5123:
5206:
5187:
6421:
45:
5314:
5276:
8646:
8354:
5998:
5871:
1724:
1711:
1145:
1135:
1105:
1095:
1073:
1063:
1028:
1006:
996:
961:
929:
919:
894:
862:
847:
827:
795:
785:
775:
755:
723:
703:
693:
683:
651:
631:
587:
567:
557:
520:
495:
485:
453:
428:
391:
361:
329:
267:
232:
6113:
12341:
11763:-irreducible). For example, in Pic. 2, the elements 2, 3, 4, and 5 are join irreducible, while 12, 15, 20, and 30 are meet irreducible. Depending on definition, the bottom element 1 and top element 60 may or may not be considered join irreducible and meet irreducible, respectively. In the lattice of
4805:
8257:
Every poset that is a complete semilattice is also a complete lattice. Related to this result is the interesting phenomenon that there are various competing notions of homomorphism for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete
6410:
partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2, 3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility (12 and 18 do not divide each other). Likewise the pair 12,
9778:
Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via
4695:
4361:
This is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, that is, for finite subsets
5122:
8253:
its subsets have both a join and a meet. In particular, every complete lattice is a bounded lattice. While bounded lattice homomorphisms in general preserve only finite joins and meets, complete lattice homomorphisms are required to preserve arbitrary joins and meets.
3285:. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the
4700:
1455:
1384:
4568:
5841:
complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property that distinguishes arithmetic lattices from
5060:
By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as
3701:
4593:
4495:
2887:. In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.
5205:
1699:
1313:
4844:
Every lattice can be embedded into a bounded lattice by adding a greatest and a least element. Furthermore, every non-empty finite lattice is bounded, by taking the join (respectively, meet) of all elements, denoted by
7071:
6692:
9191:
1584:
1540:
6528:
1496:
7156:
9933:
8962:
8555:
3518:
10524:
8840:
8632:
4298:
4252:
3941:
10449:
7309:
4965:
2774:
1721:
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
9466:
6902:
9288:
6843:
5027:
4904:
12611:
2705:
1654:
1551:
1507:
1466:
1395:
1324:
1268:
6776:
3566:
5658:
11846:
6609:
7362:
11694:
4359:
4840:
3993:
11544:
7690:
7606:
4330:
2782:
argument that every non-empty finite subset of a lattice has a least upper bound and a greatest lower bound. With additional assumptions, further conclusions may be possible; see
10840:
8476:
8432:
3144:
3101:
1257:
7532:
7475:
6408:
4500:
3424:
2448:
9055:
4031:
3606:
3380:
2938:
5275:
1643:
10010:
5615:
2639:
2599:
10631:
11803:
8098:
5186:
2830:
12055:
11165:
11027:
10906:
10862:
10311:
7505:
7448:
5263:
9981:
9102:
8717:
8214:
6271:
4183:
3279:
3213:
2534:
9609:
6350:
5170:
4427:
11579:
11501:
10789:
7905:
6981:
6309:
4152:
3601:
3247:
3182:
2556:
2512:
2402:
1390:
12194:
11246:
9375:
8130:
7879:
6946:
5956:
5306:
4121:
3747:
3342:
3053:
11872:
11761:
11648:
11217:
11110:
10705:
10596:
10172:
9721:
9635:
9223:
8993:
For an overview of stronger notions of distributivity that are appropriate for complete lattices and that are used to define more special classes of lattices such as
8863:
8761:
8388:
7931:
6195:
5731:
3706:
One can now check that the relation ≤ introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations
3001:
2478:
12304: – lattice of all clones (sets of logical connectives closed under composition and containing all projections) on a two-element set {0, 1}, ordered by inclusion
12020:
11994:
11191:
10932:
10140:
8261:"Partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.
4078:
1797:
1319:
11737:
10563:
10377:
10350:
10259:
9701:
8883:
8741:
6711:
6547:
5690:
3889:
3826:
3724:
3444:
3319:
2981:
6224:
6071:
4300:
and therefore every element of a poset is both an upper bound and a lower bound of the empty set. This implies that the join of an empty set is the least element
1878:
11308:
1849:
1823:
1611:
1228:
12125:
11622:
11443:
11366:
11331:
11081:
10977:
10676:
10199:
10108:
9678:
9540:
8339:
8316:
8173:
8066:
7997:
7954:
7787:
6094:
5979:
5774:
5547:
5520:
5474:
5451:
5349:
5102:
4423:
3804:
2857:
12219:
12165:
12145:
12102:
12075:
11964:
11944:
11924:
11900:
11713:
11599:
11462:
11420:
11389:
11279:
11130:
11054:
10952:
10745:
10725:
10653:
10279:
10229:
10074:
10054:
10034:
9875:
9855:
9835:
9815:
9655:
9580:
9560:
9517:
9497:
8150:
8039:
8019:
7974:
7847:
7827:
7807:
7764:
7744:
7710:
7626:
7250:
7230:
7210:
7178:
6985:
6445:
6157:
6137:
6042:
6022:
5915:
5895:
5813:
5793:
5751:
5494:
5417:
5393:
5369:
5079:
4588:
4400:
4380:
4205:
4098:
4051:
3868:
3464:
3021:
2958:
2877:
1758:
4800:{\displaystyle \bigwedge (A\cup \varnothing )=\left(\bigwedge A\right)\wedge \left(\bigwedge \varnothing \right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A,}
4187:
A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. For every element
12618:
3148:
The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together. These are called
5313:
8229:
We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.
7075:
9884:
3751:
Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.
1649:
8890:
1263:
3469:
8768:
7370:
binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.
6411:
18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility (2 and 3 do not divide each other).
9937:
In general, some elements of a bounded lattice might not have a complement, and others might have more than one complement. For example, the set
2788:
for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable
13179:
4690:{\displaystyle \bigvee (A\cup \varnothing )=\left(\bigvee A\right)\vee \left(\bigvee \varnothing \right)=\left(\bigvee A\right)\vee 0=\bigvee A}
3896:
10313:
If the pseudo-complement of every element of a
Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra.
6614:
2216:
9107:
7255:
1546:
5104:
respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded.
13135:
Schlimm, Dirk (November 2011). "On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others".
1502:
6450:
1461:
7381:. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), although an
9683:
A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with
5560:
in their usual order form an unbounded lattice, under the operations of "min" and "max". 1 is bottom; there is no top (see Pic. 4).
13967:
132:
8483:
3522:
13950:
12498:
7378:
7367:
11882:. Every join-prime element is also join irreducible, and every meet-prime element is also meet irreducible. The converse holds if
10454:
8560:
4257:
13480:
4214:
7313:
13316:
10382:
9019:
For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice
9735:, it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class of
7188:. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a
2710:
13072:
13052:
13013:
8990:. Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).
4909:
12244:
of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.
9382:
5053:
is the only defining identity that is peculiar to lattice theory. A bounded lattice can also be thought of as a commutative
13797:
13214:
6848:
9228:
6792:
4970:
13087:
12548:
4848:
3105:
3062:
2644:
9304:(shown in Pic. 11). Besides distributive lattices, examples of modular lattices are the lattice of submodules of a
12999:
12438:
6226:
is a lattice if and only if it has at most one element. In particular the two-element discrete poset is not a lattice.
2209:
13933:
13792:
12956:
12939:
12775:
12696:
12641:
3289:
of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same
88:
66:
8295:. A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element
6715:
59:
13787:
5620:
11808:
8978:; they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it does not have a
13423:
13093:
12387:
Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.
8998:
6551:
4156:
3186:
13505:
13110:
12229:
However, many sources and mathematical communities use the term "atomic" to mean "atomistic" as defined above.
4125:
3155:
125:
12876:
Continuous posets, prime spectra of completely distributive complete lattices, and
Hausdorff compactifications
7366:
In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function
3344:
Since the commutative, associative and absorption laws can easily be verified for these operations, they make
2360:
A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure.
14005:
13824:
13744:
13239:
13034:
11653:
9354:. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank function
9002:
5041:. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative
2202:
17:
13609:
13538:
13418:
13229:
13086:
R. Freese, J. Jezek, and J. B. Nation, 1985. "Free
Lattices". Mathematical Surveys and Monographs Vol. 42.
12516:
4810:
4335:
3948:
11506:
7635:
7551:
13512:
13500:
13463:
13438:
13413:
13367:
13336:
13234:
12542:
12448:
12319:
10811:
10084:
8437:
8393:
4303:
2784:
2330:
2071:
1233:
8287:
has a join (that is, a least upper bound). Such lattices provide the most direct generalization of the
6357:
3388:
2415:
13809:
13443:
13433:
13309:
13005:
12688:
10211:
are an example of distributive lattices where some members might be lacking complements. Every element
9022:
3998:
3347:
2905:
1617:
13782:
13448:
12433:
12265:
9986:
8966:
A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, is called a
5570:
2604:
2564:
2280:
118:
13149:
10601:
6420:
13714:
13341:
12198:
11776:
9780:
8994:
8071:
7510:
7453:
2806:
1192:
1016:
185:
53:
12025:
11402:
We now define some order-theoretic notions of importance to lattice theory. In the following, let
11135:
10982:
10867:
10845:
10284:
7488:
7431:
5221:
1450:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
14000:
13962:
13945:
12468:
12458:
12417:
10175:
9940:
9066:
8681:
8178:
6232:
5696:
3252:
2961:
2797:
2517:
2296:
2162:
12655:
11767:
with the usual order, each element is join irreducible, but none is completely join irreducible.
9585:
6314:
6311:
so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in
5134:
13874:
13490:
13144:
11549:
11474:
10762:
8675:
7884:
6951:
6276:
5215:
3571:
3286:
3220:
2779:
2539:
2491:
2481:
2375:
2310:
1379:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
1162:
155:
70:
12750:
12715:
12170:
11222:
9357:
8103:
7852:
6919:
5920:
5858:
Further examples of lattices are given for each of the additional properties discussed below.
5826:) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical
5287:
4103:
3729:
3324:
3026:
13852:
13687:
13678:
13547:
13428:
13382:
13346:
13302:
12964:
12792:
12734:
12526:
12081:
11851:
11746:
11627:
11196:
11086:
10681:
10568:
9706:
9614:
9202:
8848:
8746:
8367:
7910:
6174:
5710:
5700:
2986:
2457:
2369:
2292:
2248:
1717:
indicates that the column's property is always true the row's term (at the very left), while
1172:
165:
11999:
11973:
11170:
10911:
10145:
5827:
4060:
1767:
13940:
13899:
13889:
13879:
13624:
13587:
13577:
13557:
13542:
12478:
12453:
12397:
12280:
12276:
12237:
12233:
11722:
10541:
10355:
10328:
10116:
10077:
10076:(see Pic. 11). A bounded lattice for which every element has a complement is called a
9686:
9305:
8868:
8726:
8665:
6696:
6532:
5982:
5663:
4563:{\displaystyle \bigwedge (A\cup B)=\left(\bigwedge A\right)\wedge \left(\bigwedge B\right)}
3874:
3811:
3709:
3429:
3304:
2966:
2149:
2141:
2113:
2108:
2099:
2056:
1998:
1738:
949:
224:
10238:
6200:
6047:
3696:{\displaystyle a=a\wedge b{\text{ implies }}b=b\vee (b\wedge a)=(a\wedge b)\vee b=a\vee b}
1854:
8:
13867:
13778:
13724:
13683:
13673:
13562:
13495:
13458:
12503:
12422:
12362:
11967:
11287:
9351:
9345:
9333:
9329:
9325:
8670:
Since lattices come with two binary operations, it is natural to ask whether one of them
2900:
2303:
2167:
2157:
2008:
1908:
1900:
1891:
1881:
A term's definition may require additional properties that are not listed in this table.
1828:
1802:
1761:
1590:
1207:
1197:
608:
190:
107:
13124:
12107:
11604:
11425:
11348:
11313:
11063:
10959:
10658:
10181:
10090:
9660:
9522:
8321:
8298:
8155:
8048:
7979:
7936:
7769:
6076:
5961:
5756:
5529:
5502:
5456:
5433:
5331:
5084:
5045:
having the same domain. For a bounded lattice, these semigroups are in fact commutative
4405:
3786:
2839:
13979:
13906:
13759:
13668:
13658:
13599:
13517:
13453:
13162:
13061:
13040:
12844:
12809:
12763:
12407:
12392:
12324:
12313: – mathematical object formed by an order on the way of parenthesing an expression
12204:
12150:
12130:
12087:
12060:
11949:
11929:
11909:
11885:
11698:
11584:
11447:
11405:
11374:
11338:
11264:
11115:
11039:
10937:
10730:
10710:
10638:
10264:
10214:
10059:
10039:
10019:
9860:
9840:
9820:
9800:
9761:
9640:
9565:
9545:
9502:
9482:
9317:
8288:
8135:
8024:
8004:
7959:
7832:
7812:
7792:
7749:
7729:
7695:
7611:
7382:
7235:
7215:
7195:
7163:
6430:
6142:
6122:
6027:
6007:
5900:
5880:
5838:
5798:
5778:
5736:
5550:
5523:
5479:
5402:
5378:
5354:
5064:
4573:
4385:
4365:
4190:
4083:
4036:
3853:
3449:
3006:
2943:
2862:
2793:
2485:
2451:
2276:
2264:
2256:
1973:
1964:
1922:
1743:
1187:
1167:
1157:
1083:
180:
160:
150:
13819:
13916:
13894:
13754:
13739:
13719:
13522:
13248:
13195:
13068:
13048:
13009:
12952:
12947:
12935:
12771:
12692:
12637:
12301:
11368:
11057:
10954:
10232:
9770:
8270:
7374:
6097:
5843:
5557:
5108:
5054:
2789:
2318:
13166:
12978:
13729:
13582:
13190:
13175:
13154:
13026:
12995:
12930:
12879:
12836:
12801:
12630:
12473:
12412:
12296:
9792:
9736:
9472:
8238:
7389:
5847:
5819:
5564:
5424:
5396:
4490:{\displaystyle \bigvee (A\cup B)=\left(\bigvee A\right)\vee \left(\bigvee B\right)}
4054:
3765:
2833:
2558:
2345:
2244:
1993:
882:
815:
2018:
13911:
13694:
13572:
13552:
13468:
13362:
12493:
12259:
10208:
9752:
9748:
9321:
9313:
9014:
8224:
7422:
5834:
5038:
2338:
2326:
2079:
2066:
2003:
1940:
540:
111:
31:
9739:, consisting of posets where every element can be obtained as the supremum of a
13829:
13814:
13804:
13663:
13641:
13619:
12402:
12310:
9817:
be a bounded lattice with greatest element 1 and least element 0. Two elements
9723:
exchanged, "covers" exchanged with "is covered by", and inequalities reversed.
8671:
7546:
5050:
4208:
2792:
between related partially ordered sets—an approach of special interest for the
2284:
2127:
1694:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
411:
13158:
7421:
is a bijective lattice endomorphism. Lattices and their homomorphisms form a
1308:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
13994:
13928:
13884:
13862:
13734:
13604:
13592:
13397:
13251:
12989:
The standard contemporary introductory text, somewhat harder than the above:
12483:
12462:
12443:
12253:
11874:, although this is unusual. This too can be generalized to obtain the notion
9732:
5851:
5177:
5173:
3832:
3290:
2013:
1978:
1935:
1182:
1177:
349:
175:
170:
8970:. The only non-distributive lattices with fewer than 6 elements are called M
5476:
ordered by inclusion, is also a lattice, and will be bounded if and only if
3945:
A bounded lattice may also be defined as an algebraic structure of the form
13749:
13631:
13614:
13532:
13372:
13325:
12488:
12366:
12286:
11764:
11310:
The free semilattice is defined to consist of all of the finite subsets of
11256:
10802:
10793:
9740:
9332:
with the ordering "is more specific than" is a non-modular lattice used in
7181:
5704:
5695:
The natural numbers also form a lattice under the operations of taking the
2314:
2288:
2240:
2236:
2187:
2118:
1952:
13100:. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press.
7066:{\displaystyle f\left(a\vee _{L}b\right)=f(a)\vee _{M}f(b),{\text{ and }}}
13955:
13648:
13527:
13392:
12270:
8292:
7396:
7185:
6687:{\displaystyle f(u)\wedge f(v)=u^{\prime }\wedge u^{\prime }=u^{\prime }}
3282:
2409:
2322:
2313:. Since the two definitions are equivalent, lattice theory draws on both
2177:
2172:
2061:
2051:
2025:
473:
9186:{\displaystyle (a\wedge c)\vee (b\wedge c)=((a\wedge c)\vee b)\wedge c.}
1579:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
13923:
13857:
13698:
12883:
12848:
12813:
9297:
8645:
8353:
6167:
Most partially ordered sets are not lattices, including the following.
5419:
itself and the empty set. In this lattice, the supremum is provided by
5042:
1927:
671:
9350:
A finite lattice is modular if and only if it is both upper and lower
1535:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
13974:
13847:
13653:
13275:
13256:
12427:
12355:
11334:
9751:
of a poset for obtaining these directed sets, then the poset is even
9744:
7406:
7385:
6523:{\displaystyle f(u)\vee f(v)=u^{\prime }\vee u^{\prime }=u^{\prime }}
5420:
5372:
2268:
2182:
1988:
1945:
1913:
1491:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
13180:"Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler"
12840:
12805:
5107:
The algebraic interpretation of lattices plays an essential role in
3603:
The laws of absorption ensure that both definitions are equivalent:
13769:
13636:
13387:
12351:
12222:
11029:
The value of the rank function for a lattice element is called its
10202:
8674:
over the other, that is, whether one or the other of the following
6782:
5997:
4100:(the lattice's top) is the identity element for the meet operation
3301:
An order-theoretic lattice gives rise to the two binary operations
2252:
1983:
288:
8258:
meet-semilattices, or as join-complete or meet-complete lattices.
3385:
The converse is also true. Given an algebraically defined lattice
13281:
12902:
5870:
2260:
12890:
7151:{\displaystyle f\left(a\wedge _{L}b\right)=f(a)\wedge _{M}f(b).}
2561:. Both operations are monotone with respect to the given order:
13294:
13266:
12241:
9928:{\displaystyle x\vee y=1\quad {\text{ and }}\quad x\wedge y=0.}
8480:
The labelled elements also violate the distributivity equation
6447:
between lattices that preserves neither joins nor meets, since
5046:
3023:
satisfying the following axiomatic identities for all elements
2272:
1917:
13285:
sequence A006966 (Number of unlabeled lattices with
10747:
have the same length, then the lattice is said to satisfy the
8957:{\displaystyle a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c).}
8550:{\displaystyle c\wedge (a\vee b)=(c\wedge a)\vee (c\wedge b),}
6112:
11650:. When the first condition is generalized to arbitrary joins
3513:{\displaystyle a\leq b{\text{ if }}a=a\wedge b,{\text{ or }}}
2307:
12951:, Lecture Notes in Mathematics 1533, Springer Verlag, 1992.
12855:
12279: – Nonempty, upper-bounded, downward-closed subset and
10519:{\displaystyle x_{0}<x_{1}<x_{2}<\ldots <x_{n}.}
8835:{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c).}
8627:{\displaystyle c\vee (a\wedge b)=(c\vee a)\wedge (c\vee b).}
7766:
that is a lattice with the same meet and join operations as
5850:. Both of these classes of complete lattices are studied in
5322:
Lattice of nonnegative integer pairs, ordered componentwise.
4293:{\displaystyle {\text{ for all }}a\in \varnothing ,a\leq x,}
13284:
12726:
12724:
9747:
the element. If one can additionally restrict these to the
8360:
Smallest non-modular (and hence non-distributive) lattice N
5032:
4247:{\displaystyle {\text{ for all }}a\in \varnothing ,x\leq a}
10805:
for an alternative meaning), if it can be equipped with a
10201:
called complementation, introduces an analogue of logical
7415:
is a lattice homomorphism from a lattice to itself, and a
3936:{\displaystyle 0\leq x\leq 1\;{\text{ for every }}x\in L.}
10444:{\displaystyle \left\{x_{0},x_{1},\ldots ,x_{n}\right\},}
5692:
is the bottom element; there is no top (see Pic. 5).
3296:
2836:– they are undefined if their value is not in the subset
12983:
Lattice Theory: First concepts and distributive lattices
12878:. Continuous Lattices. Vol. 871. pp. 159–208.
12721:
7304:{\displaystyle f\left(0_{L}\right)=0_{M},{\text{ and }}}
6273:
partially ordered by divisibility is a lattice, the set
4960:{\textstyle 0=\bigwedge L=a_{1}\land \cdots \land a_{n}}
2769:{\displaystyle a_{1}\wedge b_{1}\leq a_{2}\wedge b_{2}.}
12330:
9755:. Both concepts can be applied to lattices as follows:
9296:
A lattice is modular if and only if it does not have a
11397:
10534:, or one less than its number of elements. A chain is
10241:
10148:
10119:
10083:
A complemented lattice that is also distributive is a
9991:
9471:
Another equivalent (for graded lattices) condition is
9461:{\displaystyle r(x)+r(y)\geq r(x\wedge y)+r(x\vee y).}
7377:
with respect to the associated ordering relation; see
4912:
4851:
4338:
4332:
and the meet of the empty set is the greatest element
4306:
13131:
Textbook with numerous attributions in the footnotes.
12551:
12207:
12173:
12153:
12133:
12110:
12090:
12063:
12028:
12002:
11976:
11952:
11932:
11912:
11888:
11854:
11811:
11779:
11749:
11725:
11701:
11656:
11630:
11607:
11587:
11552:
11509:
11477:
11450:
11428:
11408:
11377:
11351:
11316:
11290:
11267:
11225:
11199:
11173:
11138:
11118:
11089:
11066:
11042:
10985:
10962:
10940:
10914:
10870:
10848:
10814:
10765:
10733:
10713:
10684:
10661:
10641:
10604:
10571:
10544:
10457:
10385:
10358:
10331:
10287:
10267:
10217:
10184:
10093:
10062:
10042:
10022:
9989:
9943:
9887:
9863:
9843:
9823:
9803:
9786:
9709:
9689:
9663:
9643:
9617:
9588:
9568:
9548:
9525:
9505:
9485:
9385:
9360:
9231:
9205:
9199:
This condition is equivalent to the following axiom:
9110:
9069:
9025:
8893:
8871:
8851:
8771:
8749:
8729:
8684:
8563:
8486:
8440:
8396:
8370:
8324:
8301:
8181:
8158:
8138:
8106:
8074:
8051:
8027:
8007:
7982:
7962:
7939:
7913:
7887:
7855:
7835:
7815:
7795:
7772:
7752:
7732:
7698:
7638:
7614:
7554:
7513:
7491:
7456:
7434:
7316:
7258:
7238:
7218:
7198:
7166:
7078:
6988:
6954:
6922:
6897:{\displaystyle \left(M,\vee _{M},\wedge _{M}\right),}
6851:
6795:
6786:
6718:
6699:
6617:
6554:
6535:
6453:
6433:
6360:
6317:
6279:
6235:
6203:
6177:
6145:
6125:
6079:
6050:
6030:
6010:
5964:
5923:
5903:
5883:
5801:
5781:
5759:
5739:
5713:
5666:
5623:
5573:
5532:
5505:
5482:
5459:
5436:
5405:
5381:
5357:
5334:
5290:
5224:
5176:. The name "lattice" is suggested by the form of the
5137:
5087:
5067:
4973:
4813:
4703:
4596:
4576:
4503:
4430:
4408:
4388:
4368:
4260:
4217:
4193:
4159:
4128:
4106:
4086:
4063:
4039:
4001:
3951:
3899:
3877:
3856:
3814:
3789:
3732:
3712:
3609:
3574:
3525:
3472:
3452:
3432:
3391:
3350:
3327:
3307:
3255:
3223:
3189:
3158:
3108:
3065:
3029:
3009:
2989:
2969:
2946:
2908:
2865:
2842:
2809:
2713:
2647:
2607:
2567:
2542:
2520:
2494:
2460:
2418:
2378:
1857:
1831:
1805:
1770:
1746:
1652:
1620:
1593:
1549:
1505:
1464:
1393:
1322:
1266:
1236:
1210:
12963:
Elementary texts recommended for those with limited
12928:
12629:
12315:
Pages displaying wikidata descriptions as a fallback
12306:
Pages displaying wikidata descriptions as a fallback
12291:
Pages displaying wikidata descriptions as a fallback
10113:
In the case that the complement is unique, we write
10012:
does not have a complement. In the bounded lattice N
9765:
is a complete lattice that is continuous as a poset.
9283:{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge c.}
6838:{\displaystyle \left(L,\vee _{L},\wedge _{L}\right)}
5022:{\displaystyle L=\left\{a_{1},\ldots ,a_{n}\right\}}
4899:{\textstyle 1=\bigvee L=a_{1}\lor \cdots \lor a_{n}}
12606:{\displaystyle a=a\vee (a\wedge (a\vee a))=a\vee a}
9774:
is a complete lattice that is algebraic as a poset.
2700:{\displaystyle a_{1}\vee b_{1}\leq a_{2}\vee b_{2}}
13060:
13033:, 3rd ed. Vol. 25 of AMS Colloquium Publications.
12605:
12213:
12188:
12159:
12139:
12119:
12096:
12069:
12049:
12014:
11988:
11958:
11938:
11918:
11894:
11866:
11840:
11797:
11755:
11731:
11707:
11688:
11642:
11616:
11593:
11573:
11538:
11495:
11456:
11437:
11414:
11383:
11360:
11325:
11302:
11273:
11240:
11211:
11185:
11159:
11124:
11104:
11075:
11048:
11021:
10971:
10946:
10926:
10900:
10856:
10834:
10783:
10739:
10719:
10699:
10670:
10647:
10625:
10590:
10557:
10518:
10443:
10371:
10344:
10316:
10305:
10273:
10253:
10223:
10193:
10166:
10134:
10102:
10068:
10048:
10028:
10004:
9983:with its usual ordering is a bounded lattice, and
9975:
9927:
9869:
9849:
9829:
9809:
9715:
9695:
9672:
9649:
9629:
9603:
9574:
9554:
9534:
9511:
9491:
9460:
9369:
9282:
9217:
9185:
9096:
9049:
8956:
8877:
8857:
8834:
8755:
8735:
8711:
8626:
8549:
8470:
8426:
8382:
8333:
8310:
8208:
8167:
8144:
8124:
8092:
8060:
8033:
8013:
7991:
7968:
7948:
7925:
7899:
7873:
7841:
7821:
7801:
7781:
7758:
7738:
7704:
7684:
7620:
7600:
7526:
7499:
7469:
7442:
7356:
7303:
7244:
7224:
7204:
7172:
7150:
7065:
6975:
6940:
6896:
6837:
6770:
6705:
6686:
6603:
6541:
6522:
6439:
6402:
6344:
6303:
6265:
6218:
6189:
6151:
6131:
6088:
6065:
6036:
6016:
5973:
5950:
5909:
5889:
5807:
5787:
5768:
5745:
5725:
5684:
5652:
5609:
5541:
5514:
5488:
5468:
5445:
5411:
5387:
5363:
5343:
5300:
5257:
5164:
5096:
5073:
5021:
4959:
4898:
4834:
4799:
4689:
4582:
4562:
4489:
4417:
4394:
4374:
4353:
4324:
4292:
4246:
4199:
4177:
4146:
4115:
4092:
4072:
4045:
4025:
3987:
3935:
3883:
3862:
3820:
3798:
3741:
3718:
3695:
3595:
3560:
3512:
3458:
3438:
3418:
3374:
3336:
3313:
3273:
3241:
3207:
3176:
3138:
3095:
3047:
3015:
2995:
2975:
2952:
2932:
2871:
2851:
2824:
2768:
2699:
2633:
2593:
2550:
2528:
2506:
2472:
2442:
2396:
1872:
1843:
1817:
1791:
1752:
1693:
1637:
1605:
1578:
1534:
1490:
1449:
1378:
1307:
1251:
1222:
12293:(generalization to non-commutative join and meet)
11333:with the semilattice operation given by ordinary
8652:Smallest non-distributive (but modular) lattice M
13992:
13246:
13207:
12770:, Cambridge University Press, pp. 103–104,
10087:. For a distributive lattice, the complement of
5037:Lattices have some connections to the family of
1469:
13208:Garrett Birkhoff (1967). James C. Abbot (ed.).
13108:
12273: – Order whose elements are all comparable
10231:of a Heyting algebra has, on the other hand, a
5195:Lattice of integer divisors of 60, ordered by "
12993:
12861:
12826:
12789:
12746:
12730:
12711:
11371:gave a construction based on polynomials over
10261:The pseudo-complement is the greatest element
9726:
6771:{\displaystyle 0^{\prime }=f(0)=f(u\wedge v).}
5815:is top. Pic. 2 shows a finite sublattice.
3561:{\displaystyle a\leq b{\text{ if }}b=a\vee b,}
13310:
7192:(usually called just "lattice homomorphism")
5653:{\displaystyle a\leq c{\text{ and }}b\leq d.}
2251:in which every pair of elements has a unique
2210:
1734:in the "Antisymmetric" column, respectively.
126:
13109:Štĕpánka Bilová (2001). Eduard Fuchs (ed.).
11841:{\displaystyle x\leq a{\text{ or }}x\leq b.}
9970:
9944:
7679:
7673:
7667:
7652:
7595:
7589:
7583:
7568:
7373:Any homomorphism of lattices is necessarily
6397:
6361:
6336:
6318:
6298:
6280:
6260:
6236:
6171:A discrete poset, meaning a poset such that
5249:
5225:
5156:
5138:
2960:and two binary, commutative and associative
2431:
2419:
12827:Philip Whitman (1942). "Free Lattices II".
8264:
6785:between two lattices flows easily from the
6604:{\displaystyle 1^{\prime }=f(1)=f(u\vee v)}
5861:
2363:
13968:Positive cone of a partially ordered group
13317:
13303:
12790:Philip Whitman (1941). "Free Lattices I".
7357:{\displaystyle f\left(1_{L}\right)=1_{M}.}
3915:
2217:
2203:
133:
119:
13194:
13148:
12682:
12613:and dually". Birkhoff attributes this to
10850:
10828:
8218:
7516:
7493:
7459:
7436:
7252:should also have the following property:
6789:algebraic definition. Given two lattices
5291:
5284:Lattice of positive integers, ordered by
2890:
2547:
2543:
2525:
2521:
89:Learn how and when to remove this message
13951:Positive cone of an ordered vector space
13174:
13122:
12873:
12614:
12538:
8644:
8352:
6419:
6415:
6111:
5996:
5869:
5567:of the natural numbers, ordered so that
5453:the collection of all finite subsets of
5033:Connection to other algebraic structures
2325:include lattices, which in turn include
2235:is an abstract structure studied in the
52:This article includes a list of general
13134:
13058:
12908:
12896:
12762:
12522:
11689:{\displaystyle \bigvee _{i\in I}a_{i},}
4807:which is consistent with the fact that
3382:into a lattice in the algebraic sense.
2488:(i.e. greatest lower bound, denoted by
2263:(also called a greatest lower bound or
14:
13993:
13067:(Second ed.). Basel: Birkhäuser.
4354:{\textstyle \bigwedge \varnothing =1.}
3297:Connection between the two definitions
2302:Lattices can also be characterized as
27:Set whose pairs have minima and maxima
13298:
13247:
12653:
7849:such that for every pair of elements
5846:, for which the compacts only form a
4835:{\displaystyle A\cup \varnothing =A.}
3988:{\displaystyle (L,\vee ,\wedge ,0,1)}
3763:is a lattice that additionally has a
12434:Algebraizations of first-order logic
12334:
12331:Applications that use lattice theory
11926:have a bottom element 0. An element
11539:{\displaystyle x=a{\text{ or }}x=b.}
8678:laws holds for every three elements
7685:{\displaystyle f^{-1}\{f(1)\}=\{1\}}
7601:{\displaystyle f^{-1}\{f(0)\}=\{0\}}
4325:{\textstyle \bigvee \varnothing =0,}
3703:and dually for the other direction.
2454:(i.e. least upper bound, denoted by
2255:(also called a least upper bound or
1737:All definitions tacitly require the
38:
13203:On applications of lattice theory:
13171:Summary of the history of lattices.
13112:Lattice theory — its birth and life
13088:Mathematical Association of America
11398:Important lattice-theoretic notions
10864:, compatible with the ordering (so
10835:{\displaystyle r:L\to \mathbb {N} }
8471:{\displaystyle (b\vee a)\wedge c=c}
8427:{\displaystyle b\vee (a\wedge c)=b}
7409:lattice homomorphism. Similarly, a
3139:{\displaystyle a\wedge (a\vee b)=a}
3096:{\displaystyle a\vee (a\wedge b)=a}
2341:as well as algebraic descriptions.
1252:{\displaystyle S\neq \varnothing :}
24:
13478:Properties & Types (
13104:On the history of lattice theory:
13001:Introduction to Lattices and Order
12924:Monographs available free online:
12768:Enumerative Combinatorics (vol. 1)
12439:Semantics of programming languages
11739:-irreducible). The dual notion is
10242:
10149:
10120:
9787:Complements and pseudo-complements
6724:
6679:
6666:
6653:
6560:
6515:
6502:
6489:
6403:{\displaystyle \{1,2,3,12,18,36\}}
3878:
3815:
3754:
3419:{\displaystyle (L,\vee ,\wedge ),}
2443:{\displaystyle \{a,b\}\subseteq L}
2283:. Another example is given by the
58:it lacks sufficient corresponding
25:
14017:
13934:Positive cone of an ordered field
13221:
9339:
9050:{\displaystyle (L,\vee ,\wedge )}
8478:, so the modular law is violated.
8344:
7395:Given the standard definition of
5823:
5553:, is a lattice (see Pic. 3).
5351:the collection of all subsets of
4820:
4752:
4716:
4645:
4609:
4342:
4310:
4272:
4229:
4026:{\displaystyle (L,\vee ,\wedge )}
3375:{\displaystyle (L,\vee ,\wedge )}
2933:{\displaystyle (L,\vee ,\wedge )}
2408:if it is both a join- and a meet-
1243:
13788:Ordered topological vector space
13324:
13196:10.24355/dbbs.084-200908140200-2
12339:
11337:. The free semilattice has the
11250:
10754:
8999:completely distributive lattices
8979:
5312:
5274:
5204:
5185:
5121:
5057:without the distributive axiom.
2348:that studies lattices is called
2291:, for which the supremum is the
2275:, for which the supremum is the
1722:
1709:
1638:{\displaystyle {\text{not }}aRa}
1143:
1133:
1103:
1093:
1071:
1061:
1026:
1004:
994:
959:
927:
917:
892:
860:
845:
825:
793:
783:
773:
753:
721:
701:
691:
681:
649:
629:
585:
565:
555:
518:
493:
483:
451:
426:
389:
359:
327:
265:
230:
43:
13118:. Prometheus. pp. 250–257.
12945:Jipsen, Peter, and Henry Rose,
12867:
12820:
12783:
12687:(2nd ed.). New York City:
12256: – Concept in order theory
10749:Jordan–Dedekind chain condition
10317:Jordan–Dedekind chain condition
10005:{\displaystyle {\tfrac {1}{2}}}
9909:
9903:
8232:
5610:{\displaystyle (a,b)\leq (c,d)}
5423:and the infimum is provided by
5399:to obtain a lattice bounded by
5039:group-like algebraic structures
3426:one can define a partial order
2634:{\displaystyle b_{1}\leq b_{2}}
2594:{\displaystyle a_{1}\leq a_{2}}
2412:, i.e. each two-element subset
2271:of a set, partially ordered by
12931:A Course in Universal Algebra.
12756:
12740:
12705:
12676:
12663:UCLA Department of Mathematics
12647:
12623:
12588:
12585:
12573:
12564:
12532:
12354:format but may read better as
12262: – Concept in mathematics
11422:be an element of some lattice
11010:
11004:
10995:
10989:
10895:
10889:
10880:
10874:
10824:
10778:
10766:
10626:{\displaystyle 1\leq i\leq n.}
9881:of each other if and only if:
9452:
9440:
9431:
9419:
9410:
9404:
9395:
9389:
9268:
9256:
9250:
9238:
9171:
9162:
9150:
9147:
9141:
9129:
9123:
9111:
9104:the following identity holds:
9044:
9026:
9003:distributivity in order theory
8948:
8936:
8930:
8918:
8912:
8900:
8826:
8814:
8808:
8796:
8790:
8778:
8618:
8606:
8600:
8588:
8582:
8570:
8541:
8529:
8523:
8511:
8505:
8493:
8453:
8441:
8415:
8403:
8277:conditionally complete lattice
7715:
7664:
7658:
7580:
7574:
7142:
7136:
7120:
7114:
7052:
7046:
7030:
7024:
6932:
6762:
6750:
6741:
6735:
6642:
6636:
6627:
6621:
6598:
6586:
6577:
6571:
6478:
6472:
6463:
6457:
5679:
5667:
5604:
5592:
5586:
5574:
4719:
4707:
4612:
4600:
4519:
4507:
4446:
4434:
4053:(the lattice's bottom) is the
4020:
4002:
3982:
3952:
3672:
3660:
3654:
3642:
3410:
3392:
3369:
3351:
3268:
3256:
3236:
3224:
3217:These axioms assert that both
3127:
3115:
3084:
3072:
2927:
2909:
2796:approach to lattices, and for
2391:
2379:
2267:). An example is given by the
1730:in the "Symmetric" column and
1666:
1411:
1356:
1285:
13:
1:
13745:Series-parallel partial order
13271:, course notes, revised 2017.
13210:What can Lattices do for you?
13035:American Mathematical Society
12918:
12632:A Course in Universal Algebra
12240:refer to particular kinds of
12084:if for every nonzero element
11798:{\displaystyle x\leq a\vee b}
9008:
8093:{\displaystyle x\leq z\leq y}
7527:{\displaystyle \mathbb {L} '}
7470:{\displaystyle \mathbb {L} '}
7212:between two bounded lattices
2825:{\displaystyle H\subseteq L,}
2803:Given a subset of a lattice,
2355:
1731:
1718:
1128:
1123:
1118:
1113:
1088:
1056:
1051:
1046:
1041:
1036:
1021:
989:
984:
979:
974:
969:
954:
942:
937:
912:
907:
902:
887:
875:
870:
855:
840:
835:
820:
808:
803:
768:
763:
748:
736:
731:
716:
711:
676:
664:
659:
644:
639:
624:
619:
614:
600:
595:
580:
575:
550:
545:
533:
528:
513:
508:
503:
478:
466:
461:
446:
441:
436:
421:
416:
404:
399:
384:
379:
374:
369:
354:
342:
337:
322:
317:
312:
307:
302:
297:
280:
275:
260:
255:
250:
245:
240:
13424:Cantor's isomorphism theorem
13045:Algebraic Theory of Lattices
12874:Hoffmann, Rudolf-E. (1981).
12747:Davey & Priestley (2002)
12731:Davey & Priestley (2002)
12712:Davey & Priestley (2002)
12685:Set Theory and Metric Spaces
12050:{\displaystyle 0<y<x.}
11996:and there exists no element
11281:may be used to generate the
11160:{\displaystyle y>z>x.}
11022:{\displaystyle r(y)=r(x)+1.}
10901:{\displaystyle r(x)<r(y)}
10857:{\displaystyle \mathbb {Z} }
10306:{\displaystyle x\wedge y=0.}
8279:is a lattice in which every
7500:{\displaystyle \mathbb {L} }
7443:{\displaystyle \mathbb {L} }
7190:bounded-lattice homomorphism
6781:The appropriate notion of a
5258:{\displaystyle \{1,2,3,4\},}
5029:is the set of all elements.
7:
13464:Szpilrajn extension theorem
13439:Hausdorff maximal principle
13414:Boolean prime ideal theorem
13235:Encyclopedia of Mathematics
12911:, p. 234, after Def.1.
12449:Ontology (computer science)
12289: – Algebraic Structure
12247:
11848:Again some authors require
11717:completely join irreducible
11112:but there does not exist a
10110:when it exists, is unique.
9976:{\displaystyle \{0,1/2,1\}}
9727:Continuity and algebraicity
9097:{\displaystyle a,b,c\in L,}
8712:{\displaystyle a,b,c\in L,}
8209:{\displaystyle x,y,z\in L.}
7399:as invertible morphisms, a
6266:{\displaystyle \{1,2,3,6\}}
6159:have no common upper bound.
5114:
4178:{\displaystyle a\wedge 1=a}
3274:{\displaystyle (L,\wedge )}
3208:{\displaystyle a\wedge a=a}
2859:The resulting structure on
2785:Completeness (order theory)
2529:{\displaystyle \,\wedge \,}
10:
14022:
13810:Topological vector lattice
13187:Braunschweiger Festschrift
13123:Birkhoff, Garrett (1948).
13043:and Crawley, Peter, 1973.
13006:Cambridge University Press
12899:, p. 246, Exercise 3.
12862:Davey & Priestley 2002
12683:Kaplansky, Irving (1972).
11254:
10036:has two complements, viz.
9790:
9604:{\displaystyle x\wedge y,}
9343:
9012:
8663:
8268:
8236:
8222:
7392:is also order-preserving.
6345:{\displaystyle \{2,3,6\}.}
5165:{\displaystyle \{x,y,z\},}
2832:meet and join restrict to
29:
13840:
13768:
13707:
13477:
13406:
13355:
13332:
13276:"Lattice Theory Homepage"
13159:10.1007/s11229-009-9667-9
12971:Donnellan, Thomas, 1968.
12499:Regular language learning
12266:Orthocomplemented lattice
11574:{\displaystyle a,b\in L.}
11496:{\displaystyle x=a\vee b}
10784:{\displaystyle (L,\leq )}
9781:Scott information systems
7900:{\displaystyle a\wedge b}
7388:is a homomorphism if its
7379:Limit preserving function
6976:{\displaystyle a,b\in L:}
6304:{\displaystyle \{1,2,3\}}
6044:have common upper bounds
5917:have common lower bounds
4147:{\displaystyle a\vee 0=a}
3596:{\displaystyle a,b\in L.}
3242:{\displaystyle (L,\vee )}
3177:{\displaystyle a\vee a=a}
2551:{\displaystyle \,\vee \,}
2514:). This definition makes
2507:{\displaystyle a\wedge b}
2397:{\displaystyle (L,\leq )}
13419:Cantor–Bernstein theorem
13059:Grätzer, George (2003).
12509:
12189:{\displaystyle a\leq x;}
11241:{\displaystyle x\neq y.}
10707:all maximal chains from
10174:The corresponding unary
9370:{\displaystyle r\colon }
9330:set of first-order terms
8265:Conditional completeness
8125:{\displaystyle x,y\in M}
7874:{\displaystyle a,b\in M}
6941:{\displaystyle f:L\to M}
6096:but none of them is the
5981:but none of them is the
5951:{\displaystyle 0,d,g,h,}
5862:Examples of non-lattices
5301:{\displaystyle \,\leq ,}
4116:{\displaystyle \wedge .}
3783:element, and denoted by
3742:{\displaystyle \wedge .}
3337:{\displaystyle \wedge .}
3048:{\displaystyle a,b\in L}
2364:As partially ordered set
30:Not to be confused with
13963:Partially ordered group
13783:Specialization preorder
13268:Notes on Lattice Theory
13230:"Lattice-ordered group"
12459:Formal concept analysis
12418:Abstract interpretation
12363:converting this article
12320:Young–Fibonacci lattice
12236:and the dual notion of
11867:{\displaystyle x\neq 0}
11756:{\displaystyle \wedge }
11643:{\displaystyle x\neq 0}
11212:{\displaystyle x\leq y}
11105:{\displaystyle y>x,}
10700:{\displaystyle x<y,}
10591:{\displaystyle x_{i-1}}
10167:{\textstyle \lnot y=x.}
9716:{\displaystyle \wedge }
9630:{\displaystyle x\vee y}
9218:{\displaystyle a\leq c}
8858:{\displaystyle \wedge }
8756:{\displaystyle \wedge }
8383:{\displaystyle b\leq c}
8285:that has an upper bound
7926:{\displaystyle a\vee b}
6190:{\displaystyle x\leq y}
5726:{\displaystyle a\leq b}
5707:as the order relation:
5697:greatest common divisor
4057:for the join operation
2996:{\displaystyle \wedge }
2798:formal concept analysis
2473:{\displaystyle a\vee b}
2297:greatest common divisor
2295:and the infimum is the
2287:, partially ordered by
2279:and the infimum is the
73:more precise citations.
13449:Kruskal's tree theorem
13444:Knaster–Tarski theorem
13434:Dushnik–Miller theorem
13063:General Lattice Theory
12689:AMS Chelsea Publishing
12607:
12215:
12190:
12161:
12141:
12121:
12098:
12071:
12051:
12016:
12015:{\displaystyle y\in L}
11990:
11989:{\displaystyle 0<x}
11960:
11940:
11920:
11896:
11868:
11842:
11799:
11757:
11733:
11709:
11690:
11644:
11618:
11595:
11575:
11540:
11497:
11458:
11439:
11416:
11385:
11362:
11327:
11304:
11275:
11242:
11213:
11187:
11186:{\displaystyle y>x}
11161:
11126:
11106:
11077:
11050:
11023:
10973:
10948:
10928:
10927:{\displaystyle x<y}
10902:
10858:
10836:
10785:
10741:
10721:
10701:
10672:
10649:
10627:
10592:
10559:
10520:
10445:
10373:
10346:
10307:
10275:
10255:
10225:
10195:
10168:
10136:
10135:{\textstyle \lnot x=y}
10104:
10070:
10050:
10030:
10006:
9977:
9929:
9871:
9851:
9831:
9811:
9717:
9697:
9674:
9651:
9631:
9605:
9576:
9556:
9536:
9513:
9493:
9462:
9371:
9284:
9219:
9187:
9098:
9051:
8958:
8879:
8859:
8836:
8757:
8737:
8713:
8657:
8634:
8628:
8551:
8472:
8428:
8384:
8335:
8312:
8219:Properties of lattices
8210:
8169:
8146:
8126:
8094:
8062:
8035:
8015:
7993:
7970:
7950:
7927:
7901:
7875:
7843:
7823:
7803:
7783:
7760:
7740:
7706:
7686:
7622:
7602:
7528:
7501:
7485:. A homomorphism from
7471:
7444:
7358:
7305:
7246:
7226:
7206:
7184:of the two underlying
7174:
7152:
7067:
6977:
6942:
6898:
6839:
6778:
6772:
6707:
6688:
6605:
6543:
6524:
6441:
6404:
6346:
6305:
6267:
6220:
6191:
6160:
6153:
6133:
6101:
6090:
6067:
6038:
6018:
5986:
5975:
5952:
5911:
5891:
5809:
5789:
5770:
5747:
5727:
5686:
5654:
5611:
5543:
5522:the collection of all
5516:
5490:
5470:
5447:
5413:
5389:
5365:
5345:
5302:
5259:
5166:
5098:
5075:
5023:
4961:
4900:
4836:
4801:
4691:
4584:
4564:
4491:
4419:
4396:
4376:
4355:
4326:
4294:
4248:
4201:
4179:
4148:
4117:
4094:
4074:
4073:{\displaystyle \vee ,}
4047:
4027:
3989:
3937:
3885:
3864:
3822:
3800:
3743:
3720:
3697:
3597:
3562:
3514:
3460:
3440:
3420:
3376:
3338:
3315:
3275:
3243:
3209:
3178:
3140:
3097:
3049:
3017:
2997:
2977:
2954:
2940:, consisting of a set
2934:
2891:As algebraic structure
2873:
2853:
2826:
2770:
2701:
2635:
2595:
2552:
2530:
2508:
2474:
2444:
2398:
1874:
1845:
1819:
1793:
1792:{\displaystyle a,b,c,}
1754:
1695:
1639:
1607:
1580:
1536:
1492:
1451:
1380:
1309:
1253:
1224:
13022:Advanced monographs:
12965:mathematical maturity
12948:Varieties of Lattices
12829:Annals of Mathematics
12793:Annals of Mathematics
12654:Baker, Kirby (2010).
12608:
12216:
12191:
12162:
12142:
12127:there exists an atom
12122:
12099:
12072:
12052:
12017:
11991:
11961:
11941:
11921:
11897:
11878:. The dual notion is
11876:completely join prime
11869:
11843:
11800:
11758:
11734:
11732:{\displaystyle \vee }
11710:
11691:
11645:
11624:some authors require
11619:
11601:has a bottom element
11596:
11576:
11541:
11498:
11459:
11440:
11417:
11386:
11363:
11328:
11305:
11276:
11243:
11214:
11188:
11162:
11127:
11107:
11078:
11051:
11024:
10974:
10949:
10934:) such that whenever
10929:
10903:
10859:
10837:
10786:
10742:
10722:
10702:
10673:
10650:
10628:
10593:
10560:
10558:{\displaystyle x_{i}}
10521:
10446:
10374:
10372:{\displaystyle x_{n}}
10347:
10345:{\displaystyle x_{0}}
10308:
10276:
10256:
10254:{\textstyle \lnot x.}
10226:
10205:into lattice theory.
10196:
10169:
10137:
10105:
10071:
10051:
10031:
10007:
9978:
9930:
9872:
9852:
9832:
9812:
9743:of elements that are
9718:
9698:
9696:{\displaystyle \vee }
9675:
9652:
9632:
9606:
9577:
9557:
9537:
9514:
9494:
9463:
9372:
9320:, and the lattice of
9285:
9220:
9188:
9099:
9063:if, for all elements
9052:
8959:
8880:
8878:{\displaystyle \vee }
8860:
8837:
8758:
8738:
8736:{\displaystyle \vee }
8714:
8648:
8629:
8557:but satisfy its dual
8552:
8473:
8429:
8385:
8356:
8336:
8313:
8223:Further information:
8211:
8170:
8147:
8127:
8095:
8063:
8036:
8016:
7994:
7971:
7951:
7928:
7902:
7876:
7844:
7824:
7804:
7784:
7761:
7741:
7707:
7687:
7623:
7603:
7529:
7502:
7477:be two lattices with
7472:
7445:
7359:
7306:
7247:
7227:
7207:
7175:
7153:
7068:
6978:
6943:
6899:
6840:
6773:
6708:
6706:{\displaystyle \neq }
6689:
6606:
6544:
6542:{\displaystyle \neq }
6525:
6442:
6423:
6416:Morphisms of lattices
6405:
6347:
6306:
6268:
6221:
6192:
6154:
6134:
6115:
6091:
6068:
6039:
6019:
6000:
5976:
5953:
5912:
5892:
5873:
5810:
5790:
5771:
5748:
5728:
5701:least common multiple
5687:
5685:{\displaystyle (0,0)}
5655:
5612:
5544:
5517:
5491:
5471:
5448:
5414:
5395:) can be ordered via
5390:
5366:
5346:
5303:
5260:
5167:
5099:
5076:
5024:
4962:
4901:
4837:
4802:
4692:
4590:to be the empty set,
4585:
4565:
4492:
4420:
4397:
4377:
4356:
4327:
4295:
4249:
4202:
4180:
4149:
4118:
4095:
4075:
4048:
4028:
3990:
3938:
3918: for every
3886:
3884:{\displaystyle \bot }
3865:
3823:
3821:{\displaystyle \top }
3801:
3744:
3721:
3719:{\displaystyle \vee }
3698:
3598:
3563:
3515:
3461:
3441:
3439:{\displaystyle \leq }
3421:
3377:
3339:
3316:
3314:{\displaystyle \vee }
3276:
3244:
3210:
3179:
3141:
3098:
3050:
3018:
2998:
2978:
2976:{\displaystyle \vee }
2955:
2935:
2874:
2854:
2827:
2771:
2702:
2636:
2596:
2553:
2531:
2509:
2475:
2445:
2399:
2370:partially ordered set
2337:structures all admit
2293:least common multiple
2249:partially ordered set
1875:
1846:
1820:
1794:
1755:
1696:
1640:
1608:
1581:
1537:
1493:
1452:
1381:
1310:
1254:
1225:
1204:Definitions, for all
14006:Algebraic structures
13941:Ordered vector space
12549:
12479:Ordinal optimization
12454:Multiple inheritance
12398:Lattice of subgroups
12205:
12201:if every element of
12171:
12151:
12131:
12108:
12088:
12061:
12026:
12000:
11974:
11950:
11930:
11910:
11886:
11852:
11809:
11777:
11747:
11723:
11699:
11654:
11628:
11605:
11585:
11550:
11507:
11475:
11448:
11426:
11406:
11375:
11349:
11314:
11288:
11265:
11223:
11197:
11171:
11136:
11116:
11087:
11064:
11040:
10983:
10960:
10938:
10912:
10868:
10846:
10812:
10763:
10731:
10711:
10682:
10659:
10639:
10602:
10569:
10542:
10455:
10383:
10356:
10329:
10285:
10265:
10239:
10215:
10182:
10146:
10117:
10091:
10078:complemented lattice
10060:
10040:
10020:
9987:
9941:
9885:
9861:
9841:
9821:
9801:
9707:
9687:
9661:
9641:
9615:
9586:
9566:
9546:
9523:
9503:
9483:
9383:
9358:
9229:
9203:
9108:
9067:
9023:
8968:distributive lattice
8891:
8869:
8849:
8769:
8747:
8727:
8682:
8666:Distributive lattice
8561:
8484:
8438:
8394:
8368:
8322:
8318:its minimum element
8299:
8243:A poset is called a
8179:
8156:
8136:
8104:
8072:
8049:
8025:
8005:
7980:
7960:
7937:
7911:
7885:
7853:
7833:
7813:
7793:
7770:
7750:
7730:
7696:
7636:
7612:
7552:
7511:
7489:
7454:
7432:
7418:lattice automorphism
7412:lattice endomorphism
7314:
7256:
7236:
7216:
7196:
7164:
7076:
6986:
6952:
6920:
6906:lattice homomorphism
6849:
6793:
6716:
6697:
6615:
6552:
6533:
6451:
6431:
6358:
6315:
6277:
6233:
6219:{\displaystyle x=y,}
6201:
6175:
6143:
6123:
6077:
6066:{\displaystyle d,e,}
6048:
6028:
6008:
5983:greatest lower bound
5962:
5921:
5901:
5881:
5799:
5779:
5757:
5737:
5711:
5664:
5621:
5571:
5530:
5503:
5480:
5457:
5434:
5403:
5379:
5355:
5332:
5288:
5222:
5135:
5085:
5065:
4971:
4910:
4849:
4811:
4701:
4594:
4574:
4501:
4428:
4406:
4386:
4366:
4336:
4304:
4258:
4215:
4191:
4157:
4126:
4104:
4084:
4061:
4037:
3999:
3949:
3897:
3875:
3854:
3812:
3787:
3730:
3710:
3607:
3572:
3523:
3470:
3450:
3430:
3389:
3348:
3325:
3305:
3253:
3221:
3187:
3156:
3106:
3063:
3027:
3007:
2987:
2967:
2944:
2906:
2863:
2840:
2807:
2711:
2645:
2605:
2565:
2540:
2518:
2492:
2458:
2416:
2376:
2304:algebraic structures
2114:Group with operators
2057:Complemented lattice
1892:Algebraic structures
1873:{\displaystyle aRc.}
1855:
1829:
1803:
1768:
1744:
1739:homogeneous relation
1650:
1618:
1591:
1547:
1503:
1462:
1391:
1320:
1264:
1234:
1208:
950:Strict partial order
225:Equivalence relation
13779:Alexandrov topology
13725:Lexicographic order
13684:Well-quasi-ordering
12656:"Complete Lattices"
12504:Analogical modeling
12423:Subsumption lattice
11741:meet irreducibility
11303:{\displaystyle FX.}
9346:Semimodular lattice
9334:automated reasoning
7976:is a sublattice of
7402:lattice isomorphism
6119:Non-lattice poset:
6004:Non-lattice poset:
5877:Non-lattice poset:
4263: for all
4220: for all
3627: implies
2901:algebraic structure
2306:satisfying certain
2247:. It consists of a
2168:Composition algebra
1928:Quasigroup and loop
1844:{\displaystyle bRc}
1818:{\displaystyle aRb}
1606:{\displaystyle aRa}
1223:{\displaystyle a,b}
609:Well-quasi-ordering
13760:Transitive closure
13720:Converse/Transpose
13429:Dilworth's theorem
13249:Weisstein, Eric W.
13082:On free lattices:
13041:Robert P. Dilworth
12884:10.1007/BFb0089907
12764:Stanley, Richard P
12603:
12408:Invariant subspace
12393:Pointless topology
12365:, if appropriate.
12325:0,1-simple lattice
12211:
12186:
12157:
12137:
12120:{\displaystyle L,}
12117:
12094:
12067:
12047:
12012:
11986:
11956:
11936:
11916:
11892:
11864:
11838:
11795:
11753:
11729:
11705:
11686:
11672:
11640:
11617:{\displaystyle 0,}
11614:
11591:
11571:
11536:
11493:
11454:
11438:{\displaystyle L.}
11435:
11412:
11381:
11361:{\displaystyle X,}
11358:
11339:universal property
11326:{\displaystyle X,}
11323:
11300:
11271:
11238:
11209:
11183:
11157:
11122:
11102:
11076:{\displaystyle x,}
11073:
11046:
11036:A lattice element
11019:
10972:{\displaystyle x,}
10969:
10944:
10924:
10898:
10854:
10832:
10781:
10737:
10717:
10697:
10671:{\displaystyle y,}
10668:
10645:
10623:
10588:
10555:
10516:
10441:
10369:
10342:
10303:
10271:
10251:
10221:
10194:{\displaystyle L,}
10191:
10164:
10142:and equivalently,
10132:
10103:{\displaystyle x,}
10100:
10066:
10046:
10026:
10002:
10000:
9973:
9925:
9867:
9847:
9827:
9807:
9762:continuous lattice
9713:
9693:
9673:{\displaystyle y.}
9670:
9647:
9627:
9601:
9572:
9552:
9535:{\displaystyle L,}
9532:
9509:
9489:
9458:
9367:
9312:), the lattice of
9280:
9215:
9183:
9094:
9047:
8954:
8875:
8855:
8845:Distributivity of
8832:
8753:
8733:
8723:Distributivity of
8709:
8658:
8635:
8624:
8547:
8468:
8424:
8380:
8334:{\displaystyle 0,}
8331:
8311:{\displaystyle 1,}
8308:
8289:completeness axiom
8206:
8168:{\displaystyle M,}
8165:
8142:
8122:
8090:
8061:{\displaystyle L,}
8058:
8031:
8011:
7992:{\displaystyle L.}
7989:
7966:
7949:{\displaystyle M,}
7946:
7923:
7897:
7871:
7839:
7819:
7799:
7782:{\displaystyle L.}
7779:
7756:
7736:
7702:
7682:
7618:
7598:
7524:
7497:
7467:
7440:
7354:
7301:
7242:
7222:
7202:
7170:
7148:
7063:
6973:
6948:such that for all
6938:
6894:
6835:
6779:
6768:
6703:
6684:
6601:
6539:
6520:
6437:
6400:
6342:
6301:
6263:
6216:
6187:
6161:
6149:
6129:
6102:
6089:{\displaystyle f,}
6086:
6063:
6034:
6014:
5987:
5974:{\displaystyle i,}
5971:
5948:
5907:
5887:
5844:algebraic lattices
5805:
5785:
5769:{\displaystyle b.}
5766:
5743:
5723:
5682:
5650:
5607:
5542:{\displaystyle A,}
5539:
5515:{\displaystyle A,}
5512:
5486:
5469:{\displaystyle A,}
5466:
5446:{\displaystyle A,}
5443:
5427:(see Pic. 1).
5409:
5385:
5361:
5344:{\displaystyle A,}
5341:
5298:
5255:
5162:
5097:{\displaystyle 1,}
5094:
5071:
5019:
4957:
4896:
4832:
4797:
4687:
4580:
4560:
4487:
4418:{\displaystyle L,}
4415:
4392:
4372:
4351:
4322:
4290:
4244:
4197:
4175:
4144:
4113:
4090:
4070:
4043:
4023:
3985:
3933:
3881:
3860:
3818:
3799:{\displaystyle 1,}
3796:
3739:
3716:
3693:
3593:
3558:
3510:
3456:
3436:
3416:
3372:
3334:
3311:
3271:
3239:
3205:
3174:
3136:
3093:
3055:(sometimes called
3045:
3013:
2993:
2973:
2950:
2930:
2869:
2852:{\displaystyle H.}
2849:
2822:
2794:category theoretic
2790:Galois connections
2766:
2697:
2631:
2591:
2548:
2526:
2504:
2470:
2440:
2394:
2239:subdisciplines of
1870:
1841:
1815:
1789:
1750:
1691:
1689:
1635:
1603:
1576:
1574:
1532:
1530:
1488:
1486:
1447:
1445:
1376:
1374:
1305:
1303:
1249:
1220:
1084:Strict total order
13988:
13987:
13946:Partially ordered
13755:Symmetric closure
13740:Reflexive closure
13483:
13215:Table of contents
13176:Dedekind, Richard
13074:978-3-7643-6996-5
13053:978-0-13-022269-5
13047:. Prentice-Hall.
13015:978-0-521-78451-1
12934:Springer-Verlag.
12749:, Theorem 10.21,
12636:Springer-Verlag.
12465:(theory and tool)
12384:
12383:
12214:{\displaystyle L}
12160:{\displaystyle L}
12140:{\displaystyle a}
12097:{\displaystyle x}
12070:{\displaystyle L}
11959:{\displaystyle L}
11939:{\displaystyle x}
11919:{\displaystyle L}
11895:{\displaystyle L}
11824:
11708:{\displaystyle x}
11657:
11594:{\displaystyle L}
11522:
11457:{\displaystyle x}
11415:{\displaystyle x}
11384:{\displaystyle X}
11274:{\displaystyle X}
11125:{\displaystyle z}
11049:{\displaystyle y}
10947:{\displaystyle y}
10740:{\displaystyle y}
10720:{\displaystyle x}
10648:{\displaystyle x}
10635:If for any pair,
10530:of this chain is
10274:{\displaystyle y}
10233:pseudo-complement
10224:{\displaystyle z}
10069:{\displaystyle c}
10049:{\displaystyle b}
10029:{\displaystyle a}
9999:
9907:
9870:{\displaystyle L}
9850:{\displaystyle y}
9830:{\displaystyle x}
9810:{\displaystyle L}
9771:algebraic lattice
9737:continuous posets
9650:{\displaystyle x}
9575:{\displaystyle y}
9555:{\displaystyle x}
9512:{\displaystyle y}
9492:{\displaystyle x}
8662:
8661:
8639:
8638:
8271:Dedekind complete
8175:for all elements
8145:{\displaystyle z}
8043:convex sublattice
8034:{\displaystyle L}
8014:{\displaystyle M}
7969:{\displaystyle M}
7842:{\displaystyle L}
7822:{\displaystyle M}
7809:is a lattice and
7802:{\displaystyle L}
7759:{\displaystyle L}
7739:{\displaystyle L}
7705:{\displaystyle f}
7621:{\displaystyle f}
7299:
7245:{\displaystyle M}
7225:{\displaystyle L}
7205:{\displaystyle f}
7173:{\displaystyle f}
7061:
6440:{\displaystyle f}
6229:Although the set
6165:
6164:
6152:{\displaystyle d}
6132:{\displaystyle c}
6106:
6105:
6098:least upper bound
6037:{\displaystyle c}
6017:{\displaystyle b}
5991:
5990:
5910:{\displaystyle b}
5890:{\displaystyle a}
5808:{\displaystyle 0}
5788:{\displaystyle 1}
5746:{\displaystyle a}
5636:
5558:positive integers
5489:{\displaystyle A}
5412:{\displaystyle A}
5388:{\displaystyle A}
5364:{\displaystyle A}
5109:universal algebra
5074:{\displaystyle 0}
4583:{\displaystyle B}
4395:{\displaystyle B}
4375:{\displaystyle A}
4264:
4221:
4207:of a poset it is
4200:{\displaystyle x}
4093:{\displaystyle 1}
4046:{\displaystyle 0}
3919:
3863:{\displaystyle 0}
3628:
3568:for all elements
3538:
3508:
3485:
3459:{\displaystyle L}
3016:{\displaystyle L}
2953:{\displaystyle L}
2872:{\displaystyle H}
2834:partial functions
2778:It follows by an
2559:binary operations
2344:The sub-field of
2319:universal algebra
2227:
2226:
1886:
1885:
1753:{\displaystyle R}
1704:
1703:
1676:
1624:
1570:
1526:
1482:
1430:
1339:
1017:Strict weak order
203:Total, Semiconnex
99:
98:
91:
16:(Redirected from
14013:
13730:Linear extension
13479:
13459:Mirsky's theorem
13319:
13312:
13305:
13296:
13295:
13283:
13262:
13261:
13243:
13213:
13199:
13198:
13184:
13170:
13152:
13130:
13119:
13117:
13094:Johnstone, P. T.
13078:
13066:
13027:Garrett Birkhoff
13018:
12996:Priestley, H. A.
12985:. W. H. Freeman.
12912:
12906:
12900:
12894:
12888:
12887:
12871:
12865:
12859:
12853:
12852:
12824:
12818:
12817:
12787:
12781:
12780:
12760:
12754:
12751:pp. 238–239
12744:
12738:
12733:, Theorem 4.10,
12728:
12719:
12714:, Exercise 4.1,
12709:
12703:
12702:
12680:
12674:
12673:
12671:
12669:
12660:
12651:
12645:
12627:
12621:
12612:
12610:
12609:
12604:
12536:
12530:
12520:
12474:Information flow
12413:Closure operator
12379:
12376:
12370:
12361:You can help by
12343:
12342:
12335:
12316:
12307:
12297:Eulerian lattice
12292:
12220:
12218:
12217:
12212:
12195:
12193:
12192:
12187:
12166:
12164:
12163:
12158:
12146:
12144:
12143:
12138:
12126:
12124:
12123:
12118:
12103:
12101:
12100:
12095:
12076:
12074:
12073:
12068:
12056:
12054:
12053:
12048:
12021:
12019:
12018:
12013:
11995:
11993:
11992:
11987:
11965:
11963:
11962:
11957:
11945:
11943:
11942:
11937:
11925:
11923:
11922:
11917:
11902:is distributive.
11901:
11899:
11898:
11893:
11873:
11871:
11870:
11865:
11847:
11845:
11844:
11839:
11825:
11822:
11804:
11802:
11801:
11796:
11762:
11760:
11759:
11754:
11738:
11736:
11735:
11730:
11714:
11712:
11711:
11706:
11695:
11693:
11692:
11687:
11682:
11681:
11671:
11649:
11647:
11646:
11641:
11623:
11621:
11620:
11615:
11600:
11598:
11597:
11592:
11580:
11578:
11577:
11572:
11545:
11543:
11542:
11537:
11523:
11520:
11502:
11500:
11499:
11494:
11469:Join irreducible
11463:
11461:
11460:
11455:
11444:
11442:
11441:
11436:
11421:
11419:
11418:
11413:
11393:
11390:
11388:
11387:
11382:
11367:
11365:
11364:
11359:
11332:
11330:
11329:
11324:
11309:
11307:
11306:
11301:
11283:free semilattice
11280:
11278:
11277:
11272:
11247:
11245:
11244:
11239:
11218:
11216:
11215:
11210:
11192:
11190:
11189:
11184:
11166:
11164:
11163:
11158:
11131:
11129:
11128:
11123:
11111:
11109:
11108:
11103:
11082:
11080:
11079:
11074:
11060:another element
11055:
11053:
11052:
11047:
11028:
11026:
11025:
11020:
10978:
10976:
10975:
10970:
10953:
10951:
10950:
10945:
10933:
10931:
10930:
10925:
10907:
10905:
10904:
10899:
10863:
10861:
10860:
10855:
10853:
10841:
10839:
10838:
10833:
10831:
10790:
10788:
10787:
10782:
10746:
10744:
10743:
10738:
10726:
10724:
10723:
10718:
10706:
10704:
10703:
10698:
10677:
10675:
10674:
10669:
10654:
10652:
10651:
10646:
10632:
10630:
10629:
10624:
10597:
10595:
10594:
10589:
10587:
10586:
10564:
10562:
10561:
10556:
10554:
10553:
10525:
10523:
10522:
10517:
10512:
10511:
10493:
10492:
10480:
10479:
10467:
10466:
10450:
10448:
10447:
10442:
10437:
10433:
10432:
10431:
10413:
10412:
10400:
10399:
10378:
10376:
10375:
10370:
10368:
10367:
10351:
10349:
10348:
10343:
10341:
10340:
10312:
10310:
10309:
10304:
10280:
10278:
10277:
10272:
10260:
10258:
10257:
10252:
10230:
10228:
10227:
10222:
10209:Heyting algebras
10200:
10198:
10197:
10192:
10173:
10171:
10170:
10165:
10141:
10139:
10138:
10133:
10109:
10107:
10106:
10101:
10075:
10073:
10072:
10067:
10055:
10053:
10052:
10047:
10035:
10033:
10032:
10027:
10011:
10009:
10008:
10003:
10001:
9992:
9982:
9980:
9979:
9974:
9960:
9934:
9932:
9931:
9926:
9908:
9905:
9876:
9874:
9873:
9868:
9856:
9854:
9853:
9848:
9836:
9834:
9833:
9828:
9816:
9814:
9813:
9808:
9793:pseudocomplement
9749:compact elements
9722:
9720:
9719:
9714:
9702:
9700:
9699:
9694:
9679:
9677:
9676:
9671:
9656:
9654:
9653:
9648:
9636:
9634:
9633:
9628:
9610:
9608:
9607:
9602:
9581:
9579:
9578:
9573:
9561:
9559:
9558:
9553:
9541:
9539:
9538:
9533:
9518:
9516:
9515:
9510:
9498:
9496:
9495:
9490:
9467:
9465:
9464:
9459:
9376:
9374:
9373:
9368:
9322:normal subgroups
9314:two-sided ideals
9289:
9287:
9286:
9281:
9224:
9222:
9221:
9216:
9195:Modular identity
9192:
9190:
9189:
9184:
9103:
9101:
9100:
9095:
9056:
9054:
9053:
9048:
8963:
8961:
8960:
8955:
8884:
8882:
8881:
8876:
8864:
8862:
8861:
8856:
8841:
8839:
8838:
8833:
8762:
8760:
8759:
8754:
8742:
8740:
8739:
8734:
8718:
8716:
8715:
8710:
8641:
8640:
8633:
8631:
8630:
8625:
8556:
8554:
8553:
8548:
8477:
8475:
8474:
8469:
8433:
8431:
8430:
8425:
8389:
8387:
8386:
8381:
8349:
8348:
8340:
8338:
8337:
8332:
8317:
8315:
8314:
8309:
8246:complete lattice
8239:Complete lattice
8215:
8213:
8212:
8207:
8174:
8172:
8171:
8166:
8151:
8149:
8148:
8143:
8131:
8129:
8128:
8123:
8099:
8097:
8096:
8091:
8067:
8065:
8064:
8059:
8040:
8038:
8037:
8032:
8020:
8018:
8017:
8012:
7998:
7996:
7995:
7990:
7975:
7973:
7972:
7967:
7955:
7953:
7952:
7947:
7932:
7930:
7929:
7924:
7906:
7904:
7903:
7898:
7880:
7878:
7877:
7872:
7848:
7846:
7845:
7840:
7828:
7826:
7825:
7820:
7808:
7806:
7805:
7800:
7788:
7786:
7785:
7780:
7765:
7763:
7762:
7757:
7745:
7743:
7742:
7737:
7711:
7709:
7708:
7703:
7691:
7689:
7688:
7683:
7651:
7650:
7627:
7625:
7624:
7619:
7607:
7605:
7604:
7599:
7567:
7566:
7533:
7531:
7530:
7525:
7523:
7519:
7506:
7504:
7503:
7498:
7496:
7476:
7474:
7473:
7468:
7466:
7462:
7449:
7447:
7446:
7441:
7439:
7383:order-preserving
7363:
7361:
7360:
7355:
7350:
7349:
7337:
7333:
7332:
7310:
7308:
7307:
7302:
7300:
7297:
7292:
7291:
7279:
7275:
7274:
7251:
7249:
7248:
7243:
7231:
7229:
7228:
7223:
7211:
7209:
7208:
7203:
7179:
7177:
7176:
7171:
7157:
7155:
7154:
7149:
7132:
7131:
7107:
7103:
7099:
7098:
7072:
7070:
7069:
7064:
7062:
7059:
7042:
7041:
7017:
7013:
7009:
7008:
6982:
6980:
6979:
6974:
6947:
6945:
6944:
6939:
6903:
6901:
6900:
6895:
6890:
6886:
6885:
6884:
6872:
6871:
6844:
6842:
6841:
6836:
6834:
6830:
6829:
6828:
6816:
6815:
6777:
6775:
6774:
6769:
6728:
6727:
6712:
6710:
6709:
6704:
6693:
6691:
6690:
6685:
6683:
6682:
6670:
6669:
6657:
6656:
6610:
6608:
6607:
6602:
6564:
6563:
6548:
6546:
6545:
6540:
6529:
6527:
6526:
6521:
6519:
6518:
6506:
6505:
6493:
6492:
6446:
6444:
6443:
6438:
6409:
6407:
6406:
6401:
6351:
6349:
6348:
6343:
6310:
6308:
6307:
6302:
6272:
6270:
6269:
6264:
6225:
6223:
6222:
6217:
6196:
6194:
6193:
6188:
6158:
6156:
6155:
6150:
6138:
6136:
6135:
6130:
6108:
6107:
6095:
6093:
6092:
6087:
6072:
6070:
6069:
6064:
6043:
6041:
6040:
6035:
6023:
6021:
6020:
6015:
5993:
5992:
5980:
5978:
5977:
5972:
5957:
5955:
5954:
5949:
5916:
5914:
5913:
5908:
5896:
5894:
5893:
5888:
5866:
5865:
5848:join-semilattice
5835:compact elements
5820:complete lattice
5814:
5812:
5811:
5806:
5794:
5792:
5791:
5786:
5775:
5773:
5772:
5767:
5752:
5750:
5749:
5744:
5732:
5730:
5729:
5724:
5691:
5689:
5688:
5683:
5659:
5657:
5656:
5651:
5637:
5634:
5616:
5614:
5613:
5608:
5565:Cartesian square
5548:
5546:
5545:
5540:
5521:
5519:
5518:
5513:
5495:
5493:
5492:
5487:
5475:
5473:
5472:
5467:
5452:
5450:
5449:
5444:
5425:set intersection
5418:
5416:
5415:
5410:
5397:subset inclusion
5394:
5392:
5391:
5386:
5370:
5368:
5367:
5362:
5350:
5348:
5347:
5342:
5316:
5307:
5305:
5304:
5299:
5278:
5264:
5262:
5261:
5256:
5208:
5189:
5171:
5169:
5168:
5163:
5125:
5103:
5101:
5100:
5095:
5080:
5078:
5077:
5072:
5028:
5026:
5025:
5020:
5018:
5014:
5013:
5012:
4994:
4993:
4966:
4964:
4963:
4958:
4956:
4955:
4937:
4936:
4905:
4903:
4902:
4897:
4895:
4894:
4876:
4875:
4841:
4839:
4838:
4833:
4806:
4804:
4803:
4798:
4778:
4774:
4759:
4755:
4740:
4736:
4696:
4694:
4693:
4688:
4671:
4667:
4652:
4648:
4633:
4629:
4589:
4587:
4586:
4581:
4569:
4567:
4566:
4561:
4559:
4555:
4540:
4536:
4496:
4494:
4493:
4488:
4486:
4482:
4467:
4463:
4424:
4422:
4421:
4416:
4401:
4399:
4398:
4393:
4381:
4379:
4378:
4373:
4360:
4358:
4357:
4352:
4331:
4329:
4328:
4323:
4299:
4297:
4296:
4291:
4265:
4262:
4253:
4251:
4250:
4245:
4222:
4219:
4206:
4204:
4203:
4198:
4184:
4182:
4181:
4176:
4153:
4151:
4150:
4145:
4122:
4120:
4119:
4114:
4099:
4097:
4096:
4091:
4079:
4077:
4076:
4071:
4055:identity element
4052:
4050:
4049:
4044:
4032:
4030:
4029:
4024:
3994:
3992:
3991:
3986:
3942:
3940:
3939:
3934:
3920:
3917:
3892:
3890:
3888:
3887:
3882:
3869:
3867:
3866:
3861:
3829:
3827:
3825:
3824:
3819:
3805:
3803:
3802:
3797:
3767:greatest element
3748:
3746:
3745:
3740:
3725:
3723:
3722:
3717:
3702:
3700:
3699:
3694:
3629:
3626:
3602:
3600:
3599:
3594:
3567:
3565:
3564:
3559:
3539:
3536:
3519:
3517:
3516:
3511:
3509:
3506:
3486:
3483:
3465:
3463:
3462:
3457:
3445:
3443:
3442:
3437:
3425:
3423:
3422:
3417:
3381:
3379:
3378:
3373:
3343:
3341:
3340:
3335:
3320:
3318:
3317:
3312:
3280:
3278:
3277:
3272:
3248:
3246:
3245:
3240:
3214:
3212:
3211:
3206:
3183:
3181:
3180:
3175:
3145:
3143:
3142:
3137:
3102:
3100:
3099:
3094:
3054:
3052:
3051:
3046:
3022:
3020:
3019:
3014:
3002:
3000:
2999:
2994:
2982:
2980:
2979:
2974:
2959:
2957:
2956:
2951:
2939:
2937:
2936:
2931:
2885:
2884:
2878:
2876:
2875:
2870:
2858:
2856:
2855:
2850:
2831:
2829:
2828:
2823:
2775:
2773:
2772:
2767:
2762:
2761:
2749:
2748:
2736:
2735:
2723:
2722:
2706:
2704:
2703:
2698:
2696:
2695:
2683:
2682:
2670:
2669:
2657:
2656:
2640:
2638:
2637:
2632:
2630:
2629:
2617:
2616:
2600:
2598:
2597:
2592:
2590:
2589:
2577:
2576:
2557:
2555:
2554:
2549:
2535:
2533:
2532:
2527:
2513:
2511:
2510:
2505:
2479:
2477:
2476:
2471:
2449:
2447:
2446:
2441:
2403:
2401:
2400:
2395:
2346:abstract algebra
2331:Boolean algebras
2245:abstract algebra
2219:
2212:
2205:
1994:Commutative ring
1923:Rack and quandle
1888:
1887:
1879:
1877:
1876:
1871:
1850:
1848:
1847:
1842:
1824:
1822:
1821:
1816:
1798:
1796:
1795:
1790:
1759:
1757:
1756:
1751:
1733:
1729:
1726:
1725:
1720:
1716:
1713:
1712:
1700:
1698:
1697:
1692:
1690:
1677:
1674:
1644:
1642:
1641:
1636:
1625:
1622:
1612:
1610:
1609:
1604:
1585:
1583:
1582:
1577:
1575:
1571:
1568:
1541:
1539:
1538:
1533:
1531:
1527:
1524:
1497:
1495:
1494:
1489:
1487:
1483:
1480:
1456:
1454:
1453:
1448:
1446:
1431:
1428:
1405:
1385:
1383:
1382:
1377:
1375:
1366:
1340:
1337:
1314:
1312:
1311:
1306:
1304:
1289:
1270:
1258:
1256:
1255:
1250:
1229:
1227:
1226:
1221:
1150:
1147:
1146:
1140:
1137:
1136:
1130:
1125:
1120:
1115:
1110:
1107:
1106:
1100:
1097:
1096:
1090:
1078:
1075:
1074:
1068:
1065:
1064:
1058:
1053:
1048:
1043:
1038:
1033:
1030:
1029:
1023:
1011:
1008:
1007:
1001:
998:
997:
991:
986:
981:
976:
971:
966:
963:
962:
956:
944:
939:
934:
931:
930:
924:
921:
920:
914:
909:
904:
899:
896:
895:
889:
883:Meet-semilattice
877:
872:
867:
864:
863:
857:
852:
849:
848:
842:
837:
832:
829:
828:
822:
816:Join-semilattice
810:
805:
800:
797:
796:
790:
787:
786:
780:
777:
776:
770:
765:
760:
757:
756:
750:
738:
733:
728:
725:
724:
718:
713:
708:
705:
704:
698:
695:
694:
688:
685:
684:
678:
666:
661:
656:
653:
652:
646:
641:
636:
633:
632:
626:
621:
616:
611:
602:
597:
592:
589:
588:
582:
577:
572:
569:
568:
562:
559:
558:
552:
547:
535:
530:
525:
522:
521:
515:
510:
505:
500:
497:
496:
490:
487:
486:
480:
468:
463:
458:
455:
454:
448:
443:
438:
433:
430:
429:
423:
418:
406:
401:
396:
393:
392:
386:
381:
376:
371:
366:
363:
362:
356:
344:
339:
334:
331:
330:
324:
319:
314:
309:
304:
299:
294:
292:
282:
277:
272:
269:
268:
262:
257:
252:
247:
242:
237:
234:
233:
227:
145:
144:
135:
128:
121:
114:
112:binary relations
103:
102:
94:
87:
83:
80:
74:
69:this article by
60:inline citations
47:
46:
39:
21:
14021:
14020:
14016:
14015:
14014:
14012:
14011:
14010:
13991:
13990:
13989:
13984:
13980:Young's lattice
13836:
13764:
13703:
13553:Heyting algebra
13501:Boolean algebra
13473:
13454:Laver's theorem
13402:
13368:Boolean algebra
13363:Binary relation
13351:
13328:
13323:
13293:
13228:
13224:
13219:
13212:. Van Nostrand.
13182:
13150:10.1.1.594.8898
13129:(2nd ed.).
13115:
13075:
13016:
12979:Grätzer, George
12921:
12916:
12915:
12907:
12903:
12895:
12891:
12872:
12868:
12860:
12856:
12841:10.2307/1968883
12825:
12821:
12806:10.2307/1969001
12788:
12784:
12778:
12761:
12757:
12745:
12741:
12729:
12722:
12710:
12706:
12699:
12681:
12677:
12667:
12665:
12658:
12652:
12648:
12628:
12624:
12550:
12547:
12546:
12537:
12533:
12521:
12517:
12512:
12494:Knowledge space
12380:
12374:
12371:
12360:
12344:
12340:
12333:
12314:
12305:
12290:
12260:Map of lattices
12250:
12232:The notions of
12206:
12203:
12202:
12172:
12169:
12168:
12152:
12149:
12148:
12132:
12129:
12128:
12109:
12106:
12105:
12089:
12086:
12085:
12062:
12059:
12058:
12027:
12024:
12023:
12001:
11998:
11997:
11975:
11972:
11971:
11951:
11948:
11947:
11931:
11928:
11927:
11911:
11908:
11907:
11887:
11884:
11883:
11853:
11850:
11849:
11821:
11810:
11807:
11806:
11778:
11775:
11774:
11748:
11745:
11744:
11724:
11721:
11720:
11700:
11697:
11696:
11677:
11673:
11661:
11655:
11652:
11651:
11629:
11626:
11625:
11606:
11603:
11602:
11586:
11583:
11582:
11551:
11548:
11547:
11519:
11508:
11505:
11504:
11476:
11473:
11472:
11449:
11446:
11445:
11427:
11424:
11423:
11407:
11404:
11403:
11400:
11391:
11376:
11373:
11372:
11350:
11347:
11346:
11315:
11312:
11311:
11289:
11286:
11285:
11266:
11263:
11262:
11259:
11253:
11224:
11221:
11220:
11198:
11195:
11194:
11172:
11169:
11168:
11137:
11134:
11133:
11117:
11114:
11113:
11088:
11085:
11084:
11065:
11062:
11061:
11041:
11038:
11037:
10984:
10981:
10980:
10961:
10958:
10957:
10939:
10936:
10935:
10913:
10910:
10909:
10869:
10866:
10865:
10849:
10847:
10844:
10843:
10827:
10813:
10810:
10809:
10764:
10761:
10760:
10757:
10732:
10729:
10728:
10712:
10709:
10708:
10683:
10680:
10679:
10660:
10657:
10656:
10640:
10637:
10636:
10603:
10600:
10599:
10576:
10572:
10570:
10567:
10566:
10549:
10545:
10543:
10540:
10539:
10507:
10503:
10488:
10484:
10475:
10471:
10462:
10458:
10456:
10453:
10452:
10427:
10423:
10408:
10404:
10395:
10391:
10390:
10386:
10384:
10381:
10380:
10363:
10359:
10357:
10354:
10353:
10336:
10332:
10330:
10327:
10326:
10319:
10286:
10283:
10282:
10266:
10263:
10262:
10240:
10237:
10236:
10235:, also denoted
10216:
10213:
10212:
10183:
10180:
10179:
10147:
10144:
10143:
10118:
10115:
10114:
10092:
10089:
10088:
10085:Boolean algebra
10061:
10058:
10057:
10041:
10038:
10037:
10021:
10018:
10017:
10015:
9990:
9988:
9985:
9984:
9956:
9942:
9939:
9938:
9906: and
9904:
9886:
9883:
9882:
9862:
9859:
9858:
9842:
9839:
9838:
9822:
9819:
9818:
9802:
9799:
9798:
9795:
9789:
9729:
9708:
9705:
9704:
9688:
9685:
9684:
9662:
9659:
9658:
9642:
9639:
9638:
9616:
9613:
9612:
9587:
9584:
9583:
9567:
9564:
9563:
9547:
9544:
9543:
9524:
9521:
9520:
9504:
9501:
9500:
9484:
9481:
9480:
9384:
9381:
9380:
9359:
9356:
9355:
9348:
9342:
9303:
9300:isomorphic to N
9295:
9293:
9230:
9227:
9226:
9204:
9201:
9200:
9198:
9196:
9109:
9106:
9105:
9068:
9065:
9064:
9061:
9024:
9021:
9020:
9017:
9015:Modular lattice
9011:
8989:
8985:
8982:isomorphic to M
8977:
8973:
8892:
8889:
8888:
8870:
8867:
8866:
8850:
8847:
8846:
8770:
8767:
8766:
8748:
8745:
8744:
8728:
8725:
8724:
8683:
8680:
8679:
8668:
8655:
8562:
8559:
8558:
8485:
8482:
8481:
8479:
8439:
8436:
8435:
8395:
8392:
8391:
8369:
8366:
8365:
8363:
8347:
8323:
8320:
8319:
8300:
8297:
8296:
8273:
8267:
8247:
8241:
8235:
8227:
8225:Map of lattices
8221:
8180:
8177:
8176:
8157:
8154:
8153:
8137:
8134:
8133:
8105:
8102:
8101:
8073:
8070:
8069:
8050:
8047:
8046:
8026:
8023:
8022:
8006:
8003:
8002:
7981:
7978:
7977:
7961:
7958:
7957:
7938:
7935:
7934:
7912:
7909:
7908:
7886:
7883:
7882:
7854:
7851:
7850:
7834:
7831:
7830:
7829:is a subset of
7814:
7811:
7810:
7794:
7791:
7790:
7771:
7768:
7767:
7751:
7748:
7747:
7746:is a subset of
7731:
7728:
7727:
7724:
7718:
7697:
7694:
7693:
7643:
7639:
7637:
7634:
7633:
7613:
7610:
7609:
7559:
7555:
7553:
7550:
7549:
7515:
7514:
7512:
7509:
7508:
7492:
7490:
7487:
7486:
7458:
7457:
7455:
7452:
7451:
7435:
7433:
7430:
7429:
7419:
7413:
7403:
7345:
7341:
7328:
7324:
7320:
7315:
7312:
7311:
7298: and
7296:
7287:
7283:
7270:
7266:
7262:
7257:
7254:
7253:
7237:
7234:
7233:
7217:
7214:
7213:
7197:
7194:
7193:
7165:
7162:
7161:
7127:
7123:
7094:
7090:
7086:
7082:
7077:
7074:
7073:
7060: and
7058:
7037:
7033:
7004:
7000:
6996:
6992:
6987:
6984:
6983:
6953:
6950:
6949:
6921:
6918:
6917:
6880:
6876:
6867:
6863:
6856:
6852:
6850:
6847:
6846:
6824:
6820:
6811:
6807:
6800:
6796:
6794:
6791:
6790:
6723:
6719:
6717:
6714:
6713:
6698:
6695:
6694:
6678:
6674:
6665:
6661:
6652:
6648:
6616:
6613:
6612:
6559:
6555:
6553:
6550:
6549:
6534:
6531:
6530:
6514:
6510:
6501:
6497:
6488:
6484:
6452:
6449:
6448:
6432:
6429:
6428:
6418:
6359:
6356:
6355:
6316:
6313:
6312:
6278:
6275:
6274:
6234:
6231:
6230:
6202:
6199:
6198:
6176:
6173:
6172:
6144:
6141:
6140:
6124:
6121:
6120:
6078:
6075:
6074:
6049:
6046:
6045:
6029:
6026:
6025:
6009:
6006:
6005:
5963:
5960:
5959:
5922:
5919:
5918:
5902:
5899:
5898:
5882:
5879:
5878:
5864:
5800:
5797:
5796:
5780:
5777:
5776:
5758:
5755:
5754:
5738:
5735:
5734:
5712:
5709:
5708:
5665:
5662:
5661:
5635: and
5633:
5622:
5619:
5618:
5572:
5569:
5568:
5531:
5528:
5527:
5504:
5501:
5500:
5481:
5478:
5477:
5458:
5455:
5454:
5435:
5432:
5431:
5404:
5401:
5400:
5380:
5377:
5376:
5356:
5353:
5352:
5333:
5330:
5329:
5323:
5317:
5308:
5289:
5286:
5285:
5279:
5270:
5223:
5220:
5219:
5209:
5200:
5190:
5181:
5136:
5133:
5132:
5126:
5117:
5086:
5083:
5082:
5066:
5063:
5062:
5035:
5008:
5004:
4989:
4985:
4984:
4980:
4972:
4969:
4968:
4951:
4947:
4932:
4928:
4911:
4908:
4907:
4890:
4886:
4871:
4867:
4850:
4847:
4846:
4812:
4809:
4808:
4767:
4763:
4748:
4744:
4729:
4725:
4702:
4699:
4698:
4660:
4656:
4641:
4637:
4622:
4618:
4595:
4592:
4591:
4575:
4572:
4571:
4548:
4544:
4529:
4525:
4502:
4499:
4498:
4475:
4471:
4456:
4452:
4429:
4426:
4425:
4407:
4404:
4403:
4387:
4384:
4383:
4367:
4364:
4363:
4337:
4334:
4333:
4305:
4302:
4301:
4261:
4259:
4256:
4255:
4218:
4216:
4213:
4212:
4192:
4189:
4188:
4158:
4155:
4154:
4127:
4124:
4123:
4105:
4102:
4101:
4085:
4082:
4081:
4062:
4059:
4058:
4038:
4035:
4034:
4000:
3997:
3996:
3950:
3947:
3946:
3916:
3898:
3895:
3894:
3876:
3873:
3872:
3871:
3855:
3852:
3851:
3848:
3842:
3835:
3813:
3810:
3809:
3807:
3788:
3785:
3784:
3781:
3775:
3768:
3761:bounded lattice
3757:
3755:Bounded lattice
3731:
3728:
3727:
3711:
3708:
3707:
3625:
3608:
3605:
3604:
3573:
3570:
3569:
3535:
3524:
3521:
3520:
3505:
3482:
3471:
3468:
3467:
3451:
3448:
3447:
3431:
3428:
3427:
3390:
3387:
3386:
3349:
3346:
3345:
3326:
3323:
3322:
3306:
3303:
3302:
3299:
3254:
3251:
3250:
3222:
3219:
3218:
3188:
3185:
3184:
3157:
3154:
3153:
3150:idempotent laws
3107:
3104:
3103:
3064:
3061:
3060:
3057:absorption laws
3028:
3025:
3024:
3008:
3005:
3004:
2988:
2985:
2984:
2968:
2965:
2964:
2945:
2942:
2941:
2907:
2904:
2903:
2893:
2883:partial lattice
2882:
2881:
2864:
2861:
2860:
2841:
2838:
2837:
2808:
2805:
2804:
2757:
2753:
2744:
2740:
2731:
2727:
2718:
2714:
2712:
2709:
2708:
2691:
2687:
2678:
2674:
2665:
2661:
2652:
2648:
2646:
2643:
2642:
2625:
2621:
2612:
2608:
2606:
2603:
2602:
2585:
2581:
2572:
2568:
2566:
2563:
2562:
2541:
2538:
2537:
2519:
2516:
2515:
2493:
2490:
2489:
2459:
2456:
2455:
2417:
2414:
2413:
2377:
2374:
2373:
2366:
2358:
2339:order-theoretic
2285:natural numbers
2259:) and a unique
2229:
2228:
2223:
2194:
2193:
2192:
2163:Non-associative
2145:
2134:
2133:
2123:
2103:
2092:
2091:
2080:Map of lattices
2076:
2072:Boolean algebra
2067:Heyting algebra
2041:
2030:
2029:
2023:
2004:Integral domain
1968:
1957:
1956:
1950:
1904:
1880:
1856:
1853:
1852:
1830:
1827:
1826:
1804:
1801:
1800:
1769:
1766:
1765:
1745:
1742:
1741:
1735:
1727:
1723:
1714:
1710:
1688:
1687:
1673:
1670:
1669:
1653:
1651:
1648:
1647:
1621:
1619:
1616:
1615:
1592:
1589:
1588:
1573:
1572:
1567:
1564:
1563:
1550:
1548:
1545:
1544:
1529:
1528:
1523:
1520:
1519:
1506:
1504:
1501:
1500:
1485:
1484:
1479:
1476:
1475:
1465:
1463:
1460:
1459:
1444:
1443:
1432:
1427:
1415:
1414:
1406:
1404:
1394:
1392:
1389:
1388:
1373:
1372:
1367:
1365:
1353:
1352:
1341:
1338: and
1336:
1323:
1321:
1318:
1317:
1302:
1301:
1290:
1288:
1282:
1281:
1267:
1265:
1262:
1261:
1235:
1232:
1231:
1209:
1206:
1205:
1148:
1144:
1138:
1134:
1108:
1104:
1098:
1094:
1076:
1072:
1066:
1062:
1031:
1027:
1009:
1005:
999:
995:
964:
960:
932:
928:
922:
918:
897:
893:
865:
861:
850:
846:
830:
826:
798:
794:
788:
784:
778:
774:
758:
754:
726:
722:
706:
702:
696:
692:
686:
682:
654:
650:
634:
630:
607:
590:
586:
570:
566:
560:
556:
541:Prewellordering
523:
519:
498:
494:
488:
484:
456:
452:
431:
427:
394:
390:
364:
360:
332:
328:
290:
287:
270:
266:
235:
231:
223:
215:
139:
106:
95:
84:
78:
75:
65:Please help to
64:
48:
44:
35:
32:Lattice (group)
28:
23:
22:
15:
12:
11:
5:
14019:
14009:
14008:
14003:
14001:Lattice theory
13986:
13985:
13983:
13982:
13977:
13972:
13971:
13970:
13960:
13959:
13958:
13953:
13948:
13938:
13937:
13936:
13926:
13921:
13920:
13919:
13914:
13907:Order morphism
13904:
13903:
13902:
13892:
13887:
13882:
13877:
13872:
13871:
13870:
13860:
13855:
13850:
13844:
13842:
13838:
13837:
13835:
13834:
13833:
13832:
13827:
13825:Locally convex
13822:
13817:
13807:
13805:Order topology
13802:
13801:
13800:
13798:Order topology
13795:
13785:
13775:
13773:
13766:
13765:
13763:
13762:
13757:
13752:
13747:
13742:
13737:
13732:
13727:
13722:
13717:
13711:
13709:
13705:
13704:
13702:
13701:
13691:
13681:
13676:
13671:
13666:
13661:
13656:
13651:
13646:
13645:
13644:
13634:
13629:
13628:
13627:
13622:
13617:
13612:
13610:Chain-complete
13602:
13597:
13596:
13595:
13590:
13585:
13580:
13575:
13565:
13560:
13555:
13550:
13545:
13535:
13530:
13525:
13520:
13515:
13510:
13509:
13508:
13498:
13493:
13487:
13485:
13475:
13474:
13472:
13471:
13466:
13461:
13456:
13451:
13446:
13441:
13436:
13431:
13426:
13421:
13416:
13410:
13408:
13404:
13403:
13401:
13400:
13395:
13390:
13385:
13380:
13375:
13370:
13365:
13359:
13357:
13353:
13352:
13350:
13349:
13344:
13339:
13333:
13330:
13329:
13322:
13321:
13314:
13307:
13299:
13292:
13291:
13279:
13274:Ralph Freese,
13272:
13263:
13244:
13225:
13223:
13222:External links
13220:
13218:
13217:
13201:
13200:
13172:
13132:
13126:Lattice Theory
13120:
13102:
13101:
13091:
13080:
13079:
13073:
13056:
13038:
13031:Lattice Theory
13020:
13019:
13014:
12994:Davey, B. A.;
12987:
12986:
12976:
12973:Lattice Theory
12961:
12960:
12943:
12922:
12920:
12917:
12914:
12913:
12901:
12889:
12866:
12854:
12835:(1): 104–115.
12819:
12800:(1): 325–329.
12782:
12776:
12755:
12739:
12720:
12704:
12697:
12691:. p. 14.
12675:
12646:
12622:
12602:
12599:
12596:
12593:
12590:
12587:
12584:
12581:
12578:
12575:
12572:
12569:
12566:
12563:
12560:
12557:
12554:
12531:
12514:
12513:
12511:
12508:
12507:
12506:
12501:
12496:
12491:
12486:
12481:
12476:
12471:
12466:
12456:
12451:
12446:
12441:
12436:
12431:
12425:
12420:
12415:
12410:
12405:
12403:Spectral space
12400:
12395:
12382:
12381:
12347:
12345:
12338:
12332:
12329:
12328:
12327:
12322:
12317:
12311:Tamari lattice
12308:
12302:Post's lattice
12299:
12294:
12284:
12283:(dual notions)
12274:
12268:
12263:
12257:
12249:
12246:
12227:
12226:
12210:
12196:
12185:
12182:
12179:
12176:
12156:
12136:
12116:
12113:
12093:
12066:
12046:
12043:
12040:
12037:
12034:
12031:
12011:
12008:
12005:
11985:
11982:
11979:
11955:
11935:
11915:
11904:
11903:
11891:
11863:
11860:
11857:
11837:
11834:
11831:
11828:
11823: or
11820:
11817:
11814:
11794:
11791:
11788:
11785:
11782:
11768:
11752:
11728:
11704:
11685:
11680:
11676:
11670:
11667:
11664:
11660:
11639:
11636:
11633:
11613:
11610:
11590:
11570:
11567:
11564:
11561:
11558:
11555:
11535:
11532:
11529:
11526:
11521: or
11518:
11515:
11512:
11492:
11489:
11486:
11483:
11480:
11453:
11434:
11431:
11411:
11399:
11396:
11380:
11357:
11354:
11322:
11319:
11299:
11296:
11293:
11270:
11255:Main article:
11252:
11249:
11237:
11234:
11231:
11228:
11208:
11205:
11202:
11182:
11179:
11176:
11156:
11153:
11150:
11147:
11144:
11141:
11121:
11101:
11098:
11095:
11092:
11072:
11069:
11045:
11018:
11015:
11012:
11009:
11006:
11003:
11000:
10997:
10994:
10991:
10988:
10968:
10965:
10943:
10923:
10920:
10917:
10897:
10894:
10891:
10888:
10885:
10882:
10879:
10876:
10873:
10852:
10830:
10826:
10823:
10820:
10817:
10780:
10777:
10774:
10771:
10768:
10756:
10753:
10736:
10716:
10696:
10693:
10690:
10687:
10667:
10664:
10644:
10622:
10619:
10616:
10613:
10610:
10607:
10585:
10582:
10579:
10575:
10552:
10548:
10515:
10510:
10506:
10502:
10499:
10496:
10491:
10487:
10483:
10478:
10474:
10470:
10465:
10461:
10440:
10436:
10430:
10426:
10422:
10419:
10416:
10411:
10407:
10403:
10398:
10394:
10389:
10366:
10362:
10339:
10335:
10318:
10315:
10302:
10299:
10296:
10293:
10290:
10270:
10250:
10247:
10244:
10220:
10190:
10187:
10163:
10160:
10157:
10154:
10151:
10131:
10128:
10125:
10122:
10099:
10096:
10065:
10045:
10025:
10016:, the element
10013:
9998:
9995:
9972:
9969:
9966:
9963:
9959:
9955:
9952:
9949:
9946:
9924:
9921:
9918:
9915:
9912:
9902:
9899:
9896:
9893:
9890:
9866:
9846:
9826:
9806:
9788:
9785:
9776:
9775:
9766:
9728:
9725:
9712:
9692:
9681:
9680:
9669:
9666:
9646:
9626:
9623:
9620:
9600:
9597:
9594:
9591:
9571:
9551:
9531:
9528:
9508:
9488:
9475:'s condition:
9469:
9468:
9457:
9454:
9451:
9448:
9445:
9442:
9439:
9436:
9433:
9430:
9427:
9424:
9421:
9418:
9415:
9412:
9409:
9406:
9403:
9400:
9397:
9394:
9391:
9388:
9366:
9363:
9344:Main article:
9341:
9340:Semimodularity
9338:
9301:
9291:
9279:
9276:
9273:
9270:
9267:
9264:
9261:
9258:
9255:
9252:
9249:
9246:
9243:
9240:
9237:
9234:
9214:
9211:
9208:
9194:
9182:
9179:
9176:
9173:
9170:
9167:
9164:
9161:
9158:
9155:
9152:
9149:
9146:
9143:
9140:
9137:
9134:
9131:
9128:
9125:
9122:
9119:
9116:
9113:
9093:
9090:
9087:
9084:
9081:
9078:
9075:
9072:
9059:
9046:
9043:
9040:
9037:
9034:
9031:
9028:
9013:Main article:
9010:
9007:
8987:
8983:
8975:
8971:
8953:
8950:
8947:
8944:
8941:
8938:
8935:
8932:
8929:
8926:
8923:
8920:
8917:
8914:
8911:
8908:
8905:
8902:
8899:
8896:
8886:
8885:
8874:
8854:
8831:
8828:
8825:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8798:
8795:
8792:
8789:
8786:
8783:
8780:
8777:
8774:
8764:
8763:
8752:
8732:
8708:
8705:
8702:
8699:
8696:
8693:
8690:
8687:
8664:Main article:
8660:
8659:
8653:
8637:
8636:
8623:
8620:
8617:
8614:
8611:
8608:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8584:
8581:
8578:
8575:
8572:
8569:
8566:
8546:
8543:
8540:
8537:
8534:
8531:
8528:
8525:
8522:
8519:
8516:
8513:
8510:
8507:
8504:
8501:
8498:
8495:
8492:
8489:
8467:
8464:
8461:
8458:
8455:
8452:
8449:
8446:
8443:
8423:
8420:
8417:
8414:
8411:
8408:
8405:
8402:
8399:
8379:
8376:
8373:
8361:
8346:
8345:Distributivity
8343:
8330:
8327:
8307:
8304:
8286:
8282:
8269:Main article:
8266:
8263:
8252:
8245:
8237:Main article:
8234:
8231:
8220:
8217:
8205:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8164:
8161:
8141:
8121:
8118:
8115:
8112:
8109:
8089:
8086:
8083:
8080:
8077:
8057:
8054:
8044:
8030:
8010:
7988:
7985:
7965:
7945:
7942:
7922:
7919:
7916:
7896:
7893:
7890:
7870:
7867:
7864:
7861:
7858:
7838:
7818:
7798:
7778:
7775:
7755:
7735:
7722:
7717:
7714:
7712:separates 1).
7701:
7681:
7678:
7675:
7672:
7669:
7666:
7663:
7660:
7657:
7654:
7649:
7646:
7642:
7617:
7597:
7594:
7591:
7588:
7585:
7582:
7579:
7576:
7573:
7570:
7565:
7562:
7558:
7547:if and only if
7522:
7518:
7495:
7465:
7461:
7438:
7417:
7411:
7401:
7353:
7348:
7344:
7340:
7336:
7331:
7327:
7323:
7319:
7295:
7290:
7286:
7282:
7278:
7273:
7269:
7265:
7261:
7241:
7221:
7201:
7169:
7147:
7144:
7141:
7138:
7135:
7130:
7126:
7122:
7119:
7116:
7113:
7110:
7106:
7102:
7097:
7093:
7089:
7085:
7081:
7057:
7054:
7051:
7048:
7045:
7040:
7036:
7032:
7029:
7026:
7023:
7020:
7016:
7012:
7007:
7003:
6999:
6995:
6991:
6972:
6969:
6966:
6963:
6960:
6957:
6937:
6934:
6931:
6928:
6925:
6916:is a function
6893:
6889:
6883:
6879:
6875:
6870:
6866:
6862:
6859:
6855:
6833:
6827:
6823:
6819:
6814:
6810:
6806:
6803:
6799:
6767:
6764:
6761:
6758:
6755:
6752:
6749:
6746:
6743:
6740:
6737:
6734:
6731:
6726:
6722:
6702:
6681:
6677:
6673:
6668:
6664:
6660:
6655:
6651:
6647:
6644:
6641:
6638:
6635:
6632:
6629:
6626:
6623:
6620:
6600:
6597:
6594:
6591:
6588:
6585:
6582:
6579:
6576:
6573:
6570:
6567:
6562:
6558:
6538:
6517:
6513:
6509:
6504:
6500:
6496:
6491:
6487:
6483:
6480:
6477:
6474:
6471:
6468:
6465:
6462:
6459:
6456:
6436:
6427:Monotonic map
6417:
6414:
6413:
6412:
6399:
6396:
6393:
6390:
6387:
6384:
6381:
6378:
6375:
6372:
6369:
6366:
6363:
6352:
6341:
6338:
6335:
6332:
6329:
6326:
6323:
6320:
6300:
6297:
6294:
6291:
6288:
6285:
6282:
6262:
6259:
6256:
6253:
6250:
6247:
6244:
6241:
6238:
6227:
6215:
6212:
6209:
6206:
6186:
6183:
6180:
6163:
6162:
6148:
6128:
6104:
6103:
6085:
6082:
6062:
6059:
6056:
6053:
6033:
6013:
5989:
5988:
5970:
5967:
5947:
5944:
5941:
5938:
5935:
5932:
5929:
5926:
5906:
5886:
5863:
5860:
5856:
5855:
5831:
5816:
5804:
5784:
5765:
5762:
5742:
5722:
5719:
5716:
5693:
5681:
5678:
5675:
5672:
5669:
5649:
5646:
5643:
5640:
5632:
5629:
5626:
5606:
5603:
5600:
5597:
5594:
5591:
5588:
5585:
5582:
5579:
5576:
5561:
5554:
5538:
5535:
5511:
5508:
5497:
5485:
5465:
5462:
5442:
5439:
5428:
5408:
5384:
5360:
5340:
5337:
5325:
5324:
5318:
5311:
5309:
5297:
5294:
5280:
5273:
5271:
5254:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5230:
5227:
5210:
5203:
5201:
5191:
5184:
5182:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5127:
5120:
5116:
5113:
5093:
5090:
5070:
5051:absorption law
5034:
5031:
5017:
5011:
5007:
5003:
5000:
4997:
4992:
4988:
4983:
4979:
4976:
4954:
4950:
4946:
4943:
4940:
4935:
4931:
4927:
4924:
4921:
4918:
4915:
4906:(respectively
4893:
4889:
4885:
4882:
4879:
4874:
4870:
4866:
4863:
4860:
4857:
4854:
4831:
4828:
4825:
4822:
4819:
4816:
4796:
4793:
4790:
4787:
4784:
4781:
4777:
4773:
4770:
4766:
4762:
4758:
4754:
4751:
4747:
4743:
4739:
4735:
4732:
4728:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4686:
4683:
4680:
4677:
4674:
4670:
4666:
4663:
4659:
4655:
4651:
4647:
4644:
4640:
4636:
4632:
4628:
4625:
4621:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4579:
4558:
4554:
4551:
4547:
4543:
4539:
4535:
4532:
4528:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4485:
4481:
4478:
4474:
4470:
4466:
4462:
4459:
4455:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4414:
4411:
4391:
4371:
4350:
4347:
4344:
4341:
4321:
4318:
4315:
4312:
4309:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4209:vacuously true
4196:
4174:
4171:
4168:
4165:
4162:
4143:
4140:
4137:
4134:
4131:
4112:
4109:
4089:
4069:
4066:
4042:
4033:is a lattice,
4022:
4019:
4016:
4013:
4010:
4007:
4004:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3932:
3929:
3926:
3923:
3914:
3911:
3908:
3905:
3902:
3893:which satisfy
3880:
3859:
3846:
3840:
3833:
3817:
3795:
3792:
3779:
3773:
3766:
3756:
3753:
3738:
3735:
3715:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3624:
3621:
3618:
3615:
3612:
3592:
3589:
3586:
3583:
3580:
3577:
3557:
3554:
3551:
3548:
3545:
3542:
3537: if
3534:
3531:
3528:
3507: or
3504:
3501:
3498:
3495:
3492:
3489:
3484: if
3481:
3478:
3475:
3455:
3435:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3333:
3330:
3310:
3298:
3295:
3270:
3267:
3264:
3261:
3258:
3238:
3235:
3232:
3229:
3226:
3204:
3201:
3198:
3195:
3192:
3173:
3170:
3167:
3164:
3161:
3151:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3058:
3044:
3041:
3038:
3035:
3032:
3012:
2992:
2972:
2949:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2892:
2889:
2868:
2848:
2845:
2821:
2818:
2815:
2812:
2765:
2760:
2756:
2752:
2747:
2743:
2739:
2734:
2730:
2726:
2721:
2717:
2694:
2690:
2686:
2681:
2677:
2673:
2668:
2664:
2660:
2655:
2651:
2628:
2624:
2620:
2615:
2611:
2588:
2584:
2580:
2575:
2571:
2546:
2524:
2503:
2500:
2497:
2469:
2466:
2463:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2393:
2390:
2387:
2384:
2381:
2365:
2362:
2357:
2354:
2350:lattice theory
2225:
2224:
2222:
2221:
2214:
2207:
2199:
2196:
2195:
2191:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2154:
2153:
2152:
2146:
2140:
2139:
2136:
2135:
2132:
2131:
2128:Linear algebra
2122:
2121:
2116:
2111:
2105:
2104:
2098:
2097:
2094:
2093:
2090:
2089:
2086:Lattice theory
2082:
2075:
2074:
2069:
2064:
2059:
2054:
2049:
2043:
2042:
2036:
2035:
2032:
2031:
2022:
2021:
2016:
2011:
2006:
2001:
1996:
1991:
1986:
1981:
1976:
1970:
1969:
1963:
1962:
1959:
1958:
1949:
1948:
1943:
1938:
1932:
1931:
1930:
1925:
1920:
1911:
1905:
1899:
1898:
1895:
1894:
1884:
1883:
1869:
1866:
1863:
1860:
1840:
1837:
1834:
1814:
1811:
1808:
1788:
1785:
1782:
1779:
1776:
1773:
1749:
1706:
1705:
1702:
1701:
1686:
1683:
1680:
1672:
1671:
1668:
1665:
1662:
1659:
1656:
1655:
1645:
1634:
1631:
1628:
1613:
1602:
1599:
1596:
1586:
1566:
1565:
1562:
1559:
1556:
1553:
1552:
1542:
1522:
1521:
1518:
1515:
1512:
1509:
1508:
1498:
1478:
1477:
1474:
1471:
1468:
1467:
1457:
1442:
1439:
1436:
1433:
1429: or
1426:
1423:
1420:
1417:
1416:
1413:
1410:
1407:
1403:
1400:
1397:
1396:
1386:
1371:
1368:
1364:
1361:
1358:
1355:
1354:
1351:
1348:
1345:
1342:
1335:
1332:
1329:
1326:
1325:
1315:
1300:
1297:
1294:
1291:
1287:
1284:
1283:
1280:
1277:
1274:
1271:
1269:
1259:
1248:
1245:
1242:
1239:
1219:
1216:
1213:
1201:
1200:
1195:
1190:
1185:
1180:
1175:
1170:
1165:
1160:
1155:
1152:
1151:
1141:
1131:
1126:
1121:
1116:
1111:
1101:
1091:
1086:
1080:
1079:
1069:
1059:
1054:
1049:
1044:
1039:
1034:
1024:
1019:
1013:
1012:
1002:
992:
987:
982:
977:
972:
967:
957:
952:
946:
945:
940:
935:
925:
915:
910:
905:
900:
890:
885:
879:
878:
873:
868:
858:
853:
843:
838:
833:
823:
818:
812:
811:
806:
801:
791:
781:
771:
766:
761:
751:
746:
740:
739:
734:
729:
719:
714:
709:
699:
689:
679:
674:
668:
667:
662:
657:
647:
642:
637:
627:
622:
617:
612:
604:
603:
598:
593:
583:
578:
573:
563:
553:
548:
543:
537:
536:
531:
526:
516:
511:
506:
501:
491:
481:
476:
470:
469:
464:
459:
449:
444:
439:
434:
424:
419:
414:
412:Total preorder
408:
407:
402:
397:
387:
382:
377:
372:
367:
357:
352:
346:
345:
340:
335:
325:
320:
315:
310:
305:
300:
295:
284:
283:
278:
273:
263:
258:
253:
248:
243:
238:
228:
220:
219:
217:
212:
210:
208:
206:
204:
201:
199:
197:
194:
193:
188:
183:
178:
173:
168:
163:
158:
153:
148:
141:
140:
138:
137:
130:
123:
115:
101:
100:
97:
96:
51:
49:
42:
26:
9:
6:
4:
3:
2:
14018:
14007:
14004:
14002:
13999:
13998:
13996:
13981:
13978:
13976:
13973:
13969:
13966:
13965:
13964:
13961:
13957:
13954:
13952:
13949:
13947:
13944:
13943:
13942:
13939:
13935:
13932:
13931:
13930:
13929:Ordered field
13927:
13925:
13922:
13918:
13915:
13913:
13910:
13909:
13908:
13905:
13901:
13898:
13897:
13896:
13893:
13891:
13888:
13886:
13885:Hasse diagram
13883:
13881:
13878:
13876:
13873:
13869:
13866:
13865:
13864:
13863:Comparability
13861:
13859:
13856:
13854:
13851:
13849:
13846:
13845:
13843:
13839:
13831:
13828:
13826:
13823:
13821:
13818:
13816:
13813:
13812:
13811:
13808:
13806:
13803:
13799:
13796:
13794:
13791:
13790:
13789:
13786:
13784:
13780:
13777:
13776:
13774:
13771:
13767:
13761:
13758:
13756:
13753:
13751:
13748:
13746:
13743:
13741:
13738:
13736:
13735:Product order
13733:
13731:
13728:
13726:
13723:
13721:
13718:
13716:
13713:
13712:
13710:
13708:Constructions
13706:
13700:
13696:
13692:
13689:
13685:
13682:
13680:
13677:
13675:
13672:
13670:
13667:
13665:
13662:
13660:
13657:
13655:
13652:
13650:
13647:
13643:
13640:
13639:
13638:
13635:
13633:
13630:
13626:
13623:
13621:
13618:
13616:
13613:
13611:
13608:
13607:
13606:
13605:Partial order
13603:
13601:
13598:
13594:
13593:Join and meet
13591:
13589:
13586:
13584:
13581:
13579:
13576:
13574:
13571:
13570:
13569:
13566:
13564:
13561:
13559:
13556:
13554:
13551:
13549:
13546:
13544:
13540:
13536:
13534:
13531:
13529:
13526:
13524:
13521:
13519:
13516:
13514:
13511:
13507:
13504:
13503:
13502:
13499:
13497:
13494:
13492:
13491:Antisymmetric
13489:
13488:
13486:
13482:
13476:
13470:
13467:
13465:
13462:
13460:
13457:
13455:
13452:
13450:
13447:
13445:
13442:
13440:
13437:
13435:
13432:
13430:
13427:
13425:
13422:
13420:
13417:
13415:
13412:
13411:
13409:
13405:
13399:
13398:Weak ordering
13396:
13394:
13391:
13389:
13386:
13384:
13383:Partial order
13381:
13379:
13376:
13374:
13371:
13369:
13366:
13364:
13361:
13360:
13358:
13354:
13348:
13345:
13343:
13340:
13338:
13335:
13334:
13331:
13327:
13320:
13315:
13313:
13308:
13306:
13301:
13300:
13297:
13290:
13288:
13280:
13277:
13273:
13270:
13269:
13265:J.B. Nation,
13264:
13259:
13258:
13253:
13250:
13245:
13241:
13237:
13236:
13231:
13227:
13226:
13216:
13211:
13206:
13205:
13204:
13197:
13192:
13188:
13181:
13177:
13173:
13168:
13164:
13160:
13156:
13151:
13146:
13142:
13138:
13133:
13128:
13127:
13121:
13114:
13113:
13107:
13106:
13105:
13099:
13095:
13092:
13089:
13085:
13084:
13083:
13076:
13070:
13065:
13064:
13057:
13054:
13050:
13046:
13042:
13039:
13036:
13032:
13028:
13025:
13024:
13023:
13017:
13011:
13007:
13003:
13002:
12997:
12992:
12991:
12990:
12984:
12980:
12977:
12974:
12970:
12969:
12968:
12966:
12958:
12957:0-387-56314-8
12954:
12950:
12949:
12944:
12941:
12940:3-540-90578-2
12937:
12933:
12932:
12927:
12926:
12925:
12910:
12905:
12898:
12893:
12885:
12881:
12877:
12870:
12864:, p. 53.
12863:
12858:
12850:
12846:
12842:
12838:
12834:
12830:
12823:
12815:
12811:
12807:
12803:
12799:
12795:
12794:
12786:
12779:
12777:0-521-66351-2
12773:
12769:
12765:
12759:
12752:
12748:
12743:
12736:
12732:
12727:
12725:
12717:
12713:
12708:
12700:
12698:9780821826942
12694:
12690:
12686:
12679:
12664:
12657:
12650:
12643:
12642:3-540-90578-2
12639:
12635:
12633:
12626:
12620:
12616:
12615:Dedekind 1897
12600:
12597:
12594:
12591:
12582:
12579:
12576:
12570:
12567:
12561:
12558:
12555:
12552:
12544:
12540:
12539:Birkhoff 1948
12535:
12528:
12524:
12519:
12515:
12505:
12502:
12500:
12497:
12495:
12492:
12490:
12487:
12485:
12484:Quantum logic
12482:
12480:
12477:
12475:
12472:
12470:
12467:
12464:
12463:Lattice Miner
12460:
12457:
12455:
12452:
12450:
12447:
12445:
12444:Domain theory
12442:
12440:
12437:
12435:
12432:
12429:
12426:
12424:
12421:
12419:
12416:
12414:
12411:
12409:
12406:
12404:
12401:
12399:
12396:
12394:
12391:
12390:
12389:
12388:
12378:
12369:is available.
12368:
12364:
12358:
12357:
12353:
12348:This article
12346:
12337:
12336:
12326:
12323:
12321:
12318:
12312:
12309:
12303:
12300:
12298:
12295:
12288:
12285:
12282:
12278:
12275:
12272:
12269:
12267:
12264:
12261:
12258:
12255:
12254:Join and meet
12252:
12251:
12245:
12243:
12239:
12235:
12230:
12224:
12208:
12200:
12197:
12183:
12180:
12177:
12174:
12154:
12134:
12114:
12111:
12091:
12083:
12080:
12079:
12078:
12064:
12044:
12041:
12038:
12035:
12032:
12029:
12009:
12006:
12003:
11983:
11980:
11977:
11969:
11953:
11933:
11913:
11889:
11881:
11877:
11861:
11858:
11855:
11835:
11832:
11829:
11826:
11818:
11815:
11812:
11792:
11789:
11786:
11783:
11780:
11772:
11769:
11766:
11750:
11742:
11726:
11718:
11702:
11683:
11678:
11674:
11668:
11665:
11662:
11658:
11637:
11634:
11631:
11611:
11608:
11588:
11568:
11565:
11562:
11559:
11556:
11553:
11533:
11530:
11527:
11524:
11516:
11513:
11510:
11490:
11487:
11484:
11481:
11478:
11470:
11467:
11466:
11465:
11451:
11432:
11429:
11409:
11395:
11378:
11370:
11355:
11352:
11344:
11340:
11336:
11320:
11317:
11297:
11294:
11291:
11284:
11268:
11258:
11251:Free lattices
11248:
11235:
11232:
11229:
11226:
11206:
11203:
11200:
11180:
11177:
11174:
11154:
11151:
11148:
11145:
11142:
11139:
11119:
11099:
11096:
11093:
11090:
11070:
11067:
11059:
11043:
11034:
11032:
11016:
11013:
11007:
11001:
10998:
10992:
10986:
10966:
10963:
10956:
10941:
10921:
10918:
10915:
10892:
10886:
10883:
10877:
10871:
10842:sometimes to
10821:
10818:
10815:
10808:
10807:rank function
10804:
10800:
10796:
10795:
10775:
10772:
10769:
10755:Graded/ranked
10752:
10750:
10734:
10714:
10694:
10691:
10688:
10685:
10665:
10662:
10642:
10633:
10620:
10617:
10614:
10611:
10608:
10605:
10583:
10580:
10577:
10573:
10550:
10546:
10537:
10533:
10529:
10513:
10508:
10504:
10500:
10497:
10494:
10489:
10485:
10481:
10476:
10472:
10468:
10463:
10459:
10438:
10434:
10428:
10424:
10420:
10417:
10414:
10409:
10405:
10401:
10396:
10392:
10387:
10364:
10360:
10337:
10333:
10324:
10314:
10300:
10297:
10294:
10291:
10288:
10268:
10248:
10245:
10234:
10218:
10210:
10206:
10204:
10188:
10185:
10177:
10161:
10158:
10155:
10152:
10129:
10126:
10123:
10111:
10097:
10094:
10086:
10081:
10079:
10063:
10043:
10023:
9996:
9993:
9967:
9964:
9961:
9957:
9953:
9950:
9947:
9935:
9922:
9919:
9916:
9913:
9910:
9900:
9897:
9894:
9891:
9888:
9880:
9864:
9844:
9824:
9804:
9794:
9784:
9782:
9773:
9772:
9767:
9764:
9763:
9758:
9757:
9756:
9754:
9750:
9746:
9742:
9738:
9734:
9733:domain theory
9724:
9710:
9690:
9667:
9664:
9644:
9624:
9621:
9618:
9598:
9595:
9592:
9589:
9569:
9549:
9529:
9526:
9506:
9486:
9478:
9477:
9476:
9474:
9455:
9449:
9446:
9443:
9437:
9434:
9428:
9425:
9422:
9416:
9413:
9407:
9401:
9398:
9392:
9386:
9379:
9378:
9377:
9364:
9361:
9353:
9347:
9337:
9335:
9331:
9327:
9323:
9319:
9315:
9311:
9307:
9299:
9277:
9274:
9271:
9265:
9262:
9259:
9253:
9247:
9244:
9241:
9235:
9232:
9212:
9209:
9206:
9180:
9177:
9174:
9168:
9165:
9159:
9156:
9153:
9144:
9138:
9135:
9132:
9126:
9120:
9117:
9114:
9091:
9088:
9085:
9082:
9079:
9076:
9073:
9070:
9062:
9041:
9038:
9035:
9032:
9029:
9016:
9006:
9004:
9000:
8996:
8991:
8981:
8969:
8964:
8951:
8945:
8942:
8939:
8933:
8927:
8924:
8921:
8915:
8909:
8906:
8903:
8897:
8894:
8872:
8852:
8844:
8843:
8842:
8829:
8823:
8820:
8817:
8811:
8805:
8802:
8799:
8793:
8787:
8784:
8781:
8775:
8772:
8750:
8730:
8722:
8721:
8720:
8706:
8703:
8700:
8697:
8694:
8691:
8688:
8685:
8677:
8673:
8667:
8651:
8650:Pic. 10:
8647:
8643:
8642:
8621:
8615:
8612:
8609:
8603:
8597:
8594:
8591:
8585:
8579:
8576:
8573:
8567:
8564:
8544:
8538:
8535:
8532:
8526:
8520:
8517:
8514:
8508:
8502:
8499:
8496:
8490:
8487:
8465:
8462:
8459:
8456:
8450:
8447:
8444:
8421:
8418:
8412:
8409:
8406:
8400:
8397:
8377:
8374:
8371:
8359:
8358:Pic. 11:
8355:
8351:
8350:
8342:
8328:
8325:
8305:
8302:
8294:
8290:
8284:
8280:
8278:
8272:
8262:
8259:
8255:
8250:
8248:
8240:
8230:
8226:
8216:
8203:
8200:
8197:
8194:
8191:
8188:
8185:
8182:
8162:
8159:
8139:
8132:implies that
8119:
8116:
8113:
8110:
8107:
8087:
8084:
8081:
8078:
8075:
8055:
8052:
8042:
8028:
8021:of a lattice
8008:
8001:A sublattice
7999:
7986:
7983:
7963:
7943:
7940:
7920:
7917:
7914:
7894:
7891:
7888:
7868:
7865:
7862:
7859:
7856:
7836:
7816:
7796:
7776:
7773:
7753:
7733:
7726:of a lattice
7725:
7713:
7699:
7676:
7670:
7661:
7655:
7647:
7644:
7640:
7631:
7615:
7592:
7586:
7577:
7571:
7563:
7560:
7556:
7548:
7545:
7541:
7537:
7520:
7484:
7480:
7463:
7426:
7424:
7420:
7414:
7408:
7404:
7398:
7393:
7391:
7387:
7384:
7380:
7376:
7371:
7369:
7364:
7351:
7346:
7342:
7338:
7334:
7329:
7325:
7321:
7317:
7293:
7288:
7284:
7280:
7276:
7271:
7267:
7263:
7259:
7239:
7219:
7199:
7191:
7187:
7183:
7167:
7158:
7145:
7139:
7133:
7128:
7124:
7117:
7111:
7108:
7104:
7100:
7095:
7091:
7087:
7083:
7079:
7055:
7049:
7043:
7038:
7034:
7027:
7021:
7018:
7014:
7010:
7005:
7001:
6997:
6993:
6989:
6970:
6967:
6964:
6961:
6958:
6955:
6935:
6929:
6926:
6923:
6915:
6911:
6907:
6891:
6887:
6881:
6877:
6873:
6868:
6864:
6860:
6857:
6853:
6831:
6825:
6821:
6817:
6812:
6808:
6804:
6801:
6797:
6788:
6784:
6765:
6759:
6756:
6753:
6747:
6744:
6738:
6732:
6729:
6720:
6700:
6675:
6671:
6662:
6658:
6649:
6645:
6639:
6633:
6630:
6624:
6618:
6595:
6592:
6589:
6583:
6580:
6574:
6568:
6565:
6556:
6536:
6511:
6507:
6498:
6494:
6485:
6481:
6475:
6469:
6466:
6460:
6454:
6434:
6426:
6422:
6394:
6391:
6388:
6385:
6382:
6379:
6376:
6373:
6370:
6367:
6364:
6353:
6339:
6333:
6330:
6327:
6324:
6321:
6295:
6292:
6289:
6286:
6283:
6257:
6254:
6251:
6248:
6245:
6242:
6239:
6228:
6213:
6210:
6207:
6204:
6184:
6181:
6178:
6170:
6169:
6168:
6146:
6126:
6118:
6114:
6110:
6109:
6099:
6083:
6080:
6060:
6057:
6054:
6051:
6031:
6011:
6003:
5999:
5995:
5994:
5984:
5968:
5965:
5945:
5942:
5939:
5936:
5933:
5930:
5927:
5924:
5904:
5884:
5876:
5872:
5868:
5867:
5859:
5853:
5852:domain theory
5849:
5845:
5840:
5836:
5832:
5829:
5825:
5821:
5817:
5802:
5782:
5763:
5760:
5740:
5720:
5717:
5714:
5706:
5702:
5698:
5694:
5676:
5673:
5670:
5647:
5644:
5641:
5638:
5630:
5627:
5624:
5601:
5598:
5595:
5589:
5583:
5580:
5577:
5566:
5562:
5559:
5555:
5552:
5536:
5533:
5525:
5509:
5506:
5498:
5483:
5463:
5460:
5440:
5437:
5429:
5426:
5422:
5406:
5398:
5382:
5374:
5358:
5338:
5335:
5327:
5326:
5321:
5315:
5310:
5295:
5292:
5283:
5277:
5272:
5268:
5252:
5246:
5243:
5240:
5237:
5234:
5231:
5228:
5217:
5213:
5207:
5202:
5198:
5194:
5188:
5183:
5180:depicting it.
5179:
5178:Hasse diagram
5175:
5174:set inclusion
5159:
5153:
5150:
5147:
5144:
5141:
5130:
5124:
5119:
5118:
5112:
5110:
5105:
5091:
5088:
5068:
5058:
5056:
5052:
5048:
5044:
5040:
5030:
5015:
5009:
5005:
5001:
4998:
4995:
4990:
4986:
4981:
4977:
4974:
4952:
4948:
4944:
4941:
4938:
4933:
4929:
4925:
4922:
4919:
4916:
4913:
4891:
4887:
4883:
4880:
4877:
4872:
4868:
4864:
4861:
4858:
4855:
4852:
4842:
4829:
4826:
4823:
4817:
4814:
4794:
4791:
4788:
4785:
4782:
4779:
4775:
4771:
4768:
4764:
4760:
4756:
4749:
4745:
4741:
4737:
4733:
4730:
4726:
4722:
4713:
4710:
4704:
4684:
4681:
4678:
4675:
4672:
4668:
4664:
4661:
4657:
4653:
4649:
4642:
4638:
4634:
4630:
4626:
4623:
4619:
4615:
4606:
4603:
4597:
4577:
4570:hold. Taking
4556:
4552:
4549:
4545:
4541:
4537:
4533:
4530:
4526:
4522:
4516:
4513:
4510:
4504:
4483:
4479:
4476:
4472:
4468:
4464:
4460:
4457:
4453:
4449:
4443:
4440:
4437:
4431:
4412:
4409:
4389:
4369:
4348:
4345:
4339:
4319:
4316:
4313:
4307:
4287:
4284:
4281:
4278:
4275:
4269:
4266:
4241:
4238:
4235:
4232:
4226:
4223:
4210:
4194:
4185:
4172:
4169:
4166:
4163:
4160:
4141:
4138:
4135:
4132:
4129:
4110:
4107:
4087:
4067:
4064:
4056:
4040:
4017:
4014:
4011:
4008:
4005:
3979:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3943:
3930:
3927:
3924:
3921:
3912:
3909:
3906:
3903:
3900:
3857:
3850:, denoted by
3849:
3843:
3838:(also called
3837:
3836:
3834:least element
3793:
3790:
3782:
3776:
3771:(also called
3770:
3769:
3762:
3752:
3749:
3736:
3733:
3713:
3704:
3690:
3687:
3684:
3681:
3678:
3675:
3669:
3666:
3663:
3657:
3651:
3648:
3645:
3639:
3636:
3633:
3630:
3622:
3619:
3616:
3613:
3610:
3590:
3587:
3584:
3581:
3578:
3575:
3555:
3552:
3549:
3546:
3543:
3540:
3532:
3529:
3526:
3502:
3499:
3496:
3493:
3490:
3487:
3479:
3476:
3473:
3453:
3433:
3413:
3407:
3404:
3401:
3398:
3395:
3383:
3366:
3363:
3360:
3357:
3354:
3331:
3328:
3308:
3294:
3292:
3291:partial order
3288:
3284:
3265:
3262:
3259:
3233:
3230:
3227:
3215:
3202:
3199:
3196:
3193:
3190:
3171:
3168:
3165:
3162:
3159:
3149:
3146:
3133:
3130:
3124:
3121:
3118:
3112:
3109:
3090:
3087:
3081:
3078:
3075:
3069:
3066:
3056:
3042:
3039:
3036:
3033:
3030:
3010:
2990:
2970:
2963:
2947:
2924:
2921:
2918:
2915:
2912:
2902:
2898:
2888:
2886:
2866:
2846:
2843:
2835:
2819:
2816:
2813:
2810:
2801:
2799:
2795:
2791:
2787:
2786:
2781:
2776:
2763:
2758:
2754:
2750:
2745:
2741:
2737:
2732:
2728:
2724:
2719:
2715:
2692:
2688:
2684:
2679:
2675:
2671:
2666:
2662:
2658:
2653:
2649:
2641:implies that
2626:
2622:
2618:
2613:
2609:
2586:
2582:
2578:
2573:
2569:
2560:
2544:
2522:
2501:
2498:
2495:
2487:
2483:
2467:
2464:
2461:
2453:
2437:
2434:
2428:
2425:
2422:
2411:
2407:
2388:
2385:
2382:
2371:
2361:
2353:
2351:
2347:
2342:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2309:
2305:
2300:
2298:
2294:
2290:
2286:
2282:
2278:
2274:
2270:
2266:
2262:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2220:
2215:
2213:
2208:
2206:
2201:
2200:
2198:
2197:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2155:
2151:
2148:
2147:
2143:
2138:
2137:
2130:
2129:
2125:
2124:
2120:
2117:
2115:
2112:
2110:
2107:
2106:
2101:
2096:
2095:
2088:
2087:
2083:
2081:
2078:
2077:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2053:
2050:
2048:
2045:
2044:
2039:
2034:
2033:
2028:
2027:
2020:
2017:
2015:
2014:Division ring
2012:
2010:
2007:
2005:
2002:
2000:
1997:
1995:
1992:
1990:
1987:
1985:
1982:
1980:
1977:
1975:
1972:
1971:
1966:
1961:
1960:
1955:
1954:
1947:
1944:
1942:
1939:
1937:
1936:Abelian group
1934:
1933:
1929:
1926:
1924:
1921:
1919:
1915:
1912:
1910:
1907:
1906:
1902:
1897:
1896:
1893:
1890:
1889:
1882:
1867:
1864:
1861:
1858:
1838:
1835:
1832:
1812:
1809:
1806:
1786:
1783:
1780:
1777:
1774:
1771:
1763:
1747:
1740:
1708:
1707:
1684:
1681:
1678:
1663:
1660:
1657:
1646:
1632:
1629:
1626:
1614:
1600:
1597:
1594:
1587:
1560:
1557:
1554:
1543:
1516:
1513:
1510:
1499:
1472:
1458:
1440:
1437:
1434:
1424:
1421:
1418:
1408:
1401:
1398:
1387:
1369:
1362:
1359:
1349:
1346:
1343:
1333:
1330:
1327:
1316:
1298:
1295:
1292:
1278:
1275:
1272:
1260:
1246:
1240:
1237:
1217:
1214:
1211:
1203:
1202:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1179:
1176:
1174:
1171:
1169:
1166:
1164:
1163:Antisymmetric
1161:
1159:
1156:
1154:
1153:
1142:
1132:
1127:
1122:
1117:
1112:
1102:
1092:
1087:
1085:
1082:
1081:
1070:
1060:
1055:
1050:
1045:
1040:
1035:
1025:
1020:
1018:
1015:
1014:
1003:
993:
988:
983:
978:
973:
968:
958:
953:
951:
948:
947:
941:
936:
926:
916:
911:
906:
901:
891:
886:
884:
881:
880:
874:
869:
859:
854:
844:
839:
834:
824:
819:
817:
814:
813:
807:
802:
792:
782:
772:
767:
762:
752:
747:
745:
742:
741:
735:
730:
720:
715:
710:
700:
690:
680:
675:
673:
672:Well-ordering
670:
669:
663:
658:
648:
643:
638:
628:
623:
618:
613:
610:
606:
605:
599:
594:
584:
579:
574:
564:
554:
549:
544:
542:
539:
538:
532:
527:
517:
512:
507:
502:
492:
482:
477:
475:
472:
471:
465:
460:
450:
445:
440:
435:
425:
420:
415:
413:
410:
409:
403:
398:
388:
383:
378:
373:
368:
358:
353:
351:
350:Partial order
348:
347:
341:
336:
326:
321:
316:
311:
306:
301:
296:
293:
286:
285:
279:
274:
264:
259:
254:
249:
244:
239:
229:
226:
222:
221:
218:
213:
211:
209:
207:
205:
202:
200:
198:
196:
195:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
156:Antisymmetric
154:
152:
149:
147:
146:
143:
142:
136:
131:
129:
124:
122:
117:
116:
113:
109:
105:
104:
93:
90:
82:
72:
68:
62:
61:
55:
50:
41:
40:
37:
33:
19:
18:Lattice order
13772:& Orders
13750:Star product
13679:Well-founded
13632:Prefix order
13588:Distributive
13578:Complemented
13567:
13548:Foundational
13513:Completeness
13469:Zorn's lemma
13377:
13373:Cyclic order
13356:Key concepts
13326:Order theory
13286:
13267:
13255:
13233:
13209:
13202:
13186:
13143:(1): 47–68.
13140:
13136:
13125:
13111:
13103:
13098:Stone spaces
13097:
13081:
13062:
13044:
13030:
13021:
13000:
12988:
12982:
12972:
12962:
12946:
12929:
12923:
12909:Grätzer 2003
12904:
12897:Grätzer 2003
12892:
12875:
12869:
12857:
12832:
12828:
12822:
12797:
12791:
12785:
12767:
12758:
12742:
12707:
12684:
12678:
12666:. Retrieved
12662:
12649:
12631:
12625:
12534:
12523:Grätzer 2003
12518:
12489:Median graph
12469:Bloom filter
12386:
12385:
12372:
12367:Editing help
12349:
12287:Skew lattice
12231:
12228:
11905:
11879:
11875:
11770:
11765:real numbers
11740:
11716:
11468:
11401:
11343:free lattice
11342:
11282:
11260:
11257:Free lattice
11035:
11030:
10806:
10803:Ranked poset
10798:
10797:, sometimes
10792:
10758:
10748:
10634:
10535:
10531:
10527:
10322:
10320:
10207:
10112:
10082:
9936:
9878:
9796:
9777:
9769:
9760:
9741:directed set
9730:
9682:
9637:covers both
9470:
9349:
9309:
9058:
9018:
8992:
8967:
8965:
8887:
8765:
8669:
8649:
8357:
8293:real numbers
8276:
8274:
8260:
8256:
8244:
8242:
8233:Completeness
8228:
8000:
7789:That is, if
7721:
7719:
7629:
7543:
7539:
7535:
7482:
7478:
7427:
7416:
7410:
7400:
7397:isomorphisms
7394:
7372:
7365:
7189:
7186:semilattices
7182:homomorphism
7159:
6913:
6909:
6905:
6780:
6425:Pic. 9:
6424:
6166:
6117:Pic. 6:
6116:
6002:Pic. 7:
6001:
5875:Pic. 8:
5874:
5857:
5705:divisibility
5499:For any set
5430:For any set
5371:(called the
5328:For any set
5320:Pic. 5:
5319:
5282:Pic. 4:
5281:
5266:
5265:ordered by "
5212:Pic. 3:
5211:
5196:
5193:Pic. 2:
5192:
5129:Pic. 1:
5128:
5106:
5059:
5036:
4843:
4186:
3944:
3845:
3839:
3831:
3778:
3772:
3764:
3760:
3758:
3750:
3705:
3384:
3300:
3283:semilattices
3216:
3147:
2896:
2894:
2880:
2879:is called a
2802:
2783:
2777:
2405:
2404:is called a
2367:
2359:
2349:
2343:
2335:lattice-like
2334:
2323:Semilattices
2315:order theory
2301:
2289:divisibility
2281:intersection
2241:order theory
2237:mathematical
2232:
2230:
2188:Hopf algebra
2126:
2119:Vector space
2085:
2084:
2046:
2037:
2024:
1953:Group theory
1951:
1916: /
1736:
1173:Well-founded
743:
291:(Quasiorder)
166:Well-founded
85:
76:
57:
36:
13956:Riesz space
13917:Isomorphism
13793:Normal cone
13715:Composition
13649:Semilattice
13558:Homogeneous
13543:Equivalence
13393:Total order
12975:. Pergamon.
12716:p. 104
12271:Total order
12077:is called:
11464:is called:
11394:s members.
11345:over a set
11056:is said to
9879:complements
9582:both cover
9352:semimodular
9292:Modular law
8672:distributes
8152:belongs to
7716:Sublattices
5833:The set of
5795:is bottom;
5549:ordered by
5214:Lattice of
5131:Subsets of
4402:of a poset
3466:by setting
2410:semilattice
2173:Lie algebra
2158:Associative
2062:Total order
2052:Semilattice
2026:Ring theory
1193:Irreflexive
474:Total order
186:Irreflexive
71:introducing
13995:Categories
13924:Order type
13858:Cofinality
13699:Well-order
13674:Transitive
13563:Idempotent
13496:Asymmetric
12919:References
12735:p. 89
12617:, p.
12541:, p.
12525:, p.
12375:March 2017
12167:such that
12022:such that
11880:meet prime
11771:Join prime
11715:is called
11341:. For the
11132:such that
10791:is called
10759:A lattice
10281:such that
9791:See also:
9298:sublattice
9009:Modularity
8980:sublattice
7723:sublattice
7628:separates
7544:separating
7534:is called
7405:is just a
7368:preserving
5839:arithmetic
5822:(also see
5551:refinement
5524:partitions
5496:is finite.
5216:partitions
5043:semigroups
3995:such that
2962:operations
2356:Definition
2311:identities
1764:: for all
1762:transitive
1198:Asymmetric
191:Asymmetric
108:Transitive
54:references
13975:Upper set
13912:Embedding
13848:Antichain
13669:Tolerance
13659:Symmetric
13654:Semiorder
13600:Reflexive
13518:Connected
13289:elements)
13257:MathWorld
13252:"Lattice"
13240:EMS Press
13145:CiteSeerX
12598:∨
12580:∨
12571:∧
12562:∨
12545:. "since
12428:Fuzzy set
12225:of atoms.
12199:Atomistic
12178:≤
12007:∈
11859:≠
11830:≤
11816:≤
11790:∨
11784:≤
11751:∧
11727:∨
11666:∈
11659:⋁
11635:≠
11563:∈
11488:∨
11335:set union
11230:≠
11204:≤
10908:whenever
10825:→
10801:(but see
10776:≤
10615:≤
10609:≤
10581:−
10498:…
10418:…
10379:is a set
10292:∧
10243:¬
10176:operation
10150:¬
10121:¬
9914:∧
9892:∨
9753:algebraic
9745:way-below
9711:∧
9691:∨
9622:∨
9593:∧
9479:for each
9447:∨
9426:∧
9414:≥
9365::
9272:∧
9263:∨
9245:∧
9236:∨
9210:≤
9175:∧
9166:∨
9157:∧
9136:∧
9127:∨
9118:∧
9086:∈
9042:∧
9036:∨
8943:∧
8934:∨
8925:∧
8907:∨
8898:∧
8873:∨
8853:∧
8821:∨
8812:∧
8803:∨
8785:∧
8776:∨
8751:∧
8731:∨
8701:∈
8613:∨
8604:∧
8595:∨
8577:∧
8568:∨
8536:∧
8527:∨
8518:∧
8500:∨
8491:∧
8457:∧
8448:∨
8410:∧
8401:∨
8375:≤
8341:or both.
8198:∈
8117:∈
8085:≤
8079:≤
7918:∨
7892:∧
7866:∈
7645:−
7561:−
7407:bijective
7386:bijection
7125:∧
7092:∧
7035:∨
7002:∨
6965:∈
6933:→
6878:∧
6865:∨
6822:∧
6809:∨
6757:∧
6725:′
6701:≠
6680:′
6667:′
6659:∧
6654:′
6631:∧
6593:∨
6561:′
6537:≠
6516:′
6503:′
6495:∨
6490:′
6467:∨
6182:≤
5718:≤
5660:The pair
5642:≤
5628:≤
5590:≤
5421:set union
5373:power set
5293:≤
4999:…
4945:∧
4942:⋯
4939:∧
4920:⋀
4884:∨
4881:⋯
4878:∨
4859:⋁
4821:∅
4818:∪
4789:⋀
4780:∧
4769:⋀
4753:∅
4750:⋀
4742:∧
4731:⋀
4717:∅
4714:∪
4705:⋀
4682:⋁
4673:∨
4662:⋁
4646:∅
4643:⋁
4635:∨
4624:⋁
4610:∅
4607:∪
4598:⋁
4550:⋀
4542:∧
4531:⋀
4514:∪
4505:⋀
4477:⋁
4469:∨
4458:⋁
4441:∪
4432:⋁
4343:∅
4340:⋀
4311:∅
4308:⋁
4282:≤
4273:∅
4270:∈
4239:≤
4230:∅
4227:∈
4164:∧
4133:∨
4108:∧
4065:∨
4018:∧
4012:∨
3968:∧
3962:∨
3925:∈
3910:≤
3904:≤
3879:⊥
3816:⊤
3734:∧
3714:∨
3688:∨
3676:∨
3667:∧
3649:∧
3640:∨
3620:∧
3585:∈
3550:∨
3530:≤
3497:∧
3477:≤
3434:≤
3408:∧
3402:∨
3367:∧
3361:∨
3329:∧
3309:∨
3266:∧
3234:∨
3194:∧
3163:∨
3122:∨
3113:∧
3079:∧
3070:∨
3040:∈
2991:∧
2971:∨
2925:∧
2919:∨
2814:⊆
2780:induction
2751:∧
2738:≤
2725:∧
2685:∨
2672:≤
2659:∨
2619:≤
2579:≤
2545:∨
2523:∧
2499:∧
2465:∨
2435:⊆
2389:≤
2308:axiomatic
2273:inclusion
2269:power set
2183:Bialgebra
1989:Near-ring
1946:Lie group
1914:Semigroup
1675:not
1667:⇒
1623:not
1558:∧
1514:∨
1412:⇒
1402:≠
1357:⇒
1286:⇒
1244:∅
1241:≠
1188:Reflexive
1183:Has meets
1178:Has joins
1168:Connected
1158:Symmetric
289:Preorder
216:reflexive
181:Reflexive
176:Has meets
171:Has joins
161:Connected
151:Symmetric
13770:Topology
13637:Preorder
13620:Eulerian
13583:Complete
13533:Directed
13523:Covering
13388:Preorder
13347:Category
13342:Glossary
13178:(1897),
13167:11012081
13137:Synthese
13096:, 1982.
13029:, 1967.
12998:(2002),
12981:, 1971.
12766:(1997),
12248:See also
12223:supremum
11805:implies
11546:for all
11503:implies
11261:Any set
10598:for all
10203:negation
9473:Birkhoff
9225:implies
8281:nonempty
7521:′
7464:′
7423:category
7375:monotone
6783:morphism
6354:The set
6197:implies
5828:examples
5753:divides
5115:Examples
4967:) where
2372:(poset)
2333:. These
2253:supremum
2019:Lie ring
1984:Semiring
1732:✗
1719:✗
1129:✗
1124:✗
1119:✗
1114:✗
1089:✗
1057:✗
1052:✗
1047:✗
1042:✗
1037:✗
1022:✗
990:✗
985:✗
980:✗
975:✗
970:✗
955:✗
943:✗
938:✗
913:✗
908:✗
903:✗
888:✗
876:✗
871:✗
856:✗
841:✗
836:✗
821:✗
809:✗
804:✗
769:✗
764:✗
749:✗
737:✗
732:✗
717:✗
712:✗
677:✗
665:✗
660:✗
645:✗
640:✗
625:✗
620:✗
615:✗
601:✗
596:✗
581:✗
576:✗
551:✗
546:✗
534:✗
529:✗
514:✗
509:✗
504:✗
479:✗
467:✗
462:✗
447:✗
442:✗
437:✗
422:✗
417:✗
405:✗
400:✗
385:✗
380:✗
375:✗
370:✗
355:✗
343:✗
338:✗
323:✗
318:✗
313:✗
308:✗
303:✗
298:✗
281:✗
276:✗
261:✗
256:✗
251:✗
246:✗
241:✗
79:May 2009
13875:Duality
13853:Cofinal
13841:Related
13820:Fréchet
13697:)
13573:Bounded
13568:Lattice
13541:)
13539:Partial
13407:Results
13378:Lattice
13242:, 2001
12849:1968883
12814:1969001
12242:subsets
12238:filters
11369:Whitman
10565:covers
10536:maximal
9310:modular
9308:(hence
9060:modular
8291:of the
8283:subset
7933:are in
7390:inverse
5703:, with
5267:refines
5197:divides
5047:monoids
3841:minimum
3774:maximum
2897:lattice
2406:lattice
2327:Heyting
2261:infimum
2233:lattice
2150:Algebra
2142:Algebra
2047:Lattice
2038:Lattice
744:Lattice
67:improve
13900:Subnet
13880:Filter
13830:Normed
13815:Banach
13781:&
13688:Better
13625:Strict
13615:Graded
13506:topics
13337:Topics
13165:
13147:
13071:
13051:
13012:
12955:
12938:
12847:
12812:
12774:
12695:
12668:8 June
12640:
12430:theory
12350:is in
12281:filter
12234:ideals
12082:Atomic
11966:is an
11193:means
11167:Here,
10955:covers
10799:ranked
10794:graded
10678:where
10528:length
10451:where
9328:. The
9306:module
9001:, see
8995:frames
8390:, but
7632:) and
5837:of an
5818:Every
5172:under
5049:. The
3870:or by
3847:bottom
3830:and a
2899:is an
2482:dually
2480:) and
2450:has a
2178:Graded
2109:Module
2100:Module
1999:Domain
1918:Monoid
1569:exists
1525:exists
1481:exists
110:
56:, but
13890:Ideal
13868:Graph
13664:Total
13642:Total
13528:Dense
13183:(PDF)
13163:S2CID
13116:(PDF)
12845:JSTOR
12810:JSTOR
12659:(PDF)
12510:Notes
12356:prose
12277:Ideal
12221:is a
12057:Then
11392:'
11058:cover
10979:then
10325:from
10323:chain
10178:over
9611:then
9326:group
9324:of a
9316:of a
8974:and N
8865:over
8743:over
8041:is a
7956:then
7881:both
7180:is a
7160:Thus
6908:from
6787:above
5824:below
4211:that
3844:, or
3777:, or
2277:union
2144:-like
2102:-like
2040:-like
2009:Field
1967:-like
1941:Magma
1909:Group
1903:-like
1901:Group
1851:then
214:Anti-
13481:list
13282:OEIS
13069:ISBN
13049:ISBN
13010:ISBN
12953:ISBN
12936:ISBN
12772:ISBN
12693:ISBN
12670:2022
12638:ISBN
12461:and
12352:list
12039:<
12033:<
11981:<
11968:atom
11906:Let
11719:(or
11219:and
11178:>
11149:>
11143:>
11094:>
11031:rank
10919:<
10884:<
10689:<
10655:and
10526:The
10501:<
10495:<
10482:<
10469:<
10056:and
9877:are
9837:and
9797:Let
9703:and
9657:and
9562:and
9499:and
9318:ring
8997:and
8986:or N
8676:dual
8434:and
8100:and
7907:and
7481:and
7450:and
7428:Let
7232:and
6845:and
6611:and
6139:and
6073:and
6024:and
5958:and
5897:and
5699:and
5563:The
5556:The
5081:and
4697:and
4497:and
4382:and
4254:and
4080:and
3726:and
3321:and
3287:dual
3281:are
3249:and
2983:and
2707:and
2601:and
2536:and
2486:meet
2452:join
2329:and
2317:and
2265:meet
2257:join
2243:and
1974:Ring
1965:Ring
1825:and
1230:and
13895:Net
13695:Pre
13191:doi
13155:doi
13141:183
12880:doi
12837:doi
12802:doi
12147:of
12104:of
11970:if
11946:of
11773:if
11581:If
11471:if
11083:if
10727:to
10538:if
10352:to
9857:of
9768:An
9731:In
9542:if
9519:in
9057:is
8251:all
8249:if
8068:if
8045:of
7507:to
6912:to
5733:if
5617:if
5526:of
5375:of
5218:of
5055:rig
3808:by
3806:or
3780:top
3446:on
3059:):
3003:on
1979:Rng
1799:if
1760:be
1470:min
13997::
13254:.
13238:,
13232:,
13189:,
13185:,
13161:.
13153:.
13139:.
13008:,
13004:,
12967::
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12833:43
12831:.
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12723:^
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12543:18
12527:52
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10080:.
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86:(
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77:(
63:.
34:.
20:)
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