1642:
1629:
1063:
1053:
1023:
1013:
991:
981:
946:
924:
914:
879:
847:
837:
812:
780:
765:
745:
713:
703:
693:
673:
641:
621:
611:
601:
569:
549:
505:
485:
475:
438:
413:
403:
371:
346:
309:
279:
247:
185:
150:
4404:. Specifically, a complete join-semilattice requires that the homomorphisms preserve all joins, but contrary to the situation we find for completeness properties, this does not require that homomorphisms preserve all meets. On the other hand, we can conclude that every such mapping is the lower adjoint of some
4388:
Nowadays, the term "complete semilattice" has no generally accepted meaning, and various mutually inconsistent definitions exist. If completeness is taken to require the existence of all infinite joins, or all infinite meets, whichever the case may be, as well as finite ones, this immediately leads
4297:
Surprisingly, there is a notion of "distributivity" applicable to semilattices, even though distributivity conventionally requires the interaction of two binary operations. This notion requires but a single operation, and generalizes the distributivity condition for lattices. A join-semilattice is
4430:
joins. If such a structure has also a greatest element (the meet of the empty set), it is also a complete lattice. Thus a complete semilattice turns out to be "a complete lattice possibly lacking a top". This definition is of interest specifically in
1888:, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order.
1373:
1302:
3842:
1895:
is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by corresponding
1617:
1231:
3360:
3971:
2953:
By induction on the number of elements, any non-empty finite meet semilattice has a least element and any non-empty finite join semilattice has a greatest element. (In neither case will the semilattice necessarily be
1502:
1458:
1414:
4368:
Distributive meet-semilattices are defined dually. These definitions are justified by the fact that any distributive join-semilattice in which binary meets exist is a distributive lattice. See the entry
4207:
1639:
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
2919:
Hence the two definitions may be used interchangeably, depending on which one is more convenient for a particular purpose. A similar conclusion holds for join-semilattices and the dual ordering ≥.
4712:). The case of free meet-semilattices is dual, using the opposite subset inclusion as an ordering. For join-semilattices with bottom, we just add the empty set to the above collection of subsets.
3260:
1572:
1469:
1425:
1384:
1313:
1242:
1186:
4694:
4400:
Nevertheless, the literature on occasion still takes complete join- or meet-semilattices to be complete lattices. In this case, "completeness" denotes a restriction on the scope of the
2661:
2500:
4239:
4167:
4422:. A complete meet-semilattice in this sense is arguably the "most complete" meet-semilattice that is not necessarily a complete lattice. Indeed, a complete meet-semilattice has all
3687:
1175:
1561:
4287:
4263:
4108:
3751:
3630:
3574:
3216:
3040:
3002:
3286:
4023:
3783:
3719:
3606:
1308:
1715:
1237:
3386:
2417:
argument shows that the existence of all possible pairwise suprema (infima), as per the definition, implies the existence of all non-empty finite suprema (infima).
4052:
3931:
3788:
3184:
1796:
1767:
1741:
1529:
1146:
4076:
3991:
3902:
3882:
3862:
3658:
1676:
4715:
In addition, semilattices often serve as generators for free objects within other categories. Notably, both the forgetful functors from the category of
4832:
It is often the case that standard treatments of lattice theory define a semilattice, if that, and then say no more. See the references in the entries
2965:, hence in particular a meet-semilattice and join-semilattice: any two distinct elements have a greater and lesser one, which are their meet and join.
1567:
1181:
3291:
3936:
2447:
for more discussion on this subject. That article also discusses how we may rephrase the above definition in terms of the existence of suitable
2231:
3008:
are a bounded join-semilattice, with least element 0, although they have no greatest element: they are the smallest infinite well-ordered set.
1464:
4408:. The corresponding (unique) upper adjoint will then be a homomorphism of complete meet-semilattices. This gives rise to a number of useful
4393:. For why the existence of all possible infinite joins entails the existence of all possible infinite meets (and vice versa), see the entry
1420:
1379:
50:
3046:. It has a least element (the empty word), which is an annihilator element of the meet operation, but no greatest (identity) element.
3537:
3518:
4172:
3042:
is a (generally unbounded) meet-semilattice. Consider for example the set of finite words over some alphabet, ordered by the
3517:
In the order-theoretic formulation, these conditions just state that a homomorphism of join-semilattices is a function that
3156:
defines a join-semilattice, with join read as binary fusion. This semilattice is bounded from above by the world individual.
3221:
4719:
and frame-homomorphisms, and from the category of distributive lattices and lattice-homomorphisms, have a left adjoint.
4782:
2224:
4822:
4796:
2927:
Semilattices are employed to construct other order structures, or in conjunction with other completeness properties.
3529:
and 0 with 1—transforms this definition of a join-semilattice homomorphism into its meet-semilattice equivalent.
4624:
4412:
between the categories of all complete semilattices with morphisms preserving all meets or joins, respectively.
4450:
Cardinality-restricted notions of completeness for semilattices have been rarely considered in the literature.
43:
2634:
2473:
4874:
4370:
4212:
4113:
2217:
3663:
4394:
2444:
2086:
1151:
4814:
4788:
1535:
4409:
3549:
2718:. One can be ambivalent about the particular choice of symbol for the operation, and speak simply of
36:
4268:
4244:
4081:
3724:
3611:
3555:
3195:
4854:
4806:
4716:
1110:
934:
103:
3022:
2985:
1368:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
4869:
2177:
3265:
4419:
3996:
3756:
3692:
3579:
2547:
2429:
2414:
1843:
1297:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
1080:
73:
3837:{\displaystyle \operatorname {Id} \ f\colon \operatorname {Id} \ S\to \operatorname {Id} \ T}
2258:
1821:
1635:
indicates that the column's property is always true the row's term (at the very left), while
1090:
83:
1685:
4734:
4377:
3365:
2962:
2164:
2156:
2128:
2123:
2114:
2071:
2013:
1859:
1656:
867:
142:
4028:
3907:
3169:
2935:
is both a join- and a meet-semilattice. The interaction of these two semilattices via the
1772:
8:
2958:
2905:
may be recovered. Conversely, the order induced by the algebraically defined semilattice
2468:
2182:
2172:
2023:
1923:
1915:
1906:
1799:
A term's definition may require additional properties that are not listed in this table.
1746:
1720:
1679:
1508:
1125:
1115:
526:
108:
25:
4061:
3976:
3887:
3867:
3847:
3643:
2737:
2377:
2327:
1988:
1979:
1937:
1855:
1825:
1661:
1105:
1085:
1075:
1001:
98:
78:
68:
4840:. Moreover, there is no literature on semilattices of comparable magnitude to that on
1862:) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the
4818:
4792:
4467:
4405:
3633:
3533:
3362:. This semilattice is bounded, with the least element being the singleton partition
3016:
2980:
2503:
2448:
2250:
1829:
4731: – Mathematical ordering with upper bounds − generalization of join semilattice
4778:
4709:
4475:
4416:
4390:
3536:
with respect to the associated ordering relation. For an explanation see the entry
3130:
2969:
2819:
2674:
2512:
2437:
2403:
2008:
1885:
800:
733:
2976:
join-semilattice, as the set as a whole has a least element, hence it is bounded.
2901:
introduced in this way defines a partial ordering from which the binary operation
2033:
4471:
4459:
4436:
4055:
3637:
2947:
2943:
2932:
2842:
2452:
2262:
2100:
2094:
2081:
2061:
2052:
2018:
1955:
1892:
661:
458:
29:
4837:
3074:
2936:
2142:
1897:
1612:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
329:
1226:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
4863:
4432:
3640:-preserving complete join-homomorphisms, as follows. With a join-semilattice
2726:
2586:
2554:
2425:
2028:
1993:
1950:
1878:
1874:
1863:
1100:
1095:
267:
93:
88:
4702:
suffices to obtain the required adjunction—the morphism-part of the functor
3355:{\displaystyle \xi \vee \eta =\{P\cap Q\mid P\in \xi \ \&\ Q\in \eta \}}
4833:
4728:
4440:
4427:
4401:
3418:
3063:
3050:
3043:
2795:. In a bounded meet-semilattice, the identity 1 is the greatest element of
2202:
2133:
1967:
1833:
3966:{\displaystyle \operatorname {Id} \colon {\mathcal {S}}\to {\mathcal {A}}}
4463:
3126:
2802:
Similarly, an identity element in a join semilattice is a least element.
2730:
2610:
2192:
2187:
2076:
2040:
1882:
1809:
391:
4470:
from the category of join-semilattices (and their homomorphisms) to the
1497:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
3479:
1942:
1836:
589:
2950:, under the induced partial ordering, form a bounded join-semilattice.
1453:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
4841:
4376:
A join-semilattice is distributive if and only if the lattice of its
3397:
The above algebraic definition of a semilattice suggests a notion of
3153:
2733:
2197:
2003:
1960:
1928:
1409:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
4742: – Algebraic ring that need not have additive negative elements
4739:
4078:, and with every compactness-preserving complete join-homomorphism
3398:
2833:
is an algebraic meet-semilattice. Conversely, the meet-semilattice
2357:
1998:
206:
3521:
and least elements, if such there be. The obvious dual—replacing
2295:
1870:
4763:, Graduate Texts in Mathematics Volume 26, Springer 1976, p. 57
4499:
3501:
2741:
1932:
1839:
4426:
meets (which is equivalent to being bounded complete) and all
3192:
is a join-semilattice. In fact, the partial order is given by
3121:
are in fact the same set. Commutativity and associativity of
2451:
between related posets — an approach of special interest for
2747:
A partial order is induced on a meet-semilattice by setting
2771:. For a join-semilattice, the order is induced by setting
4495:
is constructed by taking the collection of all non-empty
4415:
Another usage of "complete meet-semilattice" refers to a
2939:
is what truly distinguishes a lattice from a semilattice.
2805:
4202:{\displaystyle K\colon {\mathcal {A}}\to {\mathcal {S}}}
4110:
between algebraic lattices we associate the restriction
2714:
in the definition just given, the structure is called a
4627:
3543:
3532:
Note that any semilattice homomorphism is necessarily
3401:
between two semilattices. Given two join-semilattices
3019:(with the single root as the least element) of height
4271:
4247:
4215:
4175:
4116:
4084:
4064:
4031:
3999:
3979:
3939:
3910:
3890:
3870:
3850:
3791:
3759:
3727:
3695:
3666:
3646:
3614:
3582:
3558:
3368:
3294:
3268:
3224:
3198:
3172:
3025:
2988:
2740:. A bounded semilattice is an idempotent commutative
2637:
2476:
1775:
1749:
1723:
1688:
1664:
1570:
1538:
1511:
1467:
1423:
1382:
1311:
1240:
1184:
1154:
1128:
3255:{\displaystyle \forall Q\in \eta ,\exists P\in \xi }
2443:
Other properties may be assumed; see the article on
4688:
4281:
4257:
4233:
4201:
4161:
4102:
4070:
4046:
4017:
3985:
3965:
3925:
3896:
3876:
3856:
3836:
3777:
3745:
3713:
3681:
3652:
3624:
3600:
3568:
3380:
3354:
3280:
3254:
3210:
3178:
3073:because a semilattice captures the essence of set
3034:
2996:
2655:
2494:
1790:
1761:
1735:
1709:
1670:
1611:
1555:
1523:
1496:
1452:
1408:
1367:
1296:
1225:
1169:
1140:
4861:
4708:can be derived from general considerations (see
3504:homomorphism, i.e. we additionally require that
1387:
4776:
3108:Two sets differing only in one or both of the:
2244:
2225:
1652:in the "Antisymmetric" column, respectively.
44:
4680:
4651:
4292:
3375:
3369:
3349:
3307:
2650:
2638:
2489:
2477:
4689:{\textstyle f'(A)=\bigvee \{f(s)|s\in A\}.}
4458:This section presupposes some knowledge of
3973:. Conversely, with every algebraic lattice
3288:and the join of two partitions is given by
3660:with zero, we associate its ideal lattice
2232:
2218:
51:
37:
4570:(more formally, to the underlying set of
2990:
1854:) is a partially ordered set which has a
4383:
3392:
3147:because a set is not a member of itself.
3113:Order in which their members are listed;
2656:{\displaystyle \langle S,\land \rangle }
2495:{\displaystyle \langle S,\land \rangle }
4805:
4478:. Therefore, the free join-semilattice
4443:. Hence Scott domains have been called
4241:defines a category equivalence between
4234:{\displaystyle (\operatorname {Id} ,K)}
4162:{\displaystyle K(f)\colon K(A)\to K(B)}
2854:in the following way: for all elements
2458:
2356:Replacing "greatest lower bound" with "
4862:
4509:ordered by subset inclusion. Clearly,
3682:{\displaystyle \operatorname {Id} \ S}
3421:of (join-) semilattices is a function
2806:Connection between the two definitions
4466:semilattices exist. For example, the
3494:both include a least element 0, then
3482:associated with each semilattice. If
4453:
3785:-semilattices, we associate the map
3576:of join-semilattices with zero with
3116:Multiplicity of one or more members,
2810:An order theoretic meet-semilattice
2314:The greatest lower bound of the set
1655:All definitions tacitly require the
4389:to partial orders that are in fact
4380:(under inclusion) is distributive.
3544:Equivalence with algebraic lattices
2360:" results in the dual concept of a
1170:{\displaystyle S\neq \varnothing :}
13:
4853:Jipsen's algebra structures page:
4784:Introduction to Lattices and Order
4274:
4250:
4194:
4184:
3958:
3948:
3617:
3561:
3478:is just a homomorphism of the two
3334:
3240:
3225:
14:
4886:
4847:
4474:of sets (and functions) admits a
1869:Semilattices can also be defined
1161:
4576:) induces a unique homomorphism
3608:-homomorphisms and the category
1640:
1627:
1556:{\displaystyle {\text{not }}aRa}
1061:
1051:
1021:
1011:
989:
979:
944:
922:
912:
877:
845:
835:
810:
778:
763:
743:
711:
701:
691:
671:
639:
619:
609:
599:
567:
547:
503:
483:
473:
436:
411:
401:
369:
344:
307:
277:
245:
183:
148:
3129:, (2). This semilattice is the
3066:of a semilattice with base set
2913:coincides with that induced by
2455:investigations of the concept.
4753:
4696:Now the obvious uniqueness of
4667:
4663:
4657:
4642:
4636:
4582:between the join-semilattices
4282:{\displaystyle {\mathcal {A}}}
4258:{\displaystyle {\mathcal {S}}}
4228:
4216:
4189:
4156:
4150:
4144:
4141:
4135:
4126:
4120:
4103:{\displaystyle f\colon A\to B}
4094:
4041:
4035:
4012:
4000:
3953:
3920:
3914:
3819:
3772:
3760:
3746:{\displaystyle f\colon S\to T}
3737:
3708:
3696:
3625:{\displaystyle {\mathcal {A}}}
3595:
3583:
3569:{\displaystyle {\mathcal {S}}}
3211:{\displaystyle \xi \leq \eta }
1648:in the "Symmetric" column and
1584:
1329:
1274:
1203:
1:
4770:
4371:distributivity (order theory)
3166:the collection of partitions
2440:, the meet of the empty set.
2428:, the join of the empty set.
1649:
1636:
1046:
1041:
1036:
1031:
1006:
974:
969:
964:
959:
954:
939:
907:
902:
897:
892:
887:
872:
860:
855:
830:
825:
820:
805:
793:
788:
773:
758:
753:
738:
726:
721:
686:
681:
666:
654:
649:
634:
629:
594:
582:
577:
562:
557:
542:
537:
532:
518:
513:
498:
493:
468:
463:
451:
446:
431:
426:
421:
396:
384:
379:
364:
359:
354:
339:
334:
322:
317:
302:
297:
292:
287:
272:
260:
255:
240:
235:
230:
225:
220:
215:
198:
193:
178:
173:
168:
163:
158:
3035:{\displaystyle \leq \omega }
2997:{\displaystyle \mathbb {N} }
2522:, such that for all members
2445:completeness in order theory
7:
4722:
4395:completeness (order theory)
2922:
2364:. The least upper bound of
10:
4891:
4815:Cambridge University Press
4789:Cambridge University Press
3281:{\displaystyle Q\subset P}
2245:Order-theoretic definition
4462:. In various situations,
4435:, where bounded complete
4293:Distributive semilattices
4169:. This defines a functor
4018:{\displaystyle (\vee ,0)}
3933:. This defines a functor
3778:{\displaystyle (\vee ,0)}
3714:{\displaystyle (\vee ,0)}
3601:{\displaystyle (\vee ,0)}
3004:, with their usual order
4746:
3884:associates the ideal of
2432:, a meet-semilattice is
16:Partial order with joins
4532:that takes any element
4690:
4564:to a join-semilattice
4445:algebraic semilattices
4283:
4259:
4235:
4203:
4163:
4104:
4072:
4048:
4019:
3987:
3967:
3927:
3898:
3878:
3858:
3844:, that with any ideal
3838:
3779:
3747:
3715:
3683:
3654:
3626:
3602:
3570:
3548:There is a well-known
3538:preservation of limits
3519:preserves binary joins
3382:
3356:
3282:
3256:
3212:
3180:
3152:Classical extensional
3056:Membership in any set
3053:is a meet-semilattice.
3036:
2998:
2848:that partially orders
2736:; i.e., a commutative
2657:
2496:
2420:A join-semilattice is
1792:
1763:
1737:
1711:
1710:{\displaystyle a,b,c,}
1672:
1613:
1557:
1525:
1498:
1454:
1410:
1369:
1298:
1227:
1171:
1142:
4691:
4544:to the singleton set
4515:can be embedded into
4410:categorical dualities
4384:Complete semilattices
4284:
4260:
4236:
4204:
4164:
4105:
4073:
4049:
4020:
3988:
3968:
3928:
3899:
3879:
3859:
3839:
3780:
3748:
3716:
3684:
3655:
3627:
3603:
3571:
3552:between the category
3393:Semilattice morphisms
3383:
3381:{\displaystyle \{S\}}
3357:
3283:
3257:
3213:
3181:
3140:It is not bounded by
3037:
2999:
2658:
2497:
1822:partially ordered set
1793:
1764:
1738:
1712:
1673:
1614:
1558:
1526:
1499:
1455:
1411:
1370:
1299:
1228:
1172:
1143:
1122:Definitions, for all
4875:Algebraic structures
4735:List of order topics
4625:
4439:cpos are studied as
4269:
4245:
4213:
4173:
4114:
4082:
4062:
4047:{\displaystyle K(A)}
4029:
3997:
3977:
3937:
3926:{\displaystyle f(I)}
3908:
3888:
3868:
3848:
3789:
3757:
3725:
3693:
3664:
3644:
3612:
3580:
3556:
3366:
3292:
3266:
3222:
3196:
3179:{\displaystyle \xi }
3170:
3023:
2986:
2963:distributive lattice
2635:
2474:
2459:Algebraic definition
2402:. Meet and join are
2296:greatest lower bound
2129:Group with operators
2072:Complemented lattice
1907:Algebraic structures
1873:: join and meet are
1860:greatest lower bound
1791:{\displaystyle aRc.}
1773:
1747:
1721:
1686:
1662:
1657:homogeneous relation
1568:
1536:
1509:
1465:
1421:
1380:
1309:
1238:
1182:
1152:
1126:
868:Strict partial order
143:Equivalence relation
4787:(second ed.).
3062:can be taken as a
2959:totally ordered set
2725:A semilattice is a
2631:A meet-semilattice
2469:algebraic structure
2183:Composition algebra
1943:Quasigroup and loop
1762:{\displaystyle bRc}
1736:{\displaystyle aRb}
1524:{\displaystyle aRa}
1141:{\displaystyle a,b}
527:Well-quasi-ordering
4811:Topology via Logic
4761:Algebraic theories
4686:
4552:Then any function
4279:
4255:
4231:
4199:
4159:
4100:
4068:
4044:
4015:
3983:
3963:
3923:
3894:
3874:
3854:
3834:
3775:
3743:
3711:
3679:
3650:
3634:algebraic lattices
3622:
3598:
3566:
3378:
3352:
3278:
3252:
3208:
3176:
3032:
3015:Any single-rooted
2994:
2653:
2492:
2453:category theoretic
2449:Galois connections
1788:
1759:
1733:
1707:
1668:
1609:
1607:
1553:
1521:
1494:
1492:
1450:
1448:
1406:
1404:
1365:
1363:
1294:
1292:
1223:
1221:
1167:
1138:
1002:Strict total order
4468:forgetful functor
4454:Free semilattices
4406:Galois connection
4391:complete lattices
4071:{\displaystyle A}
3993:we associate the
3986:{\displaystyle A}
3897:{\displaystyle T}
3877:{\displaystyle S}
3857:{\displaystyle I}
3830:
3815:
3800:
3675:
3653:{\displaystyle S}
3500:should also be a
3339:
3333:
2404:binary operations
2358:least upper bound
2276:For all elements
2259:partially ordered
2242:
2241:
1886:binary operations
1852:lower semilattice
1830:least upper bound
1818:upper semilattice
1804:
1803:
1671:{\displaystyle R}
1622:
1621:
1594:
1542:
1488:
1444:
1400:
1348:
1257:
935:Strict weak order
121:Total, Semiconnex
4882:
4828:
4802:
4779:Priestley, H. A.
4764:
4757:
4710:adjoint functors
4707:
4701:
4695:
4693:
4692:
4687:
4670:
4635:
4620:
4614:
4599:
4592:
4581:
4575:
4569:
4563:
4557:
4551:
4543:
4537:
4531:
4525:
4514:
4508:
4494:
4488:
4417:bounded complete
4367:
4352:
4342:
4332:
4318:
4312:
4288:
4286:
4285:
4280:
4278:
4277:
4264:
4262:
4261:
4256:
4254:
4253:
4240:
4238:
4237:
4232:
4208:
4206:
4205:
4200:
4198:
4197:
4188:
4187:
4168:
4166:
4165:
4160:
4109:
4107:
4106:
4101:
4077:
4075:
4074:
4069:
4056:compact elements
4053:
4051:
4050:
4045:
4024:
4022:
4021:
4016:
3992:
3990:
3989:
3984:
3972:
3970:
3969:
3964:
3962:
3961:
3952:
3951:
3932:
3930:
3929:
3924:
3903:
3901:
3900:
3895:
3883:
3881:
3880:
3875:
3863:
3861:
3860:
3855:
3843:
3841:
3840:
3835:
3828:
3813:
3798:
3784:
3782:
3781:
3776:
3752:
3750:
3749:
3744:
3720:
3718:
3717:
3712:
3688:
3686:
3685:
3680:
3673:
3659:
3657:
3656:
3651:
3631:
3629:
3628:
3623:
3621:
3620:
3607:
3605:
3604:
3599:
3575:
3573:
3572:
3567:
3565:
3564:
3528:
3524:
3513:
3499:
3493:
3487:
3477:
3468:
3434:
3416:
3408:
3387:
3385:
3384:
3379:
3361:
3359:
3358:
3353:
3337:
3331:
3287:
3285:
3284:
3279:
3261:
3259:
3258:
3253:
3217:
3215:
3214:
3209:
3191:
3185:
3183:
3182:
3177:
3165:
3146:
3139:
3131:free semilattice
3124:
3107:
3096:
3086:
3072:
3061:
3041:
3039:
3038:
3033:
3007:
3003:
3001:
3000:
2995:
2993:
2970:well-ordered set
2946:of an algebraic
2944:compact elements
2916:
2912:
2904:
2900:
2894:
2879:
2865:
2859:
2853:
2847:
2841:gives rise to a
2840:
2832:
2824:
2820:binary operation
2818:gives rise to a
2817:
2801:
2794:
2780:
2770:
2756:
2716:join-semilattice
2713:
2705:
2699:
2692:
2686:
2675:identity element
2672:
2662:
2660:
2659:
2654:
2627:
2607:
2583:
2545:
2538:
2532:
2517:
2513:binary operation
2510:
2502:consisting of a
2501:
2499:
2498:
2493:
2465:meet-semilattice
2438:greatest element
2412:
2401:
2391:
2385:
2375:
2362:join-semilattice
2353:
2342:
2335:
2325:
2309:
2293:
2287:
2281:
2270:meet-semilattice
2267:
2257:
2234:
2227:
2220:
2009:Commutative ring
1938:Rack and quandle
1903:
1902:
1866:and vice versa.
1848:meet-semilattice
1814:join-semilattice
1797:
1795:
1794:
1789:
1768:
1766:
1765:
1760:
1742:
1740:
1739:
1734:
1716:
1714:
1713:
1708:
1677:
1675:
1674:
1669:
1651:
1647:
1644:
1643:
1638:
1634:
1631:
1630:
1618:
1616:
1615:
1610:
1608:
1595:
1592:
1562:
1560:
1559:
1554:
1543:
1540:
1530:
1528:
1527:
1522:
1503:
1501:
1500:
1495:
1493:
1489:
1486:
1459:
1457:
1456:
1451:
1449:
1445:
1442:
1415:
1413:
1412:
1407:
1405:
1401:
1398:
1374:
1372:
1371:
1366:
1364:
1349:
1346:
1323:
1303:
1301:
1300:
1295:
1293:
1284:
1258:
1255:
1232:
1230:
1229:
1224:
1222:
1207:
1188:
1176:
1174:
1173:
1168:
1147:
1145:
1144:
1139:
1068:
1065:
1064:
1058:
1055:
1054:
1048:
1043:
1038:
1033:
1028:
1025:
1024:
1018:
1015:
1014:
1008:
996:
993:
992:
986:
983:
982:
976:
971:
966:
961:
956:
951:
948:
947:
941:
929:
926:
925:
919:
916:
915:
909:
904:
899:
894:
889:
884:
881:
880:
874:
862:
857:
852:
849:
848:
842:
839:
838:
832:
827:
822:
817:
814:
813:
807:
801:Meet-semilattice
795:
790:
785:
782:
781:
775:
770:
767:
766:
760:
755:
750:
747:
746:
740:
734:Join-semilattice
728:
723:
718:
715:
714:
708:
705:
704:
698:
695:
694:
688:
683:
678:
675:
674:
668:
656:
651:
646:
643:
642:
636:
631:
626:
623:
622:
616:
613:
612:
606:
603:
602:
596:
584:
579:
574:
571:
570:
564:
559:
554:
551:
550:
544:
539:
534:
529:
520:
515:
510:
507:
506:
500:
495:
490:
487:
486:
480:
477:
476:
470:
465:
453:
448:
443:
440:
439:
433:
428:
423:
418:
415:
414:
408:
405:
404:
398:
386:
381:
376:
373:
372:
366:
361:
356:
351:
348:
347:
341:
336:
324:
319:
314:
311:
310:
304:
299:
294:
289:
284:
281:
280:
274:
262:
257:
252:
249:
248:
242:
237:
232:
227:
222:
217:
212:
210:
200:
195:
190:
187:
186:
180:
175:
170:
165:
160:
155:
152:
151:
145:
63:
62:
53:
46:
39:
32:
30:binary relations
21:
20:
4890:
4889:
4885:
4884:
4883:
4881:
4880:
4879:
4860:
4859:
4850:
4825:
4807:Vickers, Steven
4799:
4773:
4768:
4767:
4758:
4754:
4749:
4725:
4703:
4697:
4666:
4628:
4626:
4623:
4622:
4616:
4601:
4594:
4583:
4577:
4571:
4565:
4559:
4553:
4545:
4539:
4533:
4527:
4516:
4510:
4503:
4490:
4479:
4460:category theory
4456:
4386:
4354:
4344:
4334:
4320:
4314:
4303:
4295:
4273:
4272:
4270:
4267:
4266:
4249:
4248:
4246:
4243:
4242:
4214:
4211:
4210:
4193:
4192:
4183:
4182:
4174:
4171:
4170:
4115:
4112:
4111:
4083:
4080:
4079:
4063:
4060:
4059:
4030:
4027:
4026:
3998:
3995:
3994:
3978:
3975:
3974:
3957:
3956:
3947:
3946:
3938:
3935:
3934:
3909:
3906:
3905:
3889:
3886:
3885:
3869:
3866:
3865:
3849:
3846:
3845:
3790:
3787:
3786:
3758:
3755:
3754:
3726:
3723:
3722:
3694:
3691:
3690:
3665:
3662:
3661:
3645:
3642:
3641:
3616:
3615:
3613:
3610:
3609:
3581:
3578:
3577:
3560:
3559:
3557:
3554:
3553:
3546:
3526:
3522:
3508:
3495:
3489:
3483:
3473:
3439:
3422:
3410:
3402:
3395:
3367:
3364:
3363:
3293:
3290:
3289:
3267:
3264:
3263:
3223:
3220:
3219:
3197:
3194:
3193:
3187:
3171:
3168:
3167:
3160:
3141:
3134:
3122:
3098:
3088:
3078:
3067:
3057:
3024:
3021:
3020:
3005:
2989:
2987:
2984:
2983:
2981:natural numbers
2925:
2914:
2906:
2902:
2898:
2881:
2880:if and only if
2867:
2861:
2855:
2849:
2845:
2843:binary relation
2834:
2826:
2822:
2811:
2808:
2796:
2782:
2772:
2758:
2748:
2711:
2703:
2694:
2688:
2678:
2668:
2636:
2633:
2632:
2615:
2591:
2559:
2540:
2534:
2523:
2515:
2506:
2475:
2472:
2471:
2461:
2407:
2393:
2387:
2381:
2365:
2344:
2337:
2331:
2315:
2299:
2289:
2283:
2277:
2265:
2263:binary relation
2253:
2247:
2238:
2209:
2208:
2207:
2178:Non-associative
2160:
2149:
2148:
2138:
2118:
2107:
2106:
2095:Map of lattices
2091:
2087:Boolean algebra
2082:Heyting algebra
2056:
2045:
2044:
2038:
2019:Integral domain
1983:
1972:
1971:
1965:
1919:
1898:absorption laws
1806:
1805:
1798:
1774:
1771:
1770:
1748:
1745:
1744:
1722:
1719:
1718:
1687:
1684:
1683:
1663:
1660:
1659:
1653:
1645:
1641:
1632:
1628:
1606:
1605:
1591:
1588:
1587:
1571:
1569:
1566:
1565:
1539:
1537:
1534:
1533:
1510:
1507:
1506:
1491:
1490:
1485:
1482:
1481:
1468:
1466:
1463:
1462:
1447:
1446:
1441:
1438:
1437:
1424:
1422:
1419:
1418:
1403:
1402:
1397:
1394:
1393:
1383:
1381:
1378:
1377:
1362:
1361:
1350:
1345:
1333:
1332:
1324:
1322:
1312:
1310:
1307:
1306:
1291:
1290:
1285:
1283:
1271:
1270:
1259:
1256: and
1254:
1241:
1239:
1236:
1235:
1220:
1219:
1208:
1206:
1200:
1199:
1185:
1183:
1180:
1179:
1153:
1150:
1149:
1127:
1124:
1123:
1066:
1062:
1056:
1052:
1026:
1022:
1016:
1012:
994:
990:
984:
980:
949:
945:
927:
923:
917:
913:
882:
878:
850:
846:
840:
836:
815:
811:
783:
779:
768:
764:
748:
744:
716:
712:
706:
702:
696:
692:
676:
672:
644:
640:
624:
620:
614:
610:
604:
600:
572:
568:
552:
548:
525:
508:
504:
488:
484:
478:
474:
459:Prewellordering
441:
437:
416:
412:
406:
402:
374:
370:
349:
345:
312:
308:
282:
278:
250:
246:
208:
205:
188:
184:
153:
149:
141:
133:
57:
24:
17:
12:
11:
5:
4888:
4878:
4877:
4872:
4870:Lattice theory
4858:
4857:
4849:
4848:External links
4846:
4838:lattice theory
4830:
4829:
4823:
4803:
4797:
4777:Davey, B. A.;
4772:
4769:
4766:
4765:
4751:
4750:
4748:
4745:
4744:
4743:
4737:
4732:
4724:
4721:
4685:
4682:
4679:
4676:
4673:
4669:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4634:
4631:
4455:
4452:
4385:
4382:
4294:
4291:
4276:
4252:
4230:
4227:
4224:
4221:
4218:
4196:
4191:
4186:
4181:
4178:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4099:
4096:
4093:
4090:
4087:
4067:
4043:
4040:
4037:
4034:
4014:
4011:
4008:
4005:
4002:
3982:
3960:
3955:
3950:
3945:
3942:
3922:
3919:
3916:
3913:
3893:
3873:
3853:
3833:
3827:
3824:
3821:
3818:
3812:
3809:
3806:
3803:
3797:
3794:
3774:
3771:
3768:
3765:
3762:
3742:
3739:
3736:
3733:
3730:
3721:-homomorphism
3710:
3707:
3704:
3701:
3698:
3678:
3672:
3669:
3649:
3619:
3597:
3594:
3591:
3588:
3585:
3563:
3545:
3542:
3515:
3514:
3470:
3469:
3394:
3391:
3390:
3389:
3377:
3374:
3371:
3351:
3348:
3345:
3342:
3336:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3277:
3274:
3271:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3207:
3204:
3201:
3175:
3157:
3149:
3148:
3118:
3117:
3114:
3110:
3109:
3075:extensionality
3054:
3047:
3031:
3028:
3013:
3012:
3011:
3010:
3009:
2992:
2955:
2951:
2940:
2937:absorption law
2924:
2921:
2807:
2804:
2702:If the symbol
2652:
2649:
2646:
2643:
2640:
2629:
2628:
2613:
2608:
2589:
2584:
2557:
2546:the following
2491:
2488:
2485:
2482:
2479:
2460:
2457:
2376:is called the
2326:is called the
2312:
2311:
2246:
2243:
2240:
2239:
2237:
2236:
2229:
2222:
2214:
2211:
2210:
2206:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2169:
2168:
2167:
2161:
2155:
2154:
2151:
2150:
2147:
2146:
2143:Linear algebra
2137:
2136:
2131:
2126:
2120:
2119:
2113:
2112:
2109:
2108:
2105:
2104:
2101:Lattice theory
2097:
2090:
2089:
2084:
2079:
2074:
2069:
2064:
2058:
2057:
2051:
2050:
2047:
2046:
2037:
2036:
2031:
2026:
2021:
2016:
2011:
2006:
2001:
1996:
1991:
1985:
1984:
1978:
1977:
1974:
1973:
1964:
1963:
1958:
1953:
1947:
1946:
1945:
1940:
1935:
1926:
1920:
1914:
1913:
1910:
1909:
1802:
1801:
1787:
1784:
1781:
1778:
1758:
1755:
1752:
1732:
1729:
1726:
1706:
1703:
1700:
1697:
1694:
1691:
1667:
1624:
1623:
1620:
1619:
1604:
1601:
1598:
1590:
1589:
1586:
1583:
1580:
1577:
1574:
1573:
1563:
1552:
1549:
1546:
1531:
1520:
1517:
1514:
1504:
1484:
1483:
1480:
1477:
1474:
1471:
1470:
1460:
1440:
1439:
1436:
1433:
1430:
1427:
1426:
1416:
1396:
1395:
1392:
1389:
1386:
1385:
1375:
1360:
1357:
1354:
1351:
1347: or
1344:
1341:
1338:
1335:
1334:
1331:
1328:
1325:
1321:
1318:
1315:
1314:
1304:
1289:
1286:
1282:
1279:
1276:
1273:
1272:
1269:
1266:
1263:
1260:
1253:
1250:
1247:
1244:
1243:
1233:
1218:
1215:
1212:
1209:
1205:
1202:
1201:
1198:
1195:
1192:
1189:
1187:
1177:
1166:
1163:
1160:
1157:
1137:
1134:
1131:
1119:
1118:
1113:
1108:
1103:
1098:
1093:
1088:
1083:
1078:
1073:
1070:
1069:
1059:
1049:
1044:
1039:
1034:
1029:
1019:
1009:
1004:
998:
997:
987:
977:
972:
967:
962:
957:
952:
942:
937:
931:
930:
920:
910:
905:
900:
895:
890:
885:
875:
870:
864:
863:
858:
853:
843:
833:
828:
823:
818:
808:
803:
797:
796:
791:
786:
776:
771:
761:
756:
751:
741:
736:
730:
729:
724:
719:
709:
699:
689:
684:
679:
669:
664:
658:
657:
652:
647:
637:
632:
627:
617:
607:
597:
592:
586:
585:
580:
575:
565:
560:
555:
545:
540:
535:
530:
522:
521:
516:
511:
501:
496:
491:
481:
471:
466:
461:
455:
454:
449:
444:
434:
429:
424:
419:
409:
399:
394:
388:
387:
382:
377:
367:
362:
357:
352:
342:
337:
332:
330:Total preorder
326:
325:
320:
315:
305:
300:
295:
290:
285:
275:
270:
264:
263:
258:
253:
243:
238:
233:
228:
223:
218:
213:
202:
201:
196:
191:
181:
176:
171:
166:
161:
156:
146:
138:
137:
135:
130:
128:
126:
124:
122:
119:
117:
115:
112:
111:
106:
101:
96:
91:
86:
81:
76:
71:
66:
59:
58:
56:
55:
48:
41:
33:
19:
18:
15:
9:
6:
4:
3:
2:
4887:
4876:
4873:
4871:
4868:
4867:
4865:
4856:
4855:Semilattices.
4852:
4851:
4845:
4843:
4839:
4835:
4826:
4824:0-521-36062-5
4820:
4816:
4812:
4808:
4804:
4800:
4798:0-521-78451-4
4794:
4790:
4786:
4785:
4780:
4775:
4774:
4762:
4759:E. G. Manes,
4756:
4752:
4741:
4738:
4736:
4733:
4730:
4727:
4726:
4720:
4718:
4713:
4711:
4706:
4700:
4683:
4677:
4674:
4671:
4660:
4654:
4648:
4645:
4639:
4632:
4629:
4619:
4612:
4608:
4604:
4597:
4590:
4586:
4580:
4574:
4568:
4562:
4556:
4549:
4542:
4536:
4530:
4526:by a mapping
4523:
4519:
4513:
4506:
4501:
4498:
4493:
4486:
4482:
4477:
4473:
4469:
4465:
4461:
4451:
4448:
4446:
4442:
4441:Scott domains
4438:
4434:
4433:domain theory
4429:
4425:
4421:
4418:
4413:
4411:
4407:
4403:
4402:homomorphisms
4398:
4396:
4392:
4381:
4379:
4374:
4372:
4365:
4361:
4357:
4351:
4347:
4341:
4337:
4331:
4327:
4323:
4317:
4310:
4306:
4301:
4290:
4225:
4222:
4219:
4179:
4176:
4153:
4147:
4138:
4132:
4129:
4123:
4117:
4097:
4091:
4088:
4085:
4065:
4057:
4038:
4032:
4025:-semilattice
4009:
4006:
4003:
3980:
3943:
3940:
3917:
3911:
3904:generated by
3891:
3871:
3851:
3831:
3825:
3822:
3816:
3810:
3807:
3804:
3801:
3795:
3792:
3769:
3766:
3763:
3740:
3734:
3731:
3728:
3705:
3702:
3699:
3676:
3670:
3667:
3647:
3639:
3635:
3592:
3589:
3586:
3551:
3541:
3539:
3535:
3530:
3520:
3511:
3507:
3506:
3505:
3503:
3498:
3492:
3486:
3481:
3476:
3466:
3462:
3458:
3454:
3450:
3446:
3442:
3438:
3437:
3436:
3433:
3429:
3425:
3420:
3414:
3406:
3400:
3372:
3346:
3343:
3340:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3304:
3301:
3298:
3295:
3275:
3272:
3269:
3249:
3246:
3243:
3237:
3234:
3231:
3228:
3205:
3202:
3199:
3190:
3173:
3163:
3158:
3155:
3151:
3150:
3144:
3137:
3132:
3128:
3120:
3119:
3115:
3112:
3111:
3105:
3101:
3095:
3091:
3085:
3081:
3076:
3070:
3065:
3060:
3055:
3052:
3048:
3045:
3029:
3026:
3018:
3014:
2982:
2978:
2977:
2975:
2972:is further a
2971:
2967:
2966:
2964:
2960:
2956:
2952:
2949:
2945:
2941:
2938:
2934:
2930:
2929:
2928:
2920:
2917:
2910:
2897:The relation
2895:
2892:
2888:
2884:
2878:
2874:
2870:
2864:
2858:
2852:
2844:
2838:
2830:
2821:
2815:
2803:
2799:
2793:
2789:
2785:
2779:
2775:
2769:
2765:
2761:
2755:
2751:
2745:
2743:
2739:
2735:
2732:
2728:
2723:
2721:
2717:
2709:
2700:
2697:
2691:
2685:
2681:
2676:
2671:
2666:
2647:
2644:
2641:
2626:
2622:
2618:
2614:
2612:
2609:
2606:
2602:
2598:
2594:
2590:
2588:
2587:Commutativity
2585:
2582:
2578:
2574:
2570:
2566:
2562:
2558:
2556:
2555:Associativity
2553:
2552:
2551:
2549:
2543:
2537:
2530:
2526:
2521:
2514:
2509:
2505:
2486:
2483:
2480:
2470:
2466:
2456:
2454:
2450:
2446:
2441:
2439:
2435:
2431:
2427:
2426:least element
2423:
2418:
2416:
2410:
2405:
2400:
2396:
2390:
2384:
2379:
2373:
2369:
2363:
2359:
2354:
2351:
2347:
2340:
2334:
2329:
2323:
2319:
2307:
2303:
2297:
2292:
2286:
2280:
2275:
2274:
2273:
2271:
2264:
2260:
2256:
2252:
2235:
2230:
2228:
2223:
2221:
2216:
2215:
2213:
2212:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2170:
2166:
2163:
2162:
2158:
2153:
2152:
2145:
2144:
2140:
2139:
2135:
2132:
2130:
2127:
2125:
2122:
2121:
2116:
2111:
2110:
2103:
2102:
2098:
2096:
2093:
2092:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2059:
2054:
2049:
2048:
2043:
2042:
2035:
2032:
2030:
2029:Division ring
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2005:
2002:
2000:
1997:
1995:
1992:
1990:
1987:
1986:
1981:
1976:
1975:
1970:
1969:
1962:
1959:
1957:
1954:
1952:
1951:Abelian group
1949:
1948:
1944:
1941:
1939:
1936:
1934:
1930:
1927:
1925:
1922:
1921:
1917:
1912:
1911:
1908:
1905:
1904:
1901:
1899:
1894:
1889:
1887:
1884:
1880:
1876:
1872:
1871:algebraically
1867:
1865:
1864:inverse order
1861:
1857:
1853:
1849:
1845:
1841:
1838:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1800:
1785:
1782:
1779:
1776:
1756:
1753:
1750:
1730:
1727:
1724:
1704:
1701:
1698:
1695:
1692:
1689:
1681:
1665:
1658:
1626:
1625:
1602:
1599:
1596:
1581:
1578:
1575:
1564:
1550:
1547:
1544:
1532:
1518:
1515:
1512:
1505:
1478:
1475:
1472:
1461:
1434:
1431:
1428:
1417:
1390:
1376:
1358:
1355:
1352:
1342:
1339:
1336:
1326:
1319:
1316:
1305:
1287:
1280:
1277:
1267:
1264:
1261:
1251:
1248:
1245:
1234:
1216:
1213:
1210:
1196:
1193:
1190:
1178:
1164:
1158:
1155:
1135:
1132:
1129:
1121:
1120:
1117:
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1081:Antisymmetric
1079:
1077:
1074:
1072:
1071:
1060:
1050:
1045:
1040:
1035:
1030:
1020:
1010:
1005:
1003:
1000:
999:
988:
978:
973:
968:
963:
958:
953:
943:
938:
936:
933:
932:
921:
911:
906:
901:
896:
891:
886:
876:
871:
869:
866:
865:
859:
854:
844:
834:
829:
824:
819:
809:
804:
802:
799:
798:
792:
787:
777:
772:
762:
757:
752:
742:
737:
735:
732:
731:
725:
720:
710:
700:
690:
685:
680:
670:
665:
663:
660:
659:
653:
648:
638:
633:
628:
618:
608:
598:
593:
591:
590:Well-ordering
588:
587:
581:
576:
566:
561:
556:
546:
541:
536:
531:
528:
524:
523:
517:
512:
502:
497:
492:
482:
472:
467:
462:
460:
457:
456:
450:
445:
435:
430:
425:
420:
410:
400:
395:
393:
390:
389:
383:
378:
368:
363:
358:
353:
343:
338:
333:
331:
328:
327:
321:
316:
306:
301:
296:
291:
286:
276:
271:
269:
268:Partial order
266:
265:
259:
254:
244:
239:
234:
229:
224:
219:
214:
211:
204:
203:
197:
192:
182:
177:
172:
167:
162:
157:
147:
144:
140:
139:
136:
131:
129:
127:
125:
123:
120:
118:
116:
114:
113:
110:
107:
105:
102:
100:
97:
95:
92:
90:
87:
85:
82:
80:
77:
75:
74:Antisymmetric
72:
70:
67:
65:
64:
61:
60:
54:
49:
47:
42:
40:
35:
34:
31:
27:
23:
22:
4834:order theory
4831:
4810:
4783:
4760:
4755:
4729:Directed set
4714:
4704:
4698:
4621:is given by
4617:
4615:Explicitly,
4610:
4606:
4602:
4595:
4588:
4584:
4578:
4572:
4566:
4560:
4554:
4547:
4540:
4534:
4528:
4521:
4517:
4511:
4504:
4496:
4491:
4484:
4480:
4476:left adjoint
4457:
4449:
4444:
4423:
4414:
4399:
4387:
4375:
4363:
4359:
4355:
4349:
4345:
4339:
4335:
4333:there exist
4329:
4325:
4321:
4315:
4308:
4304:
4300:distributive
4299:
4296:
3547:
3531:
3516:
3509:
3496:
3490:
3484:
3474:
3471:
3464:
3460:
3456:
3452:
3448:
3444:
3440:
3431:
3427:
3423:
3419:homomorphism
3412:
3404:
3396:
3188:
3161:
3159:Given a set
3142:
3135:
3125:assure (1),
3103:
3099:
3093:
3089:
3083:
3079:
3068:
3058:
3051:Scott domain
3044:prefix order
2973:
2926:
2918:
2908:
2896:
2890:
2886:
2882:
2876:
2872:
2868:
2862:
2856:
2850:
2836:
2828:
2813:
2809:
2797:
2791:
2787:
2783:
2777:
2773:
2767:
2763:
2759:
2753:
2749:
2746:
2724:
2720:semilattices
2719:
2715:
2707:
2701:
2695:
2689:
2683:
2679:
2677:1 such that
2673:includes an
2669:
2664:
2630:
2624:
2620:
2616:
2604:
2600:
2596:
2592:
2580:
2576:
2572:
2568:
2564:
2560:
2541:
2535:
2528:
2524:
2519:
2507:
2464:
2462:
2442:
2436:if it has a
2433:
2424:if it has a
2421:
2419:
2408:
2398:
2394:
2388:
2382:
2371:
2367:
2361:
2355:
2349:
2345:
2338:
2332:
2321:
2317:
2313:
2305:
2301:
2290:
2284:
2278:
2269:
2254:
2248:
2203:Hopf algebra
2141:
2134:Vector space
2099:
2066:
2039:
1968:Group theory
1966:
1931: /
1890:
1868:
1851:
1847:
1817:
1813:
1807:
1654:
1091:Well-founded
209:(Quasiorder)
84:Well-founded
4489:over a set
4302:if for all
4209:. The pair
3638:compactness
3550:equivalence
3127:idempotence
2911:, ∧⟩
2839:, ∧⟩
2831:, ∧⟩
2816:, ≤⟩
2727:commutative
2710:, replaces
2611:Idempotency
2298:of the set
2188:Lie algebra
2173:Associative
2077:Total order
2067:Semilattice
2041:Ring theory
1879:commutative
1875:associative
1824:that has a
1810:mathematics
1111:Irreflexive
392:Total order
104:Irreflexive
4864:Categories
4842:semigroups
4771:References
4600:such that
4353:such that
3480:semigroups
3435:such that
3262:such that
2825:such that
2731:idempotent
2548:identities
2392:, denoted
1883:idempotent
1832:) for any
1682:: for all
1680:transitive
1116:Asymmetric
109:Asymmetric
26:Transitive
4675:∈
4649:⋁
4437:algebraic
4424:non-empty
4190:→
4180::
4145:→
4130::
4095:→
4089::
4004:∨
3954:→
3944::
3826:
3820:→
3811:
3805::
3796:
3764:∨
3738:→
3732::
3700:∨
3689:. With a
3671:
3587:∨
3347:η
3344:∈
3335:&
3329:ξ
3326:∈
3320:∣
3314:∩
3302:η
3299:∨
3296:ξ
3273:⊂
3250:ξ
3247:∈
3241:∃
3235:η
3232:∈
3226:∀
3206:η
3203:≤
3200:ξ
3174:ξ
3154:mereology
3030:ω
3027:≤
2954:bounded.)
2781:whenever
2757:whenever
2734:semigroup
2706:, called
2651:⟩
2648:∧
2639:⟨
2518:, called
2490:⟩
2487:∧
2478:⟨
2415:induction
2413:A simple
2198:Bialgebra
2004:Near-ring
1961:Lie group
1929:Semigroup
1593:not
1585:⇒
1541:not
1476:∧
1432:∨
1330:⇒
1320:≠
1275:⇒
1204:⇒
1162:∅
1159:≠
1106:Reflexive
1101:Has meets
1096:Has joins
1086:Connected
1076:Symmetric
207:Preorder
134:reflexive
99:Reflexive
94:Has meets
89:Has joins
79:Connected
69:Symmetric
4809:(1989).
4781:(2002).
4740:Semiring
4723:See also
4633:′
4472:category
4428:directed
4362:∨
4348:≤
4338:≤
4328:∨
4324:≤
3534:monotone
3512:(0) = 0.
3399:morphism
2923:Examples
2907:⟨
2835:⟨
2827:⟨
2812:⟨
2687:for all
2343:denoted
2034:Lie ring
1999:Semiring
1834:nonempty
1650:✗
1637:✗
1047:✗
1042:✗
1037:✗
1032:✗
1007:✗
975:✗
970:✗
965:✗
960:✗
955:✗
940:✗
908:✗
903:✗
898:✗
893:✗
888:✗
873:✗
861:✗
856:✗
831:✗
826:✗
821:✗
806:✗
794:✗
789:✗
774:✗
759:✗
754:✗
739:✗
727:✗
722:✗
687:✗
682:✗
667:✗
655:✗
650:✗
635:✗
630:✗
595:✗
583:✗
578:✗
563:✗
558:✗
543:✗
538:✗
533:✗
519:✗
514:✗
499:✗
494:✗
469:✗
464:✗
452:✗
447:✗
432:✗
427:✗
422:✗
397:✗
385:✗
380:✗
365:✗
360:✗
355:✗
340:✗
335:✗
323:✗
318:✗
303:✗
298:✗
293:✗
288:✗
273:✗
261:✗
256:✗
241:✗
236:✗
231:✗
226:✗
221:✗
216:✗
199:✗
194:✗
179:✗
174:✗
169:✗
164:✗
159:✗
4558:from a
4500:subsets
4054:of all
3087:denote
2974:bounded
2948:lattice
2933:lattice
2665:bounded
2511:with a
2434:bounded
2422:bounded
2310:exists.
2261:by the
2165:Algebra
2157:Algebra
2062:Lattice
2053:Lattice
1893:lattice
1820:) is a
662:Lattice
4821:
4795:
4717:frames
4497:finite
4378:ideals
3829:
3814:
3799:
3674:
3502:monoid
3472:Hence
3338:
3332:
3097:&
3077:. Let
2742:monoid
2682:∧ 1 =
2550:hold:
2467:is an
2430:Dually
2294:, the
2193:Graded
2124:Module
2115:Module
2014:Domain
1933:Monoid
1844:Dually
1840:subset
1837:finite
1487:exists
1443:exists
1399:exists
28:
4747:Notes
4319:with
3636:with
3525:with
3133:over
3064:model
2961:is a
2571:) = (
2268:is a
2159:-like
2117:-like
2055:-like
2024:Field
1982:-like
1956:Magma
1924:Group
1918:-like
1916:Group
1769:then
132:Anti-
4836:and
4819:ISBN
4793:ISBN
4593:and
4464:free
4343:and
4313:and
4265:and
3488:and
3459:) ∨
3451:) =
3417:, a
3415:, ∨)
3409:and
3407:, ∨)
3017:tree
2979:The
2942:The
2860:and
2738:band
2708:join
2579:) ∧
2533:and
2520:meet
2386:and
2378:join
2336:and
2328:meet
2282:and
1989:Ring
1980:Ring
1858:(or
1856:meet
1850:(or
1846:, a
1826:join
1816:(or
1812:, a
1743:and
1148:and
4699:f'
4618:f'
4607:f'
4579:f'
4538:in
4502:of
4420:cpo
4364:b'
4360:a'
4346:b'
4336:a'
4058:of
3864:of
3753:of
3632:of
3218:if
3186:of
2866:in
2693:in
2667:if
2663:is
2563:∧ (
2539:of
2504:set
2406:on
2380:of
2330:of
2288:of
2272:if
2251:set
1994:Rng
1828:(a
1808:In
1717:if
1678:be
1388:min
4866::
4844:.
4817:.
4813:.
4791:.
4609:○
4605:=
4550:}.
4447:.
4397:.
4373:.
4358:=
4307:,
4289:.
4220:Id
3941:Id
3823:Id
3808:Id
3793:Id
3668:Id
3540:.
3467:).
3447:∨
3430:→
3426::
3102:∈
3092:∈
3082:∧
3049:A
3006:≤,
2968:A
2957:A
2931:A
2915:≤.
2889:∧
2885:=
2875:≤
2871:,
2790:=
2786:∨
2776:≤
2766:=
2762:∧
2752:≤
2744:.
2729:,
2722:.
2623:=
2619:∧
2603:∧
2599:=
2595:∧
2575:∧
2567:∧
2527:,
2463:A
2397:∨
2370:,
2348:∧
2320:,
2304:,
2249:A
1900:.
1891:A
1881:,
1877:,
1842:.
4827:.
4801:.
4705:F
4684:.
4681:}
4678:A
4672:s
4668:|
4664:)
4661:s
4658:(
4655:f
4652:{
4646:=
4643:)
4640:A
4637:(
4630:f
4613:.
4611:e
4603:f
4598:,
4596:T
4591:)
4589:S
4587:(
4585:F
4573:T
4567:T
4561:S
4555:f
4548:s
4546:{
4541:S
4535:s
4529:e
4524:)
4522:S
4520:(
4518:F
4512:S
4507:,
4505:S
4492:S
4487:)
4485:S
4483:(
4481:F
4366:.
4356:x
4350:b
4340:a
4330:b
4326:a
4322:x
4316:x
4311:,
4309:b
4305:a
4275:A
4251:S
4229:)
4226:K
4223:,
4217:(
4195:S
4185:A
4177:K
4157:)
4154:B
4151:(
4148:K
4142:)
4139:A
4136:(
4133:K
4127:)
4124:f
4121:(
4118:K
4098:B
4092:A
4086:f
4066:A
4042:)
4039:A
4036:(
4033:K
4013:)
4010:0
4007:,
4001:(
3981:A
3959:A
3949:S
3921:)
3918:I
3915:(
3912:f
3892:T
3872:S
3852:I
3832:T
3817:S
3802:f
3773:)
3770:0
3767:,
3761:(
3741:T
3735:S
3729:f
3709:)
3706:0
3703:,
3697:(
3677:S
3648:S
3618:A
3596:)
3593:0
3590:,
3584:(
3562:S
3527:∨
3523:∧
3510:f
3497:f
3491:T
3485:S
3475:f
3465:y
3463:(
3461:f
3457:x
3455:(
3453:f
3449:y
3445:x
3443:(
3441:f
3432:T
3428:S
3424:f
3413:T
3411:(
3405:S
3403:(
3388:.
3376:}
3373:S
3370:{
3350:}
3341:Q
3323:P
3317:Q
3311:P
3308:{
3305:=
3276:P
3270:Q
3244:P
3238:,
3229:Q
3189:S
3164:,
3162:S
3145:,
3143:L
3138:.
3136:L
3123:∧
3106:.
3104:L
3100:b
3094:L
3090:a
3084:b
3080:a
3071:,
3069:L
3059:L
2991:N
2909:S
2903:∧
2899:≤
2893:.
2891:y
2887:x
2883:x
2877:y
2873:x
2869:S
2863:y
2857:x
2851:S
2846:≤
2837:S
2829:S
2823:∧
2814:S
2800:.
2798:S
2792:y
2788:y
2784:x
2778:y
2774:x
2768:x
2764:y
2760:x
2754:y
2750:x
2712:∧
2704:∨
2698:.
2696:S
2690:x
2684:x
2680:x
2670:S
2645:,
2642:S
2625:x
2621:x
2617:x
2605:x
2601:y
2597:y
2593:x
2581:z
2577:y
2573:x
2569:z
2565:y
2561:x
2544:,
2542:S
2536:z
2531:,
2529:y
2525:x
2516:∧
2508:S
2484:,
2481:S
2411:.
2409:S
2399:y
2395:x
2389:y
2383:x
2374:}
2372:y
2368:x
2366:{
2352:.
2350:y
2346:x
2341:,
2339:y
2333:x
2324:}
2322:y
2318:x
2316:{
2308:}
2306:y
2302:x
2300:{
2291:S
2285:y
2279:x
2266:≤
2255:S
2233:e
2226:t
2219:v
1786:.
1783:c
1780:R
1777:a
1757:c
1754:R
1751:b
1731:b
1728:R
1725:a
1705:,
1702:c
1699:,
1696:b
1693:,
1690:a
1666:R
1646:Y
1633:Y
1603:a
1600:R
1597:b
1582:b
1579:R
1576:a
1551:a
1548:R
1545:a
1519:a
1516:R
1513:a
1479:b
1473:a
1435:b
1429:a
1391:S
1359:a
1356:R
1353:b
1343:b
1340:R
1337:a
1327:b
1317:a
1288:b
1281:=
1278:a
1268:a
1265:R
1262:b
1252:b
1249:R
1246:a
1217:a
1214:R
1211:b
1197:b
1194:R
1191:a
1165::
1156:S
1136:b
1133:,
1130:a
1067:Y
1057:Y
1027:Y
1017:Y
995:Y
985:Y
950:Y
928:Y
918:Y
883:Y
851:Y
841:Y
816:Y
784:Y
769:Y
749:Y
717:Y
707:Y
697:Y
677:Y
645:Y
625:Y
615:Y
605:Y
573:Y
553:Y
509:Y
489:Y
479:Y
442:Y
417:Y
407:Y
375:Y
350:Y
313:Y
283:Y
251:Y
189:Y
154:Y
52:e
45:t
38:v
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