43:
5114:
5101:
4535:
4525:
4495:
4485:
4463:
4453:
4418:
4396:
4386:
4351:
4319:
4309:
4284:
4252:
4237:
4217:
4185:
4175:
4165:
4145:
4113:
4093:
4083:
4073:
4041:
4021:
3977:
3957:
3947:
3910:
3885:
3875:
3843:
3818:
3781:
3751:
3719:
3657:
3622:
1554:
that is totally ordered for the induced order. Typically, the partially ordered set is a set of subsets of a given set that is ordered by inclusion, and the term is used for stating properties of the set of the chains. This high number of nested levels of sets explains the usefulness of the term.
4845:
4774:
5089:
4703:
4974:
4930:
4886:
5111:
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
494:
Reflexivity (1.) already follows from connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called
5044:
4941:
4897:
4856:
4785:
4714:
4658:
3045:
1798:
1506:
are totally ordered by definition. They include the rational numbers and the real numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any
1418:
ordered by the usual "less than or equal to" (≤) or "greater than or equal to" (≥) relations is totally ordered. Hence each subset of the real numbers is totally ordered, such as the
2563:
4647:
2986:
2922:
2400:
2354:
5033:
2467:
1870:
argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has a least element. Thus every finite total order is in fact a
2710:
2637:
3448:
in the product order, this relation also holds in the lexicographic order, and so on. All three can similarly be defined for the
Cartesian product of more than two sets.
3133:
2743:
2670:
2596:
2507:
1852:
3247:
3214:
2810:
2777:
4780:
2850:
2427:
1667:
of a chain is the number of inequalities (or set inclusions) between consecutive elements of the chain; that is, the number minus one of elements in the chain. Thus a
3071:
1189:
1131:
1105:
915:
484:
458:
400:
374:
341:
315:
289:
256:
5669:
5643:
5608:
5582:
5556:
5359: – the proposition, independent of ZFC, that a nonempty unbounded complete dense total order satisfying the countable chain condition is isomorphic to the reals
5187:
4709:
3161:
3100:
2876:
1163:
1079:
967:
941:
882:
856:
830:
797:
771:
738:
2949:
1237:
1050:
1006:
138:
5268:
1265:
1209:
1026:
619:
5239:
5213:
5001:
4618:
668:
426:
187:
5148:
3181:
708:
688:
642:
596:
572:
227:
207:
161:
5039:
2203:. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation
4653:
4936:
4892:
4851:
5298:
There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a
6702:
3522:
6685:
2088:
When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if
6735:
6215:
6051:
5995:
5830:
5712:
3267:, i.e. there is an algorithm for deciding which first-order statements hold for all total orders. Using interpretability in
6532:
2295:
1609:
In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its
17:
1582:
being a set of subsets; in this case, the upperbound is obtained by proving that the union of the elements of a chain in
5895:
5770:
6668:
6527:
5966:
5940:
5905:
5778:
5738:
86:
64:
57:
6522:
2995:
1741:
3433:
6158:
6240:
5702:
3515:
2817:
Intuitively, this means that the elements of the second set are added on top of the elements of the first set.
6559:
6479:
6022:
2516:
1902:ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).
1679:
of a space is often defined or characterized as the maximal length of chains of subspaces. For example, the
6344:
6273:
6153:
1715:
6247:
6235:
6198:
6173:
6148:
6102:
6071:
6017:
6012:
4623:
2160:
2133:
1680:
1546:
is sometimes defined as a synonym for a totally ordered set, but it is generally used for referring to a
6544:
6178:
6168:
6044:
3272:
2954:
2881:
2359:
2313:
1629:
977:
Asymmetry follows from transitivity and irreflexivity; moreover, irreflexivity follows from asymmetry.
31:
5007:
3466:
6517:
6183:
3508:
2432:
1637:
5280:
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a
3480:
6449:
6076:
4582:
4406:
3575:
51:
4840:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
3295:. Here are three of these possible orders, listed such that each order is stronger than the next:
2676:
2603:
2210:
There are a number of results relating properties of the order topology to the completeness of X:
6697:
6680:
3105:
2715:
2642:
2568:
2472:
1825:
533:, but refers generally to some sort of totally ordered subsets of a given partially ordered set.
3219:
3186:
2782:
2749:
6609:
6225:
5762:
5707:. Studies in Logic and the Foundations of Mathematics. Vol. 145 (1st ed.). Elsevier.
5685:
4769:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
4552:
3545:
2823:
2405:
1982:
1915:
1703:
429:
68:
6740:
6587:
6422:
6413:
6282:
6163:
6117:
6081:
6037:
5810:
5292:
5107:
indicates that the column's property is always true the row's term (at the very left), while
4562:
3555:
3299:
3050:
2278:
is complete if and only if every bounded set that is closed in the order topology is compact.
1919:
1551:
1396:
1168:
1110:
1084:
894:
463:
437:
379:
353:
320:
294:
268:
235:
5648:
5622:
5587:
5561:
5535:
5157:
3140:
3079:
2855:
1212:
1142:
1058:
946:
920:
861:
835:
809:
776:
750:
717:
6675:
6634:
6624:
6614:
6359:
6322:
6312:
6292:
6277:
5299:
5128:
4339:
3614:
3459:
3264:
2927:
1855:
1434:) "initial example" of a totally ordered set with a certain property, (here, a total order
1222:
1035:
991:
741:
575:
123:
5244:
1250:
1194:
1011:
604:
8:
6745:
6602:
6513:
6459:
6418:
6408:
6297:
6230:
6193:
5698:
5356:
5307:
5288:
5271:
A term's definition may require additional properties that are not listed in this table.
5218:
5192:
5151:
4980:
4597:
4587:
3998:
3580:
3497:
1711:
1645:
1134:
885:
800:
647:
405:
344:
166:
6714:
6641:
6494:
6403:
6393:
6334:
6252:
6188:
5515:
5480:
5349: – generalization of the notion of prefix of a string, and of the notion of a tree
5133:
4577:
4557:
4547:
4473:
3570:
3550:
3540:
3166:
1633:
1599:
1516:
1472:
The integers form an initial non-empty totally ordered set with neither an upper nor a
1330:
970:
693:
673:
627:
581:
557:
487:
259:
212:
192:
146:
6554:
3287:
There are several ways to take two totally ordered sets and extend to an order on the
598:
in which any two distinct elements are comparable. That is, a strict total order is a
6651:
6629:
6489:
6474:
6454:
6257:
5991:
5962:
5936:
5911:
5901:
5826:
5774:
5744:
5734:
5708:
5507:
3288:
3268:
3260:
2142:
1961:
1898:
natural numbers. Hence it is common to index finite total orders or well orders with
1614:
1610:
1508:
1431:
1400:
1244:
1216:
622:
141:
5484:
2228:
is connected under the order topology if and only if it is complete and there is no
1890:
of the natural numbers ordered by <. In other words, a total order on a set with
6464:
6317:
5891:
5878:
5855:
5818:
5470:
4272:
4205:
2283:
1883:
1875:
1692:
1676:
1427:
537:
6646:
6429:
6307:
6302:
6287:
6203:
6112:
6097:
5985:
5981:
5817:. Lecture Notes in Computer Science. Vol. 2500. Springer. pp. 207–230.
5329:
4133:
3930:
3501:
2164:
2153:
1887:
1732:
1688:
1684:
1641:
1563:
1419:
1303:
599:
118:
6564:
6549:
6539:
6398:
6376:
6354:
5946:
5860:
5681:
5084:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
3801:
2082:
1879:
1820:
1495:
1307:
4698:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
6729:
6663:
6619:
6597:
6469:
6339:
6327:
6132:
5822:
5511:
5320:
5281:
4572:
4567:
3739:
3565:
3560:
3378:
3292:
3276:
2299:
2291:
2287:
1707:
1668:
1603:
1503:
1491:
114:
2119:
The order topology induced by a total order may be shown to be hereditarily
6484:
6366:
6349:
6267:
6107:
6060:
5498:
Ganapathy, Jayanthi (1992). "Maximal
Elements and Upper Bounds in Posets".
5346:
5334:
5303:
3452:
2120:
1672:
1591:
1415:
5475:
5462:
3282:
1632:
if every descending chain eventually stabilizes. For example, an order is
1578:
contains at least one maximal element. Zorn's lemma is commonly used with
1465:
The natural numbers form an initial non-empty totally ordered set with no
6690:
6383:
6262:
5340:
2188:
2168:
2146:
2138:
1965:
1911:
1710:
of polynomials. Another example is the use of "chain" as a synonym for a
1696:
1595:
1484:
1473:
1466:
102:
5519:
4969:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
3436:
of the corresponding strict total orders). This is also a partial order.
6658:
6592:
6433:
5728:
5461:
Strohmeier, Alfred; Genillard, Christian; Weber, Mats (1 August 1990).
5365:
4061:
1899:
1871:
1652:
1640:
means that every ascending chain eventually stabilizes. For example, a
1490:
The real numbers form an initial unbounded totally ordered set that is
1408:
1311:
117:
in which any two elements are comparable. That is, a total order is a
4925:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
6709:
6582:
6388:
4881:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
2176:
1958:
1480:
1404:
5846:
Macpherson, H. Dugald (2011), "A survey of homogeneous structures",
5323: – ring that satisfies the descending chain condition on ideals
2116:(in this case they happen to be identical but will not in general).
1964:
between two totally ordered sets that respects the two orders is an
6504:
6371:
6122:
5885:. Colloquium Publications. Vol. 25. Providence: Am. Math. Soc.
3678:
3440:
Each of these orders extends the next in the sense that if we have
2196:
2078:
1923:
1874:. Either by direct proof or by observing that every well order is
1867:
1423:
536:
An extension of a given partial order to a total order is called a
5987:
Relations and Graphs: Discrete
Mathematics for Computer Scientists
5790:
that is, beyond some index, all further sequence members are equal
1479:
The rational numbers form an initial totally ordered set which is
1483:
in the real numbers. Moreover, the reflexive reduction < is a
5915:
5900:. Cambridge Mathematical Textbooks. Cambridge University Press.
5748:
5733:. Cambridge Mathematical Textbooks. Cambridge University Press.
5415:
5413:
6029:
1547:
1277:
3481:
defines a strict weak order and a corresponding total preorder
2469:, which is called the sum of the two orders or sometimes just
2290:. Examples are the closed intervals of real numbers, e.g. the
1625:, depending whether the sequence is increasing or decreasing.
1566:
which asserts that, if every chain in a partially ordered set
5444:
5442:
5440:
5410:
2163:(not to be confused with being "total") do not carry over to
1731:
One may define a totally ordered set as a particular kind of
1702:"Chain" may also be used for some totally ordered subsets of
1706:
that are not partially ordered sets. An example is given by
2282:
A totally ordered set (with its order topology) which is a
5437:
5933:
Lattice theory: first concepts and distributive lattices.
5385:
5383:
1636:
if it has the descending chain condition. Similarly, the
1590:. This is the way that is generally used to prove that a
5460:
2298:(extended real number line). There are order-preserving
1430:. Each of these can be shown to be the unique (up to an
5400:
5398:
3283:
Orders on the
Cartesian product of totally ordered sets
2951:
are pairwise disjoint, then the natural total order on
1286:
is totally ordered for the restriction of the order on
30:"Linear order" redirects here. Not to be confused with
5380:
5651:
5625:
5590:
5564:
5538:
5247:
5221:
5195:
5160:
5136:
5042:
5010:
4983:
4939:
4895:
4854:
4783:
4712:
4656:
4626:
4600:
3222:
3189:
3169:
3143:
3108:
3082:
3053:
2998:
2957:
2930:
2884:
2858:
2826:
2785:
2752:
2718:
2679:
2645:
2606:
2571:
2519:
2475:
2435:
2408:
2362:
2316:
2100:
greater than we might refer to the order topology on
1828:
1744:
1450:
has the property, there is an order isomorphism from
1253:
1225:
1197:
1171:
1145:
1113:
1087:
1061:
1038:
1014:
994:
949:
923:
897:
864:
838:
812:
779:
753:
720:
696:
676:
650:
630:
607:
584:
560:
543:
466:
440:
408:
382:
356:
323:
297:
271:
238:
215:
195:
169:
149:
126:
5395:
5361:
Pages displaying wikidata descriptions as a fallback
5351:
Pages displaying wikidata descriptions as a fallback
5325:
Pages displaying wikidata descriptions as a fallback
1515:
The letters of the alphabet ordered by the standard
980:
For delimitation purposes, a total order as defined
5302:. Forgetting the location of the ends results in a
5663:
5637:
5602:
5576:
5550:
5262:
5233:
5207:
5181:
5142:
5083:
5027:
4995:
4968:
4924:
4880:
4839:
4768:
4697:
4641:
4612:
3241:
3208:
3175:
3155:
3127:
3094:
3065:
3039:
2980:
2943:
2916:
2870:
2844:
2804:
2771:
2737:
2704:
2664:
2631:
2590:
2557:
2501:
2461:
2421:
2394:
2348:
1846:
1792:
1259:
1231:
1203:
1183:
1157:
1125:
1099:
1073:
1044:
1020:
1000:
961:
935:
909:
876:
850:
824:
791:
765:
732:
702:
682:
662:
636:
613:
590:
566:
478:
452:
420:
394:
368:
335:
309:
283:
250:
221:
201:
181:
155:
132:
5610:by transitivity, which contradicts irreflexivity.
5425:
2598:holds if and only if one of the following holds:
1613:. In this case, a chain can be identified with a
6727:
5980:
5729:Brian A. Davey and Hilary Ann Priestley (1990).
5419:
4859:
2159:is not. In other words, the various concepts of
1512:ordered field is isomorphic to the real numbers.
5343: – Mathematical version of an order change
1926:being maps which respect the orders, i.e. maps
5889:
5697:
5448:
1882:one may show that every finite total order is
6045:
5124:in the "Antisymmetric" column, respectively.
3516:
3040:{\displaystyle x,y\in \bigcup _{i\in I}A_{i}}
2852:is a totally ordered index set, and for each
1793:{\displaystyle \{a\vee b,a\wedge b\}=\{a,b\}}
1655:are considered. In this case, one talks of a
2077:We can use these open intervals to define a
1894:elements induces a bijection with the first
1787:
1775:
1769:
1745:
1562:for referring to totally ordered subsets is
1052:that can be defined in two equivalent ways:
6703:Positive cone of a partially ordered group
6052:
6038:
5971:
5957:John G. Hocking and Gail S. Young (1961).
5845:
3523:
3509:
5859:
5497:
5474:
3291:, though the resulting order may only be
988:order. For each (non-strict) total order
87:Learn how and when to remove this message
6686:Positive cone of an ordered vector space
5877:
5799:Davey and Priestly 1990, Def.2.24, p. 37
5404:
1651:In other contexts, only chains that are
644:, which satisfies the following for all
163:, which satisfies the following for all
50:This article includes a list of general
5368: – Class of mathematical orderings
1648:satisfy the ascending chain condition.
509:A set equipped with a total order is a
27:Order whose elements are all comparable
14:
6728:
5945:
5389:
3476:real variables defined on a subset of
2558:{\displaystyle x,y\in A_{1}\cup A_{2}}
1861:
6033:
5922:
5808:
5680:This definition resembles that of an
5558:, assume for contradiction that also
5431:
3486:
2137:if every nonempty subset that has an
1403:of a family of totally ordered sets,
529:is sometimes defined as a synonym of
5815:Automata, Logics, and Infinite Games
5463:"Ordering of characters and strings"
5306:. Forgetting both data results in a
5127:All definitions tacitly require the
2296:affinely extended real number system
2131:A totally ordered set is said to be
36:
5925:Partially Ordered Algebraic Systems
5291:with a compatible total order is a
4642:{\displaystyle S\neq \varnothing :}
1854:. Hence a totally ordered set is a
1721:
1695:is the maximal length of chains of
1683:is the maximal length of chains of
1295:The unique order on the empty set,
24:
6213:Properties & Types (
5897:Introduction to Lattices and Order
5771:Undergraduate Texts in Mathematics
5731:Introduction to Lattices and Order
3467:examples of partially ordered sets
2924:is a linear order, where the sets
2310:For any two disjoint total orders
1905:
1671:is a chain of length zero, and an
544:Strict and non-strict total orders
56:it lacks sufficient corresponding
25:
6757:
6669:Positive cone of an ordered field
6005:
5961:Corrected reprint, Dover, 1988.
4633:
2981:{\displaystyle \bigcup _{i}A_{i}}
2917:{\displaystyle (A_{i},\leq _{i})}
2395:{\displaystyle (A_{2},\leq _{2})}
2349:{\displaystyle (A_{1},\leq _{1})}
2305:
1971:
1726:
6523:Ordered topological vector space
6059:
5353:– a downward total partial order
5112:
5099:
5028:{\displaystyle {\text{not }}aRa}
4533:
4523:
4493:
4483:
4461:
4451:
4416:
4394:
4384:
4349:
4317:
4307:
4282:
4250:
4235:
4215:
4183:
4173:
4163:
4143:
4111:
4091:
4081:
4071:
4039:
4019:
3975:
3955:
3945:
3908:
3883:
3873:
3841:
3816:
3779:
3749:
3717:
3655:
3620:
3432:) (the reflexive closure of the
3279:total orders is also decidable.
1628:A partially ordered set has the
1008:there is an associated relation
41:
5839:
5802:
5793:
5784:
5756:
5722:
5691:
5674:
3254:
2462:{\displaystyle A_{1}\cup A_{2}}
2126:
1558:A common example of the use of
1267:is a (non-strict) total order.
6736:Properties of binary relations
5972:Rosenstein, Joseph G. (1982).
5613:
5526:
5491:
5454:
5120:in the "Symmetric" column and
5056:
4801:
4746:
4675:
2911:
2885:
2839:
2827:
2389:
2363:
2343:
2317:
1735:, namely one in which we have
1675:is a chain of length one. The
1533:etc., is a strict total order.
1339:to a totally ordered set then
13:
1:
6480:Series-parallel partial order
5871:
5811:"Decidability of S1S and S2S"
5337: – Branch of mathematics
5121:
5108:
4518:
4513:
4508:
4503:
4478:
4446:
4441:
4436:
4431:
4426:
4411:
4379:
4374:
4369:
4364:
4359:
4344:
4332:
4327:
4302:
4297:
4292:
4277:
4265:
4260:
4245:
4230:
4225:
4210:
4198:
4193:
4158:
4153:
4138:
4126:
4121:
4106:
4101:
4066:
4054:
4049:
4034:
4029:
4014:
4009:
4004:
3990:
3985:
3970:
3965:
3940:
3935:
3923:
3918:
3903:
3898:
3893:
3868:
3856:
3851:
3836:
3831:
3826:
3811:
3806:
3794:
3789:
3774:
3769:
3764:
3759:
3744:
3732:
3727:
3712:
3707:
3702:
3697:
3692:
3687:
3670:
3665:
3650:
3645:
3640:
3635:
3630:
1444:for a property, if, whenever
6159:Cantor's isomorphism theorem
5984:; Ströhlein, Thomas (1993).
5420:Schmidt & Ströhlein 1993
2705:{\displaystyle x,y\in A_{2}}
2632:{\displaystyle x,y\in A_{1}}
1976:For any totally ordered set
1910:Totally ordered sets form a
1345:induces a total ordering on
7:
6199:Szpilrajn extension theorem
6174:Hausdorff maximal principle
6149:Boolean prime ideal theorem
6018:Encyclopedia of Mathematics
5990:. Berlin: Springer-Verlag.
5976:. New York: Academic Press.
5313:
3458:, each of these make it an
3381:). This is a partial order.
3128:{\displaystyle x\leq _{i}y}
2738:{\displaystyle x\leq _{2}y}
2665:{\displaystyle x\leq _{1}y}
2591:{\displaystyle x\leq _{+}y}
2502:{\displaystyle A_{1}+A_{2}}
2402:, there is a natural order
2171:a property of the relation
2152:is complete but the set of
1847:{\displaystyle a=a\wedge b}
1681:dimension of a vector space
1270:
10:
6762:
6545:Topological vector lattice
5861:10.1016/j.disc.2011.01.024
5449:Davey & Priestley 1990
3263:theory of total orders is
3242:{\displaystyle y\in A_{j}}
3209:{\displaystyle x\in A_{i}}
2805:{\displaystyle y\in A_{2}}
2772:{\displaystyle x\in A_{1}}
2199:(also called supremum) in
2145:. For example, the set of
2108:and the order topology on
1630:descending chain condition
1411:, is itself a total order.
1310:(more strongly, these are
32:Linear order (linguistics)
29:
6575:
6503:
6442:
6212:
6141:
6090:
6067:
3342:). This is a total order.
2845:{\displaystyle (I,\leq )}
2422:{\displaystyle \leq _{+}}
2214:If the order topology on
2207:to the rational numbers.
1638:ascending chain condition
1537:
1280:of a totally ordered set
490:, formerly called total).
6154:Cantor–Bernstein theorem
5823:10.1007/3-540-36387-4_12
5373:
2302:between these examples.
2167:. For example, over the
2092:is the natural numbers,
2081:on any ordered set, the
1487:on the rational numbers.
1247:of a strict total order
981:
525:are also used. The term
6698:Partially ordered group
6518:Specialization preorder
5931:George Grätzer (1971).
3066:{\displaystyle x\leq y}
1659:, often shortened as a
1184:{\displaystyle b\leq a}
1126:{\displaystyle a\neq b}
1100:{\displaystyle a\leq b}
910:{\displaystyle a\neq b}
540:of that partial order.
479:{\displaystyle b\leq a}
453:{\displaystyle a\leq b}
395:{\displaystyle b\leq a}
369:{\displaystyle a\leq b}
336:{\displaystyle a\leq c}
310:{\displaystyle b\leq c}
284:{\displaystyle a\leq b}
251:{\displaystyle a\leq a}
71:more precise citations.
6184:Kruskal's tree theorem
6179:Knaster–Tarski theorem
6169:Dushnik–Miller theorem
5953:. Princeton: Nostrand.
5935:W. H. Freeman and Co.
5763:Yiannis N. Moschovakis
5665:
5664:{\displaystyle a<a}
5639:
5638:{\displaystyle a<a}
5604:
5603:{\displaystyle a<a}
5578:
5577:{\displaystyle b<a}
5552:
5551:{\displaystyle a<b}
5467:ACM SIGAda Ada Letters
5264:
5235:
5209:
5183:
5182:{\displaystyle a,b,c,}
5144:
5085:
5029:
4997:
4970:
4926:
4882:
4841:
4770:
4699:
4643:
4614:
3243:
3210:
3177:
3157:
3156:{\displaystyle i<j}
3129:
3096:
3095:{\displaystyle i\in I}
3067:
3041:
2982:
2945:
2918:
2872:
2871:{\displaystyle i\in I}
2846:
2806:
2773:
2739:
2706:
2666:
2633:
2592:
2559:
2503:
2463:
2423:
2396:
2350:
1920:partially ordered sets
1848:
1794:
1570:has an upper bound in
1261:
1233:
1205:
1185:
1159:
1158:{\displaystyle a<b}
1127:
1101:
1075:
1074:{\displaystyle a<b}
1046:
1022:
1002:
963:
962:{\displaystyle b<a}
937:
936:{\displaystyle a<b}
911:
878:
877:{\displaystyle a<c}
852:
851:{\displaystyle b<c}
826:
825:{\displaystyle a<b}
793:
792:{\displaystyle b<a}
767:
766:{\displaystyle a<b}
734:
733:{\displaystyle a<a}
704:
684:
664:
638:
615:
592:
568:
480:
454:
422:
396:
370:
337:
311:
285:
252:
223:
203:
183:
157:
134:
6013:"Totally ordered set"
5892:Priestley, Hilary Ann
5666:
5640:
5605:
5579:
5553:
5500:Pi Mu Epsilon Journal
5476:10.1145/101120.101136
5293:totally ordered group
5265:
5236:
5210:
5184:
5145:
5086:
5030:
4998:
4971:
4927:
4883:
4842:
4771:
4700:
4644:
4615:
4594:Definitions, for all
3300:Lexicographical order
3244:
3211:
3178:
3158:
3130:
3097:
3076:Either there is some
3068:
3042:
2983:
2946:
2944:{\displaystyle A_{i}}
2919:
2873:
2847:
2807:
2774:
2740:
2707:
2667:
2634:
2593:
2560:
2504:
2464:
2424:
2397:
2351:
2236:(a gap is two points
1849:
1795:
1552:partially ordered set
1397:lexicographical order
1262:
1234:
1232:{\displaystyle \leq }
1206:
1186:
1160:
1128:
1102:
1076:
1047:
1045:{\displaystyle \leq }
1023:
1003:
1001:{\displaystyle \leq }
964:
938:
912:
879:
853:
827:
794:
768:
735:
705:
685:
665:
639:
616:
593:
569:
481:
455:
423:
397:
371:
338:
312:
286:
253:
224:
204:
184:
158:
135:
133:{\displaystyle \leq }
6676:Ordered vector space
5848:Discrete Mathematics
5809:Weyer, Mark (2002).
5649:
5623:
5588:
5562:
5536:
5300:betweenness relation
5263:{\displaystyle aRc.}
5245:
5219:
5193:
5158:
5134:
5129:homogeneous relation
5040:
5008:
4981:
4937:
4893:
4852:
4781:
4710:
4654:
4624:
4598:
4340:Strict partial order
3615:Equivalence relation
3460:ordered vector space
3273:monadic second-order
3220:
3187:
3167:
3141:
3106:
3080:
3051:
2996:
2955:
2928:
2882:
2856:
2824:
2783:
2750:
2716:
2677:
2643:
2604:
2569:
2517:
2473:
2433:
2406:
2360:
2314:
1856:distributive lattice
1826:
1742:
1663:. In this case, the
1260:{\displaystyle <}
1251:
1223:
1204:{\displaystyle <}
1195:
1169:
1143:
1111:
1085:
1059:
1036:
1021:{\displaystyle <}
1012:
992:
984:is sometimes called
947:
921:
895:
862:
836:
810:
777:
751:
718:
694:
674:
648:
628:
614:{\displaystyle <}
605:
582:
576:strict partial order
558:
519:linearly ordered set
464:
438:
406:
380:
354:
321:
295:
269:
236:
213:
193:
167:
147:
124:
18:Chain (order theory)
6514:Alexandrov topology
6460:Lexicographic order
6419:Well-quasi-ordering
5767:Notes on set theory
5704:Theory of Relations
5308:separation relation
5234:{\displaystyle bRc}
5208:{\displaystyle aRb}
4996:{\displaystyle aRa}
4613:{\displaystyle a,b}
3999:Well-quasi-ordering
3472:A real function of
2820:More generally, if
1862:Finite total orders
1617:, and is called an
1299:, is a total order.
1135:reflexive reduction
663:{\displaystyle a,b}
531:totally ordered set
511:totally ordered set
421:{\displaystyle a=b}
182:{\displaystyle a,b}
6495:Transitive closure
6455:Converse/Transpose
6164:Dilworth's theorem
5661:
5635:
5600:
5574:
5548:
5260:
5231:
5205:
5179:
5140:
5081:
5079:
5025:
4993:
4966:
4964:
4922:
4920:
4878:
4876:
4837:
4835:
4766:
4764:
4695:
4693:
4639:
4610:
4474:Strict total order
3487:Related structures
3400:) if and only if (
3239:
3206:
3173:
3153:
3137:or there are some
3125:
3092:
3063:
3037:
3026:
2978:
2967:
2941:
2914:
2868:
2842:
2802:
2769:
2735:
2702:
2662:
2629:
2588:
2555:
2499:
2459:
2419:
2392:
2346:
1980:we can define the
1968:in this category.
1844:
1790:
1331:injective function
1257:
1229:
1201:
1181:
1155:
1123:
1097:
1071:
1042:
1030:strict total order
1018:
998:
959:
933:
907:
874:
848:
822:
789:
763:
730:
700:
680:
660:
634:
611:
588:
564:
551:strict total order
515:simply ordered set
488:strongly connected
476:
450:
418:
392:
366:
333:
307:
281:
248:
219:
199:
179:
153:
130:
6723:
6722:
6681:Partially ordered
6490:Symmetric closure
6475:Reflexive closure
6218:
5997:978-3-642-77970-1
5927:. Pergamon Press.
5923:Fuchs, L (1963).
5890:Davey, Brian A.;
5879:Birkhoff, Garrett
5854:(15): 1599–1634,
5832:978-3-540-00388-5
5714:978-0-444-50542-2
5701:(December 2000).
5276:
5275:
5143:{\displaystyle R}
5094:
5093:
5066:
5014:
4960:
4916:
4872:
4820:
4729:
4407:Strict weak order
3593:Total, Semiconnex
3361:) if and only if
3318:) if and only if
3289:Cartesian product
3176:{\displaystyle I}
3011:
2958:
2197:least upper bound
2143:least upper bound
2096:is less than and
1615:monotone sequence
1509:Dedekind-complete
1432:order isomorphism
1401:Cartesian product
1245:reflexive closure
703:{\displaystyle X}
683:{\displaystyle c}
637:{\displaystyle X}
591:{\displaystyle X}
567:{\displaystyle X}
222:{\displaystyle X}
202:{\displaystyle c}
156:{\displaystyle X}
97:
96:
89:
16:(Redirected from
6753:
6465:Linear extension
6214:
6194:Mirsky's theorem
6054:
6047:
6040:
6031:
6030:
6026:
6001:
5982:Schmidt, Gunther
5977:
5974:Linear orderings
5954:
5951:Naive Set Theory
5928:
5919:
5886:
5865:
5864:
5863:
5843:
5837:
5836:
5806:
5800:
5797:
5791:
5788:
5782:
5760:
5754:
5752:
5726:
5720:
5718:
5695:
5689:
5688:, but is weaker.
5678:
5672:
5670:
5668:
5667:
5662:
5644:
5642:
5641:
5636:
5617:
5611:
5609:
5607:
5606:
5601:
5583:
5581:
5580:
5575:
5557:
5555:
5554:
5549:
5530:
5524:
5523:
5495:
5489:
5488:
5478:
5458:
5452:
5446:
5435:
5429:
5423:
5417:
5408:
5402:
5393:
5387:
5362:
5357:Suslin's problem
5352:
5326:
5269:
5267:
5266:
5261:
5240:
5238:
5237:
5232:
5214:
5212:
5211:
5206:
5188:
5186:
5185:
5180:
5149:
5147:
5146:
5141:
5123:
5119:
5116:
5115:
5110:
5106:
5103:
5102:
5090:
5088:
5087:
5082:
5080:
5067:
5064:
5034:
5032:
5031:
5026:
5015:
5012:
5002:
5000:
4999:
4994:
4975:
4973:
4972:
4967:
4965:
4961:
4958:
4931:
4929:
4928:
4923:
4921:
4917:
4914:
4887:
4885:
4884:
4879:
4877:
4873:
4870:
4846:
4844:
4843:
4838:
4836:
4821:
4818:
4795:
4775:
4773:
4772:
4767:
4765:
4756:
4730:
4727:
4704:
4702:
4701:
4696:
4694:
4679:
4660:
4648:
4646:
4645:
4640:
4619:
4617:
4616:
4611:
4540:
4537:
4536:
4530:
4527:
4526:
4520:
4515:
4510:
4505:
4500:
4497:
4496:
4490:
4487:
4486:
4480:
4468:
4465:
4464:
4458:
4455:
4454:
4448:
4443:
4438:
4433:
4428:
4423:
4420:
4419:
4413:
4401:
4398:
4397:
4391:
4388:
4387:
4381:
4376:
4371:
4366:
4361:
4356:
4353:
4352:
4346:
4334:
4329:
4324:
4321:
4320:
4314:
4311:
4310:
4304:
4299:
4294:
4289:
4286:
4285:
4279:
4273:Meet-semilattice
4267:
4262:
4257:
4254:
4253:
4247:
4242:
4239:
4238:
4232:
4227:
4222:
4219:
4218:
4212:
4206:Join-semilattice
4200:
4195:
4190:
4187:
4186:
4180:
4177:
4176:
4170:
4167:
4166:
4160:
4155:
4150:
4147:
4146:
4140:
4128:
4123:
4118:
4115:
4114:
4108:
4103:
4098:
4095:
4094:
4088:
4085:
4084:
4078:
4075:
4074:
4068:
4056:
4051:
4046:
4043:
4042:
4036:
4031:
4026:
4023:
4022:
4016:
4011:
4006:
4001:
3992:
3987:
3982:
3979:
3978:
3972:
3967:
3962:
3959:
3958:
3952:
3949:
3948:
3942:
3937:
3925:
3920:
3915:
3912:
3911:
3905:
3900:
3895:
3890:
3887:
3886:
3880:
3877:
3876:
3870:
3858:
3853:
3848:
3845:
3844:
3838:
3833:
3828:
3823:
3820:
3819:
3813:
3808:
3796:
3791:
3786:
3783:
3782:
3776:
3771:
3766:
3761:
3756:
3753:
3752:
3746:
3734:
3729:
3724:
3721:
3720:
3714:
3709:
3704:
3699:
3694:
3689:
3684:
3682:
3672:
3667:
3662:
3659:
3658:
3652:
3647:
3642:
3637:
3632:
3627:
3624:
3623:
3617:
3535:
3534:
3525:
3518:
3511:
3504:
3502:binary relations
3493:
3492:
3483:on that subset.
3248:
3246:
3245:
3240:
3238:
3237:
3215:
3213:
3212:
3207:
3205:
3204:
3182:
3180:
3179:
3174:
3162:
3160:
3159:
3154:
3134:
3132:
3131:
3126:
3121:
3120:
3101:
3099:
3098:
3093:
3072:
3070:
3069:
3064:
3046:
3044:
3043:
3038:
3036:
3035:
3025:
2987:
2985:
2984:
2979:
2977:
2976:
2966:
2950:
2948:
2947:
2942:
2940:
2939:
2923:
2921:
2920:
2915:
2910:
2909:
2897:
2896:
2877:
2875:
2874:
2869:
2851:
2849:
2848:
2843:
2811:
2809:
2808:
2803:
2801:
2800:
2778:
2776:
2775:
2770:
2768:
2767:
2744:
2742:
2741:
2736:
2731:
2730:
2711:
2709:
2708:
2703:
2701:
2700:
2671:
2669:
2668:
2663:
2658:
2657:
2638:
2636:
2635:
2630:
2628:
2627:
2597:
2595:
2594:
2589:
2584:
2583:
2564:
2562:
2561:
2556:
2554:
2553:
2541:
2540:
2508:
2506:
2505:
2500:
2498:
2497:
2485:
2484:
2468:
2466:
2465:
2460:
2458:
2457:
2445:
2444:
2428:
2426:
2425:
2420:
2418:
2417:
2401:
2399:
2398:
2393:
2388:
2387:
2375:
2374:
2355:
2353:
2352:
2347:
2342:
2341:
2329:
2328:
2284:complete lattice
2206:
2174:
2154:rational numbers
2115:
2107:
2099:
2095:
2072:
2063:
2041:
2019:
1979:
1912:full subcategory
1884:order isomorphic
1876:order isomorphic
1853:
1851:
1850:
1845:
1799:
1797:
1796:
1791:
1722:Further concepts
1693:commutative ring
1685:linear subspaces
1644:is a ring whose
1623:descending chain
1589:
1585:
1581:
1577:
1573:
1569:
1532:
1517:dictionary order
1498:(defined below).
1461:
1455:
1449:
1439:
1428:rational numbers
1409:well ordered set
1391:
1366:
1350:
1344:
1338:
1328:
1322:
1304:cardinal numbers
1298:
1291:
1285:
1266:
1264:
1263:
1258:
1243:Conversely, the
1238:
1236:
1235:
1230:
1210:
1208:
1207:
1202:
1190:
1188:
1187:
1182:
1164:
1162:
1161:
1156:
1132:
1130:
1129:
1124:
1106:
1104:
1103:
1098:
1080:
1078:
1077:
1072:
1051:
1049:
1048:
1043:
1032:associated with
1027:
1025:
1024:
1019:
1007:
1005:
1004:
999:
968:
966:
965:
960:
942:
940:
939:
934:
916:
914:
913:
908:
883:
881:
880:
875:
857:
855:
854:
849:
831:
829:
828:
823:
798:
796:
795:
790:
772:
770:
769:
764:
739:
737:
736:
731:
709:
707:
706:
701:
689:
687:
686:
681:
669:
667:
666:
661:
643:
641:
640:
635:
620:
618:
617:
612:
597:
595:
594:
589:
573:
571:
570:
565:
538:linear extension
485:
483:
482:
477:
459:
457:
456:
451:
427:
425:
424:
419:
401:
399:
398:
393:
375:
373:
372:
367:
342:
340:
339:
334:
316:
314:
313:
308:
290:
288:
287:
282:
257:
255:
254:
249:
228:
226:
225:
220:
208:
206:
205:
200:
188:
186:
185:
180:
162:
160:
159:
154:
139:
137:
136:
131:
92:
85:
81:
78:
72:
67:this article by
58:inline citations
45:
44:
37:
21:
6761:
6760:
6756:
6755:
6754:
6752:
6751:
6750:
6726:
6725:
6724:
6719:
6715:Young's lattice
6571:
6499:
6438:
6288:Heyting algebra
6236:Boolean algebra
6208:
6189:Laver's theorem
6137:
6103:Boolean algebra
6098:Binary relation
6086:
6063:
6058:
6011:
6008:
5998:
5947:Halmos, Paul R.
5908:
5874:
5869:
5868:
5844:
5840:
5833:
5807:
5803:
5798:
5794:
5789:
5785:
5761:
5757:
5741:
5727:
5723:
5715:
5696:
5692:
5679:
5675:
5650:
5647:
5646:
5624:
5621:
5620:
5618:
5614:
5589:
5586:
5585:
5563:
5560:
5559:
5537:
5534:
5533:
5531:
5527:
5496:
5492:
5459:
5455:
5447:
5438:
5430:
5426:
5418:
5411:
5403:
5396:
5388:
5381:
5376:
5371:
5360:
5350:
5330:Countryman line
5324:
5316:
5278:
5277:
5270:
5246:
5243:
5242:
5220:
5217:
5216:
5194:
5191:
5190:
5159:
5156:
5155:
5135:
5132:
5131:
5125:
5117:
5113:
5104:
5100:
5078:
5077:
5063:
5060:
5059:
5043:
5041:
5038:
5037:
5011:
5009:
5006:
5005:
4982:
4979:
4978:
4963:
4962:
4957:
4954:
4953:
4940:
4938:
4935:
4934:
4919:
4918:
4913:
4910:
4909:
4896:
4894:
4891:
4890:
4875:
4874:
4869:
4866:
4865:
4855:
4853:
4850:
4849:
4834:
4833:
4822:
4817:
4805:
4804:
4796:
4794:
4784:
4782:
4779:
4778:
4763:
4762:
4757:
4755:
4743:
4742:
4731:
4728: and
4726:
4713:
4711:
4708:
4707:
4692:
4691:
4680:
4678:
4672:
4671:
4657:
4655:
4652:
4651:
4625:
4622:
4621:
4599:
4596:
4595:
4538:
4534:
4528:
4524:
4498:
4494:
4488:
4484:
4466:
4462:
4456:
4452:
4421:
4417:
4399:
4395:
4389:
4385:
4354:
4350:
4322:
4318:
4312:
4308:
4287:
4283:
4255:
4251:
4240:
4236:
4220:
4216:
4188:
4184:
4178:
4174:
4168:
4164:
4148:
4144:
4116:
4112:
4096:
4092:
4086:
4082:
4076:
4072:
4044:
4040:
4024:
4020:
3997:
3980:
3976:
3960:
3956:
3950:
3946:
3931:Prewellordering
3913:
3909:
3888:
3884:
3878:
3874:
3846:
3842:
3821:
3817:
3784:
3780:
3754:
3750:
3722:
3718:
3680:
3677:
3660:
3656:
3625:
3621:
3613:
3605:
3529:
3496:
3489:
3451:Applied to the
3285:
3257:
3233:
3229:
3221:
3218:
3217:
3200:
3196:
3188:
3185:
3184:
3168:
3165:
3164:
3142:
3139:
3138:
3116:
3112:
3107:
3104:
3103:
3081:
3078:
3077:
3052:
3049:
3048:
3031:
3027:
3015:
2997:
2994:
2993:
2988:is defined by
2972:
2968:
2962:
2956:
2953:
2952:
2935:
2931:
2929:
2926:
2925:
2905:
2901:
2892:
2888:
2883:
2880:
2879:
2857:
2854:
2853:
2825:
2822:
2821:
2796:
2792:
2784:
2781:
2780:
2763:
2759:
2751:
2748:
2747:
2726:
2722:
2717:
2714:
2713:
2696:
2692:
2678:
2675:
2674:
2653:
2649:
2644:
2641:
2640:
2623:
2619:
2605:
2602:
2601:
2579:
2575:
2570:
2567:
2566:
2549:
2545:
2536:
2532:
2518:
2515:
2514:
2493:
2489:
2480:
2476:
2474:
2471:
2470:
2453:
2449:
2440:
2436:
2434:
2431:
2430:
2413:
2409:
2407:
2404:
2403:
2383:
2379:
2370:
2366:
2361:
2358:
2357:
2337:
2333:
2324:
2320:
2315:
2312:
2311:
2308:
2204:
2172:
2129:
2113:
2105:
2097:
2093:
2067:
2045:
2023:
1989:
1977:
1974:
1908:
1906:Category theory
1888:initial segment
1864:
1827:
1824:
1823:
1743:
1740:
1739:
1729:
1724:
1689:Krull dimension
1642:Noetherian ring
1619:ascending chain
1587:
1583:
1579:
1575:
1571:
1567:
1540:
1520:
1457:
1456:to a subset of
1451:
1445:
1435:
1420:natural numbers
1389:
1378:
1368:
1367:if and only if
1365:
1358:
1352:
1346:
1340:
1334:
1324:
1323:is any set and
1318:
1308:ordinal numbers
1296:
1287:
1281:
1273:
1252:
1249:
1248:
1224:
1221:
1220:
1196:
1193:
1192:
1170:
1167:
1166:
1144:
1141:
1140:
1112:
1109:
1108:
1086:
1083:
1082:
1060:
1057:
1056:
1037:
1034:
1033:
1013:
1010:
1009:
993:
990:
989:
948:
945:
944:
922:
919:
918:
896:
893:
892:
863:
860:
859:
837:
834:
833:
811:
808:
807:
778:
775:
774:
752:
749:
748:
719:
716:
715:
695:
692:
691:
675:
672:
671:
649:
646:
645:
629:
626:
625:
606:
603:
602:
600:binary relation
583:
580:
579:
559:
556:
555:
546:
465:
462:
461:
439:
436:
435:
407:
404:
403:
381:
378:
377:
355:
352:
351:
322:
319:
318:
296:
293:
292:
270:
267:
266:
237:
234:
233:
214:
211:
210:
194:
191:
190:
168:
165:
164:
148:
145:
144:
125:
122:
121:
119:binary relation
93:
82:
76:
73:
63:Please help to
62:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
6759:
6749:
6748:
6743:
6738:
6721:
6720:
6718:
6717:
6712:
6707:
6706:
6705:
6695:
6694:
6693:
6688:
6683:
6673:
6672:
6671:
6661:
6656:
6655:
6654:
6649:
6642:Order morphism
6639:
6638:
6637:
6627:
6622:
6617:
6612:
6607:
6606:
6605:
6595:
6590:
6585:
6579:
6577:
6573:
6572:
6570:
6569:
6568:
6567:
6562:
6560:Locally convex
6557:
6552:
6542:
6540:Order topology
6537:
6536:
6535:
6533:Order topology
6530:
6520:
6510:
6508:
6501:
6500:
6498:
6497:
6492:
6487:
6482:
6477:
6472:
6467:
6462:
6457:
6452:
6446:
6444:
6440:
6439:
6437:
6436:
6426:
6416:
6411:
6406:
6401:
6396:
6391:
6386:
6381:
6380:
6379:
6369:
6364:
6363:
6362:
6357:
6352:
6347:
6345:Chain-complete
6337:
6332:
6331:
6330:
6325:
6320:
6315:
6310:
6300:
6295:
6290:
6285:
6280:
6270:
6265:
6260:
6255:
6250:
6245:
6244:
6243:
6233:
6228:
6222:
6220:
6210:
6209:
6207:
6206:
6201:
6196:
6191:
6186:
6181:
6176:
6171:
6166:
6161:
6156:
6151:
6145:
6143:
6139:
6138:
6136:
6135:
6130:
6125:
6120:
6115:
6110:
6105:
6100:
6094:
6092:
6088:
6087:
6085:
6084:
6079:
6074:
6068:
6065:
6064:
6057:
6056:
6049:
6042:
6034:
6028:
6027:
6007:
6006:External links
6004:
6003:
6002:
5996:
5978:
5969:
5955:
5943:
5929:
5920:
5906:
5887:
5883:Lattice Theory
5873:
5870:
5867:
5866:
5838:
5831:
5801:
5792:
5783:
5755:
5739:
5721:
5713:
5699:Roland Fraïssé
5690:
5682:initial object
5673:
5660:
5657:
5654:
5634:
5631:
5628:
5612:
5599:
5596:
5593:
5573:
5570:
5567:
5547:
5544:
5541:
5525:
5506:(7): 462–464.
5490:
5453:
5436:
5424:
5409:
5394:
5378:
5377:
5375:
5372:
5370:
5369:
5363:
5354:
5344:
5338:
5332:
5327:
5317:
5315:
5312:
5274:
5273:
5259:
5256:
5253:
5250:
5230:
5227:
5224:
5204:
5201:
5198:
5178:
5175:
5172:
5169:
5166:
5163:
5139:
5096:
5095:
5092:
5091:
5076:
5073:
5070:
5062:
5061:
5058:
5055:
5052:
5049:
5046:
5045:
5035:
5024:
5021:
5018:
5003:
4992:
4989:
4986:
4976:
4956:
4955:
4952:
4949:
4946:
4943:
4942:
4932:
4912:
4911:
4908:
4905:
4902:
4899:
4898:
4888:
4868:
4867:
4864:
4861:
4858:
4857:
4847:
4832:
4829:
4826:
4823:
4819: or
4816:
4813:
4810:
4807:
4806:
4803:
4800:
4797:
4793:
4790:
4787:
4786:
4776:
4761:
4758:
4754:
4751:
4748:
4745:
4744:
4741:
4738:
4735:
4732:
4725:
4722:
4719:
4716:
4715:
4705:
4690:
4687:
4684:
4681:
4677:
4674:
4673:
4670:
4667:
4664:
4661:
4659:
4649:
4638:
4635:
4632:
4629:
4609:
4606:
4603:
4591:
4590:
4585:
4580:
4575:
4570:
4565:
4560:
4555:
4550:
4545:
4542:
4541:
4531:
4521:
4516:
4511:
4506:
4501:
4491:
4481:
4476:
4470:
4469:
4459:
4449:
4444:
4439:
4434:
4429:
4424:
4414:
4409:
4403:
4402:
4392:
4382:
4377:
4372:
4367:
4362:
4357:
4347:
4342:
4336:
4335:
4330:
4325:
4315:
4305:
4300:
4295:
4290:
4280:
4275:
4269:
4268:
4263:
4258:
4248:
4243:
4233:
4228:
4223:
4213:
4208:
4202:
4201:
4196:
4191:
4181:
4171:
4161:
4156:
4151:
4141:
4136:
4130:
4129:
4124:
4119:
4109:
4104:
4099:
4089:
4079:
4069:
4064:
4058:
4057:
4052:
4047:
4037:
4032:
4027:
4017:
4012:
4007:
4002:
3994:
3993:
3988:
3983:
3973:
3968:
3963:
3953:
3943:
3938:
3933:
3927:
3926:
3921:
3916:
3906:
3901:
3896:
3891:
3881:
3871:
3866:
3860:
3859:
3854:
3849:
3839:
3834:
3829:
3824:
3814:
3809:
3804:
3802:Total preorder
3798:
3797:
3792:
3787:
3777:
3772:
3767:
3762:
3757:
3747:
3742:
3736:
3735:
3730:
3725:
3715:
3710:
3705:
3700:
3695:
3690:
3685:
3674:
3673:
3668:
3663:
3653:
3648:
3643:
3638:
3633:
3628:
3618:
3610:
3609:
3607:
3602:
3600:
3598:
3596:
3594:
3591:
3589:
3587:
3584:
3583:
3578:
3573:
3568:
3563:
3558:
3553:
3548:
3543:
3538:
3531:
3530:
3528:
3527:
3520:
3513:
3505:
3491:
3490:
3488:
3485:
3438:
3437:
3434:direct product
3382:
3343:
3284:
3281:
3256:
3253:
3252:
3251:
3250:
3249:
3236:
3232:
3228:
3225:
3203:
3199:
3195:
3192:
3172:
3152:
3149:
3146:
3135:
3124:
3119:
3115:
3111:
3091:
3088:
3085:
3062:
3059:
3056:
3034:
3030:
3024:
3021:
3018:
3014:
3010:
3007:
3004:
3001:
2975:
2971:
2965:
2961:
2938:
2934:
2913:
2908:
2904:
2900:
2895:
2891:
2887:
2878:the structure
2867:
2864:
2861:
2841:
2838:
2835:
2832:
2829:
2815:
2814:
2813:
2812:
2799:
2795:
2791:
2788:
2766:
2762:
2758:
2755:
2745:
2734:
2729:
2725:
2721:
2699:
2695:
2691:
2688:
2685:
2682:
2672:
2661:
2656:
2652:
2648:
2626:
2622:
2618:
2615:
2612:
2609:
2587:
2582:
2578:
2574:
2552:
2548:
2544:
2539:
2535:
2531:
2528:
2525:
2522:
2496:
2492:
2488:
2483:
2479:
2456:
2452:
2448:
2443:
2439:
2416:
2412:
2391:
2386:
2382:
2378:
2373:
2369:
2365:
2345:
2340:
2336:
2332:
2327:
2323:
2319:
2307:
2306:Sums of orders
2304:
2300:homeomorphisms
2280:
2279:
2273:
2223:
2218:is connected,
2175:is that every
2128:
2125:
2083:order topology
2075:
2074:
2065:
2043:
2021:
1983:open intervals
1973:
1972:Order topology
1970:
1907:
1904:
1863:
1860:
1843:
1840:
1837:
1834:
1831:
1821:if and only if
1812:We then write
1810:
1809:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1728:
1727:Lattice theory
1725:
1723:
1720:
1708:regular chains
1611:opposite order
1604:maximal ideals
1539:
1536:
1535:
1534:
1513:
1504:Ordered fields
1501:
1500:
1499:
1496:order topology
1488:
1477:
1470:
1412:
1393:
1387:
1376:
1363:
1356:
1315:
1300:
1293:
1272:
1269:
1256:
1241:
1240:
1228:
1200:
1180:
1177:
1174:
1154:
1151:
1148:
1138:
1122:
1119:
1116:
1096:
1093:
1090:
1070:
1067:
1064:
1041:
1017:
997:
975:
974:
958:
955:
952:
932:
929:
926:
906:
903:
900:
889:
873:
870:
867:
847:
844:
841:
821:
818:
815:
804:
788:
785:
782:
762:
759:
756:
745:
729:
726:
723:
699:
679:
659:
656:
653:
633:
610:
587:
563:
552:
545:
542:
492:
491:
475:
472:
469:
449:
446:
443:
433:
417:
414:
411:
391:
388:
385:
365:
362:
359:
348:
332:
329:
326:
306:
303:
300:
280:
277:
274:
263:
247:
244:
241:
218:
198:
178:
175:
172:
152:
129:
95:
94:
49:
47:
40:
26:
9:
6:
4:
3:
2:
6758:
6747:
6744:
6742:
6739:
6737:
6734:
6733:
6731:
6716:
6713:
6711:
6708:
6704:
6701:
6700:
6699:
6696:
6692:
6689:
6687:
6684:
6682:
6679:
6678:
6677:
6674:
6670:
6667:
6666:
6665:
6664:Ordered field
6662:
6660:
6657:
6653:
6650:
6648:
6645:
6644:
6643:
6640:
6636:
6633:
6632:
6631:
6628:
6626:
6623:
6621:
6620:Hasse diagram
6618:
6616:
6613:
6611:
6608:
6604:
6601:
6600:
6599:
6598:Comparability
6596:
6594:
6591:
6589:
6586:
6584:
6581:
6580:
6578:
6574:
6566:
6563:
6561:
6558:
6556:
6553:
6551:
6548:
6547:
6546:
6543:
6541:
6538:
6534:
6531:
6529:
6526:
6525:
6524:
6521:
6519:
6515:
6512:
6511:
6509:
6506:
6502:
6496:
6493:
6491:
6488:
6486:
6483:
6481:
6478:
6476:
6473:
6471:
6470:Product order
6468:
6466:
6463:
6461:
6458:
6456:
6453:
6451:
6448:
6447:
6445:
6443:Constructions
6441:
6435:
6431:
6427:
6424:
6420:
6417:
6415:
6412:
6410:
6407:
6405:
6402:
6400:
6397:
6395:
6392:
6390:
6387:
6385:
6382:
6378:
6375:
6374:
6373:
6370:
6368:
6365:
6361:
6358:
6356:
6353:
6351:
6348:
6346:
6343:
6342:
6341:
6340:Partial order
6338:
6336:
6333:
6329:
6328:Join and meet
6326:
6324:
6321:
6319:
6316:
6314:
6311:
6309:
6306:
6305:
6304:
6301:
6299:
6296:
6294:
6291:
6289:
6286:
6284:
6281:
6279:
6275:
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6249:
6246:
6242:
6239:
6238:
6237:
6234:
6232:
6229:
6227:
6226:Antisymmetric
6224:
6223:
6221:
6217:
6211:
6205:
6202:
6200:
6197:
6195:
6192:
6190:
6187:
6185:
6182:
6180:
6177:
6175:
6172:
6170:
6167:
6165:
6162:
6160:
6157:
6155:
6152:
6150:
6147:
6146:
6144:
6140:
6134:
6133:Weak ordering
6131:
6129:
6126:
6124:
6121:
6119:
6118:Partial order
6116:
6114:
6111:
6109:
6106:
6104:
6101:
6099:
6096:
6095:
6093:
6089:
6083:
6080:
6078:
6075:
6073:
6070:
6069:
6066:
6062:
6055:
6050:
6048:
6043:
6041:
6036:
6035:
6032:
6024:
6020:
6019:
6014:
6010:
6009:
5999:
5993:
5989:
5988:
5983:
5979:
5975:
5970:
5968:
5967:0-486-65676-4
5964:
5960:
5956:
5952:
5948:
5944:
5942:
5941:0-7167-0442-0
5938:
5934:
5930:
5926:
5921:
5917:
5913:
5909:
5907:0-521-36766-2
5903:
5899:
5898:
5893:
5888:
5884:
5880:
5876:
5875:
5862:
5857:
5853:
5849:
5842:
5834:
5828:
5824:
5820:
5816:
5812:
5805:
5796:
5787:
5781:, p. 116
5780:
5779:0-387-28723-X
5776:
5773:(Birkhäuser)
5772:
5768:
5764:
5759:
5750:
5746:
5742:
5740:0-521-36766-2
5736:
5732:
5725:
5716:
5710:
5706:
5705:
5700:
5694:
5687:
5683:
5677:
5671:by asymmetry.
5658:
5655:
5652:
5632:
5629:
5626:
5616:
5597:
5594:
5591:
5571:
5568:
5565:
5545:
5542:
5539:
5529:
5521:
5517:
5513:
5509:
5505:
5501:
5494:
5486:
5482:
5477:
5472:
5468:
5464:
5457:
5450:
5445:
5443:
5441:
5433:
5428:
5422:, p. 32.
5421:
5416:
5414:
5406:
5405:Birkhoff 1967
5401:
5399:
5391:
5386:
5384:
5379:
5367:
5364:
5358:
5355:
5348:
5345:
5342:
5339:
5336:
5333:
5331:
5328:
5322:
5321:Artinian ring
5319:
5318:
5311:
5309:
5305:
5301:
5296:
5294:
5290:
5285:
5283:
5282:partial order
5272:
5257:
5254:
5251:
5248:
5228:
5225:
5222:
5202:
5199:
5196:
5176:
5173:
5170:
5167:
5164:
5161:
5153:
5137:
5130:
5098:
5097:
5074:
5071:
5068:
5053:
5050:
5047:
5036:
5022:
5019:
5016:
5004:
4990:
4987:
4984:
4977:
4950:
4947:
4944:
4933:
4906:
4903:
4900:
4889:
4862:
4848:
4830:
4827:
4824:
4814:
4811:
4808:
4798:
4791:
4788:
4777:
4759:
4752:
4749:
4739:
4736:
4733:
4723:
4720:
4717:
4706:
4688:
4685:
4682:
4668:
4665:
4662:
4650:
4636:
4630:
4627:
4607:
4604:
4601:
4593:
4592:
4589:
4586:
4584:
4581:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4554:
4553:Antisymmetric
4551:
4549:
4546:
4544:
4543:
4532:
4522:
4517:
4512:
4507:
4502:
4492:
4482:
4477:
4475:
4472:
4471:
4460:
4450:
4445:
4440:
4435:
4430:
4425:
4415:
4410:
4408:
4405:
4404:
4393:
4383:
4378:
4373:
4368:
4363:
4358:
4348:
4343:
4341:
4338:
4337:
4331:
4326:
4316:
4306:
4301:
4296:
4291:
4281:
4276:
4274:
4271:
4270:
4264:
4259:
4249:
4244:
4234:
4229:
4224:
4214:
4209:
4207:
4204:
4203:
4197:
4192:
4182:
4172:
4162:
4157:
4152:
4142:
4137:
4135:
4132:
4131:
4125:
4120:
4110:
4105:
4100:
4090:
4080:
4070:
4065:
4063:
4062:Well-ordering
4060:
4059:
4053:
4048:
4038:
4033:
4028:
4018:
4013:
4008:
4003:
4000:
3996:
3995:
3989:
3984:
3974:
3969:
3964:
3954:
3944:
3939:
3934:
3932:
3929:
3928:
3922:
3917:
3907:
3902:
3897:
3892:
3882:
3872:
3867:
3865:
3862:
3861:
3855:
3850:
3840:
3835:
3830:
3825:
3815:
3810:
3805:
3803:
3800:
3799:
3793:
3788:
3778:
3773:
3768:
3763:
3758:
3748:
3743:
3741:
3740:Partial order
3738:
3737:
3731:
3726:
3716:
3711:
3706:
3701:
3696:
3691:
3686:
3683:
3676:
3675:
3669:
3664:
3654:
3649:
3644:
3639:
3634:
3629:
3619:
3616:
3612:
3611:
3608:
3603:
3601:
3599:
3597:
3595:
3592:
3590:
3588:
3586:
3585:
3582:
3579:
3577:
3574:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3546:Antisymmetric
3544:
3542:
3539:
3537:
3536:
3533:
3532:
3526:
3521:
3519:
3514:
3512:
3507:
3506:
3503:
3499:
3495:
3494:
3484:
3482:
3479:
3475:
3470:
3468:
3463:
3461:
3457:
3454:
3449:
3447:
3443:
3435:
3431:
3427:
3423:
3419:
3415:
3411:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3380:
3379:product order
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3341:
3337:
3333:
3329:
3325:
3321:
3317:
3313:
3309:
3305:
3301:
3298:
3297:
3296:
3294:
3290:
3280:
3278:
3274:
3270:
3266:
3262:
3234:
3230:
3226:
3223:
3201:
3197:
3193:
3190:
3170:
3150:
3147:
3144:
3136:
3122:
3117:
3113:
3109:
3089:
3086:
3083:
3075:
3074:
3060:
3057:
3054:
3032:
3028:
3022:
3019:
3016:
3012:
3008:
3005:
3002:
2999:
2991:
2990:
2989:
2973:
2969:
2963:
2959:
2936:
2932:
2906:
2902:
2898:
2893:
2889:
2865:
2862:
2859:
2836:
2833:
2830:
2818:
2797:
2793:
2789:
2786:
2764:
2760:
2756:
2753:
2746:
2732:
2727:
2723:
2719:
2697:
2693:
2689:
2686:
2683:
2680:
2673:
2659:
2654:
2650:
2646:
2624:
2620:
2616:
2613:
2610:
2607:
2600:
2599:
2585:
2580:
2576:
2572:
2550:
2546:
2542:
2537:
2533:
2529:
2526:
2523:
2520:
2512:
2511:
2510:
2494:
2490:
2486:
2481:
2477:
2454:
2450:
2446:
2441:
2437:
2414:
2410:
2384:
2380:
2376:
2371:
2367:
2338:
2334:
2330:
2325:
2321:
2303:
2301:
2297:
2293:
2292:unit interval
2289:
2285:
2277:
2274:
2271:
2267:
2263:
2259:
2256:such that no
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2224:
2221:
2217:
2213:
2212:
2211:
2208:
2202:
2198:
2194:
2190:
2186:
2182:
2178:
2170:
2166:
2162:
2158:
2155:
2151:
2148:
2144:
2140:
2136:
2135:
2124:
2122:
2117:
2111:
2103:
2091:
2086:
2084:
2080:
2071:
2066:
2061:
2057:
2053:
2049:
2044:
2039:
2035:
2031:
2027:
2022:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1988:
1987:
1986:
1985:
1984:
1969:
1967:
1963:
1960:
1955:
1953:
1949:
1945:
1941:
1937:
1933:
1930:such that if
1929:
1925:
1921:
1917:
1913:
1903:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1869:
1859:
1857:
1841:
1838:
1835:
1832:
1829:
1822:
1819:
1815:
1807:
1803:
1784:
1781:
1778:
1772:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1738:
1737:
1736:
1734:
1719:
1717:
1713:
1709:
1705:
1700:
1698:
1694:
1690:
1686:
1682:
1678:
1674:
1670:
1669:singleton set
1666:
1662:
1658:
1654:
1649:
1647:
1643:
1639:
1635:
1631:
1626:
1624:
1620:
1616:
1612:
1607:
1605:
1601:
1597:
1593:
1565:
1561:
1556:
1553:
1549:
1545:
1531:
1527:
1523:
1518:
1514:
1511:
1510:
1505:
1502:
1497:
1493:
1489:
1486:
1482:
1478:
1475:
1471:
1468:
1464:
1463:
1460:
1454:
1448:
1443:
1438:
1433:
1429:
1425:
1421:
1417:
1413:
1410:
1406:
1402:
1398:
1394:
1386:
1382:
1375:
1371:
1362:
1355:
1349:
1343:
1337:
1332:
1327:
1321:
1316:
1313:
1309:
1305:
1301:
1294:
1290:
1284:
1279:
1275:
1274:
1268:
1254:
1246:
1226:
1218:
1214:
1198:
1178:
1175:
1172:
1152:
1149:
1146:
1139:
1136:
1120:
1117:
1114:
1094:
1091:
1088:
1068:
1065:
1062:
1055:
1054:
1053:
1039:
1031:
1028:, called the
1015:
995:
987:
983:
978:
972:
956:
953:
950:
930:
927:
924:
904:
901:
898:
890:
887:
871:
868:
865:
845:
842:
839:
819:
816:
813:
805:
802:
786:
783:
780:
760:
757:
754:
746:
743:
727:
724:
721:
713:
712:
711:
697:
677:
657:
654:
651:
631:
624:
608:
601:
585:
577:
561:
553:
550:
541:
539:
534:
532:
528:
524:
520:
516:
512:
507:
505:
501:
497:
489:
473:
470:
467:
447:
444:
441:
434:
431:
430:antisymmetric
415:
412:
409:
389:
386:
383:
363:
360:
357:
349:
346:
330:
327:
324:
304:
301:
298:
278:
275:
272:
264:
261:
245:
242:
239:
232:
231:
230:
216:
196:
176:
173:
170:
150:
143:
127:
120:
116:
115:partial order
112:
108:
104:
99:
91:
88:
80:
77:February 2016
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
6741:Order theory
6507:& Orders
6485:Star product
6414:Well-founded
6367:Prefix order
6323:Distributive
6313:Complemented
6283:Foundational
6248:Completeness
6204:Zorn's lemma
6127:
6108:Cyclic order
6091:Key concepts
6061:Order theory
6016:
5986:
5973:
5958:
5950:
5932:
5924:
5896:
5882:
5851:
5847:
5841:
5814:
5804:
5795:
5786:
5766:
5758:
5753:Here: p. 100
5730:
5724:
5703:
5693:
5676:
5615:
5528:
5503:
5499:
5493:
5466:
5456:
5451:, p. 3.
5434:, p. 2.
5427:
5407:, p. 2.
5347:Prefix order
5335:Order theory
5304:cyclic order
5297:
5286:
5279:
5126:
4563:Well-founded
3863:
3681:(Quasiorder)
3556:Well-founded
3477:
3473:
3471:
3464:
3455:
3453:vector space
3450:
3445:
3441:
3439:
3429:
3425:
3421:
3417:
3413:
3409:
3405:
3401:
3397:
3393:
3389:
3385:
3374:
3370:
3366:
3362:
3358:
3354:
3350:
3346:
3339:
3335:
3331:
3327:
3323:
3319:
3315:
3311:
3307:
3303:
3286:
3258:
3255:Decidability
2819:
2816:
2309:
2281:
2275:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2225:
2222:is complete.
2219:
2215:
2209:
2200:
2192:
2184:
2180:
2169:real numbers
2165:restrictions
2161:completeness
2156:
2149:
2147:real numbers
2132:
2130:
2127:Completeness
2118:
2109:
2101:
2089:
2087:
2076:
2069:
2059:
2055:
2051:
2047:
2037:
2033:
2029:
2025:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1981:
1975:
1956:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1909:
1895:
1891:
1865:
1817:
1813:
1811:
1805:
1801:
1730:
1701:
1697:prime ideals
1673:ordered pair
1664:
1660:
1657:finite chain
1656:
1650:
1634:well founded
1627:
1622:
1618:
1608:
1592:vector space
1564:Zorn's lemma
1559:
1557:
1543:
1541:
1529:
1525:
1521:
1507:
1458:
1452:
1446:
1441:
1436:
1416:real numbers
1384:
1380:
1373:
1369:
1360:
1353:
1347:
1341:
1335:
1325:
1319:
1288:
1282:
1242:
1029:
985:
979:
976:
549:
547:
535:
530:
526:
522:
518:
514:
513:; the terms
510:
508:
503:
499:
495:
493:
111:linear order
110:
106:
100:
98:
83:
74:
55:
6691:Riesz space
6652:Isomorphism
6528:Normal cone
6450:Composition
6384:Semilattice
6293:Homogeneous
6278:Equivalence
6128:Total order
5719:Here: p. 35
5390:Halmos 1968
5341:Permutation
4583:Irreflexive
3864:Total order
3576:Irreflexive
3261:first-order
2429:on the set
2189:upper bound
2139:upper bound
2112:induced by
2104:induced by
1966:isomorphism
1922:, with the
1653:finite sets
1598:and that a
1596:Hamel bases
1485:dense order
1474:lower bound
1467:upper bound
1414:The set of
1351:by setting
1312:well-orders
1302:Any set of
742:irreflexive
504:full orders
107:total order
103:mathematics
69:introducing
6746:Set theory
6730:Categories
6659:Order type
6593:Cofinality
6434:Well-order
6409:Transitive
6298:Idempotent
6231:Asymmetric
5872:References
5645:, the not
5432:Fuchs 1963
5366:Well-order
5154:: for all
5152:transitive
4588:Asymmetric
3581:Asymmetric
3498:Transitive
3275:theory of
3073:holds if:
2294:, and the
2260:satisfies
2068:(−∞, ∞) =
1900:order type
1872:well order
1704:structures
1687:, and the
1213:complement
986:non-strict
886:transitive
801:asymmetric
345:transitive
52:references
6710:Upper set
6647:Embedding
6583:Antichain
6404:Tolerance
6394:Symmetric
6389:Semiorder
6335:Reflexive
6253:Connected
6023:EMS Press
5959:Topology.
5512:0031-952X
5469:(7): 84.
5065:not
5057:⇒
5013:not
4948:∧
4904:∨
4802:⇒
4792:≠
4747:⇒
4676:⇒
4634:∅
4631:≠
4578:Reflexive
4573:Has meets
4568:Has joins
4558:Connected
4548:Symmetric
3679:Preorder
3606:reflexive
3571:Reflexive
3566:Has meets
3561:Has joins
3551:Connected
3541:Symmetric
3465:See also
3277:countable
3265:decidable
3227:∈
3194:∈
3114:≤
3087:∈
3058:≤
3020:∈
3013:⋃
3009:∈
2960:⋃
2903:≤
2863:∈
2837:≤
2790:∈
2757:∈
2724:≤
2690:∈
2651:≤
2617:∈
2577:≤
2543:∪
2530:∈
2447:∪
2411:≤
2381:≤
2335:≤
2177:non-empty
1959:bijective
1924:morphisms
1866:A simple
1839:∧
1764:∧
1752:∨
1677:dimension
1542:The term
1492:connected
1227:≤
1176:≤
1118:≠
1092:≤
1040:≤
996:≤
971:connected
902:≠
773:then not
554:on a set
471:≤
445:≤
387:≤
361:≤
328:≤
302:≤
276:≤
260:reflexive
243:≤
128:≤
6505:Topology
6372:Preorder
6355:Eulerian
6318:Complete
6268:Directed
6258:Covering
6123:Preorder
6082:Category
6077:Glossary
5949:(1968).
5916:89009753
5894:(1990).
5881:(1967).
5749:89009753
5686:category
5520:24340068
5485:38115497
5392:, Ch.14.
5314:See also
5122:✗
5109:✗
4519:✗
4514:✗
4509:✗
4504:✗
4479:✗
4447:✗
4442:✗
4437:✗
4432:✗
4427:✗
4412:✗
4380:✗
4375:✗
4370:✗
4365:✗
4360:✗
4345:✗
4333:✗
4328:✗
4303:✗
4298:✗
4293:✗
4278:✗
4266:✗
4261:✗
4246:✗
4231:✗
4226:✗
4211:✗
4199:✗
4194:✗
4159:✗
4154:✗
4139:✗
4127:✗
4122:✗
4107:✗
4102:✗
4067:✗
4055:✗
4050:✗
4035:✗
4030:✗
4015:✗
4010:✗
4005:✗
3991:✗
3986:✗
3971:✗
3966:✗
3941:✗
3936:✗
3924:✗
3919:✗
3904:✗
3899:✗
3894:✗
3869:✗
3857:✗
3852:✗
3837:✗
3832:✗
3827:✗
3812:✗
3807:✗
3795:✗
3790:✗
3775:✗
3770:✗
3765:✗
3760:✗
3745:✗
3733:✗
3728:✗
3713:✗
3708:✗
3703:✗
3698:✗
3693:✗
3688:✗
3671:✗
3666:✗
3651:✗
3646:✗
3641:✗
3636:✗
3631:✗
2187:with an
2141:, has a
2134:complete
2079:topology
2050:, ∞) = {
1916:category
1868:counting
1800:for all
1519:, e.g.,
1424:integers
1271:Examples
1217:converse
621:on some
140:on some
6610:Duality
6588:Cofinal
6576:Related
6555:Fréchet
6432:)
6308:Bounded
6303:Lattice
6276:)
6274:Partial
6142:Results
6113:Lattice
6025:, 2001
5765:(2006)
5584:. Then
4134:Lattice
3293:partial
2288:compact
2179:subset
1914:of the
1880:ordinal
1733:lattice
1574:, then
1494:in the
1442:initial
1405:indexed
1399:on the
1215:of the
1211:is the
1191:(i.e.,
1165:if not
917:, then
65:improve
6635:Subnet
6615:Filter
6565:Normed
6550:Banach
6516:&
6423:Better
6360:Strict
6350:Graded
6241:topics
6072:Topics
5994:
5965:
5939:
5914:
5904:
5829:
5777:
5747:
5737:
5711:
5518:
5510:
5483:
4959:exists
4915:exists
4871:exists
3500:
3416:) or (
3271:, the
2195:has a
2121:normal
1886:to an
1878:to an
1665:length
1646:ideals
1586:is in
1548:subset
1538:Chains
1426:, and
1278:subset
521:, and
500:connex
496:simple
54:, but
6625:Ideal
6603:Graph
6399:Total
6377:Total
6263:Dense
5684:of a
5516:JSTOR
5481:S2CID
5374:Notes
5289:group
5241:then
3604:Anti-
3412:<
3404:<
3392:) ≤ (
3377:(the
3353:) ≤ (
3322:<
3310:) ≤ (
3183:with
3102:with
2268:<
2264:<
2252:<
2248:with
2064:, and
2058:<
2036:<
2028:) = {
2024:(−∞,
2014:<
2006:<
1998:) = {
1938:then
1716:graph
1714:in a
1691:of a
1661:chain
1621:or a
1560:chain
1550:of a
1544:chain
1528:<
1524:<
1481:dense
1407:by a
1333:from
982:above
858:then
574:is a
527:chain
523:loset
502:, or
402:then
317:then
113:is a
6216:list
5992:ISBN
5963:ISBN
5937:ISBN
5912:LCCN
5902:ISBN
5827:ISBN
5775:ISBN
5745:LCCN
5735:ISBN
5709:ISBN
5656:<
5630:<
5595:<
5569:<
5543:<
5532:Let
5508:ISSN
5215:and
4620:and
3424:and
3408:and
3369:and
3334:and
3326:or (
3259:The
3148:<
2992:For
2779:and
2712:and
2639:and
2513:For
2356:and
2240:and
2114:>
2106:<
2098:>
2094:<
2010:and
1946:) ≤
1712:walk
1602:has
1600:ring
1594:has
1395:The
1379:) ≤
1276:Any
1255:<
1199:<
1150:<
1107:and
1066:<
1016:<
954:<
928:<
869:<
843:<
832:and
817:<
784:<
758:<
725:<
714:Not
670:and
609:<
376:and
291:and
189:and
105:, a
6630:Net
6430:Pre
5856:doi
5852:311
5819:doi
5619:If
5471:doi
5189:if
5150:be
4860:min
3302:: (
3269:S2S
3216:,
3163:in
2509::
2286:is
2244:in
2232:in
2230:gap
2191:in
2183:of
1962:map
1954:).
1918:of
1462:):
1440:is
1329:an
1317:If
1306:or
1219:of
1081:if
943:or
891:If
806:If
747:If
690:in
623:set
578:on
460:or
350:If
265:If
209:in
142:set
109:or
101:In
6732::
6021:,
6015:,
5910:.
5850:,
5825:.
5813:.
5769:,
5743:.
5514:.
5502:.
5479:.
5465:.
5439:^
5412:^
5397:^
5382:^
5310:.
5295:.
5287:A
5284:.
3469:.
3462:.
3444:≤
3428:=
3420:=
3373:≤
3365:≤
3338:≤
3330:=
3047:,
2565:,
2272:.)
2123:.
2085:.
2054:|
2032:|
2002:|
1994:,
1957:A
1934:≤
1858:.
1816:≤
1804:,
1718:.
1699:.
1606:.
1422:,
1359:≤
1314:).
1239:).
1137:).
973:).
888:).
803:).
744:).
710::
548:A
517:,
506:.
498:,
432:).
347:).
262:).
229::
6428:(
6425:)
6421:(
6272:(
6219:)
6053:e
6046:t
6039:v
6000:.
5918:.
5858::
5835:.
5821::
5751:.
5717:.
5659:a
5653:a
5633:a
5627:a
5598:a
5592:a
5572:a
5566:b
5546:b
5540:a
5522:.
5504:9
5487:.
5473::
5258:.
5255:c
5252:R
5249:a
5229:c
5226:R
5223:b
5203:b
5200:R
5197:a
5177:,
5174:c
5171:,
5168:b
5165:,
5162:a
5138:R
5118:Y
5105:Y
5075:a
5072:R
5069:b
5054:b
5051:R
5048:a
5023:a
5020:R
5017:a
4991:a
4988:R
4985:a
4951:b
4945:a
4907:b
4901:a
4863:S
4831:a
4828:R
4825:b
4815:b
4812:R
4809:a
4799:b
4789:a
4760:b
4753:=
4750:a
4740:a
4737:R
4734:b
4724:b
4721:R
4718:a
4689:a
4686:R
4683:b
4669:b
4666:R
4663:a
4637::
4628:S
4608:b
4605:,
4602:a
4539:Y
4529:Y
4499:Y
4489:Y
4467:Y
4457:Y
4422:Y
4400:Y
4390:Y
4355:Y
4323:Y
4313:Y
4288:Y
4256:Y
4241:Y
4221:Y
4189:Y
4179:Y
4169:Y
4149:Y
4117:Y
4097:Y
4087:Y
4077:Y
4045:Y
4025:Y
3981:Y
3961:Y
3951:Y
3914:Y
3889:Y
3879:Y
3847:Y
3822:Y
3785:Y
3755:Y
3723:Y
3661:Y
3626:Y
3524:e
3517:t
3510:v
3478:R
3474:n
3456:R
3446:y
3442:x
3430:d
3426:b
3422:c
3418:a
3414:d
3410:b
3406:c
3402:a
3398:d
3396:,
3394:c
3390:b
3388:,
3386:a
3384:(
3375:d
3371:b
3367:c
3363:a
3359:d
3357:,
3355:c
3351:b
3349:,
3347:a
3345:(
3340:d
3336:b
3332:c
3328:a
3324:c
3320:a
3316:d
3314:,
3312:c
3308:b
3306:,
3304:a
3235:j
3231:A
3224:y
3202:i
3198:A
3191:x
3171:I
3151:j
3145:i
3123:y
3118:i
3110:x
3090:I
3084:i
3061:y
3055:x
3033:i
3029:A
3023:I
3017:i
3006:y
3003:,
3000:x
2974:i
2970:A
2964:i
2937:i
2933:A
2912:)
2907:i
2899:,
2894:i
2890:A
2886:(
2866:I
2860:i
2840:)
2834:,
2831:I
2828:(
2798:2
2794:A
2787:y
2765:1
2761:A
2754:x
2733:y
2728:2
2720:x
2698:2
2694:A
2687:y
2684:,
2681:x
2660:y
2655:1
2647:x
2625:1
2621:A
2614:y
2611:,
2608:x
2586:y
2581:+
2573:x
2551:2
2547:A
2538:1
2534:A
2527:y
2524:,
2521:x
2495:2
2491:A
2487:+
2482:1
2478:A
2455:2
2451:A
2442:1
2438:A
2415:+
2390:)
2385:2
2377:,
2372:2
2368:A
2364:(
2344:)
2339:1
2331:,
2326:1
2322:A
2318:(
2276:X
2270:b
2266:c
2262:a
2258:c
2254:b
2250:a
2246:X
2242:b
2238:a
2234:X
2226:X
2220:X
2216:X
2205:≤
2201:R
2193:R
2185:R
2181:S
2173:≤
2157:Q
2150:R
2110:N
2102:N
2090:N
2073:.
2070:X
2062:}
2060:x
2056:a
2052:x
2048:a
2046:(
2042:,
2040:}
2038:b
2034:x
2030:x
2026:b
2020:,
2018:}
2016:b
2012:x
2008:x
2004:a
2000:x
1996:b
1992:a
1990:(
1978:X
1952:b
1950:(
1948:f
1944:a
1942:(
1940:f
1936:b
1932:a
1928:f
1896:k
1892:k
1842:b
1836:a
1833:=
1830:a
1818:b
1814:a
1808:.
1806:b
1802:a
1788:}
1785:b
1782:,
1779:a
1776:{
1773:=
1770:}
1767:b
1761:a
1758:,
1755:b
1749:a
1746:{
1588:X
1584:X
1580:X
1576:X
1572:X
1568:X
1530:C
1526:B
1522:A
1476:.
1469:.
1459:B
1453:A
1447:B
1437:A
1392:.
1390:)
1388:2
1385:x
1383:(
1381:f
1377:1
1374:x
1372:(
1370:f
1364:2
1361:x
1357:1
1354:x
1348:X
1342:f
1336:X
1326:f
1320:X
1297:∅
1292:.
1289:X
1283:X
1179:a
1173:b
1153:b
1147:a
1133:(
1121:b
1115:a
1095:b
1089:a
1069:b
1063:a
969:(
957:a
951:b
931:b
925:a
905:b
899:a
884:(
872:c
866:a
846:c
840:b
820:b
814:a
799:(
787:a
781:b
761:b
755:a
740:(
728:a
722:a
698:X
678:c
658:b
655:,
652:a
632:X
586:X
562:X
486:(
474:a
468:b
448:b
442:a
428:(
416:b
413:=
410:a
390:a
384:b
364:b
358:a
343:(
331:c
325:a
305:c
299:b
279:b
273:a
258:(
246:a
240:a
217:X
197:c
177:b
174:,
171:a
151:X
90:)
84:(
79:)
75:(
61:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.