
8755:
8753:
8748:
8743:
8738:
8735:
8733:
8727:
8723:
8719:
8709:
8706:
8704:
8701:
8699:
8696:
8694:
8691:
8689:
8686:
8684:
8681:
8679:
8676:
8674:
8671:
8669:
8666:
8664:
8661:
8660:
8655:
8651:
8647:
8640:
8631:
8628:
8625:
8621:
8617:
8613:
8609:
8602:
8593:
8591:
8589:
8586:
8583:
8581:
8578:
8576:
8573:
8572:
8568:
8565:
8562:
8559:
8556:
8551:
8546:
8541:
8536:
8533:
8532:
8528:
8525:
8522:
8519:
8516:
8513:
8510:
8507:
8504:
8501:
8500:
8496:
8493:
8490:
8487:
8484:
8481:
8478:
8475:
8472:
8469:
8468:
8464:
8461:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8436:
8432:
8429:
8426:
8423:
8420:
8417:
8414:
8411:
8408:
8405:
8404:
8401:
8398:
8396:
8393:
8391:
8388:
8386:
8385:Partial order
8383:
8381:
8378:
8376:
8373:
8371:
8368:
8366:
8363:
8361:
8358:
8355:
8354:
8350:
8344:
8342:
8338:
8334:
8324:
8322:
8321:
8316:
8274:
8260:
8252:
8233:
8186:
8183:
8175:
8156:
8153:
8150:
8123:
8119:
8079:
8076:
8067:
8065:
8061:
8057:
8053:
8049:
8033:
8030:
8027:
8007:
8004:
8001:
7981:
7975:
7972:
7969:
7949:
7943:
7940:
7937:
7929:
7925:
7919:
7911:
7905:
7904:
7899:
7883:
7877:
7874:
7871:
7863:
7859:
7855:
7851:
7847:
7831:
7828:
7808:
7805:
7802:
7782:
7779:
7776:
7768:
7765:
7762:
7739:
7733:
7730:
7724:
7718:
7710:
7700:
7692:
7686:
7685:
7664:
7661:
7658:
7655:
7645:
7641:
7637:
7633:
7617:
7614:
7611:
7608:
7605:
7602:
7594:
7578:
7572:
7569:
7566:
7546:
7540:
7537:
7534:
7531:
7528:
7521:
7505:
7499:
7496:
7493:
7473:
7467:
7464:
7461:
7451:
7443:
7439:
7435:
7431:
7414:
7411:
7408:
7389:
7386:
7383:
7380:
7377:
7374:
7367:, if for all
7366:
7362:
7361:
7356:
7355:
7338:
7332:
7329:
7326:
7319:, a function
7316:
7308:
7291:
7285:
7263:
7252:
7248:
7244:
7240:
7235:
7229:
7215:
7200:
7194:
7191:
7188:
7185:
7182:
7179:
7176:
7165:
7161:
7157:
7153:
7149:
7145:
7137:
7134:
7130:
7112:
7103:
7097:
7091:
7075:
7058:
7055:
7052:
7041:
7037:
7033:
7029:
7025:
7021:
7018: β₯
7017:
7013:
7009:
7005:
7001:
6997:
6993:
6989:
6985:
6981:
6978: β€
6977:
6973:
6969:
6965:
6962:, an element
6961:
6957:
6953:
6950:
6908:
6905:
6902:
6899:
6896:
6872:
6869:
6866:
6863:
6843:
6840:
6837:
6817:
6814:
6811:
6791:
6788:
6785:
6782:
6762:
6759:
6756:
6736:
6733:
6730:
6722:
6719:
6717:is the least.
6677:
6674:
6671:
6668:
6665:
6642:
6639:
6636:
6633:
6613:
6610:
6607:
6598:least element
6597:
6583:
6580:
6577:
6557:
6554:
6551:
6548:
6528:
6525:
6522:
6512:
6498:
6495:
6492:
6484:
6481:
6480:
6479:
6465:
6462:
6448:
6444:
6440:
6436:
6434:
6430:
6407:
6401:
6398:
6395:
6392:
6389:
6366:
6360:
6357:
6354:
6328:
6317:
6312:
6308:
6304:
6298:
6294:
6289:
6285:
6269:
6266:
6263:
6260:
6257:
6254:
6246:
6242:
6238:
6234:
6218:
6215:
6212:
6204:
6200:
6196:
6192:
6188:
6184:
6180:
6176:
6172:
6168:
6164:
6163:
6158:
6154:
6140:
6134:
6131:
6128:
6102:
6092:For example,
6079:
6076:
6073:
6070:
6062:
6058:
6054:
6050:
6046:
6042:
6028:
6019:
6013:
6007:
6001:
5995:
5982:
5978:
5977:
5972:
5951:
5948:
5945:
5942:
5939:
5933:
5927:
5921:
5903:
5902:
5897:
5894:
5890:
5886:
5885:
5880:
5862:
5836:
5810:
5807:
5804:
5801:
5798:
5772:
5761:
5756:
5752:
5746:
5742:
5737:
5736:
5731:
5727:
5724:
5709:
5703:
5700:
5697:
5671:
5660:
5656:
5652:
5648:
5644:
5640:
5637:
5633:
5630:
5629:
5628:
5614:
5608:
5605:
5602:
5599:
5596:
5585:
5566:
5563:
5554:
5551:
5548:
5545:
5542:
5512:
5510:
5506:
5502:
5500:
5492:
5488:
5484:
5480:
5477:
5474:
5470:
5465:
5461:
5457:
5453:
5450:
5447:
5443:
5438:
5434:
5430:
5426:
5423:
5422:
5421:
5418:
5414:
5409:
5405:by the order
5404:
5400:
5395:
5391:
5387:
5382:
5378:
5368:
5366:
5361:
5359:
5355:
5350:
5343:
5339:
5333:
5329:
5323:
5319:
5313:
5309:
5302:
5298:
5294:
5290:
5284:
5280:
5276:
5273:
5269:
5265:
5261:
5257:
5253:
5249:
5245:
5241:
5240:product order
5237:
5233:
5229:
5223:
5219:
5213:
5209:
5202:
5198:
5194:
5190:
5184:
5180:
5179:
5178:
5176:
5162:and covering
5157:
5141:
5133:
5120:
5114:
5089:
5076:
5070:
5045:
5032:
5026:
5012:
5010:
5001:
4997:
4992:
4988:
4984:
4980:
4976:
4971:
4967:
4962:
4958:
4954:
4950:
4946:
4942:
4938:
4934:
4930:
4925:
4921:
4907:
4904:
4901:
4898:
4888:
4884:
4880:
4876:
4870:
4866:
4861:
4857:
4853:
4849:
4845:
4841:
4838:
4835:; that is, a
4817:
4814:
4792:
4788:
4784:
4779:
4775:
4747:
4744:
4739:
4734:
4730:
4726:
4721:
4711:
4708:
4703:
4698:
4694:
4690:
4680:
4676:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4645:
4641:
4637:
4634:
4630:
4626:
4623:
4619:
4615:
4612:
4608:
4604:
4600:
4596:
4593:) ordered by
4592:
4588:
4584:
4581:
4577:
4552:
4548:
4522:
4519:
4515:
4511:
4510:
4509:
4502:
4498:
4489:
4486:
4470:
4450:
4442:
4423:
4420:
4417:
4406:
4405:Hasse diagram
4401:
4399:
4398:antisymmetric
4395:
4391:
4387:
4382:
4380:
4376:
4375:
4351:
4348:
4345:
4342:
4339:
4336:
4327:
4324:
4321:
4318:
4305:
4301:
4297:
4292:
4288:
4282:
4278:
4272:
4268:
4263:
4259:
4255:
4239:
4231:
4228:
4225:
4222:
4219:
4216:
4208:
4204:
4194:
4180:
4172:
4157:
4137:
4117:
4097:
4077:
4057:
4049:
4033:
4024:
4010:
3990:
3970:
3962:
3958:
3954:
3949:
3932:
3929:
3926:
3923:
3920:
3894:
3891:
3888:
3862:
3859:
3856:
3845:
3829:
3809:
3789:
3769:
3749:
3741:
3725:
3705:
3685:
3665:
3645:
3625:
3622:
3603:
3600:
3581:
3578:
3558:
3538:
3524:
3510:
3507:
3504:
3484:
3475:
3471:
3451:
3443:
3421:
3400:
3374:
3365:
3361:
3355:
3345:
3332:
3329:
3326:
3323:
3315:
3312:
3309:
3301:
3298:
3295:
3287:
3271:
3251:
3248:
3245:
3242:
3234:
3231:
3228:
3220:
3217:
3214:
3206:
3190:
3169:
3165:
3159:
3150:
3146:
3126:
3118:
3101:
3080:
3077:
3074:
3066:
3047:
3044:
3041:
3038:
3032:
3029:
3026:
3017:
3012:
2985:
2968:
2962:
2958:
2938:
2935:
2932:
2929:
2909:
2889:
2865:
2862:
2859:
2851:
2850:Hasse diagram
2847:
2843:
2839:
2835:
2831:
2828:
2824:
2815:
2813:
2808:
2789:
2786:
2783:
2763:
2760:
2757:
2737:
2734:
2731:
2723:
2720:
2705:
2702:
2699:
2679:
2676:
2673:
2665:
2662:
2646:
2642:
2639:
2636:
2632:
2620:
2619:Irreflexivity
2617:
2616:
2615:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2573:
2569:
2565:
2549:
2541:
2533:
2529:
2519:
2517:
2497:
2494:
2491:
2471:
2468:
2465:
2445:
2442:
2439:
2431:
2428:
2413:
2410:
2407:
2387:
2384:
2381:
2361:
2358:
2355:
2347:
2344:
2329:
2326:
2323:
2315:
2312:
2311:
2310:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2267:
2263:
2262:antisymmetric
2259:
2243:
2236:
2232:
2228:
2227:partial order
2224:
2216:
2212:
2202:
2200:
2195:
2191:
2190:partial order
2181:
2167:
2147:
2127:
2107:
2099:
2083:
2060:
2057:
2054:
2048:
2045:
2037:
2033:
2029:
2025:
2024:antisymmetric
2021:
2017:
2012:
2010:
2006:
2002:
1998:
1997:partial order
1994:
1991:, especially
1990:
1967:
1941:
1915:
1912:
1909:
1878:
1862:
1856:
1853:
1850:
1847:
1844:
1833:
1829:
1828:Hasse diagram
1825:
1821:
1811:
1796:
1793:
1790:
1787:
1767:
1764:
1761:
1741:
1738:
1735:
1715:
1712:
1709:
1706:
1703:
1700:
1692:
1676:
1669:
1637:
1636:
1613:
1610:
1607:
1592:
1589:
1586:
1575:
1561:
1558:
1555:
1543:
1529:
1526:
1523:
1516:
1489:
1486:
1483:
1472:
1445:
1442:
1439:
1428:
1401:
1387:
1369:
1366:
1363:
1353:
1350:
1347:
1337:
1330:
1327:
1316:
1298:
1291:
1288:
1278:
1275:
1272:
1262:
1259:
1256:
1245:
1227:
1224:
1221:
1207:
1204:
1201:
1189:
1175:
1169:
1166:
1146:
1143:
1140:
1132:
1131:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1092:Antisymmetric
1090:
1088:
1085:
1083:
1082:
1071:
1061:
1056:
1051:
1046:
1041:
1031:
1021:
1016:
1014:
1011:
1010:
999:
989:
984:
979:
974:
969:
964:
954:
949:
947:
944:
943:
932:
922:
917:
912:
907:
902:
897:
887:
882:
880:
877:
876:
870:
865:
855:
845:
840:
835:
830:
820:
815:
813:
810:
809:
803:
798:
788:
783:
773:
768:
763:
753:
748:
746:
743:
742:
736:
731:
721:
711:
701:
696:
691:
681:
676:
674:
671:
670:
664:
659:
649:
644:
639:
629:
619:
609:
604:
602:
601:Well-ordering
599:
598:
592:
587:
577:
572:
567:
557:
552:
547:
542:
539:
535:
534:
528:
523:
513:
508:
503:
493:
483:
478:
473:
471:
468:
467:
461:
456:
446:
441:
436:
431:
421:
411:
406:
404:
401:
400:
394:
389:
379:
374:
369:
364:
354:
349:
344:
342:
339:
338:
332:
327:
317:
312:
307:
302:
297:
287:
282:
280:
279:Partial order
277:
276:
270:
265:
255:
250:
245:
240:
235:
230:
225:
222:
215:
214:
208:
203:
193:
188:
183:
178:
173:
168:
158:
155:
151:
150:
147:
142:
140:
138:
136:
134:
131:
129:
127:
125:
124:
121:
118:
116:
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
88:
86:
85:Antisymmetric
83:
81:
78:
76:
75:
72:
71:
65:
60:
58:
53:
51:
46:
45:
42:
38:
34:
33:
19:
18:Partial Order
12418:Order theory
12189:& Orders
12167:Star product
12096:Well-founded
12049:Prefix order
12005:Distributive
11995:Complemented
11965:Foundational
11930:Completeness
11886:Zorn's lemma
11799:
11790:Cyclic order
11773:Key concepts
11743:Order theory
11695:
11674:
11670:
11661:
11639:
11616:
11596:
11574:
11570:
11546:
11507:
11503:
11493:
11470:
11467:Jech, Thomas
11461:
11449:
11424:
11416:
11400:
11383:Basic Posets
11382:
11376:
11366:
11364:
11357:. Retrieved
11337:
11325:
11315:
11308:
11300:
11293:. Retrieved
11284:
11277:
11267:
11263:
11258:
11254:
11248:
11244:
11238:
11234:
11229:
11225:
11221:
11217:
11215:
11208:. Retrieved
11204:
11195:
11183:. Retrieved
11178:
11171:
11159:. Retrieved
11155:
11142:
11134:
11127:. Retrieved
11121:
11114:
11098:
11088:
11084:
11074:
11058:. Springer.
11055:
11045:
11025:
10992:
10987:
10958:
10945:
10929:Zorn's lemma
10914:
10910:
10904:
10900:
10789:Graded poset
10783:Directed set
10740:
10683:
10679:
10639:
10634:
10630:
10601:
10593:
10589:
10545:An interval
10544:
10537:
10533:
10530:
10522:
10518:
10510:
10506:
10501:
10477:
10473:
10467:
10463:
10457:
10453:
10447:
10443:
10439:
10434:
10428:
10424:
10419:
10412:
10408:
10403:
10399:
10393:
10389:
10383:
10379:
10375:
10370:
10364:
10360:
10355:
10351:
10347:
10339:
10335:
10333:
10328:
10316:
10312:
10308:
10301:
10299:
10295:order-convex
10294:
10283:real numbers
10270:
10266:
10262:
10258:
10254:
10250:
10242:
10238:
10234:
10229:is a subset
10226:
10222:
10220:
10114:and for all
9895:
9861:
9854:
9774:
9770:
9763:
9759:
9755:
9751:
9649:
9626:reachability
9619:
9465:
9416:
9360:, and write
9330:
9109:
8822:
8821:is called a
8761:
8739:
8736:
8725:
8721:
8717:
8714:
8653:
8649:
8645:
8638:
8626:
8619:
8615:
8611:
8607:
8600:
8574:
8384:
8348:
8340:
8330:
8318:
8068:
8063:
8059:
8055:
8051:
7923:
7917:
7909:
7901:
7861:
7857:
7853:
7849:
7845:
7795:and in turn
7698:
7690:
7682:
7643:
7639:
7635:
7631:
7592:
7449:
7441:
7437:
7433:
7429:
7364:
7358:
7352:
7314:
7306:
7302:
7289:
7255:, but not u
7250:
7246:
7242:
7238:
7233:
7164:prime number
7159:
7155:
7151:
7141:
7132:
7039:
7035:
7031:
7027:
7023:
7019:
7015:
7011:
7007:
7003:
6999:
6995:
6991:
6987:
6983:
6979:
6975:
6971:
6967:
6963:
6959:
6955:
6454:
6432:
6315:
6310:
6306:
6302:
6296:
6292:
6287:
6283:
6244:
6240:
6236:
6232:
6202:
6198:
6194:
6190:
6186:
6182:
6178:
6174:
6170:
6166:
6160:
6156:
6060:
6056:
6052:
6048:
6044:
5974:
5899:
5889:linear order
5888:
5882:
5760:incomparable
5759:
5754:
5750:
5744:
5740:
5733:
5729:
5725:
5654:
5650:
5646:
5642:
5638:
5635:
5631:
5518:
5503:
5499:well-ordered
5496:
5490:
5486:
5482:
5478:
5472:
5468:
5463:
5459:
5455:
5451:
5445:
5441:
5436:
5432:
5428:
5424:
5416:
5412:
5407:
5402:
5398:
5393:
5389:
5385:
5380:
5376:
5374:
5362:
5351:
5348:
5341:
5337:
5331:
5327:
5321:
5317:
5311:
5307:
5300:
5296:
5292:
5288:
5271:
5267:
5263:
5259:
5255:
5251:
5247:
5243:
5231:
5227:
5221:
5217:
5211:
5207:
5200:
5196:
5192:
5188:
5172:
5118:
5074:
5030:
5009:genealogical
5006:
4999:
4995:
4990:
4986:
4982:
4974:
4969:
4965:
4960:
4956:
4940:
4936:
4932:
4928:
4886:
4882:
4878:
4874:
4868:
4864:
4859:
4855:
4847:
4843:
4681:. Formally,
4678:
4674:
4670:
4666:
4662:
4650:
4644:vector space
4633:reachability
4622:divisibility
4551:greater than
4545:, the usual
4517:
4514:real numbers
4507:
4500:
4402:
4385:
4383:
4371:
4304:incomparable
4303:
4299:
4295:
4290:
4286:
4280:
4276:
4270:
4266:
4257:
4253:
4200:
4025:
3956:
3952:
3950:
3843:
3531:Given a set
3530:
3363:
3359:
3357:
2881:
2841:
2837:
2833:
2826:
2809:
2805:
2722:Transitivity
2535:
2531:
2527:
2525:
2513:
2430:Transitivity
2346:Antisymmetry
2226:
2218:
2214:
2210:
2208:
2193:
2189:
2187:
2097:
2096:(called the
2035:
2031:
2013:
2009:total orders
2004:
1996:
1993:order theory
1986:
1823:
1665:
1102:Well-founded
220:(Quasiorder)
95:Well-founded
12373:Riesz space
12334:Isomorphism
12210:Normal cone
12132:Composition
12066:Semilattice
11975:Homogeneous
11960:Equivalence
11810:Total order
11407:, pp.
11224:): Compare
11105:, pp.
10923:Total order
10871:Semilattice
10756:Antimatroid
10437:satisfying
10373:satisfying
10338:in a poset
10293:, one uses
10287:convex sets
10273:is also in
10225:in a poset
9862:Posets are
9224:, whenever
8395:Total order
8174:prime power
8046:yields the
7595:if for all
7520:composition
7076:upper bound
6570:An element
6155:An element
6051:an element
6043:An element
5970:is a chain.
5884:total order
5377:ordinal sum
5352:Applied to
5242:: (
4985:. An event
4638:The set of
4631:ordered by
4616:The set of
4609:ordered by
4603:subsequence
4601:ordered by
4585:The set of
3953:ordered set
3348:Dual orders
2564:irreflexive
2528:irreflexive
2314:Reflexivity
1989:mathematics
1875:ordered by
1122:Irreflexive
403:Total order
115:Irreflexive
12412:Categories
12341:Order type
12275:Cofinality
12116:Well-order
12091:Transitive
11980:Idempotent
11913:Asymmetric
11677:: 134β156.
11537:References
10762:Causal set
10581:such that
10489:such that
10387:(that is,
10223:convex set
10215:See also:
9868:isomorphic
9864:equivalent
9628:orders of
8730:refers to
8715:Note that
8365:Transitive
8347:Number of
7924:isomorphic
7591:is called
7351:is called
6856:such that
6775:such that
6169:, written
5981:singletons
5893:trichotomy
5735:comparable
5636:related to
5381:linear sum
4979:light cone
4842:For a set
4394:transitive
4207:comparison
4048:complement
3288:given by:
2572:transitive
2568:asymmetric
2266:transitive
2194:non-strict
2098:ground set
2028:transitive
1693:: for all
1691:transitive
1127:Asymmetric
120:Asymmetric
37:Transitive
12392:Upper set
12329:Embedding
12265:Antichain
12086:Tolerance
12076:Symmetric
12071:Semiorder
12017:Reflexive
11935:Connected
11469:(2008) .
11295:5 January
11210:5 January
11005:Citations
10899:"neither
10877:Semiorder
10657:×
10566:∈
10531:Whenever
10279:intervals
10211:Intervals
10190:≤
10154:≤
10081:∞
10078:→
10029:∞
10026:→
9988:×
9952:≤
9805:∘
9780:empty set
9588:∗
9584:≤
9554:≤
9543:whenever
9525:∈
9476:≤
9466:extension
9444:on a set
9430:∗
9426:≤
9392:∗
9373:∗
9346:∗
9296:∗
9266:∗
9262:≤
9235:≤
9210:∗
9123:∗
9092:≤
9064:∗
9060:≤
9014:∗
8949:≤
8927:∗
8923:≤
8876:∗
8848:≤
8804:∗
8800:≤
8791:∗
8775:∗
8758:Subposets
8375:Symmetric
8370:Reflexive
8331:Sequence
8195:→
8118:power set
8088:→
8031:∘
8005:∘
7979:→
7947:→
7898:bijective
7881:→
7829:≤
7780:≤
7766:≤
7709:injective
7659:≤
7612:∈
7576:→
7544:→
7532:∘
7503:→
7471:→
7412:≤
7384:∈
7336:→
7264:≤
6841:∈
6815:∈
6760:∈
6734:∈
6637:∈
6611:≤
6581:∈
6552:∈
6526:≤
6496:∈
6478:notably:
6264:≠
6258:≠
6216:≠
6074:≠
5976:antichain
5659:symmetric
5567:⊆
5363:See also
5185::
5154:Elements
5134:×
5090:×
5046:×
4902:∈
4818:∈
4785:≤
4748:∈
4722:≤
4712:∈
4659:sequences
4640:subspaces
4611:substring
4599:sequences
4595:inclusion
4591:power set
4547:less than
4331:→
4325:×
4229:≥
4217:≤
4181:≤
4158:≤
4118:≤
4098:≤
4034:≤
4011:⪯
3991:⊑
3971:≤
3951:The term
3927:≤
3863:≤
3790:≤
3770:≥
3750:≤
3666:≤
3623:≥
3579:≤
3559:≤
3299:≤
3272:≤
3246:≠
3232:≤
3207:given by
3191:≤
3166:∪
3156:Δ
3147:≤
3102:∖
3078:×
3045:∈
3009:Δ
2982:Δ
2975:∖
2969:≤
2933:≤
2910:≤
2692:then not
2664:Asymmetry
2629:¬
2597:∈
2495:≤
2469:≤
2443:≤
2385:≤
2359:≤
2327:≤
2291:∈
2258:reflexive
2211:reflexive
2188:The term
2128:≤
2061:≤
2020:reflexive
1887:∅
1877:inclusion
1604:not
1596:⇒
1552:not
1487:∧
1443:∨
1341:⇒
1331:≠
1286:⇒
1215:⇒
1173:∅
1170:≠
1117:Reflexive
1112:Has meets
1107:Has joins
1097:Connected
1087:Symmetric
218:Preorder
145:reflexive
110:Reflexive
105:Has meets
100:Has joins
90:Connected
80:Symmetric
12187:Topology
12054:Preorder
12037:Eulerian
12000:Complete
11950:Directed
11940:Covering
11805:Preorder
11764:Category
11759:Glossary
11660:(2016).
11638:(1997).
11595:(2010).
11334:(1989).
10749:See also
10629:in
10627:supremum
10336:interval
10291:geometry
10249:and any
9703:morphism
9656:category
8823:subposet
8762:A poset
8380:Preorder
8128:divides
7858:embedded
7755:implies
7711:, since
7648:implies
7427:implies
7360:monotone
7144:integers
6451:element.
6445:greatest
6314:for any
6300:but not
5878:are not.
4993:only if
4891:for all
4862:, where
4807:for all
4492:Examples
4392:that is
3983:such as
3658:, where
3527:Notation
3364:opposite
3115:denotes
2863:≰
2562:that is
2516:preorder
2256:that is
2018:that is
1983:are not.
1661:✗
1648:✗
1058:✗
1053:✗
1048:✗
1043:✗
1018:✗
986:✗
981:✗
976:✗
971:✗
966:✗
951:✗
919:✗
914:✗
909:✗
904:✗
899:✗
884:✗
872:✗
867:✗
842:✗
837:✗
832:✗
817:✗
805:✗
800:✗
785:✗
770:✗
765:✗
750:✗
738:✗
733:✗
698:✗
693:✗
678:✗
666:✗
661:✗
646:✗
641:✗
606:✗
594:✗
589:✗
574:✗
569:✗
554:✗
549:✗
544:✗
530:✗
525:✗
510:✗
505:✗
480:✗
475:✗
463:✗
458:✗
443:✗
438:✗
433:✗
408:✗
396:✗
391:✗
376:✗
371:✗
366:✗
351:✗
346:✗
334:✗
329:✗
314:✗
309:✗
304:✗
299:✗
284:✗
272:✗
267:✗
252:✗
247:✗
242:✗
237:✗
232:✗
227:✗
210:✗
205:✗
190:✗
185:✗
180:✗
175:✗
170:✗
12292:Duality
12270:Cofinal
12258:Related
12237:FrΓ©chet
12114:)
11990:Bounded
11985:Lattice
11958:)
11956:Partial
11824:Results
11795:Lattice
11359:27 July
11185:23 July
11161:23 July
11129:24 July
10804:Lattice
10623:infimum
10491:0 <
10323:and an
10306:lattice
10269:, then
9856:posetal
9331:induced
8750:in the
8747:A000112
8708:A000110
8703:A000142
8698:A000670
8693:A001035
8688:A000798
8683:A006125
8678:A053763
8673:A006905
8668:A002416
8333:A001035
7856:can be
7365:isotone
7290:Fig. 7b
7277:v) map.
7234:Fig. 7a
7148:divides
6447:and no
6441:minimal
6437:maximal
6424:Extrema
6162:covered
5281:of the
5156:covered
5119:Fig. 4c
5075:Fig. 4b
5031:Fig. 4a
4607:strings
4587:subsets
4388:as any
4315:compare
3464:, i.e.
3440:be the
3284:is the
3063:is the
2233:β€ on a
2229:, is a
2005:partial
1830:of the
673:Lattice
12317:Subnet
12297:Filter
12247:Normed
12232:Banach
12198:&
12105:Better
12042:Strict
12032:Graded
11923:topics
11754:Topics
11646:
11624:
11603:
11554:
11481:
11437:
11389:
11350:
11062:
11033:
10619:(1, 2)
10614:(2, 3)
10610:(1, 2)
10606:(0, 1)
10495:< 1
10483:(0, 1)
10451:(i.e.
10354:, the
10321:filter
10008:limits
9972:closed
9782:) and
9758:) = {(
9048:), if
8892:subset
8538:65,536
8317:; see
7133:Fig. 6
7074:is an
6433:Fig. 5
6185:), if
6181:<:
5325:) or (
5164:(3, 3)
5160:(3, 3)
4850:, the
4653:, the
4501:Fig. 3
4374:setoid
4130:, but
3802:, and
3000:where
2978:
2972:
2966:
2827:Fig. 2
2570:, and
2532:strong
2264:, and
2026:, and
1824:Fig. 1
1498:exists
1454:exists
1410:exists
39:
12307:Ideal
12285:Graph
12081:Total
12059:Total
11945:Dense
11409:17β18
11289:(PDF)
11257:>
11237:<
11152:(PDF)
11107:14β15
10937:Notes
10913:<
10903:<
10596:]
10588:[
10525:]
10505:[
10476:<
10466:<
10456:<
10446:<
10442:<
10367:]
10359:[
10325:ideal
10304:of a
10257:, if
10179:then
9970:is a
9705:from
9081:then
8890:is a
8742:up to
8553:1,024
8548:4,096
8543:3,994
7860:into
7695:into
7446:. If
7363:, or
7357:, or
6596:is a
6511:is a
6449:least
6309:<
6305:<
6295:<
6247:with
6055:, if
5901:chain
5641:when
5462:with
5435:with
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5320:<
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5258:) if
5250:) β€ (
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4294:, or
4289:>
4284:, or
4274:, or
4269:<
3738:(the
2776:then
2724:: if
2666:: if
2534:, or
2484:then
2432:: if
2400:then
2348:: if
2217:, or
2036:poset
1999:on a
1780:then
143:Anti-
11898:list
11703:OEIS
11691:OEIS
11644:ISBN
11622:ISBN
11601:ISBN
11552:ISBN
11479:ISBN
11435:ISBN
11387:ISBN
11361:2012
11348:ISBN
11297:2022
11266:and
11228:and
11212:2022
11187:2021
11163:2021
11131:2021
11060:ISBN
11031:ISBN
10991:See
10951:here
10909:nor
10515:and
10500:The
10461:and
10411:and
10397:and
10346:For
10241:and
10062:and
9750:hom(
9678:and
9177:and
8981:and
8914:and
8752:OEIS
8663:OEIS
8337:OEIS
8062:and
8054:and
8020:and
7962:and
7920:, βΌ)
7914:and
7912:, β€)
7848:and
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7638:) βΌ
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7317:, βΌ)
7311:and
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7245:) βΌ
6924:and
6867:<
6786:>
6205:and
6177:(or
6063:and
5852:and
5788:and
5732:are
5728:and
5485:and
5475:, or
5448:, or
5401:and
5379:(or
5335:and
5315:and
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5277:the
5266:and
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4959:and
4881:) β€
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4439:, a
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4078:>
4058:>
3933:<
3895:<
3830:<
3810:>
3706:<
3646:>
3638:and
3601:<
3362:(or
3360:dual
3358:The
3313:<
3218:<
3170:<
3093:and
2959:<
2787:<
2761:<
2750:and
2735:<
2703:<
2677:<
2640:<
2458:and
2374:and
2215:weak
2030:. A
1995:, a
1957:and
1899:and
1826:The
1754:and
1159:and
12312:Net
12112:Pre
11675:280
11579:doi
11522:hdl
11512:doi
11369:...
10625:or
10334:An
10327:of
10289:of
10281:of
10253:in
10245:in
10233:of
10071:lim
10019:lim
9896:If
9768:if
9725:to
9620:In
9617:).
9488:on
9333:by
9197:in
9110:If
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8894:of
8569:15
8560:219
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