32:
5103:
5090:
4524:
4514:
4484:
4474:
4452:
4442:
4407:
4385:
4375:
4340:
4308:
4298:
4273:
4241:
4226:
4206:
4174:
4164:
4154:
4134:
4102:
4082:
4072:
4062:
4030:
4010:
3966:
3946:
3936:
3899:
3874:
3864:
3832:
3807:
3770:
3740:
3708:
3646:
3611:
1543:
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.
4834:
4763:
5078:
4692:
4963:
4919:
4875:
5100:
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
483:
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
5033:
4930:
4886:
4845:
4774:
4703:
4647:
3034:
1787:
1495:
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
1407:
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
2552:
4636:
2975:
2911:
2389:
2343:
5022:
2456:
1859:
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
2699:
2626:
3437:
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.
3122:
2732:
2659:
2585:
2496:
1841:
3236:
3203:
2799:
2766:
4769:
2839:
2416:
1656:
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
3060:
1178:
1120:
1094:
904:
473:
447:
389:
363:
330:
304:
278:
245:
5658:
5632:
5597:
5571:
5545:
5348: – the proposition, independent of ZFC, that a nonempty unbounded complete dense total order satisfying the countable chain condition is isomorphic to the reals
5176:
4698:
3150:
3089:
2865:
1152:
1068:
956:
930:
871:
845:
819:
786:
760:
727:
2938:
1226:
1039:
995:
127:
5257:
1254:
1198:
1015:
608:
5228:
5202:
4990:
4607:
657:
415:
176:
5137:
3170:
697:
677:
631:
585:
561:
216:
196:
150:
5028:
2192:. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation
4642:
4925:
4881:
4840:
5287:
There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a
6691:
3511:
6674:
2077:
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
6724:
6204:
6040:
5984:
5819:
5701:
3256:, i.e. there is an algorithm for deciding which first-order statements hold for all total orders. Using interpretability in
6521:
2284:
1598:
In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its
1571:
being a set of subsets; in this case, the upperbound is obtained by proving that the union of the elements of a chain in
5884:
5759:
6657:
6516:
5955:
5929:
5894:
5767:
5727:
75:
53:
46:
6511:
2984:
1730:
3422:
6147:
6229:
5691:
3504:
2806:
Intuitively, this means that the elements of the second set are added on top of the elements of the first set.
6548:
6468:
6011:
2505:
1891:ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).
1668:
of a space is often defined or characterized as the maximal length of chains of subspaces. For example, the
6333:
6262:
6142:
1704:
6236:
6224:
6187:
6162:
6137:
6091:
6060:
6006:
6001:
4612:
2149:
2122:
1669:
1535:
is sometimes defined as a synonym for a totally ordered set, but it is generally used for referring to a
6533:
6167:
6157:
6033:
3261:
2943:
2870:
2348:
2302:
1618:
966:
Asymmetry follows from transitivity and irreflexivity; moreover, irreflexivity follows from asymmetry.
20:
4996:
3455:
6506:
6172:
3497:
2421:
1626:
5269:
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a
3469:
6438:
6065:
4571:
4395:
3564:
40:
4829:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
3284:. Here are three of these possible orders, listed such that each order is stronger than the next:
2665:
2592:
2199:
There are a number of results relating properties of the order topology to the completeness of X:
6686:
6669:
3094:
2704:
2631:
2557:
2461:
1814:
522:, but refers generally to some sort of totally ordered subsets of a given partially ordered set.
3208:
3175:
2771:
2738:
6598:
6214:
5751:
5696:. Studies in Logic and the Foundations of Mathematics. Vol. 145 (1st ed.). Elsevier.
5674:
4758:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
4541:
3534:
2812:
2394:
1971:
1904:
1692:
418:
57:
6729:
6576:
6411:
6402:
6271:
6152:
6106:
6070:
6026:
5799:
5281:
5096:
indicates that the column's property is always true the row's term (at the very left), while
4551:
3544:
3288:
3039:
2267:
is complete if and only if every bounded set that is closed in the order topology is compact.
1908:
1540:
1385:
1157:
1099:
1073:
883:
452:
426:
368:
342:
309:
283:
257:
224:
5637:
5611:
5576:
5550:
5524:
5146:
3129:
3068:
2844:
1201:
1131:
1047:
935:
909:
850:
824:
798:
765:
739:
706:
6664:
6623:
6613:
6603:
6348:
6311:
6301:
6281:
6266:
5288:
5117:
4328:
3603:
3448:
3253:
2916:
1844:
1423:) "initial example" of a totally ordered set with a certain property, (here, a total order
1211:
1024:
980:
730:
564:
112:
5233:
1239:
1183:
1000:
593:
8:
6734:
6591:
6502:
6448:
6407:
6397:
6286:
6219:
6182:
5687:
5345:
5296:
5277:
5260:
A term's definition may require additional properties that are not listed in this table.
5207:
5181:
5140:
4969:
4586:
4576:
3987:
3569:
3486:
1700:
1634:
1123:
874:
789:
636:
394:
333:
155:
6703:
6630:
6483:
6392:
6382:
6323:
6241:
6177:
5504:
5469:
5338: – generalization of the notion of prefix of a string, and of the notion of a tree
5122:
4566:
4546:
4536:
4462:
3559:
3539:
3529:
3155:
1622:
1588:
1505:
1461:
The integers form an initial non-empty totally ordered set with neither an upper nor a
1319:
959:
682:
662:
616:
570:
546:
476:
248:
201:
181:
135:
6543:
3276:
There are several ways to take two totally ordered sets and extend to an order on the
587:
in which any two distinct elements are comparable. That is, a strict total order is a
6640:
6618:
6478:
6463:
6443:
6246:
5980:
5951:
5925:
5900:
5890:
5815:
5763:
5733:
5723:
5697:
5496:
3277:
3257:
3249:
2131:
1950:
1887:
natural numbers. Hence it is common to index finite total orders or well orders with
1603:
1599:
1497:
1420:
1389:
1233:
1205:
611:
130:
5473:
2217:
is connected under the order topology if and only if it is complete and there is no
1879:
of the natural numbers ordered by <. In other words, a total order on a set with
6453:
6306:
5880:
5867:
5844:
5807:
5459:
4261:
4194:
2272:
1872:
1864:
1681:
1665:
1416:
526:
6635:
6418:
6296:
6291:
6276:
6192:
6101:
6086:
5974:
5970:
5806:. Lecture Notes in Computer Science. Vol. 2500. Springer. pp. 207–230.
5318:
4122:
3919:
3490:
2153:
2142:
1876:
1721:
1677:
1673:
1630:
1552:
1408:
1292:
588:
107:
6553:
6538:
6528:
6387:
6365:
6343:
5935:
5849:
5670:
5073:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
3790:
2071:
1868:
1809:
1484:
1296:
4687:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
6718:
6652:
6608:
6586:
6458:
6328:
6316:
6121:
5811:
5500:
5309:
5270:
4561:
4556:
3728:
3554:
3549:
3367:
3281:
3265:
2288:
2280:
2276:
1696:
1657:
1592:
1492:
1480:
103:
2108:
The order topology induced by a total order may be shown to be hereditarily
6473:
6355:
6338:
6256:
6096:
6049:
5487:
Ganapathy, Jayanthi (1992). "Maximal
Elements and Upper Bounds in Posets".
5335:
5323:
5292:
3441:
2109:
1661:
1580:
1404:
5464:
5451:
3271:
1621:
if every descending chain eventually stabilizes. For example, an order is
1567:
contains at least one maximal element. Zorn's lemma is commonly used with
1454:
The natural numbers form an initial non-empty totally ordered set with no
6679:
6372:
6251:
5329:
2177:
2157:
2135:
2127:
1954:
1900:
1699:
of polynomials. Another example is the use of "chain" as a synonym for a
1685:
1584:
1473:
1462:
1455:
91:
5508:
4958:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
3425:
of the corresponding strict total orders). This is also a partial order.
6647:
6581:
6422:
5717:
5450:
Strohmeier, Alfred; Genillard, Christian; Weber, Mats (1 August 1990).
5354:
4050:
1888:
1860:
1641:
1629:
means that every ascending chain eventually stabilizes. For example, a
1479:
The real numbers form an initial unbounded totally ordered set that is
1397:
1300:
106:
in which any two elements are comparable. That is, a total order is a
4914:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
6698:
6571:
6377:
4870:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
2165:
1947:
1469:
1393:
5835:
Macpherson, H. Dugald (2011), "A survey of homogeneous structures",
5312: – ring that satisfies the descending chain condition on ideals
2105:(in this case they happen to be identical but will not in general).
1953:
between two totally ordered sets that respects the two orders is an
6493:
6360:
6111:
5874:. Colloquium Publications. Vol. 25. Providence: Am. Math. Soc.
3667:
3429:
Each of these orders extends the next in the sense that if we have
2185:
2067:
1912:
1863:. Either by direct proof or by observing that every well order is
1856:
1412:
525:
An extension of a given partial order to a total order is called a
5976:
Relations and Graphs: Discrete
Mathematics for Computer Scientists
5779:
that is, beyond some index, all further sequence members are equal
1468:
The rational numbers form an initial totally ordered set which is
1472:
in the real numbers. Moreover, the reflexive reduction < is a
5904:
5889:. Cambridge Mathematical Textbooks. Cambridge University Press.
5737:
5722:. Cambridge Mathematical Textbooks. Cambridge University Press.
5404:
5402:
6018:
1536:
1266:
3470:
defines a strict weak order and a corresponding total preorder
2458:, which is called the sum of the two orders or sometimes just
2279:. Examples are the closed intervals of real numbers, e.g. the
1614:, depending whether the sequence is increasing or decreasing.
1555:
which asserts that, if every chain in a partially ordered set
5433:
5431:
5429:
5399:
2152:(not to be confused with being "total") do not carry over to
1720:
One may define a totally ordered set as a particular kind of
1691:"Chain" may also be used for some totally ordered subsets of
1695:
that are not partially ordered sets. An example is given by
2271:
A totally ordered set (with its order topology) which is a
5426:
5922:
Lattice theory: first concepts and distributive lattices.
5374:
5372:
1625:
if it has the descending chain condition. Similarly, the
1579:. This is the way that is generally used to prove that a
5449:
2287:(extended real number line). There are order-preserving
1419:. Each of these can be shown to be the unique (up to an
5389:
5387:
3272:
Orders on the
Cartesian product of totally ordered sets
2940:
are pairwise disjoint, then the natural total order on
1275:
is totally ordered for the restriction of the order on
19:"Linear order" redirects here. Not to be confused with
5369:
5640:
5614:
5579:
5553:
5527:
5236:
5210:
5184:
5149:
5125:
5031:
4999:
4972:
4928:
4884:
4843:
4772:
4701:
4645:
4615:
4589:
3211:
3178:
3158:
3132:
3097:
3071:
3042:
2987:
2946:
2919:
2873:
2847:
2815:
2774:
2741:
2707:
2668:
2634:
2595:
2560:
2508:
2464:
2424:
2397:
2351:
2305:
2089:
greater than we might refer to the order topology on
1817:
1733:
1439:
has the property, there is an order isomorphism from
1242:
1214:
1186:
1160:
1134:
1102:
1076:
1050:
1027:
1003:
983:
938:
912:
886:
853:
827:
801:
768:
742:
709:
685:
665:
639:
619:
596:
573:
549:
532:
455:
429:
397:
371:
345:
312:
286:
260:
227:
204:
184:
158:
138:
115:
5384:
5350:
Pages displaying wikidata descriptions as a fallback
5340:
Pages displaying wikidata descriptions as a fallback
5314:
Pages displaying wikidata descriptions as a fallback
1504:
The letters of the alphabet ordered by the standard
969:
For delimitation purposes, a total order as defined
5291:. Forgetting the location of the ends results in a
5652:
5626:
5591:
5565:
5539:
5251:
5222:
5196:
5170:
5131:
5072:
5016:
4984:
4957:
4913:
4869:
4828:
4757:
4686:
4630:
4601:
3230:
3197:
3164:
3144:
3116:
3083:
3054:
3028:
2969:
2932:
2905:
2859:
2833:
2793:
2760:
2726:
2693:
2653:
2620:
2579:
2546:
2490:
2450:
2410:
2383:
2337:
1835:
1781:
1248:
1220:
1192:
1172:
1146:
1114:
1088:
1062:
1033:
1009:
989:
950:
924:
898:
865:
839:
813:
780:
754:
721:
691:
671:
651:
625:
602:
579:
555:
467:
441:
409:
383:
357:
324:
298:
272:
239:
210:
190:
170:
144:
121:
5599:by transitivity, which contradicts irreflexivity.
5414:
2587:holds if and only if one of the following holds:
1602:. In this case, a chain can be identified with a
6716:
5969:
5718:Brian A. Davey and Hilary Ann Priestley (1990).
5408:
4848:
2148:is not. In other words, the various concepts of
1501:ordered field is isomorphic to the real numbers.
5332: – Mathematical version of an order change
1915:being maps which respect the orders, i.e. maps
5878:
5686:
5437:
1871:one may show that every finite total order is
6034:
5113:in the "Antisymmetric" column, respectively.
3505:
3029:{\displaystyle x,y\in \bigcup _{i\in I}A_{i}}
2841:is a totally ordered index set, and for each
1782:{\displaystyle \{a\vee b,a\wedge b\}=\{a,b\}}
1644:are considered. In this case, one talks of a
2066:We can use these open intervals to define a
1883:elements induces a bijection with the first
1776:
1764:
1758:
1734:
1551:for referring to totally ordered subsets is
1041:that can be defined in two equivalent ways:
6692:Positive cone of a partially ordered group
6041:
6027:
5960:
5946:John G. Hocking and Gail S. Young (1961).
5834:
3512:
3498:
5848:
5486:
5463:
3280:, though the resulting order may only be
977:order. For each (non-strict) total order
76:Learn how and when to remove this message
6675:Positive cone of an ordered vector space
5866:
5788:Davey and Priestly 1990, Def.2.24, p. 37
5393:
1640:In other contexts, only chains that are
633:, which satisfies the following for all
152:, which satisfies the following for all
39:This article includes a list of general
5357: – Class of mathematical orderings
1637:satisfy the ascending chain condition.
498:A set equipped with a total order is a
16:Order whose elements are all comparable
6717:
5934:
5378:
3465:real variables defined on a subset of
2547:{\displaystyle x,y\in A_{1}\cup A_{2}}
1850:
6022:
5911:
5797:
5669:This definition resembles that of an
5547:, assume for contradiction that also
5420:
3475:
2126:if every nonempty subset that has an
1392:of a family of totally ordered sets,
518:is sometimes defined as a synonym of
5804:Automata, Logics, and Infinite Games
5452:"Ordering of characters and strings"
5295:. Forgetting both data results in a
5116:All definitions tacitly require the
2285:affinely extended real number system
2120:A totally ordered set is said to be
25:
5914:Partially Ordered Algebraic Systems
5280:with a compatible total order is a
4631:{\displaystyle S\neq \varnothing :}
1843:. Hence a totally ordered set is a
1710:
1684:is the maximal length of chains of
1672:is the maximal length of chains of
1284:The unique order on the empty set,
13:
6202:Properties & Types (
5886:Introduction to Lattices and Order
5760:Undergraduate Texts in Mathematics
5720:Introduction to Lattices and Order
3456:examples of partially ordered sets
2913:is a linear order, where the sets
2299:For any two disjoint total orders
1894:
1660:is a chain of length zero, and an
533:Strict and non-strict total orders
45:it lacks sufficient corresponding
14:
6746:
6658:Positive cone of an ordered field
5994:
5950:Corrected reprint, Dover, 1988.
4622:
2970:{\displaystyle \bigcup _{i}A_{i}}
2906:{\displaystyle (A_{i},\leq _{i})}
2384:{\displaystyle (A_{2},\leq _{2})}
2338:{\displaystyle (A_{1},\leq _{1})}
2294:
1960:
1715:
6512:Ordered topological vector space
6048:
5342:– a downward total partial order
5101:
5088:
5017:{\displaystyle {\text{not }}aRa}
4522:
4512:
4482:
4472:
4450:
4440:
4405:
4383:
4373:
4338:
4306:
4296:
4271:
4239:
4224:
4204:
4172:
4162:
4152:
4132:
4100:
4080:
4070:
4060:
4028:
4008:
3964:
3944:
3934:
3897:
3872:
3862:
3830:
3805:
3768:
3738:
3706:
3644:
3609:
3421:) (the reflexive closure of the
3268:total orders is also decidable.
1617:A partially ordered set has the
997:there is an associated relation
30:
5828:
5791:
5782:
5773:
5745:
5711:
5680:
5663:
3243:
2451:{\displaystyle A_{1}\cup A_{2}}
2115:
1547:A common example of the use of
1256:is a (non-strict) total order.
6725:Properties of binary relations
5961:Rosenstein, Joseph G. (1982).
5602:
5515:
5480:
5443:
5109:in the "Symmetric" column and
5045:
4790:
4735:
4664:
2900:
2874:
2828:
2816:
2378:
2352:
2332:
2306:
1724:, namely one in which we have
1664:is a chain of length one. The
1522:etc., is a strict total order.
1328:to a totally ordered set then
1:
6469:Series-parallel partial order
5860:
5800:"Decidability of S1S and S2S"
5326: – Branch of mathematics
5110:
5097:
4507:
4502:
4497:
4492:
4467:
4435:
4430:
4425:
4420:
4415:
4400:
4368:
4363:
4358:
4353:
4348:
4333:
4321:
4316:
4291:
4286:
4281:
4266:
4254:
4249:
4234:
4219:
4214:
4199:
4187:
4182:
4147:
4142:
4127:
4115:
4110:
4095:
4090:
4055:
4043:
4038:
4023:
4018:
4003:
3998:
3993:
3979:
3974:
3959:
3954:
3929:
3924:
3912:
3907:
3892:
3887:
3882:
3857:
3845:
3840:
3825:
3820:
3815:
3800:
3795:
3783:
3778:
3763:
3758:
3753:
3748:
3733:
3721:
3716:
3701:
3696:
3691:
3686:
3681:
3676:
3659:
3654:
3639:
3634:
3629:
3624:
3619:
1433:for a property, if, whenever
6148:Cantor's isomorphism theorem
5973:; Ströhlein, Thomas (1993).
5409:Schmidt & Ströhlein 1993
2694:{\displaystyle x,y\in A_{2}}
2621:{\displaystyle x,y\in A_{1}}
1965:For any totally ordered set
1899:Totally ordered sets form a
1334:induces a total ordering on
7:
6188:Szpilrajn extension theorem
6163:Hausdorff maximal principle
6138:Boolean prime ideal theorem
6007:Encyclopedia of Mathematics
5979:. Berlin: Springer-Verlag.
5965:. New York: Academic Press.
5302:
3447:, each of these make it an
3370:). This is a partial order.
3117:{\displaystyle x\leq _{i}y}
2727:{\displaystyle x\leq _{2}y}
2654:{\displaystyle x\leq _{1}y}
2580:{\displaystyle x\leq _{+}y}
2491:{\displaystyle A_{1}+A_{2}}
2391:, there is a natural order
2160:a property of the relation
2141:is complete but the set of
1836:{\displaystyle a=a\wedge b}
1670:dimension of a vector space
1259:
10:
6751:
6534:Topological vector lattice
5850:10.1016/j.disc.2011.01.024
5438:Davey & Priestley 1990
3252:theory of total orders is
3231:{\displaystyle y\in A_{j}}
3198:{\displaystyle x\in A_{i}}
2794:{\displaystyle y\in A_{2}}
2761:{\displaystyle x\in A_{1}}
2188:(also called supremum) in
2134:. For example, the set of
2097:and the order topology on
1619:descending chain condition
1400:, is itself a total order.
1299:(more strongly, these are
21:Linear order (linguistics)
18:
6564:
6492:
6431:
6201:
6130:
6079:
6056:
3331:). This is a total order.
2834:{\displaystyle (I,\leq )}
2411:{\displaystyle \leq _{+}}
2203:If the order topology on
2196:to the rational numbers.
1627:ascending chain condition
1526:
1269:of a totally ordered set
479:, formerly called total).
6143:Cantor–Bernstein theorem
5812:10.1007/3-540-36387-4_12
5362:
2291:between these examples.
2156:. For example, over the
2081:is the natural numbers,
2070:on any ordered set, the
1476:on the rational numbers.
1236:of a strict total order
970:
514:are also used. The term
6687:Partially ordered group
6507:Specialization preorder
5920:George Grätzer (1971).
3055:{\displaystyle x\leq y}
1648:, often shortened as a
1173:{\displaystyle b\leq a}
1115:{\displaystyle a\neq b}
1089:{\displaystyle a\leq b}
899:{\displaystyle a\neq b}
529:of that partial order.
468:{\displaystyle b\leq a}
442:{\displaystyle a\leq b}
384:{\displaystyle b\leq a}
358:{\displaystyle a\leq b}
325:{\displaystyle a\leq c}
299:{\displaystyle b\leq c}
273:{\displaystyle a\leq b}
240:{\displaystyle a\leq a}
60:more precise citations.
6173:Kruskal's tree theorem
6168:Knaster–Tarski theorem
6158:Dushnik–Miller theorem
5942:. Princeton: Nostrand.
5924:W. H. Freeman and Co.
5752:Yiannis N. Moschovakis
5654:
5653:{\displaystyle a<a}
5628:
5627:{\displaystyle a<a}
5593:
5592:{\displaystyle a<a}
5567:
5566:{\displaystyle b<a}
5541:
5540:{\displaystyle a<b}
5456:ACM SIGAda Ada Letters
5253:
5224:
5198:
5172:
5171:{\displaystyle a,b,c,}
5133:
5074:
5018:
4986:
4959:
4915:
4871:
4830:
4759:
4688:
4632:
4603:
3232:
3199:
3166:
3146:
3145:{\displaystyle i<j}
3118:
3085:
3084:{\displaystyle i\in I}
3056:
3030:
2971:
2934:
2907:
2861:
2860:{\displaystyle i\in I}
2835:
2795:
2762:
2728:
2695:
2655:
2622:
2581:
2548:
2492:
2452:
2412:
2385:
2339:
1909:partially ordered sets
1837:
1783:
1559:has an upper bound in
1250:
1222:
1194:
1174:
1148:
1147:{\displaystyle a<b}
1116:
1090:
1064:
1063:{\displaystyle a<b}
1035:
1011:
991:
952:
951:{\displaystyle b<a}
926:
925:{\displaystyle a<b}
900:
867:
866:{\displaystyle a<c}
841:
840:{\displaystyle b<c}
815:
814:{\displaystyle a<b}
782:
781:{\displaystyle b<a}
756:
755:{\displaystyle a<b}
723:
722:{\displaystyle a<a}
693:
673:
653:
627:
604:
581:
557:
469:
443:
411:
385:
359:
326:
300:
274:
241:
212:
192:
172:
146:
123:
6002:"Totally ordered set"
5881:Priestley, Hilary Ann
5655:
5629:
5594:
5568:
5542:
5489:Pi Mu Epsilon Journal
5465:10.1145/101120.101136
5282:totally ordered group
5254:
5225:
5199:
5173:
5134:
5075:
5019:
4987:
4960:
4916:
4872:
4831:
4760:
4689:
4633:
4604:
4583:Definitions, for all
3289:Lexicographical order
3233:
3200:
3167:
3147:
3119:
3086:
3065:Either there is some
3057:
3031:
2972:
2935:
2933:{\displaystyle A_{i}}
2908:
2862:
2836:
2796:
2763:
2729:
2696:
2656:
2623:
2582:
2549:
2493:
2453:
2413:
2386:
2340:
2225:(a gap is two points
1838:
1784:
1541:partially ordered set
1386:lexicographical order
1251:
1223:
1221:{\displaystyle \leq }
1195:
1175:
1149:
1117:
1091:
1065:
1036:
1034:{\displaystyle \leq }
1012:
992:
990:{\displaystyle \leq }
953:
927:
901:
868:
842:
816:
783:
757:
724:
694:
674:
654:
628:
605:
582:
558:
470:
444:
412:
386:
360:
327:
301:
275:
242:
213:
193:
173:
147:
124:
122:{\displaystyle \leq }
6665:Ordered vector space
5837:Discrete Mathematics
5798:Weyer, Mark (2002).
5638:
5612:
5577:
5551:
5525:
5289:betweenness relation
5252:{\displaystyle aRc.}
5234:
5208:
5182:
5147:
5123:
5118:homogeneous relation
5029:
4997:
4970:
4926:
4882:
4841:
4770:
4699:
4643:
4613:
4587:
4329:Strict partial order
3604:Equivalence relation
3449:ordered vector space
3262:monadic second-order
3209:
3176:
3156:
3130:
3095:
3069:
3040:
2985:
2944:
2917:
2871:
2845:
2813:
2772:
2739:
2705:
2666:
2632:
2593:
2558:
2506:
2462:
2422:
2395:
2349:
2303:
1845:distributive lattice
1815:
1731:
1652:. In this case, the
1249:{\displaystyle <}
1240:
1212:
1193:{\displaystyle <}
1184:
1158:
1132:
1100:
1074:
1048:
1025:
1010:{\displaystyle <}
1001:
981:
973:is sometimes called
936:
910:
884:
851:
825:
799:
766:
740:
707:
683:
663:
637:
617:
603:{\displaystyle <}
594:
571:
565:strict partial order
547:
508:linearly ordered set
453:
427:
395:
369:
343:
310:
284:
258:
225:
202:
182:
156:
136:
113:
6503:Alexandrov topology
6449:Lexicographic order
6408:Well-quasi-ordering
5756:Notes on set theory
5693:Theory of Relations
5297:separation relation
5223:{\displaystyle bRc}
5197:{\displaystyle aRb}
4985:{\displaystyle aRa}
4602:{\displaystyle a,b}
3988:Well-quasi-ordering
3461:A real function of
2809:More generally, if
1851:Finite total orders
1606:, and is called an
1288:, is a total order.
1124:reflexive reduction
652:{\displaystyle a,b}
520:totally ordered set
500:totally ordered set
410:{\displaystyle a=b}
171:{\displaystyle a,b}
6484:Transitive closure
6444:Converse/Transpose
6153:Dilworth's theorem
5650:
5624:
5589:
5563:
5537:
5249:
5220:
5194:
5168:
5129:
5070:
5068:
5014:
4982:
4955:
4953:
4911:
4909:
4867:
4865:
4826:
4824:
4755:
4753:
4684:
4682:
4628:
4599:
4463:Strict total order
3476:Related structures
3389:) if and only if (
3228:
3195:
3162:
3142:
3126:or there are some
3114:
3081:
3052:
3026:
3015:
2967:
2956:
2930:
2903:
2857:
2831:
2791:
2758:
2724:
2691:
2651:
2618:
2577:
2544:
2488:
2448:
2408:
2381:
2335:
1969:we can define the
1957:in this category.
1833:
1779:
1320:injective function
1246:
1218:
1190:
1170:
1144:
1112:
1086:
1060:
1031:
1019:strict total order
1007:
987:
948:
922:
896:
863:
837:
811:
778:
752:
719:
689:
669:
649:
623:
600:
577:
553:
540:strict total order
504:simply ordered set
477:strongly connected
465:
439:
407:
381:
355:
322:
296:
270:
237:
208:
188:
168:
142:
119:
6712:
6711:
6670:Partially ordered
6479:Symmetric closure
6464:Reflexive closure
6207:
5986:978-3-642-77970-1
5916:. Pergamon Press.
5912:Fuchs, L (1963).
5879:Davey, Brian A.;
5868:Birkhoff, Garrett
5843:(15): 1599–1634,
5821:978-3-540-00388-5
5703:978-0-444-50542-2
5690:(December 2000).
5265:
5264:
5132:{\displaystyle R}
5083:
5082:
5055:
5003:
4949:
4905:
4861:
4809:
4718:
4396:Strict weak order
3582:Total, Semiconnex
3350:) if and only if
3307:) if and only if
3278:Cartesian product
3165:{\displaystyle I}
3000:
2947:
2186:least upper bound
2132:least upper bound
2085:is less than and
1604:monotone sequence
1498:Dedekind-complete
1421:order isomorphism
1390:Cartesian product
1234:reflexive closure
692:{\displaystyle X}
672:{\displaystyle c}
626:{\displaystyle X}
580:{\displaystyle X}
556:{\displaystyle X}
211:{\displaystyle X}
191:{\displaystyle c}
145:{\displaystyle X}
86:
85:
78:
6742:
6454:Linear extension
6203:
6183:Mirsky's theorem
6043:
6036:
6029:
6020:
6019:
6015:
5990:
5971:Schmidt, Gunther
5966:
5963:Linear orderings
5943:
5940:Naive Set Theory
5917:
5908:
5875:
5854:
5853:
5852:
5832:
5826:
5825:
5795:
5789:
5786:
5780:
5777:
5771:
5749:
5743:
5741:
5715:
5709:
5707:
5684:
5678:
5677:, but is weaker.
5667:
5661:
5659:
5657:
5656:
5651:
5633:
5631:
5630:
5625:
5606:
5600:
5598:
5596:
5595:
5590:
5572:
5570:
5569:
5564:
5546:
5544:
5543:
5538:
5519:
5513:
5512:
5484:
5478:
5477:
5467:
5447:
5441:
5435:
5424:
5418:
5412:
5406:
5397:
5391:
5382:
5376:
5351:
5346:Suslin's problem
5341:
5315:
5258:
5256:
5255:
5250:
5229:
5227:
5226:
5221:
5203:
5201:
5200:
5195:
5177:
5175:
5174:
5169:
5138:
5136:
5135:
5130:
5112:
5108:
5105:
5104:
5099:
5095:
5092:
5091:
5079:
5077:
5076:
5071:
5069:
5056:
5053:
5023:
5021:
5020:
5015:
5004:
5001:
4991:
4989:
4988:
4983:
4964:
4962:
4961:
4956:
4954:
4950:
4947:
4920:
4918:
4917:
4912:
4910:
4906:
4903:
4876:
4874:
4873:
4868:
4866:
4862:
4859:
4835:
4833:
4832:
4827:
4825:
4810:
4807:
4784:
4764:
4762:
4761:
4756:
4754:
4745:
4719:
4716:
4693:
4691:
4690:
4685:
4683:
4668:
4649:
4637:
4635:
4634:
4629:
4608:
4606:
4605:
4600:
4529:
4526:
4525:
4519:
4516:
4515:
4509:
4504:
4499:
4494:
4489:
4486:
4485:
4479:
4476:
4475:
4469:
4457:
4454:
4453:
4447:
4444:
4443:
4437:
4432:
4427:
4422:
4417:
4412:
4409:
4408:
4402:
4390:
4387:
4386:
4380:
4377:
4376:
4370:
4365:
4360:
4355:
4350:
4345:
4342:
4341:
4335:
4323:
4318:
4313:
4310:
4309:
4303:
4300:
4299:
4293:
4288:
4283:
4278:
4275:
4274:
4268:
4262:Meet-semilattice
4256:
4251:
4246:
4243:
4242:
4236:
4231:
4228:
4227:
4221:
4216:
4211:
4208:
4207:
4201:
4195:Join-semilattice
4189:
4184:
4179:
4176:
4175:
4169:
4166:
4165:
4159:
4156:
4155:
4149:
4144:
4139:
4136:
4135:
4129:
4117:
4112:
4107:
4104:
4103:
4097:
4092:
4087:
4084:
4083:
4077:
4074:
4073:
4067:
4064:
4063:
4057:
4045:
4040:
4035:
4032:
4031:
4025:
4020:
4015:
4012:
4011:
4005:
4000:
3995:
3990:
3981:
3976:
3971:
3968:
3967:
3961:
3956:
3951:
3948:
3947:
3941:
3938:
3937:
3931:
3926:
3914:
3909:
3904:
3901:
3900:
3894:
3889:
3884:
3879:
3876:
3875:
3869:
3866:
3865:
3859:
3847:
3842:
3837:
3834:
3833:
3827:
3822:
3817:
3812:
3809:
3808:
3802:
3797:
3785:
3780:
3775:
3772:
3771:
3765:
3760:
3755:
3750:
3745:
3742:
3741:
3735:
3723:
3718:
3713:
3710:
3709:
3703:
3698:
3693:
3688:
3683:
3678:
3673:
3671:
3661:
3656:
3651:
3648:
3647:
3641:
3636:
3631:
3626:
3621:
3616:
3613:
3612:
3606:
3524:
3523:
3514:
3507:
3500:
3493:
3491:binary relations
3482:
3481:
3472:on that subset.
3237:
3235:
3234:
3229:
3227:
3226:
3204:
3202:
3201:
3196:
3194:
3193:
3171:
3169:
3168:
3163:
3151:
3149:
3148:
3143:
3123:
3121:
3120:
3115:
3110:
3109:
3090:
3088:
3087:
3082:
3061:
3059:
3058:
3053:
3035:
3033:
3032:
3027:
3025:
3024:
3014:
2976:
2974:
2973:
2968:
2966:
2965:
2955:
2939:
2937:
2936:
2931:
2929:
2928:
2912:
2910:
2909:
2904:
2899:
2898:
2886:
2885:
2866:
2864:
2863:
2858:
2840:
2838:
2837:
2832:
2800:
2798:
2797:
2792:
2790:
2789:
2767:
2765:
2764:
2759:
2757:
2756:
2733:
2731:
2730:
2725:
2720:
2719:
2700:
2698:
2697:
2692:
2690:
2689:
2660:
2658:
2657:
2652:
2647:
2646:
2627:
2625:
2624:
2619:
2617:
2616:
2586:
2584:
2583:
2578:
2573:
2572:
2553:
2551:
2550:
2545:
2543:
2542:
2530:
2529:
2497:
2495:
2494:
2489:
2487:
2486:
2474:
2473:
2457:
2455:
2454:
2449:
2447:
2446:
2434:
2433:
2417:
2415:
2414:
2409:
2407:
2406:
2390:
2388:
2387:
2382:
2377:
2376:
2364:
2363:
2344:
2342:
2341:
2336:
2331:
2330:
2318:
2317:
2273:complete lattice
2195:
2163:
2143:rational numbers
2104:
2096:
2088:
2084:
2061:
2052:
2030:
2008:
1968:
1901:full subcategory
1873:order isomorphic
1865:order isomorphic
1842:
1840:
1839:
1834:
1788:
1786:
1785:
1780:
1711:Further concepts
1682:commutative ring
1674:linear subspaces
1633:is a ring whose
1612:descending chain
1578:
1574:
1570:
1566:
1562:
1558:
1521:
1506:dictionary order
1487:(defined below).
1450:
1444:
1438:
1428:
1417:rational numbers
1398:well ordered set
1380:
1355:
1339:
1333:
1327:
1317:
1311:
1293:cardinal numbers
1287:
1280:
1274:
1255:
1253:
1252:
1247:
1232:Conversely, the
1227:
1225:
1224:
1219:
1199:
1197:
1196:
1191:
1179:
1177:
1176:
1171:
1153:
1151:
1150:
1145:
1121:
1119:
1118:
1113:
1095:
1093:
1092:
1087:
1069:
1067:
1066:
1061:
1040:
1038:
1037:
1032:
1021:associated with
1016:
1014:
1013:
1008:
996:
994:
993:
988:
957:
955:
954:
949:
931:
929:
928:
923:
905:
903:
902:
897:
872:
870:
869:
864:
846:
844:
843:
838:
820:
818:
817:
812:
787:
785:
784:
779:
761:
759:
758:
753:
728:
726:
725:
720:
698:
696:
695:
690:
678:
676:
675:
670:
658:
656:
655:
650:
632:
630:
629:
624:
609:
607:
606:
601:
586:
584:
583:
578:
562:
560:
559:
554:
527:linear extension
474:
472:
471:
466:
448:
446:
445:
440:
416:
414:
413:
408:
390:
388:
387:
382:
364:
362:
361:
356:
331:
329:
328:
323:
305:
303:
302:
297:
279:
277:
276:
271:
246:
244:
243:
238:
217:
215:
214:
209:
197:
195:
194:
189:
177:
175:
174:
169:
151:
149:
148:
143:
128:
126:
125:
120:
81:
74:
70:
67:
61:
56:this article by
47:inline citations
34:
33:
26:
6750:
6749:
6745:
6744:
6743:
6741:
6740:
6739:
6715:
6714:
6713:
6708:
6704:Young's lattice
6560:
6488:
6427:
6277:Heyting algebra
6225:Boolean algebra
6197:
6178:Laver's theorem
6126:
6092:Boolean algebra
6087:Binary relation
6075:
6052:
6047:
6000:
5997:
5987:
5936:Halmos, Paul R.
5897:
5863:
5858:
5857:
5833:
5829:
5822:
5796:
5792:
5787:
5783:
5778:
5774:
5750:
5746:
5730:
5716:
5712:
5704:
5685:
5681:
5668:
5664:
5639:
5636:
5635:
5613:
5610:
5609:
5607:
5603:
5578:
5575:
5574:
5552:
5549:
5548:
5526:
5523:
5522:
5520:
5516:
5485:
5481:
5448:
5444:
5436:
5427:
5419:
5415:
5407:
5400:
5392:
5385:
5377:
5370:
5365:
5360:
5349:
5339:
5319:Countryman line
5313:
5305:
5267:
5266:
5259:
5235:
5232:
5231:
5209:
5206:
5205:
5183:
5180:
5179:
5148:
5145:
5144:
5124:
5121:
5120:
5114:
5106:
5102:
5093:
5089:
5067:
5066:
5052:
5049:
5048:
5032:
5030:
5027:
5026:
5000:
4998:
4995:
4994:
4971:
4968:
4967:
4952:
4951:
4946:
4943:
4942:
4929:
4927:
4924:
4923:
4908:
4907:
4902:
4899:
4898:
4885:
4883:
4880:
4879:
4864:
4863:
4858:
4855:
4854:
4844:
4842:
4839:
4838:
4823:
4822:
4811:
4806:
4794:
4793:
4785:
4783:
4773:
4771:
4768:
4767:
4752:
4751:
4746:
4744:
4732:
4731:
4720:
4717: and
4715:
4702:
4700:
4697:
4696:
4681:
4680:
4669:
4667:
4661:
4660:
4646:
4644:
4641:
4640:
4614:
4611:
4610:
4588:
4585:
4584:
4527:
4523:
4517:
4513:
4487:
4483:
4477:
4473:
4455:
4451:
4445:
4441:
4410:
4406:
4388:
4384:
4378:
4374:
4343:
4339:
4311:
4307:
4301:
4297:
4276:
4272:
4244:
4240:
4229:
4225:
4209:
4205:
4177:
4173:
4167:
4163:
4157:
4153:
4137:
4133:
4105:
4101:
4085:
4081:
4075:
4071:
4065:
4061:
4033:
4029:
4013:
4009:
3986:
3969:
3965:
3949:
3945:
3939:
3935:
3920:Prewellordering
3902:
3898:
3877:
3873:
3867:
3863:
3835:
3831:
3810:
3806:
3773:
3769:
3743:
3739:
3711:
3707:
3669:
3666:
3649:
3645:
3614:
3610:
3602:
3594:
3518:
3485:
3478:
3440:Applied to the
3274:
3246:
3222:
3218:
3210:
3207:
3206:
3189:
3185:
3177:
3174:
3173:
3157:
3154:
3153:
3131:
3128:
3127:
3105:
3101:
3096:
3093:
3092:
3070:
3067:
3066:
3041:
3038:
3037:
3020:
3016:
3004:
2986:
2983:
2982:
2977:is defined by
2961:
2957:
2951:
2945:
2942:
2941:
2924:
2920:
2918:
2915:
2914:
2894:
2890:
2881:
2877:
2872:
2869:
2868:
2846:
2843:
2842:
2814:
2811:
2810:
2785:
2781:
2773:
2770:
2769:
2752:
2748:
2740:
2737:
2736:
2715:
2711:
2706:
2703:
2702:
2685:
2681:
2667:
2664:
2663:
2642:
2638:
2633:
2630:
2629:
2612:
2608:
2594:
2591:
2590:
2568:
2564:
2559:
2556:
2555:
2538:
2534:
2525:
2521:
2507:
2504:
2503:
2482:
2478:
2469:
2465:
2463:
2460:
2459:
2442:
2438:
2429:
2425:
2423:
2420:
2419:
2402:
2398:
2396:
2393:
2392:
2372:
2368:
2359:
2355:
2350:
2347:
2346:
2326:
2322:
2313:
2309:
2304:
2301:
2300:
2297:
2193:
2161:
2118:
2102:
2094:
2086:
2082:
2056:
2034:
2012:
1978:
1966:
1963:
1897:
1895:Category theory
1877:initial segment
1853:
1816:
1813:
1812:
1732:
1729:
1728:
1718:
1713:
1678:Krull dimension
1631:Noetherian ring
1608:ascending chain
1576:
1572:
1568:
1564:
1560:
1556:
1529:
1509:
1446:
1445:to a subset of
1440:
1434:
1424:
1409:natural numbers
1378:
1367:
1357:
1356:if and only if
1354:
1347:
1341:
1335:
1329:
1323:
1313:
1312:is any set and
1307:
1297:ordinal numbers
1285:
1276:
1270:
1262:
1241:
1238:
1237:
1213:
1210:
1209:
1185:
1182:
1181:
1159:
1156:
1155:
1133:
1130:
1129:
1101:
1098:
1097:
1075:
1072:
1071:
1049:
1046:
1045:
1026:
1023:
1022:
1002:
999:
998:
982:
979:
978:
937:
934:
933:
911:
908:
907:
885:
882:
881:
852:
849:
848:
826:
823:
822:
800:
797:
796:
767:
764:
763:
741:
738:
737:
708:
705:
704:
684:
681:
680:
664:
661:
660:
638:
635:
634:
618:
615:
614:
595:
592:
591:
589:binary relation
572:
569:
568:
548:
545:
544:
535:
454:
451:
450:
428:
425:
424:
396:
393:
392:
370:
367:
366:
344:
341:
340:
311:
308:
307:
285:
282:
281:
259:
256:
255:
226:
223:
222:
203:
200:
199:
183:
180:
179:
157:
154:
153:
137:
134:
133:
114:
111:
110:
108:binary relation
82:
71:
65:
62:
52:Please help to
51:
35:
31:
24:
17:
12:
11:
5:
6748:
6738:
6737:
6732:
6727:
6710:
6709:
6707:
6706:
6701:
6696:
6695:
6694:
6684:
6683:
6682:
6677:
6672:
6662:
6661:
6660:
6650:
6645:
6644:
6643:
6638:
6631:Order morphism
6628:
6627:
6626:
6616:
6611:
6606:
6601:
6596:
6595:
6594:
6584:
6579:
6574:
6568:
6566:
6562:
6561:
6559:
6558:
6557:
6556:
6551:
6549:Locally convex
6546:
6541:
6531:
6529:Order topology
6526:
6525:
6524:
6522:Order topology
6519:
6509:
6499:
6497:
6490:
6489:
6487:
6486:
6481:
6476:
6471:
6466:
6461:
6456:
6451:
6446:
6441:
6435:
6433:
6429:
6428:
6426:
6425:
6415:
6405:
6400:
6395:
6390:
6385:
6380:
6375:
6370:
6369:
6368:
6358:
6353:
6352:
6351:
6346:
6341:
6336:
6334:Chain-complete
6326:
6321:
6320:
6319:
6314:
6309:
6304:
6299:
6289:
6284:
6279:
6274:
6269:
6259:
6254:
6249:
6244:
6239:
6234:
6233:
6232:
6222:
6217:
6211:
6209:
6199:
6198:
6196:
6195:
6190:
6185:
6180:
6175:
6170:
6165:
6160:
6155:
6150:
6145:
6140:
6134:
6132:
6128:
6127:
6125:
6124:
6119:
6114:
6109:
6104:
6099:
6094:
6089:
6083:
6081:
6077:
6076:
6074:
6073:
6068:
6063:
6057:
6054:
6053:
6046:
6045:
6038:
6031:
6023:
6017:
6016:
5996:
5995:External links
5993:
5992:
5991:
5985:
5967:
5958:
5944:
5932:
5918:
5909:
5895:
5876:
5872:Lattice Theory
5862:
5859:
5856:
5855:
5827:
5820:
5790:
5781:
5772:
5744:
5728:
5710:
5702:
5688:Roland Fraïssé
5679:
5671:initial object
5662:
5649:
5646:
5643:
5623:
5620:
5617:
5601:
5588:
5585:
5582:
5562:
5559:
5556:
5536:
5533:
5530:
5514:
5495:(7): 462–464.
5479:
5442:
5425:
5413:
5398:
5383:
5367:
5366:
5364:
5361:
5359:
5358:
5352:
5343:
5333:
5327:
5321:
5316:
5306:
5304:
5301:
5263:
5262:
5248:
5245:
5242:
5239:
5219:
5216:
5213:
5193:
5190:
5187:
5167:
5164:
5161:
5158:
5155:
5152:
5128:
5085:
5084:
5081:
5080:
5065:
5062:
5059:
5051:
5050:
5047:
5044:
5041:
5038:
5035:
5034:
5024:
5013:
5010:
5007:
4992:
4981:
4978:
4975:
4965:
4945:
4944:
4941:
4938:
4935:
4932:
4931:
4921:
4901:
4900:
4897:
4894:
4891:
4888:
4887:
4877:
4857:
4856:
4853:
4850:
4847:
4846:
4836:
4821:
4818:
4815:
4812:
4808: or
4805:
4802:
4799:
4796:
4795:
4792:
4789:
4786:
4782:
4779:
4776:
4775:
4765:
4750:
4747:
4743:
4740:
4737:
4734:
4733:
4730:
4727:
4724:
4721:
4714:
4711:
4708:
4705:
4704:
4694:
4679:
4676:
4673:
4670:
4666:
4663:
4662:
4659:
4656:
4653:
4650:
4648:
4638:
4627:
4624:
4621:
4618:
4598:
4595:
4592:
4580:
4579:
4574:
4569:
4564:
4559:
4554:
4549:
4544:
4539:
4534:
4531:
4530:
4520:
4510:
4505:
4500:
4495:
4490:
4480:
4470:
4465:
4459:
4458:
4448:
4438:
4433:
4428:
4423:
4418:
4413:
4403:
4398:
4392:
4391:
4381:
4371:
4366:
4361:
4356:
4351:
4346:
4336:
4331:
4325:
4324:
4319:
4314:
4304:
4294:
4289:
4284:
4279:
4269:
4264:
4258:
4257:
4252:
4247:
4237:
4232:
4222:
4217:
4212:
4202:
4197:
4191:
4190:
4185:
4180:
4170:
4160:
4150:
4145:
4140:
4130:
4125:
4119:
4118:
4113:
4108:
4098:
4093:
4088:
4078:
4068:
4058:
4053:
4047:
4046:
4041:
4036:
4026:
4021:
4016:
4006:
4001:
3996:
3991:
3983:
3982:
3977:
3972:
3962:
3957:
3952:
3942:
3932:
3927:
3922:
3916:
3915:
3910:
3905:
3895:
3890:
3885:
3880:
3870:
3860:
3855:
3849:
3848:
3843:
3838:
3828:
3823:
3818:
3813:
3803:
3798:
3793:
3791:Total preorder
3787:
3786:
3781:
3776:
3766:
3761:
3756:
3751:
3746:
3736:
3731:
3725:
3724:
3719:
3714:
3704:
3699:
3694:
3689:
3684:
3679:
3674:
3663:
3662:
3657:
3652:
3642:
3637:
3632:
3627:
3622:
3617:
3607:
3599:
3598:
3596:
3591:
3589:
3587:
3585:
3583:
3580:
3578:
3576:
3573:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3537:
3532:
3527:
3520:
3519:
3517:
3516:
3509:
3502:
3494:
3480:
3479:
3477:
3474:
3427:
3426:
3423:direct product
3371:
3332:
3273:
3270:
3245:
3242:
3241:
3240:
3239:
3238:
3225:
3221:
3217:
3214:
3192:
3188:
3184:
3181:
3161:
3141:
3138:
3135:
3124:
3113:
3108:
3104:
3100:
3080:
3077:
3074:
3051:
3048:
3045:
3023:
3019:
3013:
3010:
3007:
3003:
2999:
2996:
2993:
2990:
2964:
2960:
2954:
2950:
2927:
2923:
2902:
2897:
2893:
2889:
2884:
2880:
2876:
2867:the structure
2856:
2853:
2850:
2830:
2827:
2824:
2821:
2818:
2804:
2803:
2802:
2801:
2788:
2784:
2780:
2777:
2755:
2751:
2747:
2744:
2734:
2723:
2718:
2714:
2710:
2688:
2684:
2680:
2677:
2674:
2671:
2661:
2650:
2645:
2641:
2637:
2615:
2611:
2607:
2604:
2601:
2598:
2576:
2571:
2567:
2563:
2541:
2537:
2533:
2528:
2524:
2520:
2517:
2514:
2511:
2485:
2481:
2477:
2472:
2468:
2445:
2441:
2437:
2432:
2428:
2405:
2401:
2380:
2375:
2371:
2367:
2362:
2358:
2354:
2334:
2329:
2325:
2321:
2316:
2312:
2308:
2296:
2295:Sums of orders
2293:
2289:homeomorphisms
2269:
2268:
2262:
2212:
2207:is connected,
2164:is that every
2117:
2114:
2072:order topology
2064:
2063:
2054:
2032:
2010:
1972:open intervals
1962:
1961:Order topology
1959:
1896:
1893:
1852:
1849:
1832:
1829:
1826:
1823:
1820:
1810:if and only if
1801:We then write
1799:
1798:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1717:
1716:Lattice theory
1714:
1712:
1709:
1697:regular chains
1600:opposite order
1593:maximal ideals
1528:
1525:
1524:
1523:
1502:
1493:Ordered fields
1490:
1489:
1488:
1485:order topology
1477:
1466:
1459:
1401:
1382:
1376:
1365:
1352:
1345:
1304:
1289:
1282:
1261:
1258:
1245:
1230:
1229:
1217:
1189:
1169:
1166:
1163:
1143:
1140:
1137:
1127:
1111:
1108:
1105:
1085:
1082:
1079:
1059:
1056:
1053:
1030:
1006:
986:
964:
963:
947:
944:
941:
921:
918:
915:
895:
892:
889:
878:
862:
859:
856:
836:
833:
830:
810:
807:
804:
793:
777:
774:
771:
751:
748:
745:
734:
718:
715:
712:
688:
668:
648:
645:
642:
622:
599:
576:
552:
541:
534:
531:
481:
480:
464:
461:
458:
438:
435:
432:
422:
406:
403:
400:
380:
377:
374:
354:
351:
348:
337:
321:
318:
315:
295:
292:
289:
269:
266:
263:
252:
236:
233:
230:
207:
187:
167:
164:
161:
141:
118:
84:
83:
38:
36:
29:
15:
9:
6:
4:
3:
2:
6747:
6736:
6733:
6731:
6728:
6726:
6723:
6722:
6720:
6705:
6702:
6700:
6697:
6693:
6690:
6689:
6688:
6685:
6681:
6678:
6676:
6673:
6671:
6668:
6667:
6666:
6663:
6659:
6656:
6655:
6654:
6653:Ordered field
6651:
6649:
6646:
6642:
6639:
6637:
6634:
6633:
6632:
6629:
6625:
6622:
6621:
6620:
6617:
6615:
6612:
6610:
6609:Hasse diagram
6607:
6605:
6602:
6600:
6597:
6593:
6590:
6589:
6588:
6587:Comparability
6585:
6583:
6580:
6578:
6575:
6573:
6570:
6569:
6567:
6563:
6555:
6552:
6550:
6547:
6545:
6542:
6540:
6537:
6536:
6535:
6532:
6530:
6527:
6523:
6520:
6518:
6515:
6514:
6513:
6510:
6508:
6504:
6501:
6500:
6498:
6495:
6491:
6485:
6482:
6480:
6477:
6475:
6472:
6470:
6467:
6465:
6462:
6460:
6459:Product order
6457:
6455:
6452:
6450:
6447:
6445:
6442:
6440:
6437:
6436:
6434:
6432:Constructions
6430:
6424:
6420:
6416:
6413:
6409:
6406:
6404:
6401:
6399:
6396:
6394:
6391:
6389:
6386:
6384:
6381:
6379:
6376:
6374:
6371:
6367:
6364:
6363:
6362:
6359:
6357:
6354:
6350:
6347:
6345:
6342:
6340:
6337:
6335:
6332:
6331:
6330:
6329:Partial order
6327:
6325:
6322:
6318:
6317:Join and meet
6315:
6313:
6310:
6308:
6305:
6303:
6300:
6298:
6295:
6294:
6293:
6290:
6288:
6285:
6283:
6280:
6278:
6275:
6273:
6270:
6268:
6264:
6260:
6258:
6255:
6253:
6250:
6248:
6245:
6243:
6240:
6238:
6235:
6231:
6228:
6227:
6226:
6223:
6221:
6218:
6216:
6215:Antisymmetric
6213:
6212:
6210:
6206:
6200:
6194:
6191:
6189:
6186:
6184:
6181:
6179:
6176:
6174:
6171:
6169:
6166:
6164:
6161:
6159:
6156:
6154:
6151:
6149:
6146:
6144:
6141:
6139:
6136:
6135:
6133:
6129:
6123:
6122:Weak ordering
6120:
6118:
6115:
6113:
6110:
6108:
6107:Partial order
6105:
6103:
6100:
6098:
6095:
6093:
6090:
6088:
6085:
6084:
6082:
6078:
6072:
6069:
6067:
6064:
6062:
6059:
6058:
6055:
6051:
6044:
6039:
6037:
6032:
6030:
6025:
6024:
6021:
6013:
6009:
6008:
6003:
5999:
5998:
5988:
5982:
5978:
5977:
5972:
5968:
5964:
5959:
5957:
5956:0-486-65676-4
5953:
5949:
5945:
5941:
5937:
5933:
5931:
5930:0-7167-0442-0
5927:
5923:
5919:
5915:
5910:
5906:
5902:
5898:
5896:0-521-36766-2
5892:
5888:
5887:
5882:
5877:
5873:
5869:
5865:
5864:
5851:
5846:
5842:
5838:
5831:
5823:
5817:
5813:
5809:
5805:
5801:
5794:
5785:
5776:
5770:, p. 116
5769:
5768:0-387-28723-X
5765:
5762:(Birkhäuser)
5761:
5757:
5753:
5748:
5739:
5735:
5731:
5729:0-521-36766-2
5725:
5721:
5714:
5705:
5699:
5695:
5694:
5689:
5683:
5676:
5672:
5666:
5660:by asymmetry.
5647:
5644:
5641:
5621:
5618:
5615:
5605:
5586:
5583:
5580:
5560:
5557:
5554:
5534:
5531:
5528:
5518:
5510:
5506:
5502:
5498:
5494:
5490:
5483:
5475:
5471:
5466:
5461:
5457:
5453:
5446:
5439:
5434:
5432:
5430:
5422:
5417:
5411:, p. 32.
5410:
5405:
5403:
5395:
5394:Birkhoff 1967
5390:
5388:
5380:
5375:
5373:
5368:
5356:
5353:
5347:
5344:
5337:
5334:
5331:
5328:
5325:
5322:
5320:
5317:
5311:
5310:Artinian ring
5308:
5307:
5300:
5298:
5294:
5290:
5285:
5283:
5279:
5274:
5272:
5271:partial order
5261:
5246:
5243:
5240:
5237:
5217:
5214:
5211:
5191:
5188:
5185:
5165:
5162:
5159:
5156:
5153:
5150:
5142:
5126:
5119:
5087:
5086:
5063:
5060:
5057:
5042:
5039:
5036:
5025:
5011:
5008:
5005:
4993:
4979:
4976:
4973:
4966:
4939:
4936:
4933:
4922:
4895:
4892:
4889:
4878:
4851:
4837:
4819:
4816:
4813:
4803:
4800:
4797:
4787:
4780:
4777:
4766:
4748:
4741:
4738:
4728:
4725:
4722:
4712:
4709:
4706:
4695:
4677:
4674:
4671:
4657:
4654:
4651:
4639:
4625:
4619:
4616:
4596:
4593:
4590:
4582:
4581:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4543:
4542:Antisymmetric
4540:
4538:
4535:
4533:
4532:
4521:
4511:
4506:
4501:
4496:
4491:
4481:
4471:
4466:
4464:
4461:
4460:
4449:
4439:
4434:
4429:
4424:
4419:
4414:
4404:
4399:
4397:
4394:
4393:
4382:
4372:
4367:
4362:
4357:
4352:
4347:
4337:
4332:
4330:
4327:
4326:
4320:
4315:
4305:
4295:
4290:
4285:
4280:
4270:
4265:
4263:
4260:
4259:
4253:
4248:
4238:
4233:
4223:
4218:
4213:
4203:
4198:
4196:
4193:
4192:
4186:
4181:
4171:
4161:
4151:
4146:
4141:
4131:
4126:
4124:
4121:
4120:
4114:
4109:
4099:
4094:
4089:
4079:
4069:
4059:
4054:
4052:
4051:Well-ordering
4049:
4048:
4042:
4037:
4027:
4022:
4017:
4007:
4002:
3997:
3992:
3989:
3985:
3984:
3978:
3973:
3963:
3958:
3953:
3943:
3933:
3928:
3923:
3921:
3918:
3917:
3911:
3906:
3896:
3891:
3886:
3881:
3871:
3861:
3856:
3854:
3851:
3850:
3844:
3839:
3829:
3824:
3819:
3814:
3804:
3799:
3794:
3792:
3789:
3788:
3782:
3777:
3767:
3762:
3757:
3752:
3747:
3737:
3732:
3730:
3729:Partial order
3727:
3726:
3720:
3715:
3705:
3700:
3695:
3690:
3685:
3680:
3675:
3672:
3665:
3664:
3658:
3653:
3643:
3638:
3633:
3628:
3623:
3618:
3608:
3605:
3601:
3600:
3597:
3592:
3590:
3588:
3586:
3584:
3581:
3579:
3577:
3575:
3574:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3535:Antisymmetric
3533:
3531:
3528:
3526:
3525:
3522:
3521:
3515:
3510:
3508:
3503:
3501:
3496:
3495:
3492:
3488:
3484:
3483:
3473:
3471:
3468:
3464:
3459:
3457:
3452:
3450:
3446:
3443:
3438:
3436:
3432:
3424:
3420:
3416:
3412:
3408:
3404:
3400:
3396:
3392:
3388:
3384:
3380:
3376:
3372:
3369:
3368:product order
3365:
3361:
3357:
3353:
3349:
3345:
3341:
3337:
3333:
3330:
3326:
3322:
3318:
3314:
3310:
3306:
3302:
3298:
3294:
3290:
3287:
3286:
3285:
3283:
3279:
3269:
3267:
3263:
3259:
3255:
3251:
3223:
3219:
3215:
3212:
3190:
3186:
3182:
3179:
3159:
3139:
3136:
3133:
3125:
3111:
3106:
3102:
3098:
3078:
3075:
3072:
3064:
3063:
3049:
3046:
3043:
3021:
3017:
3011:
3008:
3005:
3001:
2997:
2994:
2991:
2988:
2980:
2979:
2978:
2962:
2958:
2952:
2948:
2925:
2921:
2895:
2891:
2887:
2882:
2878:
2854:
2851:
2848:
2825:
2822:
2819:
2807:
2786:
2782:
2778:
2775:
2753:
2749:
2745:
2742:
2735:
2721:
2716:
2712:
2708:
2686:
2682:
2678:
2675:
2672:
2669:
2662:
2648:
2643:
2639:
2635:
2613:
2609:
2605:
2602:
2599:
2596:
2589:
2588:
2574:
2569:
2565:
2561:
2539:
2535:
2531:
2526:
2522:
2518:
2515:
2512:
2509:
2501:
2500:
2499:
2483:
2479:
2475:
2470:
2466:
2443:
2439:
2435:
2430:
2426:
2403:
2399:
2373:
2369:
2365:
2360:
2356:
2327:
2323:
2319:
2314:
2310:
2292:
2290:
2286:
2282:
2281:unit interval
2278:
2274:
2266:
2263:
2260:
2256:
2252:
2248:
2245:such that no
2244:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2213:
2210:
2206:
2202:
2201:
2200:
2197:
2191:
2187:
2183:
2179:
2175:
2171:
2167:
2159:
2155:
2151:
2147:
2144:
2140:
2137:
2133:
2129:
2125:
2124:
2113:
2111:
2106:
2100:
2092:
2080:
2075:
2073:
2069:
2060:
2055:
2050:
2046:
2042:
2038:
2033:
2028:
2024:
2020:
2016:
2011:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1977:
1976:
1975:
1974:
1973:
1958:
1956:
1952:
1949:
1944:
1942:
1938:
1934:
1930:
1926:
1922:
1919:such that if
1918:
1914:
1910:
1906:
1902:
1892:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1858:
1848:
1846:
1830:
1827:
1824:
1821:
1818:
1811:
1808:
1804:
1796:
1792:
1773:
1770:
1767:
1761:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1727:
1726:
1725:
1723:
1708:
1706:
1702:
1698:
1694:
1689:
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1659:
1658:singleton set
1655:
1651:
1647:
1643:
1638:
1636:
1632:
1628:
1624:
1620:
1615:
1613:
1609:
1605:
1601:
1596:
1594:
1590:
1586:
1582:
1554:
1550:
1545:
1542:
1538:
1534:
1520:
1516:
1512:
1507:
1503:
1500:
1499:
1494:
1491:
1486:
1482:
1478:
1475:
1471:
1467:
1464:
1460:
1457:
1453:
1452:
1449:
1443:
1437:
1432:
1427:
1422:
1418:
1414:
1410:
1406:
1402:
1399:
1395:
1391:
1387:
1383:
1375:
1371:
1364:
1360:
1351:
1344:
1338:
1332:
1326:
1321:
1316:
1310:
1305:
1302:
1298:
1294:
1290:
1283:
1279:
1273:
1268:
1264:
1263:
1257:
1243:
1235:
1215:
1207:
1203:
1187:
1167:
1164:
1161:
1141:
1138:
1135:
1128:
1125:
1109:
1106:
1103:
1083:
1080:
1077:
1057:
1054:
1051:
1044:
1043:
1042:
1028:
1020:
1017:, called the
1004:
984:
976:
972:
967:
961:
945:
942:
939:
919:
916:
913:
893:
890:
887:
879:
876:
860:
857:
854:
834:
831:
828:
808:
805:
802:
794:
791:
775:
772:
769:
749:
746:
743:
735:
732:
716:
713:
710:
702:
701:
700:
686:
666:
646:
643:
640:
620:
613:
597:
590:
574:
566:
550:
542:
539:
530:
528:
523:
521:
517:
513:
509:
505:
501:
496:
494:
490:
486:
478:
462:
459:
456:
436:
433:
430:
423:
420:
419:antisymmetric
404:
401:
398:
378:
375:
372:
352:
349:
346:
338:
335:
319:
316:
313:
293:
290:
287:
267:
264:
261:
253:
250:
234:
231:
228:
221:
220:
219:
205:
185:
165:
162:
159:
139:
132:
116:
109:
105:
104:partial order
101:
97:
93:
88:
80:
77:
69:
66:February 2016
59:
55:
49:
48:
42:
37:
28:
27:
22:
6730:Order theory
6496:& Orders
6474:Star product
6403:Well-founded
6356:Prefix order
6312:Distributive
6302:Complemented
6272:Foundational
6237:Completeness
6193:Zorn's lemma
6116:
6097:Cyclic order
6080:Key concepts
6050:Order theory
6005:
5975:
5962:
5947:
5939:
5921:
5913:
5885:
5871:
5840:
5836:
5830:
5803:
5793:
5784:
5775:
5755:
5747:
5742:Here: p. 100
5719:
5713:
5692:
5682:
5665:
5604:
5517:
5492:
5488:
5482:
5455:
5445:
5440:, p. 3.
5423:, p. 2.
5416:
5396:, p. 2.
5336:Prefix order
5324:Order theory
5293:cyclic order
5286:
5275:
5268:
5115:
4552:Well-founded
3852:
3670:(Quasiorder)
3545:Well-founded
3466:
3462:
3460:
3453:
3444:
3442:vector space
3439:
3434:
3430:
3428:
3418:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3363:
3359:
3355:
3351:
3347:
3343:
3339:
3335:
3328:
3324:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3275:
3247:
3244:Decidability
2808:
2805:
2298:
2270:
2264:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2218:
2214:
2211:is complete.
2208:
2204:
2198:
2189:
2181:
2173:
2169:
2158:real numbers
2154:restrictions
2150:completeness
2145:
2138:
2136:real numbers
2121:
2119:
2116:Completeness
2107:
2098:
2090:
2078:
2076:
2065:
2058:
2048:
2044:
2040:
2036:
2026:
2022:
2018:
2014:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1970:
1964:
1945:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1898:
1884:
1880:
1854:
1806:
1802:
1800:
1794:
1790:
1719:
1690:
1686:prime ideals
1662:ordered pair
1653:
1649:
1646:finite chain
1645:
1639:
1623:well founded
1616:
1611:
1607:
1597:
1581:vector space
1553:Zorn's lemma
1548:
1546:
1532:
1530:
1518:
1514:
1510:
1496:
1447:
1441:
1435:
1430:
1425:
1405:real numbers
1373:
1369:
1362:
1358:
1349:
1342:
1336:
1330:
1324:
1314:
1308:
1277:
1271:
1231:
1018:
974:
968:
965:
538:
536:
524:
519:
515:
511:
507:
503:
502:; the terms
499:
497:
492:
488:
484:
482:
100:linear order
99:
95:
89:
87:
72:
63:
44:
6680:Riesz space
6641:Isomorphism
6517:Normal cone
6439:Composition
6373:Semilattice
6282:Homogeneous
6267:Equivalence
6117:Total order
5708:Here: p. 35
5379:Halmos 1968
5330:Permutation
4572:Irreflexive
3853:Total order
3565:Irreflexive
3250:first-order
2418:on the set
2178:upper bound
2128:upper bound
2101:induced by
2093:induced by
1955:isomorphism
1911:, with the
1642:finite sets
1587:and that a
1585:Hamel bases
1474:dense order
1463:lower bound
1456:upper bound
1403:The set of
1340:by setting
1301:well-orders
1291:Any set of
731:irreflexive
493:full orders
96:total order
92:mathematics
58:introducing
6735:Set theory
6719:Categories
6648:Order type
6582:Cofinality
6423:Well-order
6398:Transitive
6287:Idempotent
6220:Asymmetric
5861:References
5634:, the not
5421:Fuchs 1963
5355:Well-order
5143:: for all
5141:transitive
4577:Asymmetric
3570:Asymmetric
3487:Transitive
3264:theory of
3062:holds if:
2283:, and the
2249:satisfies
2057:(−∞, ∞) =
1889:order type
1861:well order
1693:structures
1676:, and the
1202:complement
975:non-strict
875:transitive
790:asymmetric
334:transitive
41:references
6699:Upper set
6636:Embedding
6572:Antichain
6393:Tolerance
6383:Symmetric
6378:Semiorder
6324:Reflexive
6242:Connected
6012:EMS Press
5948:Topology.
5501:0031-952X
5458:(7): 84.
5054:not
5046:⇒
5002:not
4937:∧
4893:∨
4791:⇒
4781:≠
4736:⇒
4665:⇒
4623:∅
4620:≠
4567:Reflexive
4562:Has meets
4557:Has joins
4547:Connected
4537:Symmetric
3668:Preorder
3595:reflexive
3560:Reflexive
3555:Has meets
3550:Has joins
3540:Connected
3530:Symmetric
3454:See also
3266:countable
3254:decidable
3216:∈
3183:∈
3103:≤
3076:∈
3047:≤
3009:∈
3002:⋃
2998:∈
2949:⋃
2892:≤
2852:∈
2826:≤
2779:∈
2746:∈
2713:≤
2679:∈
2640:≤
2606:∈
2566:≤
2532:∪
2519:∈
2436:∪
2400:≤
2370:≤
2324:≤
2166:non-empty
1948:bijective
1913:morphisms
1855:A simple
1828:∧
1753:∧
1741:∨
1666:dimension
1531:The term
1481:connected
1216:≤
1165:≤
1107:≠
1081:≤
1029:≤
985:≤
960:connected
891:≠
762:then not
543:on a set
460:≤
434:≤
376:≤
350:≤
317:≤
291:≤
265:≤
249:reflexive
232:≤
117:≤
6494:Topology
6361:Preorder
6344:Eulerian
6307:Complete
6257:Directed
6247:Covering
6112:Preorder
6071:Category
6066:Glossary
5938:(1968).
5905:89009753
5883:(1990).
5870:(1967).
5738:89009753
5675:category
5509:24340068
5474:38115497
5381:, Ch.14.
5303:See also
5111:✗
5098:✗
4508:✗
4503:✗
4498:✗
4493:✗
4468:✗
4436:✗
4431:✗
4426:✗
4421:✗
4416:✗
4401:✗
4369:✗
4364:✗
4359:✗
4354:✗
4349:✗
4334:✗
4322:✗
4317:✗
4292:✗
4287:✗
4282:✗
4267:✗
4255:✗
4250:✗
4235:✗
4220:✗
4215:✗
4200:✗
4188:✗
4183:✗
4148:✗
4143:✗
4128:✗
4116:✗
4111:✗
4096:✗
4091:✗
4056:✗
4044:✗
4039:✗
4024:✗
4019:✗
4004:✗
3999:✗
3994:✗
3980:✗
3975:✗
3960:✗
3955:✗
3930:✗
3925:✗
3913:✗
3908:✗
3893:✗
3888:✗
3883:✗
3858:✗
3846:✗
3841:✗
3826:✗
3821:✗
3816:✗
3801:✗
3796:✗
3784:✗
3779:✗
3764:✗
3759:✗
3754:✗
3749:✗
3734:✗
3722:✗
3717:✗
3702:✗
3697:✗
3692:✗
3687:✗
3682:✗
3677:✗
3660:✗
3655:✗
3640:✗
3635:✗
3630:✗
3625:✗
3620:✗
2176:with an
2130:, has a
2123:complete
2068:topology
2039:, ∞) = {
1905:category
1857:counting
1789:for all
1508:, e.g.,
1413:integers
1260:Examples
1206:converse
610:on some
129:on some
6599:Duality
6577:Cofinal
6565:Related
6544:Fréchet
6421:)
6297:Bounded
6292:Lattice
6265:)
6263:Partial
6131:Results
6102:Lattice
6014:, 2001
5754:(2006)
5573:. Then
4123:Lattice
3282:partial
2277:compact
2168:subset
1903:of the
1869:ordinal
1722:lattice
1563:, then
1483:in the
1431:initial
1394:indexed
1388:on the
1204:of the
1200:is the
1180:(i.e.,
1154:if not
906:, then
54:improve
6624:Subnet
6604:Filter
6554:Normed
6539:Banach
6505:&
6412:Better
6349:Strict
6339:Graded
6230:topics
6061:Topics
5983:
5954:
5928:
5903:
5893:
5818:
5766:
5736:
5726:
5700:
5507:
5499:
5472:
4948:exists
4904:exists
4860:exists
3489:
3405:) or (
3260:, the
2184:has a
2110:normal
1875:to an
1867:to an
1654:length
1635:ideals
1575:is in
1537:subset
1527:Chains
1415:, and
1267:subset
510:, and
489:connex
485:simple
43:, but
6614:Ideal
6592:Graph
6388:Total
6366:Total
6252:Dense
5673:of a
5505:JSTOR
5470:S2CID
5363:Notes
5278:group
5230:then
3593:Anti-
3401:<
3393:<
3381:) ≤ (
3366:(the
3342:) ≤ (
3311:<
3299:) ≤ (
3172:with
3091:with
2257:<
2253:<
2241:<
2237:with
2053:, and
2047:<
2025:<
2017:) = {
2013:(−∞,
2003:<
1995:<
1987:) = {
1927:then
1705:graph
1703:in a
1680:of a
1650:chain
1610:or a
1549:chain
1539:of a
1533:chain
1517:<
1513:<
1470:dense
1396:by a
1322:from
971:above
847:then
563:is a
516:chain
512:loset
491:, or
391:then
306:then
102:is a
6205:list
5981:ISBN
5952:ISBN
5926:ISBN
5901:LCCN
5891:ISBN
5816:ISBN
5764:ISBN
5734:LCCN
5724:ISBN
5698:ISBN
5645:<
5619:<
5584:<
5558:<
5532:<
5521:Let
5497:ISSN
5204:and
4609:and
3413:and
3397:and
3358:and
3323:and
3315:or (
3248:The
3137:<
2981:For
2768:and
2701:and
2628:and
2502:For
2345:and
2229:and
2103:>
2095:<
2087:>
2083:<
1999:and
1935:) ≤
1701:walk
1591:has
1589:ring
1583:has
1384:The
1368:) ≤
1265:Any
1244:<
1188:<
1139:<
1096:and
1055:<
1005:<
943:<
917:<
858:<
832:<
821:and
806:<
773:<
747:<
714:<
703:Not
659:and
598:<
365:and
280:and
178:and
94:, a
6619:Net
6419:Pre
5845:doi
5841:311
5808:doi
5608:If
5460:doi
5178:if
5139:be
4849:min
3291:: (
3258:S2S
3205:,
3152:in
2498::
2275:is
2233:in
2221:in
2219:gap
2180:in
2172:of
1951:map
1943:).
1907:of
1451:):
1429:is
1318:an
1306:If
1295:or
1208:of
1070:if
932:or
880:If
795:If
736:If
679:in
612:set
567:on
449:or
339:If
254:If
198:in
131:set
98:or
90:In
6721::
6010:,
6004:,
5899:.
5839:,
5814:.
5802:.
5758:,
5732:.
5503:.
5491:.
5468:.
5454:.
5428:^
5401:^
5386:^
5371:^
5299:.
5284:.
5276:A
5273:.
3458:.
3451:.
3433:≤
3417:=
3409:=
3362:≤
3354:≤
3327:≤
3319:=
3036:,
2554:,
2261:.)
2112:.
2074:.
2043:|
2021:|
1991:|
1983:,
1946:A
1923:≤
1847:.
1805:≤
1793:,
1707:.
1688:.
1595:.
1411:,
1348:≤
1303:).
1228:).
1126:).
962:).
877:).
792:).
733:).
699::
537:A
506:,
495:.
487:,
421:).
336:).
251:).
218::
6417:(
6414:)
6410:(
6261:(
6208:)
6042:e
6035:t
6028:v
5989:.
5907:.
5847::
5824:.
5810::
5740:.
5706:.
5648:a
5642:a
5622:a
5616:a
5587:a
5581:a
5561:a
5555:b
5535:b
5529:a
5511:.
5493:9
5476:.
5462::
5247:.
5244:c
5241:R
5238:a
5218:c
5215:R
5212:b
5192:b
5189:R
5186:a
5166:,
5163:c
5160:,
5157:b
5154:,
5151:a
5127:R
5107:Y
5094:Y
5064:a
5061:R
5058:b
5043:b
5040:R
5037:a
5012:a
5009:R
5006:a
4980:a
4977:R
4974:a
4940:b
4934:a
4896:b
4890:a
4852:S
4820:a
4817:R
4814:b
4804:b
4801:R
4798:a
4788:b
4778:a
4749:b
4742:=
4739:a
4729:a
4726:R
4723:b
4713:b
4710:R
4707:a
4678:a
4675:R
4672:b
4658:b
4655:R
4652:a
4626::
4617:S
4597:b
4594:,
4591:a
4528:Y
4518:Y
4488:Y
4478:Y
4456:Y
4446:Y
4411:Y
4389:Y
4379:Y
4344:Y
4312:Y
4302:Y
4277:Y
4245:Y
4230:Y
4210:Y
4178:Y
4168:Y
4158:Y
4138:Y
4106:Y
4086:Y
4076:Y
4066:Y
4034:Y
4014:Y
3970:Y
3950:Y
3940:Y
3903:Y
3878:Y
3868:Y
3836:Y
3811:Y
3774:Y
3744:Y
3712:Y
3650:Y
3615:Y
3513:e
3506:t
3499:v
3467:R
3463:n
3445:R
3435:y
3431:x
3419:d
3415:b
3411:c
3407:a
3403:d
3399:b
3395:c
3391:a
3387:d
3385:,
3383:c
3379:b
3377:,
3375:a
3373:(
3364:d
3360:b
3356:c
3352:a
3348:d
3346:,
3344:c
3340:b
3338:,
3336:a
3334:(
3329:d
3325:b
3321:c
3317:a
3313:c
3309:a
3305:d
3303:,
3301:c
3297:b
3295:,
3293:a
3224:j
3220:A
3213:y
3191:i
3187:A
3180:x
3160:I
3140:j
3134:i
3112:y
3107:i
3099:x
3079:I
3073:i
3050:y
3044:x
3022:i
3018:A
3012:I
3006:i
2995:y
2992:,
2989:x
2963:i
2959:A
2953:i
2926:i
2922:A
2901:)
2896:i
2888:,
2883:i
2879:A
2875:(
2855:I
2849:i
2829:)
2823:,
2820:I
2817:(
2787:2
2783:A
2776:y
2754:1
2750:A
2743:x
2722:y
2717:2
2709:x
2687:2
2683:A
2676:y
2673:,
2670:x
2649:y
2644:1
2636:x
2614:1
2610:A
2603:y
2600:,
2597:x
2575:y
2570:+
2562:x
2540:2
2536:A
2527:1
2523:A
2516:y
2513:,
2510:x
2484:2
2480:A
2476:+
2471:1
2467:A
2444:2
2440:A
2431:1
2427:A
2404:+
2379:)
2374:2
2366:,
2361:2
2357:A
2353:(
2333:)
2328:1
2320:,
2315:1
2311:A
2307:(
2265:X
2259:b
2255:c
2251:a
2247:c
2243:b
2239:a
2235:X
2231:b
2227:a
2223:X
2215:X
2209:X
2205:X
2194:≤
2190:R
2182:R
2174:R
2170:S
2162:≤
2146:Q
2139:R
2099:N
2091:N
2079:N
2062:.
2059:X
2051:}
2049:x
2045:a
2041:x
2037:a
2035:(
2031:,
2029:}
2027:b
2023:x
2019:x
2015:b
2009:,
2007:}
2005:b
2001:x
1997:x
1993:a
1989:x
1985:b
1981:a
1979:(
1967:X
1941:b
1939:(
1937:f
1933:a
1931:(
1929:f
1925:b
1921:a
1917:f
1885:k
1881:k
1831:b
1825:a
1822:=
1819:a
1807:b
1803:a
1797:.
1795:b
1791:a
1777:}
1774:b
1771:,
1768:a
1765:{
1762:=
1759:}
1756:b
1750:a
1747:,
1744:b
1738:a
1735:{
1577:X
1573:X
1569:X
1565:X
1561:X
1557:X
1519:C
1515:B
1511:A
1465:.
1458:.
1448:B
1442:A
1436:B
1426:A
1381:.
1379:)
1377:2
1374:x
1372:(
1370:f
1366:1
1363:x
1361:(
1359:f
1353:2
1350:x
1346:1
1343:x
1337:X
1331:f
1325:X
1315:f
1309:X
1286:∅
1281:.
1278:X
1272:X
1168:a
1162:b
1142:b
1136:a
1122:(
1110:b
1104:a
1084:b
1078:a
1058:b
1052:a
958:(
946:a
940:b
920:b
914:a
894:b
888:a
873:(
861:c
855:a
835:c
829:b
809:b
803:a
788:(
776:a
770:b
750:b
744:a
729:(
717:a
711:a
687:X
667:c
647:b
644:,
641:a
621:X
575:X
551:X
475:(
463:a
457:b
437:b
431:a
417:(
405:b
402:=
399:a
379:a
373:b
353:b
347:a
332:(
320:c
314:a
294:c
288:b
268:b
262:a
247:(
235:a
229:a
206:X
186:c
166:b
163:,
160:a
140:X
79:)
73:(
68:)
64:(
50:.
23:.
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