1664:
1651:
1085:
1075:
1045:
1035:
1013:
1003:
968:
946:
936:
901:
869:
859:
834:
802:
787:
767:
735:
725:
715:
695:
663:
643:
633:
623:
591:
571:
527:
507:
497:
460:
435:
425:
393:
368:
331:
301:
269:
207:
172:
1831:
2089:, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a
5377:
5300:
2742:
1395:
2093:. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components.
5139:
1324:
7259:
1639:
2534:
1253:
6655:
and there is one morphism for objects which are related, zero otherwise. In this sense, categories "generalize" preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder
1524:
8119:
As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:
1480:
1436:
4838:
3770:
1661:
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
5078:
1594:
1491:
1447:
1406:
1335:
1264:
1208:
5495:
4565:
6493:
2686:
6244:
6055:
5648:
8497:
5841:
5083:
2370:
5989:
1197:
6867:
4312:
4131:
4106:
1583:
7346:
7292:
4344:
4284:
3792:
3543:
8319:
4198:
3670:
3489:
3460:
6077:
5907:
5517:
5195:
4909:
4492:
4466:
3641:
3615:
3589:
2808:
2786:
2616:
2392:
2276:
2228:
2116:
8686:
8462:
8368:
8280:
7556:
6212:
4440:
4412:
3168:
2167:
8978:
8433:
8193:
7619:
7593:
7527:
7461:
7435:
6900:
6820:
6744:
6702:
6570:
6426:
6372:
5958:
4987:
4961:
4879:
4169:
2575:
2445:
2328:
3245:
7654:
7099:
6121:
6099:
5929:
5749:
5724:
5680:
5559:
5221:
5009:
3877:
3705:
2764:
2661:
2302:
2138:
1330:
5771:
5702:
5539:
5425:
5403:
5295:
5161:
4745:
4633:
4380:
3901:
3853:
2870:
2842:
8715:
8651:
7060:
6325:
6270:
6183:
6015:
5801:
5605:
5247:
4778:
4718:
4521:
4081:
4053:
3822:
3730:
3101:
2931:
2903:
2474:
8622:
7393:
6296:
5455:
5035:
4935:
4689:
4025:
3979:
3515:
3411:
1737:
1259:
8776:
8592:
8537:
5579:
4262:
4230:
3139:
2987:
8560:
8235:
8152:
7490:
6141:
5885:
5861:
5372:{\displaystyle a<b\quad {\text{ if and only if }}\quad a\lesssim b\;{\text{ and }}\;a\neq b\quad \quad ({\text{assuming }}\lesssim {\text{ is antisymmetric}}).}
1818:
5273:
3013:
1789:
1763:
1551:
1168:
7373:
7319:
7196:
7166:
6653:
3924:
3434:
3271:
2254:
1997:
1974:
7143:
7119:
7019:
6999:
6979:
4653:
4605:
3999:
3953:
3563:
3385:
3353:
3321:
3291:
3073:
3053:
3033:
2951:
2681:
2636:
2076:
2056:
2017:
1951:
1698:
8749:
8403:
6766:
between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not one-to-one.
3220:
3194:
2078:, together with a partial order on the set of equivalence class. Like partial orders and equivalence relations, preorders (on a nonempty set) are never
7201:
6382:
in the directed graph. Conversely, every preorder is the reachability relationship of a directed graph (for instance, the graph that has an edge from
1589:
2485:
1203:
1486:
1442:
1401:
9008:
Schmidt, Gunther, "Relational
Mathematics", Encyclopedia of Mathematics and its Applications, vol. 132, Cambridge University Press, 2011,
1933:. For example, the divides relation is reflexive as every integer divides itself. But the divides relation is not antisymmetric, because
9709:
72:
4789:
2034:
and (non-strict) partial orders. Both of these are special cases of a preorder: an antisymmetric preorder is a partial order, and a
9692:
9742:
9222:
9058:
5843:(that is, take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "
3735:
9013:
9539:
5040:
6428:). However, many different graphs may have the same reachability preorder as each other. In the same way, reachability of
8113:
2027:" (except that, for integers, the greatest common divisor is also the greatest for the natural order of the integers).
5541:
would not be transitive (consider how equivalent non-equal elements relate). This is the reason for using the symbol "
5460:
4526:
9675:
9534:
9028:
8904:
7716:
6462:
9529:
9165:
8894:
6217:
5581:", which might cause confusion for a preorder that is not antisymmetric since it might misleadingly suggest that
3360:
6020:
9247:
65:
7682:; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.
9566:
9486:
6594:, which is when the literals in a disjunctive first-order formula are contained by another, after applying a
5610:
4568:
3328:
8467:
7262:
5806:
9351:
9280:
9160:
8805:
6942:
3170:
That this is well-defined, meaning that its defining condition does not depend on which representatives of
2333:
5963:
9254:
9242:
9205:
9180:
9155:
9109:
9078:
8994:, Studies in logic and the foundation of mathematics, vol. 102, Amsterdam, the Netherlands: Elsevier
8825:
3332:
3324:
2813:
Using this relation, it is possible to construct a partial order on the quotient set of the equivalence,
1173:
6825:
9551:
9185:
9175:
9051:
6751:
4292:
4111:
4086:
1557:
9524:
9190:
7324:
7268:
6763:
6759:
6532:
6502:
4317:
4267:
3775:
3526:
3294:
2737:{\displaystyle a\sim b\quad {\text{ if and only if }}\quad a\lesssim b\;{\text{ and }}\;b\lesssim a.}
58:
28:
8292:
6496:
4174:
3646:
3465:
3439:
9456:
9083:
6717:
6629:
6060:
5890:
5500:
5178:
4892:
4471:
4445:
3620:
3594:
3568:
2791:
2769:
2599:
2375:
2259:
2211:
2099:
1132:
956:
125:
8656:
8438:
8344:
8253:
7666:
equipped with a preorder. Every set is a class and so every preordered set is a preordered class.
7532:
6188:
4420:
4385:
3144:
2143:
1390:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
9704:
9687:
8963:
8412:
8166:
7598:
7572:
7506:
7440:
7414:
7349:
6879:
6799:
6723:
6666:
6573:
6549:
6405:
6351:
5937:
5657:
Using the construction above, multiple non-strict preorders can produce the same strict preorder
4966:
4940:
4843:
4136:
2539:
2424:
2307:
2020:
3225:
9616:
9232:
8791:
8371:
7624:
7408:
7065:
6609:
6539:
6429:
6104:
6082:
5912:
5729:
5707:
5660:
5544:
5204:
5198:
5134:{\displaystyle a\lesssim b\quad {\text{ if and only if }}\quad a<b\;{\text{ or }}\;a\sim b.}
4992:
3858:
3675:
3364:
2747:
2644:
2281:
2121:
2090:
2024:
1915:
1319:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
1102:
95:
5754:
5685:
5522:
5408:
5386:
5278:
5144:
4723:
4616:
4353:
3882:
3827:
2851:
2816:
9747:
9594:
9429:
9420:
9289:
9170:
9124:
9088:
9044:
8787:
8691:
8627:
7766:
7704:
7493:
7032:
6788:
6581:
6433:
6301:
6249:
6162:
5994:
5780:
5726:
for instance), it might not be possible to reconstruct the original non-strict preorder from
5584:
5226:
4754:
4694:
4497:
4058:
4030:
3801:
3709:
3078:
2908:
2875:
2450:
1657:
indicates that the column's property is always true the row's term (at the very left), while
1112:
105:
8601:
7378:
6275:
5434:
5014:
4914:
4668:
4004:
3958:
3494:
3390:
1707:
9682:
9641:
9631:
9621:
9366:
9329:
9319:
9299:
9284:
8876:
8796:
8722:
8565:
8510:
7781:
7771:
7559:
7396:
6595:
5564:
5164:
4659:
4584:
4580:
4235:
4203:
3795:
3106:
2960:
2954:
2639:
2031:
1678:
889:
164:
8545:
8211:
8128:
7466:
6126:
5866:
5846:
1794:
8:
9609:
9520:
9466:
9425:
9415:
9304:
9237:
9200:
8835:
7746:
7679:
7663:
7122:
6522:
5252:
4783:
4415:
2992:
2479:
2411:
2079:
1900:
1821:
A term's definition may require additional properties that are not listed in this table.
1768:
1742:
1701:
1530:
1147:
1137:
548:
130:
47:
7355:
7301:
7178:
7148:
6635:
5991:" (that is, take the inverse complement of the relation), which corresponds to defining
3906:
3416:
3253:
2236:
1979:
1956:
9721:
9648:
9501:
9410:
9400:
9341:
9259:
9195:
8943:
8890:
8809:
8800:
7756:
7751:
7500:
7128:
7104:
7022:
7004:
6984:
6964:
6903:
6528:
6506:
6148:
4638:
4590:
3984:
3929:
3548:
3370:
3356:
3338:
3306:
3276:
3058:
3038:
3018:
2936:
2666:
2621:
2418:
2407:
2061:
2041:
2035:
2002:
1936:
1896:
1683:
1127:
1107:
1097:
1023:
120:
100:
90:
20:
9561:
8376:
3199:
3173:
9658:
9636:
9496:
9481:
9461:
9264:
9024:
9009:
8900:
7295:
7026:
6919:
6772:
6660:
6591:
6144:
4287:
3387:
is that it is closed under logical consequences so that, for instance, if a sentence
2845:
2231:
8947:
4200:
where the right hand side condition is independent of the choice of representatives
9471:
9324:
8933:
8921:
8864:
7693:
7686:
7659:
6510:
1930:
1852:
1845:
822:
755:
9653:
9436:
9314:
9309:
9294:
9210:
9119:
9104:
8872:
8830:
7741:
4613:
is sometimes used for a strict partial order. That is, this is a binary relation
1892:
1856:
683:
480:
51:
7254:{\displaystyle R\backslash R={\overline {R^{\textsf {T}}\circ {\overline {R}}}}}
1867:) are shown together as a single node. The relation on equivalence classes is a
9571:
9556:
9546:
9405:
9383:
9361:
7566:
6456:
6345:
6152:
2086:
1634:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
351:
2529:{\displaystyle a\lesssim b{\text{ and }}b\lesssim c{\text{ then }}a\lesssim c}
1248:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
9736:
9670:
9626:
9604:
9476:
9346:
9334:
9139:
8868:
6625:
1911:
1868:
1834:
1122:
1117:
289:
115:
110:
2019:. It is to this preorder that "greatest" and "lowest" refer in the phrases "
1922:
9491:
9373:
9356:
9274:
9114:
9067:
8899:. Cambridge, Massachusetts/London, England: The MIT Press. pp. 182ff.
8820:
6776:
6543:
6341:
4347:
3519:
1880:
35:
19:
This article is about binary relations. For the graph vertex ordering, see
8938:
9697:
9390:
9269:
9134:
8814:
7776:
7712:
7697:
6934:
6923:
6780:
6440:
5383:
used as (nor is it equivalent to) the general definition of the relation
1876:
413:
1519:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
9665:
9599:
9440:
7708:
6949:
6915:
6452:
In computer science, one can find examples of the following preorders.
1926:
611:
34:"Quasiorder" redirects here. For irreflexive transitive relations, see
5863:" through reflexive closure; in this case the equivalence is equality
5751:
Possible (non-strict) preorders that induce the given strict preorder
1475:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
9716:
9589:
9395:
6930:
6516:
1431:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
24:
8855:
Eklund, Patrik; Gähler, Werner (1990), "Generalized Cauchy spaces",
9511:
6784:
6613:
2038:
preorder is an equivalence relation. Moreover, a preorder on a set
3247:
It is readily verified that this yields a partially ordered set.
9036:
4833:{\displaystyle a<b{\text{ and }}b<c{\text{ then }}a<c}
5652:
6348:(possibly containing cycles) gives rise to a preorder, where
4884:
4583:(while keeping transitivity) then we get the definition of a
8922:"A machine-oriented logic based on the resolution principle"
5704:
was constructed (such knowledge of the equivalence relation
2591:
8089:
8084:
8079:
8074:
8069:
8064:
8059:
8054:
8049:
8044:
6436:(preorders satisfying an additional antisymmetry property).
3250:
Conversely, from any partial order on a partition of a set
1830:
6783:. The definition of convergence via nets is important in
3331:
with certain properties (details of which can be found in
2058:
can equivalently be defined as an equivalence relation on
6101:
are in general not transitive; however, if they are then
2788:
is reflexive; transitive by applying the transitivity of
6918:, or any type of structure-preserving function, such as
6762:
of a topological space in this way. That is, there is a
4574:
3772:; that is, two sentences are equivalent with respect to
3765:{\displaystyle A\Leftarrow B{\text{ and }}B\Leftarrow A}
3297:
between preorders and pairs (partition, partial order).
8507:. One may choose to extend the definition to all pairs
4264:
of the equivalence classes. All that has been said of
7208:
6659:
Alternately, a preordered set can be understood as an
8966:
8752:
8725:
8694:
8659:
8630:
8604:
8568:
8548:
8513:
8470:
8441:
8415:
8379:
8347:
8295:
8256:
8214:
8169:
8131:
7674:
Preorders play a pivotal role in several situations:
7627:
7601:
7575:
7535:
7509:
7469:
7443:
7417:
7381:
7358:
7327:
7304:
7271:
7204:
7181:
7151:
7131:
7107:
7068:
7035:
7007:
6987:
6967:
6882:
6828:
6802:
6726:
6669:
6638:
6552:
6465:
6408:
6354:
6304:
6278:
6252:
6220:
6191:
6165:
6129:
6107:
6085:
6063:
6023:
5997:
5966:
5940:
5915:
5893:
5869:
5849:
5809:
5783:
5757:
5732:
5710:
5688:
5663:
5613:
5587:
5567:
5547:
5525:
5503:
5463:
5437:
5411:
5389:
5303:
5281:
5255:
5229:
5207:
5181:
5147:
5086:
5073:{\displaystyle a\lesssim b{\text{ and not }}a\sim b;}
5043:
5017:
4995:
4969:
4943:
4917:
4895:
4846:
4792:
4757:
4726:
4697:
4671:
4641:
4619:
4593:
4529:
4500:
4474:
4448:
4423:
4388:
4356:
4320:
4295:
4270:
4238:
4206:
4177:
4139:
4114:
4089:
4061:
4033:
4007:
3987:
3961:
3932:
3909:
3885:
3861:
3830:
3804:
3778:
3738:
3712:
3678:
3649:
3623:
3597:
3571:
3551:
3529:
3497:
3468:
3442:
3419:
3393:
3373:
3341:
3309:
3279:
3256:
3228:
3202:
3176:
3147:
3109:
3081:
3061:
3041:
3021:
2995:
2963:
2939:
2911:
2878:
2854:
2819:
2794:
2772:
2750:
2689:
2669:
2647:
2624:
2602:
2542:
2488:
2453:
2427:
2378:
2336:
2310:
2284:
2262:
2239:
2214:
2146:
2124:
2102:
2064:
2044:
2005:
1982:
1959:
1939:
1797:
1771:
1745:
1710:
1686:
1592:
1560:
1533:
1489:
1445:
1404:
1333:
1262:
1206:
1176:
1150:
7062:
The transitive closure indicates path connection in
6374:
in the preorder if and only if there is a path from
7171:
Left residual preorder induced by a binary relation
6720:gives rise to a preorder on its points by defining
6495:. The corresponding equivalence relation is called
5171:strict partial order can be constructed this way.
2580:A set that is equipped with a preorder is called a
23:. For purchase orders for unreleased products, see
8992:Set Theory, An Introduction to Independence Proofs
8972:
8770:
8743:
8709:
8680:
8645:
8616:
8586:
8554:
8531:
8491:
8456:
8427:
8397:
8362:
8313:
8274:
8229:
8187:
8146:
7648:
7613:
7587:
7550:
7521:
7484:
7455:
7429:
7387:
7367:
7340:
7313:
7286:
7253:
7190:
7160:
7137:
7113:
7093:
7054:
7013:
6993:
6973:
6894:
6861:
6814:
6738:
6696:
6647:
6564:
6487:
6420:
6366:
6319:
6290:
6264:
6238:
6206:
6177:
6135:
6115:
6093:
6071:
6049:
6009:
5983:
5952:
5923:
5901:
5879:
5855:
5835:
5795:
5765:
5743:
5718:
5696:
5674:
5642:
5599:
5573:
5553:
5533:
5511:
5489:
5449:
5419:
5397:
5371:
5289:
5267:
5241:
5215:
5189:
5155:
5133:
5072:
5029:
5003:
4981:
4955:
4929:
4903:
4873:
4832:
4772:
4739:
4712:
4683:
4647:
4627:
4599:
4559:
4515:
4486:
4460:
4434:
4406:
4374:
4338:
4306:
4278:
4256:
4224:
4192:
4163:
4125:
4100:
4075:
4047:
4019:
3993:
3973:
3947:
3918:
3895:
3871:
3847:
3816:
3786:
3764:
3724:
3699:
3664:
3635:
3609:
3583:
3557:
3537:
3509:
3483:
3454:
3428:
3405:
3379:
3347:
3315:
3285:
3265:
3239:
3214:
3188:
3162:
3133:
3095:
3067:
3047:
3027:
3007:
2981:
2945:
2925:
2897:
2864:
2836:
2802:
2780:
2758:
2736:
2675:
2655:
2630:
2610:
2569:
2528:
2468:
2439:
2386:
2364:
2322:
2296:
2270:
2248:
2222:
2200:. Occasionally, the notation ← or → is also used.
2161:
2132:
2110:
2070:
2050:
2011:
1991:
1968:
1945:
1859:. Equivalence classes (sets of elements such that
1812:
1783:
1757:
1731:
1692:
1633:
1577:
1545:
1518:
1474:
1430:
1389:
1318:
1247:
1191:
1162:
5561:" instead of the "less than or equal to" symbol "
5519:is not antisymmetric then the resulting relation
9734:
6779:preorder, that is, each pair of elements has an
6432:, directed graphs with no cycles, gives rise to
5490:{\displaystyle a\lesssim b{\text{ and }}a\neq b}
5201:(and thus a partial order) then the equivalence
4911:gives rise to a strict partial order defined by
4560:{\displaystyle \left(S/\sim ,\Leftarrow \right)}
3824:is commonly denoted with its own special symbol
2096:As a binary relation, a preorder may be denoted
1409:
6488:{\displaystyle f:\mathbb {N} \to \mathbb {N} }
4108:which will also be denoted by the same symbol
9052:
7733:-element binary relations of different types
6758:. Every finite preorder can be formed as the
1914:, but not quite, as they are not necessarily
1674:in the "Antisymmetric" column, respectively.
66:
8854:
7678:Every preorder can be given a topology, the
6239:{\displaystyle \,\lesssim \;\;=\;\;\leq \,}
5653:Preorders induced by a strict partial order
3222:are chosen, follows from the definition of
9710:Positive cone of a partially ordered group
9059:
9045:
8817:– preorder that is antisymmetric and total
6624:is a preorder. Such categories are called
6231:
6230:
6226:
6225:
6050:{\displaystyle a<b{\text{ nor }}b<a}
5335:
5329:
5118:
5112:
4885:Strict partial order induced by a preorder
4567:is consequently also a directed set. See
4041:
4037:
3867:
3863:
3838:
3834:
3273:it is possible to construct a preorder on
2721:
2715:
73:
59:
8937:
8542:Using the corresponding strict relation "
7278:
7226:
6481:
6473:
6235:
6221:
6112:
6108:
6090:
6086:
6068:
6064:
5920:
5916:
5898:
5894:
5870:
5762:
5758:
5740:
5733:
5715:
5711:
5693:
5689:
5671:
5664:
5530:
5526:
5508:
5504:
5416:
5412:
5394:
5390:
5286:
5282:
5275:) and so in this case, the definition of
5212:
5208:
5186:
5182:
5152:
5148:
5000:
4996:
4900:
4896:
4727:
4624:
4620:
4431:
4424:
4303:
4296:
4275:
4271:
4122:
4115:
4097:
4090:
3886:
3868:
3862:
3783:
3779:
3534:
3530:
3236:
3229:
2855:
2799:
2795:
2777:
2773:
2755:
2751:
2652:
2648:
2607:
2603:
2592:Preorders as partial orders on partitions
2383:
2379:
2355:
2267:
2263:
2219:
2215:
2129:
2125:
2107:
2103:
9693:Positive cone of an ordered vector space
9018:
8919:
6787:, where preorders cannot be replaced by
1829:
16:Reflexive and transitive binary relation
5643:{\displaystyle a<b{\text{ or }}a=b.}
5379:But importantly, this new condition is
3798:. This particular equivalence relation
1921:A natural example of a preorder is the
1907:is meant to suggest that preorders are
9735:
8889:
8688:An open interval may be empty even if
8492:{\displaystyle a\lesssim x\lesssim b.}
7723:
7402:
5836:{\displaystyle a<b{\text{ or }}a=b}
5682:so without more information about how
9040:
8989:
4575:Relationship to strict partial orders
2810:twice; and symmetric by definition.
2365:{\displaystyle (a,b)\in \,\lesssim .}
8562:", one can also define the interval
8242:7 partitions with two classes (4 of
6513:are preorders on complexity classes.
6151:(formerly called total); that is, a
5984:{\displaystyle {\text{ not }}b<a}
3903:The equivalence class of a sentence
1677:All definitions tacitly require the
8539:The extra intervals are all empty.
8239:1 partition of 4, giving 1 preorder
8156:1 partition of 3, giving 1 preorder
8114:Stirling numbers of the second kind
6447:
1192:{\displaystyle S\neq \varnothing :}
13:
9220:Properties & Types (
8967:
6862:{\displaystyle f(x)\lesssim f(y),}
6791:without losing important features.
6602:
6459:causes a preorder over functions
2085:A preorder can be visualized as a
14:
9759:
9676:Positive cone of an ordered field
8980:" does not mean "set difference".
7001:can be extended to a preorder on
6873:is a function into some preorder.
6123:is an equivalence; in that case "
4989:. Using the equivalence relation
4307:{\displaystyle \,\Rightarrow .\,}
3981:that are logically equivalent to
2030:Preorders are closely related to
1183:
9530:Ordered topological vector space
9066:
8499:It contains at least the points
7685:Preorders may be used to define
6956:
6519:relations are usually preorders.
4579:If reflexivity is replaced with
4126:{\displaystyle \,\Leftarrow ,\,}
4101:{\displaystyle \,\Leftarrow ,\,}
3413:logically implies some sentence
3367:. One of the many properties of
2766:is reflexive since the preorder
1925:"x divides y" between integers,
1662:
1649:
1578:{\displaystyle {\text{not }}aRa}
1083:
1073:
1043:
1033:
1011:
1001:
966:
944:
934:
899:
867:
857:
832:
800:
785:
765:
733:
723:
713:
693:
661:
641:
631:
621:
589:
569:
525:
505:
495:
458:
433:
423:
391:
366:
329:
299:
267:
205:
170:
8896:Types and Programming Languages
7341:{\displaystyle {\overline {R}}}
7287:{\displaystyle R^{\textsf {T}}}
6914:. Injection may be replaced by
6525:are preorders (hence the name).
6334:
5346:
5345:
5319:
5313:
5102:
5096:
4414:denotes the sentence formed by
4339:{\displaystyle (S,\Leftarrow )}
4286:so far can also be said of its
4279:{\displaystyle \,\Leftarrow \,}
3787:{\displaystyle \,\Leftarrow \,}
3538:{\displaystyle \,\Leftarrow \,}
2705:
2699:
9743:Properties of binary relations
9019:Schröder, Bernd S. W. (2002),
8983:
8954:
8913:
8883:
8847:
8765:
8753:
8738:
8726:
8581:
8569:
8526:
8514:
8392:
8380:
8331:I.e., together, 355 preorders.
8314:{\displaystyle 6\times 19=114}
6853:
6847:
6838:
6832:
6688:
6682:
6676:
6632:correspond to the elements of
6477:
5363:
5347:
4549:
4478:
4452:
4333:
4330:
4321:
4297:
4272:
4251:
4245:
4219:
4213:
4193:{\displaystyle A\Leftarrow B,}
4181:
4158:
4152:
4149:
4146:
4140:
4116:
4091:
4038:
3939:
3933:
3864:
3835:
3780:
3756:
3742:
3665:{\displaystyle A\Leftarrow C.}
3653:
3627:
3601:
3575:
3531:
3484:{\displaystyle B\Leftarrow A,}
3472:
3455:{\displaystyle A\Rightarrow B}
3446:
3209:
3203:
3183:
3177:
3128:
3122:
3116:
3110:
2976:
2970:
2872:If the preorder is denoted by
2349:
2337:
1670:in the "Symmetric" column and
1606:
1351:
1296:
1225:
1:
9487:Series-parallel partial order
9021:Ordered Sets: An Introduction
9002:
8205:I.e., together, 29 preorders.
7198:the complemented composition
6952:, according to common models.
6663:, enriched over the category
6214:The converse holds (that is,
6072:{\displaystyle \,\lesssim \,}
5902:{\displaystyle \,\lesssim \,}
5512:{\displaystyle \,\lesssim \,}
5190:{\displaystyle \,\lesssim \,}
4904:{\displaystyle \,\lesssim \,}
4487:{\displaystyle B\Leftarrow C}
4461:{\displaystyle A\Leftarrow C}
3636:{\displaystyle B\Leftarrow C}
3610:{\displaystyle A\Leftarrow B}
3584:{\displaystyle A\Leftarrow A}
2803:{\displaystyle \,\lesssim \,}
2781:{\displaystyle \,\lesssim \,}
2611:{\displaystyle \,\lesssim \,}
2387:{\displaystyle \,\lesssim \,}
2271:{\displaystyle \,\lesssim \,}
2223:{\displaystyle \,\lesssim \,}
2203:
2111:{\displaystyle \,\lesssim \,}
1671:
1658:
1068:
1063:
1058:
1053:
1028:
996:
991:
986:
981:
976:
961:
929:
924:
919:
914:
909:
894:
882:
877:
852:
847:
842:
827:
815:
810:
795:
780:
775:
760:
748:
743:
708:
703:
688:
676:
671:
656:
651:
616:
604:
599:
584:
579:
564:
559:
554:
540:
535:
520:
515:
490:
485:
473:
468:
453:
448:
443:
418:
406:
401:
386:
381:
376:
361:
356:
344:
339:
324:
319:
314:
309:
294:
282:
277:
262:
257:
252:
247:
242:
237:
220:
215:
200:
195:
190:
185:
180:
9166:Cantor's isomorphism theorem
8681:{\displaystyle a<x<b.}
8457:{\displaystyle x\lesssim b,}
8363:{\displaystyle a\lesssim b,}
8275:{\displaystyle 7\times 3=21}
7551:{\displaystyle b\lesssim a,}
7499:On the other hand, if it is
7333:
7261:forms a preorder called the
7246:
7240:
6443:relation is also a preorder.
6207:{\displaystyle a\lesssim b.}
6147:. The resulting preorder is
4607:. For this reason, the term
4435:{\displaystyle \,\wedge ,\,}
4407:{\displaystyle C:=A\wedge B}
3163:{\displaystyle x\lesssim y.}
2414:; that is, if it satisfies:
2162:{\displaystyle a\lesssim b,}
7:
9206:Szpilrajn extension theorem
9181:Hausdorff maximal principle
9156:Boolean prime ideal theorem
8973:{\displaystyle \backslash }
8826:Category of preordered sets
8781:
8428:{\displaystyle a\lesssim x}
8336:
8188:{\displaystyle 3\times 3=9}
7614:{\displaystyle b\lesssim a}
7588:{\displaystyle a\lesssim b}
7522:{\displaystyle a\lesssim b}
7456:{\displaystyle b\lesssim a}
7430:{\displaystyle a\lesssim b}
7101:if and only if there is an
6895:{\displaystyle x\lesssim y}
6815:{\displaystyle x\lesssim y}
6739:{\displaystyle x\lesssim y}
6697:{\displaystyle 2=(0\to 1).}
6565:{\displaystyle s\lesssim t}
6421:{\displaystyle x\lesssim y}
6367:{\displaystyle x\lesssim y}
6329:
5953:{\displaystyle a\lesssim b}
5080:and so the following holds
4982:{\displaystyle b\lesssim a}
4956:{\displaystyle a\lesssim b}
4874:{\displaystyle a,b,c\in P.}
4164:{\displaystyle \Leftarrow }
3361:Zermelo–Fraenkel set theory
2570:{\displaystyle a,b,c\in P.}
2440:{\displaystyle a\lesssim a}
2323:{\displaystyle a\lesssim b}
10:
9764:
9552:Topological vector lattice
8778:can be defined similarly.
6533:abstract rewriting systems
6246:) if and only if whenever
5497:) because if the preorder
5316: if and only if
5099: if and only if
4523:The partially ordered set
3955:consists of all sentences
3591:always holds and whenever
3333:the article on the subject
3240:{\displaystyle \,\sim .\,}
2702: if and only if
2230:be a binary relation on a
33:
18:
9582:
9510:
9449:
9219:
9148:
9097:
9074:
8857:Mathematische Nachrichten
7649:{\displaystyle a,b\in P.}
7094:{\displaystyle R:xR^{+}y}
6764:one-to-one correspondence
6116:{\displaystyle \,\sim \,}
6094:{\displaystyle \,\sim \,}
5924:{\displaystyle \,\sim \,}
5744:{\displaystyle \,<.\,}
5719:{\displaystyle \,\sim \,}
5675:{\displaystyle \,<,\,}
5554:{\displaystyle \lesssim }
5216:{\displaystyle \,\sim \,}
5004:{\displaystyle \,\sim \,}
4569:Lindenbaum–Tarski algebra
4055:). The partial order on
3872:{\displaystyle \,\iff \,}
3700:{\displaystyle A,B\in S,}
3436:which will be written as
3295:one-to-one correspondence
3103:it is possible to define
2759:{\displaystyle \,\sim \,}
2656:{\displaystyle \,\sim \,}
2297:{\displaystyle P\times P}
2133:{\displaystyle \,\leq \,}
29:Preorder (disambiguation)
9161:Cantor–Bernstein theorem
8920:Robinson, J. A. (1965).
8869:10.1002/mana.19901470123
8841:
7175:Given a binary relation
6876:The relation defined by
6796:The relation defined by
6718:finite topological space
6707:
5766:{\displaystyle \,<\,}
5697:{\displaystyle \,<\,}
5534:{\displaystyle \,<\,}
5420:{\displaystyle \,<\,}
5398:{\displaystyle \,<\,}
5290:{\displaystyle \,<\,}
5156:{\displaystyle \,<\,}
4740:{\displaystyle \,a<a}
4628:{\displaystyle \,<\,}
4375:{\displaystyle A,B\in S}
3896:{\displaystyle \,\sim .}
3848:{\displaystyle A\iff B,}
3794:if and only if they are
2865:{\displaystyle \,\sim .}
2844:which is the set of all
2837:{\displaystyle S/\sim ,}
9705:Partially ordered group
9525:Specialization preorder
8990:Kunen, Kenneth (1980),
8853:For "proset", see e.g.
8710:{\displaystyle a<b.}
8646:{\displaystyle x<b,}
7669:
7055:{\displaystyle R^{+=}.}
6933:relation for countable
6760:specialization preorder
6430:directed acyclic graphs
6320:{\displaystyle b<a.}
6265:{\displaystyle a\neq b}
6178:{\displaystyle a\leq b}
6010:{\displaystyle a\sim b}
5796:{\displaystyle a\leq b}
5773:include the following:
5600:{\displaystyle a\leq b}
5242:{\displaystyle a\sim b}
4773:{\displaystyle a\in P,}
4713:{\displaystyle a\in P;}
4571:for a related example.
4516:{\displaystyle C\in S.}
4076:{\displaystyle S/\sim }
4048:{\displaystyle A\iff B}
3879:may be used instead of
3817:{\displaystyle A\sim B}
3725:{\displaystyle A\sim B}
3643:both hold then so does
3096:{\displaystyle S/\sim }
2926:{\displaystyle S/\sim }
2898:{\displaystyle R^{+=},}
2744:The resulting relation
2469:{\displaystyle a\in P,}
2256:so that by definition,
2021:greatest common divisor
9191:Kruskal's tree theorem
9186:Knaster–Tarski theorem
9176:Dushnik–Miller theorem
9023:, Boston: Birkhäuser,
8974:
8772:
8745:
8711:
8682:
8647:
8618:
8617:{\displaystyle a<x}
8588:
8556:
8533:
8493:
8458:
8429:
8399:
8364:
8328:, giving 219 preorders
8315:
8276:
8231:
8189:
8148:
7703:Preorders are used in
7692:Preorders provide the
7650:
7615:
7589:
7552:
7523:
7486:
7457:
7431:
7407:If a preorder is also
7389:
7388:{\displaystyle \circ }
7369:
7342:
7315:
7288:
7255:
7192:
7162:
7139:
7115:
7095:
7056:
7015:
6995:
6975:
6961:Every binary relation
6896:
6863:
6816:
6789:partially ordered sets
6740:
6698:
6649:
6566:
6540:encompassment preorder
6497:asymptotic equivalence
6489:
6434:partially ordered sets
6422:
6368:
6321:
6292:
6291:{\displaystyle a<b}
6266:
6240:
6208:
6179:
6137:
6117:
6095:
6073:
6051:
6011:
5985:
5954:
5925:
5903:
5881:
5857:
5837:
5797:
5767:
5745:
5720:
5698:
5676:
5644:
5601:
5575:
5555:
5535:
5513:
5491:
5451:
5450:{\displaystyle a<b}
5421:
5399:
5373:
5360: is antisymmetric
5291:
5269:
5243:
5223:is equality (that is,
5217:
5191:
5157:
5135:
5074:
5031:
5030:{\displaystyle a<b}
5005:
4983:
4957:
4931:
4930:{\displaystyle a<b}
4905:
4875:
4834:
4774:
4741:
4714:
4685:
4684:{\displaystyle a<a}
4649:
4629:
4601:
4561:
4517:
4488:
4462:
4436:
4408:
4376:
4340:
4308:
4280:
4258:
4226:
4194:
4165:
4127:
4102:
4077:
4049:
4021:
4020:{\displaystyle B\in S}
3995:
3975:
3974:{\displaystyle B\in S}
3949:
3920:
3897:
3873:
3849:
3818:
3788:
3766:
3726:
3701:
3666:
3637:
3611:
3585:
3559:
3539:
3511:
3510:{\displaystyle B\in S}
3485:
3456:
3430:
3407:
3406:{\displaystyle A\in S}
3381:
3349:
3317:
3287:
3267:
3241:
3216:
3190:
3164:
3135:
3097:
3069:
3049:
3029:
3009:
2983:
2957:equivalence classes:
2947:
2927:
2899:
2866:
2838:
2804:
2782:
2760:
2738:
2677:
2657:
2632:
2612:
2571:
2530:
2470:
2441:
2388:
2366:
2324:
2298:
2272:
2250:
2224:
2163:
2134:
2112:
2091:directed acyclic graph
2072:
2052:
2025:lowest common multiple
2013:
1993:
1970:
1947:
1872:
1814:
1785:
1759:
1733:
1732:{\displaystyle a,b,c,}
1694:
1635:
1579:
1547:
1520:
1476:
1432:
1391:
1320:
1249:
1193:
1164:
27:. For other uses, see
8975:
8939:10.1145/321250.321253
8773:
8771:{\displaystyle (a,b]}
8746:
8744:{\displaystyle [a,b)}
8712:
8683:
8648:
8619:
8594:as the set of points
8589:
8587:{\displaystyle (a,b)}
8557:
8534:
8532:{\displaystyle (a,b)}
8494:
8459:
8430:
8405:is the set of points
8400:
8365:
8316:
8277:
8232:
8202:, giving 19 preorders
8190:
8149:
7696:for certain types of
7651:
7616:
7590:
7553:
7524:
7487:
7458:
7432:
7390:
7370:
7343:
7316:
7289:
7256:
7193:
7163:
7140:
7116:
7096:
7057:
7016:
6996:
6976:
6902:if there exists some
6897:
6864:
6817:
6741:
6699:
6650:
6582:substitution instance
6567:
6490:
6423:
6369:
6322:
6293:
6267:
6241:
6209:
6180:
6138:
6118:
6096:
6074:
6052:
6012:
5986:
5955:
5926:
5904:
5882:
5858:
5838:
5798:
5768:
5746:
5721:
5699:
5677:
5645:
5602:
5576:
5574:{\displaystyle \leq }
5556:
5536:
5514:
5492:
5452:
5422:
5400:
5374:
5297:can be restated as:
5292:
5270:
5244:
5218:
5192:
5158:
5136:
5075:
5032:
5006:
4984:
4958:
4932:
4906:
4876:
4835:
4775:
4742:
4715:
4686:
4662:or anti-reflexivity:
4650:
4630:
4602:
4562:
4518:
4489:
4463:
4437:
4409:
4377:
4341:
4309:
4281:
4259:
4257:{\displaystyle B\in }
4227:
4225:{\displaystyle A\in }
4195:
4166:
4128:
4103:
4078:
4050:
4022:
3996:
3976:
3950:
3921:
3898:
3874:
3850:
3819:
3789:
3767:
3727:
3702:
3672:Furthermore, for any
3667:
3638:
3612:
3586:
3560:
3540:
3512:
3486:
3457:
3431:
3408:
3382:
3350:
3318:
3288:
3268:
3242:
3217:
3191:
3165:
3136:
3134:{\displaystyle \leq }
3098:
3070:
3050:
3030:
3010:
2984:
2982:{\displaystyle x\in }
2948:
2928:
2900:
2867:
2839:
2805:
2783:
2761:
2739:
2678:
2658:
2633:
2613:
2572:
2531:
2471:
2442:
2389:
2367:
2325:
2299:
2273:
2251:
2225:
2164:
2135:
2113:
2073:
2053:
2032:equivalence relations
2014:
1994:
1971:
1948:
1833:
1815:
1786:
1760:
1734:
1695:
1636:
1580:
1548:
1521:
1477:
1433:
1392:
1321:
1250:
1194:
1165:
1144:Definitions, for all
9683:Ordered vector space
8964:
8797:Equivalence relation
8750:
8723:
8692:
8657:
8628:
8602:
8566:
8555:{\displaystyle <}
8546:
8511:
8468:
8439:
8413:
8377:
8345:
8293:
8254:
8230:{\displaystyle n=4:}
8212:
8167:
8147:{\displaystyle n=3:}
8129:
7782:Equivalence relation
7625:
7599:
7573:
7560:equivalence relation
7533:
7507:
7485:{\displaystyle a=b,}
7467:
7441:
7415:
7397:relation composition
7379:
7356:
7325:
7302:
7269:
7202:
7179:
7149:
7129:
7105:
7066:
7033:
7005:
6985:
6965:
6880:
6826:
6800:
6724:
6667:
6636:
6620:to any other object
6550:
6523:Simulation preorders
6463:
6406:
6352:
6344:relationship in any
6302:
6276:
6250:
6218:
6189:
6163:
6136:{\displaystyle <}
6127:
6105:
6083:
6061:
6021:
5995:
5964:
5938:
5913:
5891:
5880:{\displaystyle \,=,}
5867:
5856:{\displaystyle <}
5847:
5807:
5781:
5755:
5730:
5708:
5686:
5661:
5611:
5585:
5565:
5545:
5523:
5501:
5461:
5435:
5409:
5387:
5301:
5279:
5253:
5227:
5205:
5179:
5165:strict partial order
5145:
5084:
5041:
5015:
4993:
4967:
4941:
4915:
4893:
4844:
4790:
4755:
4724:
4695:
4669:
4639:
4617:
4591:
4585:strict partial order
4527:
4498:
4472:
4446:
4421:
4386:
4354:
4318:
4293:
4268:
4236:
4204:
4175:
4137:
4133:is characterized by
4112:
4087:
4059:
4031:
4005:
3985:
3959:
3930:
3907:
3883:
3859:
3828:
3802:
3796:logically equivalent
3776:
3736:
3710:
3676:
3647:
3621:
3595:
3569:
3549:
3527:
3495:
3466:
3440:
3417:
3391:
3371:
3339:
3327:, which is a set of
3307:
3277:
3254:
3226:
3200:
3174:
3145:
3107:
3079:
3059:
3039:
3019:
2993:
2961:
2937:
2909:
2876:
2852:
2817:
2792:
2770:
2748:
2687:
2667:
2645:
2640:equivalence relation
2622:
2600:
2540:
2486:
2451:
2425:
2376:
2334:
2330:is used in place of
2308:
2282:
2260:
2237:
2212:
2144:
2122:
2100:
2062:
2042:
2003:
1980:
1957:
1937:
1813:{\displaystyle aRc.}
1795:
1769:
1743:
1708:
1684:
1679:homogeneous relation
1590:
1558:
1531:
1487:
1443:
1402:
1331:
1260:
1204:
1174:
1148:
890:Strict partial order
165:Equivalence relation
9521:Alexandrov topology
9467:Lexicographic order
9426:Well-quasi-ordering
8891:Pierce, Benjamin C.
8836:Well-quasi-ordering
8808:– preorder that is
8799:– preorder that is
8790:– preorder that is
7734:
7724:Number of preorders
7680:Alexandrov topology
7403:Related definitions
6529:Reduction relations
6057:"; these relations
5268:{\displaystyle a=b}
5055: and not
4416:logical conjunction
4314:The preordered set
3855:and so this symbol
3365:zeroth-order theory
3293:itself. There is a
3075:. In any case, on
3008:{\displaystyle x=y}
2846:equivalence classes
1929:, or elements of a
1784:{\displaystyle bRc}
1758:{\displaystyle aRb}
1546:{\displaystyle aRa}
1163:{\displaystyle a,b}
549:Well-quasi-ordering
9502:Transitive closure
9462:Converse/Transpose
9171:Dilworth's theorem
8970:
8960:In this context, "
8768:
8741:
8707:
8678:
8643:
8614:
8584:
8552:
8529:
8489:
8454:
8425:
8395:
8360:
8311:
8272:
8227:
8185:
8144:
7728:
7646:
7611:
7585:
7548:
7519:
7482:
7453:
7427:
7385:
7368:{\displaystyle R,}
7365:
7338:
7314:{\displaystyle R,}
7311:
7284:
7251:
7191:{\displaystyle R,}
7188:
7161:{\displaystyle y.}
7158:
7135:
7111:
7091:
7052:
7023:transitive closure
7011:
6991:
6971:
6892:
6859:
6812:
6736:
6712:Further examples:
6694:
6648:{\displaystyle P,}
6645:
6562:
6507:many-one (mapping)
6485:
6418:
6364:
6317:
6288:
6262:
6236:
6204:
6175:
6133:
6113:
6091:
6069:
6047:
6007:
5981:
5950:
5921:
5899:
5877:
5853:
5833:
5793:
5763:
5741:
5716:
5694:
5672:
5640:
5597:
5571:
5551:
5531:
5509:
5487:
5447:
5417:
5395:
5369:
5287:
5265:
5239:
5213:
5187:
5153:
5131:
5070:
5027:
5011:introduced above,
5001:
4979:
4953:
4927:
4901:
4871:
4830:
4770:
4737:
4710:
4681:
4645:
4625:
4597:
4557:
4513:
4484:
4458:
4432:
4404:
4372:
4336:
4304:
4276:
4254:
4222:
4190:
4161:
4123:
4098:
4073:
4045:
4017:
3991:
3971:
3945:
3919:{\displaystyle A,}
3916:
3893:
3869:
3845:
3814:
3784:
3762:
3722:
3697:
3662:
3633:
3607:
3581:
3555:
3535:
3507:
3481:
3452:
3429:{\displaystyle B,}
3426:
3403:
3377:
3357:first-order theory
3345:
3313:
3283:
3266:{\displaystyle S,}
3263:
3237:
3212:
3186:
3160:
3131:
3093:
3065:
3045:
3025:
3005:
2979:
2943:
2923:
2895:
2862:
2834:
2800:
2778:
2756:
2734:
2673:
2653:
2638:one may define an
2628:
2608:
2567:
2526:
2466:
2437:
2384:
2362:
2320:
2294:
2278:is some subset of
2268:
2249:{\displaystyle P,}
2246:
2220:
2159:
2130:
2108:
2068:
2048:
2009:
1992:{\displaystyle -1}
1989:
1969:{\displaystyle -1}
1966:
1943:
1873:
1810:
1781:
1755:
1729:
1690:
1631:
1629:
1575:
1543:
1516:
1514:
1472:
1470:
1428:
1426:
1387:
1385:
1316:
1314:
1245:
1243:
1189:
1160:
1024:Strict total order
21:depth-first search
9730:
9729:
9688:Partially ordered
9497:Symmetric closure
9482:Reflexive closure
9225:
9014:978-0-521-76268-7
8095:
8094:
7687:interior algebras
7336:
7296:converse relation
7280:
7249:
7243:
7228:
7138:{\displaystyle x}
7114:{\displaystyle R}
7027:reflexive closure
7014:{\displaystyle S}
6994:{\displaystyle S}
6974:{\displaystyle R}
6920:ring homomorphism
6750:belongs to every
6661:enriched category
6612:with at most one
6592:Theta-subsumption
6511:Turing reductions
6145:strict weak order
6036:
5970:
5822:
5626:
5476:
5361:
5353:
5333:
5317:
5116:
5100:
5056:
4819:
4805:
4648:{\displaystyle P}
4600:{\displaystyle P}
4288:converse relation
3994:{\displaystyle A}
3948:{\displaystyle ,}
3751:
3558:{\displaystyle S}
3545:is a preorder on
3523:). The relation
3491:then necessarily
3380:{\displaystyle S}
3348:{\displaystyle S}
3335:). For instance,
3316:{\displaystyle S}
3286:{\displaystyle S}
3068:{\displaystyle y}
3048:{\displaystyle R}
3028:{\displaystyle x}
2946:{\displaystyle R}
2719:
2703:
2676:{\displaystyle S}
2631:{\displaystyle S}
2596:Given a preorder
2515:
2501:
2304:and the notation
2169:one may say that
2140:. In words, when
2071:{\displaystyle X}
2051:{\displaystyle X}
2012:{\displaystyle 1}
1946:{\displaystyle 1}
1826:
1825:
1693:{\displaystyle R}
1644:
1643:
1616:
1564:
1510:
1466:
1422:
1370:
1279:
957:Strict weak order
143:Total, Semiconnex
9755:
9472:Linear extension
9221:
9201:Mirsky's theorem
9061:
9054:
9047:
9038:
9037:
9033:
8997:
8995:
8987:
8981:
8979:
8977:
8976:
8971:
8958:
8952:
8951:
8941:
8917:
8911:
8910:
8887:
8881:
8879:
8851:
8777:
8775:
8774:
8769:
8748:
8747:
8742:
8716:
8714:
8713:
8708:
8687:
8685:
8684:
8679:
8652:
8650:
8649:
8644:
8623:
8621:
8620:
8615:
8593:
8591:
8590:
8585:
8561:
8559:
8558:
8553:
8538:
8536:
8535:
8530:
8498:
8496:
8495:
8490:
8463:
8461:
8460:
8455:
8434:
8432:
8431:
8426:
8404:
8402:
8401:
8398:{\displaystyle }
8396:
8369:
8367:
8366:
8361:
8327:
8320:
8318:
8317:
8312:
8288:
8285:6 partitions of
8281:
8279:
8278:
8273:
8249:
8245:
8236:
8234:
8233:
8228:
8201:
8194:
8192:
8191:
8186:
8162:
8159:3 partitions of
8153:
8151:
8150:
8145:
8111:
8039:
8026:
8025:
8005:
7988:
7987:
7936:
7931:
7926:
7921:
7735:
7727:
7694:Kripke semantics
7660:preordered class
7655:
7653:
7652:
7647:
7620:
7618:
7617:
7612:
7594:
7592:
7591:
7586:
7557:
7555:
7554:
7549:
7528:
7526:
7525:
7520:
7491:
7489:
7488:
7483:
7462:
7460:
7459:
7454:
7436:
7434:
7433:
7428:
7394:
7392:
7391:
7386:
7374:
7372:
7371:
7366:
7347:
7345:
7344:
7339:
7337:
7329:
7320:
7318:
7317:
7312:
7293:
7291:
7290:
7285:
7283:
7282:
7281:
7260:
7258:
7257:
7252:
7250:
7245:
7244:
7236:
7231:
7230:
7229:
7218:
7197:
7195:
7194:
7189:
7167:
7165:
7164:
7159:
7144:
7142:
7141:
7136:
7120:
7118:
7117:
7112:
7100:
7098:
7097:
7092:
7087:
7086:
7061:
7059:
7058:
7053:
7048:
7047:
7020:
7018:
7017:
7012:
7000:
6998:
6997:
6992:
6980:
6978:
6977:
6972:
6901:
6899:
6898:
6893:
6868:
6866:
6865:
6860:
6821:
6819:
6818:
6813:
6745:
6743:
6742:
6737:
6703:
6701:
6700:
6695:
6654:
6652:
6651:
6646:
6616:from any object
6571:
6569:
6568:
6563:
6494:
6492:
6491:
6486:
6484:
6476:
6457:Asymptotic order
6448:Computer science
6427:
6425:
6424:
6419:
6401:
6373:
6371:
6370:
6365:
6326:
6324:
6323:
6318:
6297:
6295:
6294:
6289:
6271:
6269:
6268:
6263:
6245:
6243:
6242:
6237:
6213:
6211:
6210:
6205:
6184:
6182:
6181:
6176:
6142:
6140:
6139:
6134:
6122:
6120:
6119:
6114:
6100:
6098:
6097:
6092:
6078:
6076:
6075:
6070:
6056:
6054:
6053:
6048:
6037:
6034:
6016:
6014:
6013:
6008:
5990:
5988:
5987:
5982:
5971:
5968:
5959:
5957:
5956:
5951:
5930:
5928:
5927:
5922:
5908:
5906:
5905:
5900:
5886:
5884:
5883:
5878:
5862:
5860:
5859:
5854:
5842:
5840:
5839:
5834:
5823:
5820:
5802:
5800:
5799:
5794:
5772:
5770:
5769:
5764:
5750:
5748:
5747:
5742:
5725:
5723:
5722:
5717:
5703:
5701:
5700:
5695:
5681:
5679:
5678:
5673:
5649:
5647:
5646:
5641:
5627:
5624:
5606:
5604:
5603:
5598:
5580:
5578:
5577:
5572:
5560:
5558:
5557:
5552:
5540:
5538:
5537:
5532:
5518:
5516:
5515:
5510:
5496:
5494:
5493:
5488:
5477:
5474:
5456:
5454:
5453:
5448:
5426:
5424:
5423:
5418:
5404:
5402:
5401:
5396:
5378:
5376:
5375:
5370:
5362:
5359:
5354:
5351:
5334:
5331:
5318:
5315:
5296:
5294:
5293:
5288:
5274:
5272:
5271:
5266:
5248:
5246:
5245:
5240:
5222:
5220:
5219:
5214:
5196:
5194:
5193:
5188:
5162:
5160:
5159:
5154:
5140:
5138:
5137:
5132:
5117:
5114:
5101:
5098:
5079:
5077:
5076:
5071:
5057:
5054:
5036:
5034:
5033:
5028:
5010:
5008:
5007:
5002:
4988:
4986:
4985:
4980:
4962:
4960:
4959:
4954:
4936:
4934:
4933:
4928:
4910:
4908:
4907:
4902:
4880:
4878:
4877:
4872:
4839:
4837:
4836:
4831:
4820:
4818: then
4817:
4806:
4803:
4779:
4777:
4776:
4771:
4746:
4744:
4743:
4738:
4719:
4717:
4716:
4711:
4690:
4688:
4687:
4682:
4655:that satisfies:
4654:
4652:
4651:
4646:
4634:
4632:
4631:
4626:
4606:
4604:
4603:
4598:
4566:
4564:
4563:
4558:
4556:
4552:
4542:
4522:
4520:
4519:
4514:
4493:
4491:
4490:
4485:
4467:
4465:
4464:
4459:
4441:
4439:
4438:
4433:
4413:
4411:
4410:
4405:
4381:
4379:
4378:
4373:
4345:
4343:
4342:
4337:
4313:
4311:
4310:
4305:
4285:
4283:
4282:
4277:
4263:
4261:
4260:
4255:
4231:
4229:
4228:
4223:
4199:
4197:
4196:
4191:
4170:
4168:
4167:
4162:
4132:
4130:
4129:
4124:
4107:
4105:
4104:
4099:
4082:
4080:
4079:
4074:
4069:
4054:
4052:
4051:
4046:
4026:
4024:
4023:
4018:
4000:
3998:
3997:
3992:
3980:
3978:
3977:
3972:
3954:
3952:
3951:
3946:
3925:
3923:
3922:
3917:
3902:
3900:
3899:
3894:
3878:
3876:
3875:
3870:
3854:
3852:
3851:
3846:
3823:
3821:
3820:
3815:
3793:
3791:
3790:
3785:
3771:
3769:
3768:
3763:
3752:
3749:
3731:
3729:
3728:
3723:
3706:
3704:
3703:
3698:
3671:
3669:
3668:
3663:
3642:
3640:
3639:
3634:
3616:
3614:
3613:
3608:
3590:
3588:
3587:
3582:
3564:
3562:
3561:
3556:
3544:
3542:
3541:
3536:
3516:
3514:
3513:
3508:
3490:
3488:
3487:
3482:
3461:
3459:
3458:
3453:
3435:
3433:
3432:
3427:
3412:
3410:
3409:
3404:
3386:
3384:
3383:
3378:
3354:
3352:
3351:
3346:
3322:
3320:
3319:
3314:
3292:
3290:
3289:
3284:
3272:
3270:
3269:
3264:
3246:
3244:
3243:
3238:
3221:
3219:
3218:
3215:{\displaystyle }
3213:
3195:
3193:
3192:
3189:{\displaystyle }
3187:
3169:
3167:
3166:
3161:
3140:
3138:
3137:
3132:
3102:
3100:
3099:
3094:
3089:
3074:
3072:
3071:
3066:
3054:
3052:
3051:
3046:
3034:
3032:
3031:
3026:
3014:
3012:
3011:
3006:
2988:
2986:
2985:
2980:
2952:
2950:
2949:
2944:
2932:
2930:
2929:
2924:
2919:
2904:
2902:
2901:
2896:
2891:
2890:
2871:
2869:
2868:
2863:
2843:
2841:
2840:
2835:
2827:
2809:
2807:
2806:
2801:
2787:
2785:
2784:
2779:
2765:
2763:
2762:
2757:
2743:
2741:
2740:
2735:
2720:
2717:
2704:
2701:
2682:
2680:
2679:
2674:
2662:
2660:
2659:
2654:
2637:
2635:
2634:
2629:
2617:
2615:
2614:
2609:
2576:
2574:
2573:
2568:
2535:
2533:
2532:
2527:
2516:
2514: then
2513:
2502:
2499:
2475:
2473:
2472:
2467:
2446:
2444:
2443:
2438:
2393:
2391:
2390:
2385:
2371:
2369:
2368:
2363:
2329:
2327:
2326:
2321:
2303:
2301:
2300:
2295:
2277:
2275:
2274:
2269:
2255:
2253:
2252:
2247:
2229:
2227:
2226:
2221:
2168:
2166:
2165:
2160:
2139:
2137:
2136:
2131:
2117:
2115:
2114:
2109:
2077:
2075:
2074:
2069:
2057:
2055:
2054:
2049:
2018:
2016:
2015:
2010:
1998:
1996:
1995:
1990:
1975:
1973:
1972:
1967:
1952:
1950:
1949:
1944:
1931:commutative ring
1923:divides relation
1879:, especially in
1837:of the preorder
1819:
1817:
1816:
1811:
1790:
1788:
1787:
1782:
1764:
1762:
1761:
1756:
1738:
1736:
1735:
1730:
1699:
1697:
1696:
1691:
1673:
1669:
1666:
1665:
1660:
1656:
1653:
1652:
1640:
1638:
1637:
1632:
1630:
1617:
1614:
1584:
1582:
1581:
1576:
1565:
1562:
1552:
1550:
1549:
1544:
1525:
1523:
1522:
1517:
1515:
1511:
1508:
1481:
1479:
1478:
1473:
1471:
1467:
1464:
1437:
1435:
1434:
1429:
1427:
1423:
1420:
1396:
1394:
1393:
1388:
1386:
1371:
1368:
1345:
1325:
1323:
1322:
1317:
1315:
1306:
1280:
1277:
1254:
1252:
1251:
1246:
1244:
1229:
1210:
1198:
1196:
1195:
1190:
1169:
1167:
1166:
1161:
1090:
1087:
1086:
1080:
1077:
1076:
1070:
1065:
1060:
1055:
1050:
1047:
1046:
1040:
1037:
1036:
1030:
1018:
1015:
1014:
1008:
1005:
1004:
998:
993:
988:
983:
978:
973:
970:
969:
963:
951:
948:
947:
941:
938:
937:
931:
926:
921:
916:
911:
906:
903:
902:
896:
884:
879:
874:
871:
870:
864:
861:
860:
854:
849:
844:
839:
836:
835:
829:
823:Meet-semilattice
817:
812:
807:
804:
803:
797:
792:
789:
788:
782:
777:
772:
769:
768:
762:
756:Join-semilattice
750:
745:
740:
737:
736:
730:
727:
726:
720:
717:
716:
710:
705:
700:
697:
696:
690:
678:
673:
668:
665:
664:
658:
653:
648:
645:
644:
638:
635:
634:
628:
625:
624:
618:
606:
601:
596:
593:
592:
586:
581:
576:
573:
572:
566:
561:
556:
551:
542:
537:
532:
529:
528:
522:
517:
512:
509:
508:
502:
499:
498:
492:
487:
475:
470:
465:
462:
461:
455:
450:
445:
440:
437:
436:
430:
427:
426:
420:
408:
403:
398:
395:
394:
388:
383:
378:
373:
370:
369:
363:
358:
346:
341:
336:
333:
332:
326:
321:
316:
311:
306:
303:
302:
296:
284:
279:
274:
271:
270:
264:
259:
254:
249:
244:
239:
234:
232:
222:
217:
212:
209:
208:
202:
197:
192:
187:
182:
177:
174:
173:
167:
85:
84:
75:
68:
61:
54:
52:binary relations
43:
42:
9763:
9762:
9758:
9757:
9756:
9754:
9753:
9752:
9733:
9732:
9731:
9726:
9722:Young's lattice
9578:
9506:
9445:
9295:Heyting algebra
9243:Boolean algebra
9215:
9196:Laver's theorem
9144:
9110:Boolean algebra
9105:Binary relation
9093:
9070:
9065:
9031:
9005:
9000:
8988:
8984:
8965:
8962:
8961:
8959:
8955:
8918:
8914:
8907:
8888:
8884:
8852:
8848:
8844:
8831:Prewellordering
8784:
8751:
8724:
8721:
8720:
8693:
8690:
8689:
8658:
8655:
8654:
8629:
8626:
8625:
8603:
8600:
8599:
8567:
8564:
8563:
8547:
8544:
8543:
8512:
8509:
8508:
8469:
8466:
8465:
8440:
8437:
8436:
8414:
8411:
8410:
8378:
8375:
8374:
8346:
8343:
8342:
8339:
8334:
8325:
8324:1 partition of
8294:
8291:
8290:
8286:
8255:
8252:
8251:
8247:
8243:
8213:
8210:
8209:
8199:
8198:1 partition of
8168:
8165:
8164:
8160:
8130:
8127:
8126:
8098:
8024:
8018:
8017:
8016:
8014:
7986:
7980:
7979:
7978:
7976:
7934:
7929:
7924:
7919:
7738:Elements
7726:
7672:
7626:
7623:
7622:
7600:
7597:
7596:
7574:
7571:
7570:
7534:
7531:
7530:
7508:
7505:
7504:
7468:
7465:
7464:
7442:
7439:
7438:
7416:
7413:
7412:
7405:
7380:
7377:
7376:
7357:
7354:
7353:
7328:
7326:
7323:
7322:
7303:
7300:
7299:
7277:
7276:
7272:
7270:
7267:
7266:
7235:
7225:
7224:
7220:
7219:
7217:
7203:
7200:
7199:
7180:
7177:
7176:
7150:
7147:
7146:
7130:
7127:
7126:
7106:
7103:
7102:
7082:
7078:
7067:
7064:
7063:
7040:
7036:
7034:
7031:
7030:
7006:
7003:
7002:
6986:
6983:
6982:
6966:
6963:
6962:
6959:
6935:total orderings
6881:
6878:
6877:
6827:
6824:
6823:
6801:
6798:
6797:
6746:if and only if
6725:
6722:
6721:
6710:
6668:
6665:
6664:
6637:
6634:
6633:
6605:
6603:Category theory
6551:
6548:
6547:
6503:Polynomial-time
6480:
6472:
6464:
6461:
6460:
6450:
6407:
6404:
6403:
6391:
6390:for every pair
6353:
6350:
6349:
6337:
6332:
6303:
6300:
6299:
6277:
6274:
6273:
6251:
6248:
6247:
6219:
6216:
6215:
6190:
6187:
6186:
6164:
6161:
6160:
6128:
6125:
6124:
6106:
6103:
6102:
6084:
6081:
6080:
6062:
6059:
6058:
6035: nor
6033:
6022:
6019:
6018:
5996:
5993:
5992:
5969: not
5967:
5965:
5962:
5961:
5939:
5936:
5935:
5931:are not needed.
5914:
5911:
5910:
5892:
5889:
5888:
5887:so the symbols
5868:
5865:
5864:
5848:
5845:
5844:
5819:
5808:
5805:
5804:
5782:
5779:
5778:
5756:
5753:
5752:
5731:
5728:
5727:
5709:
5706:
5705:
5687:
5684:
5683:
5662:
5659:
5658:
5655:
5623:
5612:
5609:
5608:
5586:
5583:
5582:
5566:
5563:
5562:
5546:
5543:
5542:
5524:
5521:
5520:
5502:
5499:
5498:
5475: and
5473:
5462:
5459:
5458:
5457:if and only if
5436:
5433:
5432:
5410:
5407:
5406:
5388:
5385:
5384:
5358:
5350:
5332: and
5330:
5314:
5302:
5299:
5298:
5280:
5277:
5276:
5254:
5251:
5250:
5249:if and only if
5228:
5225:
5224:
5206:
5203:
5202:
5180:
5177:
5176:
5146:
5143:
5142:
5113:
5097:
5085:
5082:
5081:
5053:
5042:
5039:
5038:
5037:if and only if
5016:
5013:
5012:
4994:
4991:
4990:
4968:
4965:
4964:
4942:
4939:
4938:
4937:if and only if
4916:
4913:
4912:
4894:
4891:
4890:
4887:
4845:
4842:
4841:
4816:
4804: and
4802:
4791:
4788:
4787:
4756:
4753:
4752:
4725:
4722:
4721:
4696:
4693:
4692:
4670:
4667:
4666:
4640:
4637:
4636:
4618:
4615:
4614:
4610:strict preorder
4592:
4589:
4588:
4577:
4538:
4534:
4530:
4528:
4525:
4524:
4499:
4496:
4495:
4473:
4470:
4469:
4447:
4444:
4443:
4422:
4419:
4418:
4387:
4384:
4383:
4355:
4352:
4351:
4319:
4316:
4315:
4294:
4291:
4290:
4269:
4266:
4265:
4237:
4234:
4233:
4205:
4202:
4201:
4176:
4173:
4172:
4171:if and only if
4138:
4135:
4134:
4113:
4110:
4109:
4088:
4085:
4084:
4065:
4060:
4057:
4056:
4032:
4029:
4028:
4006:
4003:
4002:
3986:
3983:
3982:
3960:
3957:
3956:
3931:
3928:
3927:
3908:
3905:
3904:
3884:
3881:
3880:
3860:
3857:
3856:
3829:
3826:
3825:
3803:
3800:
3799:
3777:
3774:
3773:
3750: and
3748:
3737:
3734:
3733:
3732:if and only if
3711:
3708:
3707:
3677:
3674:
3673:
3648:
3645:
3644:
3622:
3619:
3618:
3596:
3593:
3592:
3570:
3567:
3566:
3550:
3547:
3546:
3528:
3525:
3524:
3496:
3493:
3492:
3467:
3464:
3463:
3441:
3438:
3437:
3418:
3415:
3414:
3392:
3389:
3388:
3372:
3369:
3368:
3363:) or a simpler
3340:
3337:
3336:
3308:
3305:
3304:
3278:
3275:
3274:
3255:
3252:
3251:
3227:
3224:
3223:
3201:
3198:
3197:
3175:
3172:
3171:
3146:
3143:
3142:
3141:if and only if
3108:
3105:
3104:
3085:
3080:
3077:
3076:
3060:
3057:
3056:
3040:
3037:
3036:
3020:
3017:
3016:
2994:
2991:
2990:
2989:if and only if
2962:
2959:
2958:
2938:
2935:
2934:
2915:
2910:
2907:
2906:
2883:
2879:
2877:
2874:
2873:
2853:
2850:
2849:
2823:
2818:
2815:
2814:
2793:
2790:
2789:
2771:
2768:
2767:
2749:
2746:
2745:
2718: and
2716:
2700:
2688:
2685:
2684:
2668:
2665:
2664:
2646:
2643:
2642:
2623:
2620:
2619:
2601:
2598:
2597:
2594:
2541:
2538:
2537:
2512:
2500: and
2498:
2487:
2484:
2483:
2452:
2449:
2448:
2426:
2423:
2422:
2377:
2374:
2373:
2335:
2332:
2331:
2309:
2306:
2305:
2283:
2280:
2279:
2261:
2258:
2257:
2238:
2235:
2234:
2213:
2210:
2209:
2206:
2145:
2142:
2141:
2123:
2120:
2119:
2101:
2098:
2097:
2063:
2060:
2059:
2043:
2040:
2039:
2004:
2001:
2000:
1981:
1978:
1977:
1958:
1955:
1954:
1938:
1935:
1934:
1893:binary relation
1857:natural numbers
1828:
1827:
1820:
1796:
1793:
1792:
1770:
1767:
1766:
1744:
1741:
1740:
1709:
1706:
1705:
1685:
1682:
1681:
1675:
1667:
1663:
1654:
1650:
1628:
1627:
1613:
1610:
1609:
1593:
1591:
1588:
1587:
1561:
1559:
1556:
1555:
1532:
1529:
1528:
1513:
1512:
1507:
1504:
1503:
1490:
1488:
1485:
1484:
1469:
1468:
1463:
1460:
1459:
1446:
1444:
1441:
1440:
1425:
1424:
1419:
1416:
1415:
1405:
1403:
1400:
1399:
1384:
1383:
1372:
1367:
1355:
1354:
1346:
1344:
1334:
1332:
1329:
1328:
1313:
1312:
1307:
1305:
1293:
1292:
1281:
1278: and
1276:
1263:
1261:
1258:
1257:
1242:
1241:
1230:
1228:
1222:
1221:
1207:
1205:
1202:
1201:
1175:
1172:
1171:
1149:
1146:
1145:
1088:
1084:
1078:
1074:
1048:
1044:
1038:
1034:
1016:
1012:
1006:
1002:
971:
967:
949:
945:
939:
935:
904:
900:
872:
868:
862:
858:
837:
833:
805:
801:
790:
786:
770:
766:
738:
734:
728:
724:
718:
714:
698:
694:
666:
662:
646:
642:
636:
632:
626:
622:
594:
590:
574:
570:
547:
530:
526:
510:
506:
500:
496:
481:Prewellordering
463:
459:
438:
434:
428:
424:
396:
392:
371:
367:
334:
330:
304:
300:
272:
268:
230:
227:
210:
206:
175:
171:
163:
155:
79:
46:
39:
32:
17:
12:
11:
5:
9761:
9751:
9750:
9745:
9728:
9727:
9725:
9724:
9719:
9714:
9713:
9712:
9702:
9701:
9700:
9695:
9690:
9680:
9679:
9678:
9668:
9663:
9662:
9661:
9656:
9649:Order morphism
9646:
9645:
9644:
9634:
9629:
9624:
9619:
9614:
9613:
9612:
9602:
9597:
9592:
9586:
9584:
9580:
9579:
9577:
9576:
9575:
9574:
9569:
9567:Locally convex
9564:
9559:
9549:
9547:Order topology
9544:
9543:
9542:
9540:Order topology
9537:
9527:
9517:
9515:
9508:
9507:
9505:
9504:
9499:
9494:
9489:
9484:
9479:
9474:
9469:
9464:
9459:
9453:
9451:
9447:
9446:
9444:
9443:
9433:
9423:
9418:
9413:
9408:
9403:
9398:
9393:
9388:
9387:
9386:
9376:
9371:
9370:
9369:
9364:
9359:
9354:
9352:Chain-complete
9344:
9339:
9338:
9337:
9332:
9327:
9322:
9317:
9307:
9302:
9297:
9292:
9287:
9277:
9272:
9267:
9262:
9257:
9252:
9251:
9250:
9240:
9235:
9229:
9227:
9217:
9216:
9214:
9213:
9208:
9203:
9198:
9193:
9188:
9183:
9178:
9173:
9168:
9163:
9158:
9152:
9150:
9146:
9145:
9143:
9142:
9137:
9132:
9127:
9122:
9117:
9112:
9107:
9101:
9099:
9095:
9094:
9092:
9091:
9086:
9081:
9075:
9072:
9071:
9064:
9063:
9056:
9049:
9041:
9035:
9034:
9029:
9016:
9004:
9001:
8999:
8998:
8982:
8969:
8953:
8912:
8905:
8882:
8845:
8843:
8840:
8839:
8838:
8833:
8828:
8823:
8818:
8812:
8806:Total preorder
8803:
8794:
8783:
8780:
8767:
8764:
8761:
8758:
8755:
8740:
8737:
8734:
8731:
8728:
8706:
8703:
8700:
8697:
8677:
8674:
8671:
8668:
8665:
8662:
8642:
8639:
8636:
8633:
8613:
8610:
8607:
8583:
8580:
8577:
8574:
8571:
8551:
8528:
8525:
8522:
8519:
8516:
8488:
8485:
8482:
8479:
8476:
8473:
8453:
8450:
8447:
8444:
8424:
8421:
8418:
8394:
8391:
8388:
8385:
8382:
8359:
8356:
8353:
8350:
8338:
8335:
8333:
8332:
8330:
8329:
8322:
8310:
8307:
8304:
8301:
8298:
8283:
8271:
8268:
8265:
8262:
8259:
8240:
8226:
8223:
8220:
8217:
8206:
8204:
8203:
8196:
8184:
8181:
8178:
8175:
8172:
8157:
8143:
8140:
8137:
8134:
8122:
8093:
8092:
8087:
8082:
8077:
8072:
8067:
8062:
8057:
8052:
8047:
8041:
8040:
8019:
8012:
8006:
7981:
7974:
7972:
7970:
7967:
7964:
7962:
7959:
7953:
7952:
7949:
7946:
7943:
7940:
7937:
7932:
7927:
7922:
7917:
7913:
7912:
7909:
7906:
7903:
7900:
7897:
7894:
7891:
7888:
7885:
7881:
7880:
7877:
7874:
7871:
7868:
7865:
7862:
7859:
7856:
7853:
7849:
7848:
7845:
7842:
7839:
7836:
7833:
7830:
7827:
7824:
7821:
7817:
7816:
7813:
7810:
7807:
7804:
7801:
7798:
7795:
7792:
7789:
7785:
7784:
7779:
7774:
7772:Total preorder
7769:
7764:
7759:
7754:
7749:
7744:
7739:
7725:
7722:
7721:
7720:
7701:
7690:
7683:
7671:
7668:
7645:
7642:
7639:
7636:
7633:
7630:
7610:
7607:
7604:
7584:
7581:
7578:
7565:A preorder is
7558:then it is an
7547:
7544:
7541:
7538:
7518:
7515:
7512:
7503:, that is, if
7481:
7478:
7475:
7472:
7452:
7449:
7446:
7426:
7423:
7420:
7404:
7401:
7384:
7364:
7361:
7335:
7332:
7310:
7307:
7275:
7248:
7242:
7239:
7234:
7223:
7216:
7213:
7210:
7207:
7187:
7184:
7157:
7154:
7134:
7110:
7090:
7085:
7081:
7077:
7074:
7071:
7051:
7046:
7043:
7039:
7021:by taking the
7010:
6990:
6970:
6958:
6955:
6954:
6953:
6943:total preorder
6939:
6938:
6927:
6891:
6888:
6885:
6874:
6858:
6855:
6852:
6849:
6846:
6843:
6840:
6837:
6834:
6831:
6811:
6808:
6805:
6793:
6792:
6768:
6767:
6735:
6732:
6729:
6709:
6706:
6705:
6704:
6693:
6690:
6687:
6684:
6681:
6678:
6675:
6672:
6657:
6644:
6641:
6604:
6601:
6600:
6599:
6598:to the former.
6589:
6561:
6558:
6555:
6542:on the set of
6536:
6526:
6520:
6514:
6500:
6483:
6479:
6475:
6471:
6468:
6449:
6446:
6445:
6444:
6437:
6417:
6414:
6411:
6363:
6360:
6357:
6346:directed graph
6336:
6333:
6331:
6328:
6316:
6313:
6310:
6307:
6287:
6284:
6281:
6261:
6258:
6255:
6234:
6229:
6224:
6203:
6200:
6197:
6194:
6174:
6171:
6168:
6157:
6156:
6153:total preorder
6132:
6111:
6089:
6067:
6046:
6043:
6040:
6032:
6029:
6026:
6006:
6003:
6000:
5980:
5977:
5974:
5949:
5946:
5943:
5932:
5919:
5897:
5876:
5873:
5852:
5832:
5829:
5826:
5821: or
5818:
5815:
5812:
5792:
5789:
5786:
5761:
5739:
5736:
5714:
5692:
5670:
5667:
5654:
5651:
5639:
5636:
5633:
5630:
5625: or
5622:
5619:
5616:
5596:
5593:
5590:
5570:
5550:
5529:
5507:
5486:
5483:
5480:
5472:
5469:
5466:
5446:
5443:
5440:
5430:
5415:
5393:
5382:
5368:
5365:
5357:
5352:assuming
5349:
5344:
5341:
5338:
5328:
5325:
5322:
5312:
5309:
5306:
5285:
5264:
5261:
5258:
5238:
5235:
5232:
5211:
5185:
5174:
5170:
5151:
5130:
5127:
5124:
5121:
5115: or
5111:
5108:
5105:
5095:
5092:
5089:
5069:
5066:
5063:
5060:
5052:
5049:
5046:
5026:
5023:
5020:
4999:
4978:
4975:
4972:
4952:
4949:
4946:
4926:
4923:
4920:
4899:
4886:
4883:
4882:
4881:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4829:
4826:
4823:
4815:
4812:
4809:
4801:
4798:
4795:
4781:
4769:
4766:
4763:
4760:
4750:
4736:
4733:
4730:
4709:
4706:
4703:
4700:
4680:
4677:
4674:
4665:
4644:
4623:
4611:
4596:
4576:
4573:
4555:
4551:
4548:
4545:
4541:
4537:
4533:
4512:
4509:
4506:
4503:
4483:
4480:
4477:
4457:
4454:
4451:
4430:
4427:
4403:
4400:
4397:
4394:
4391:
4371:
4368:
4365:
4362:
4359:
4335:
4332:
4329:
4326:
4323:
4302:
4299:
4274:
4253:
4250:
4247:
4244:
4241:
4221:
4218:
4215:
4212:
4209:
4189:
4186:
4183:
4180:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4121:
4118:
4096:
4093:
4072:
4068:
4064:
4044:
4040:
4036:
4016:
4013:
4010:
4001:(that is, all
3990:
3970:
3967:
3964:
3944:
3941:
3938:
3935:
3915:
3912:
3892:
3889:
3866:
3844:
3841:
3837:
3833:
3813:
3810:
3807:
3782:
3761:
3758:
3755:
3747:
3744:
3741:
3721:
3718:
3715:
3696:
3693:
3690:
3687:
3684:
3681:
3661:
3658:
3655:
3652:
3632:
3629:
3626:
3606:
3603:
3600:
3580:
3577:
3574:
3554:
3533:
3506:
3503:
3500:
3480:
3477:
3474:
3471:
3451:
3448:
3445:
3425:
3422:
3402:
3399:
3396:
3376:
3344:
3312:
3302:
3282:
3262:
3259:
3235:
3232:
3211:
3208:
3205:
3185:
3182:
3179:
3159:
3156:
3153:
3150:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3092:
3088:
3084:
3064:
3044:
3024:
3004:
3001:
2998:
2978:
2975:
2972:
2969:
2966:
2942:
2933:is the set of
2922:
2918:
2914:
2894:
2889:
2886:
2882:
2861:
2858:
2833:
2830:
2826:
2822:
2798:
2776:
2754:
2733:
2730:
2727:
2724:
2714:
2711:
2708:
2698:
2695:
2692:
2672:
2651:
2627:
2606:
2593:
2590:
2582:preordered set
2578:
2577:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2525:
2522:
2519:
2511:
2508:
2505:
2497:
2494:
2491:
2477:
2465:
2462:
2459:
2456:
2436:
2433:
2430:
2404:
2398:
2382:
2361:
2358:
2354:
2351:
2348:
2345:
2342:
2339:
2319:
2316:
2313:
2293:
2290:
2287:
2266:
2245:
2242:
2218:
2205:
2202:
2195:
2185:
2175:
2158:
2155:
2152:
2149:
2128:
2106:
2087:directed graph
2067:
2047:
2008:
1988:
1985:
1965:
1962:
1942:
1912:partial orders
1906:
1824:
1823:
1809:
1806:
1803:
1800:
1780:
1777:
1774:
1754:
1751:
1748:
1728:
1725:
1722:
1719:
1716:
1713:
1689:
1646:
1645:
1642:
1641:
1626:
1623:
1620:
1612:
1611:
1608:
1605:
1602:
1599:
1596:
1595:
1585:
1574:
1571:
1568:
1553:
1542:
1539:
1536:
1526:
1506:
1505:
1502:
1499:
1496:
1493:
1492:
1482:
1462:
1461:
1458:
1455:
1452:
1449:
1448:
1438:
1418:
1417:
1414:
1411:
1408:
1407:
1397:
1382:
1379:
1376:
1373:
1369: or
1366:
1363:
1360:
1357:
1356:
1353:
1350:
1347:
1343:
1340:
1337:
1336:
1326:
1311:
1308:
1304:
1301:
1298:
1295:
1294:
1291:
1288:
1285:
1282:
1275:
1272:
1269:
1266:
1265:
1255:
1240:
1237:
1234:
1231:
1227:
1224:
1223:
1220:
1217:
1214:
1211:
1209:
1199:
1188:
1185:
1182:
1179:
1159:
1156:
1153:
1141:
1140:
1135:
1130:
1125:
1120:
1115:
1110:
1105:
1100:
1095:
1092:
1091:
1081:
1071:
1066:
1061:
1056:
1051:
1041:
1031:
1026:
1020:
1019:
1009:
999:
994:
989:
984:
979:
974:
964:
959:
953:
952:
942:
932:
927:
922:
917:
912:
907:
897:
892:
886:
885:
880:
875:
865:
855:
850:
845:
840:
830:
825:
819:
818:
813:
808:
798:
793:
783:
778:
773:
763:
758:
752:
751:
746:
741:
731:
721:
711:
706:
701:
691:
686:
680:
679:
674:
669:
659:
654:
649:
639:
629:
619:
614:
608:
607:
602:
597:
587:
582:
577:
567:
562:
557:
552:
544:
543:
538:
533:
523:
518:
513:
503:
493:
488:
483:
477:
476:
471:
466:
456:
451:
446:
441:
431:
421:
416:
410:
409:
404:
399:
389:
384:
379:
374:
364:
359:
354:
352:Total preorder
348:
347:
342:
337:
327:
322:
317:
312:
307:
297:
292:
286:
285:
280:
275:
265:
260:
255:
250:
245:
240:
235:
224:
223:
218:
213:
203:
198:
193:
188:
183:
178:
168:
160:
159:
157:
152:
150:
148:
146:
144:
141:
139:
137:
134:
133:
128:
123:
118:
113:
108:
103:
98:
93:
88:
81:
80:
78:
77:
70:
63:
55:
41:
40:
15:
9:
6:
4:
3:
2:
9760:
9749:
9746:
9744:
9741:
9740:
9738:
9723:
9720:
9718:
9715:
9711:
9708:
9707:
9706:
9703:
9699:
9696:
9694:
9691:
9689:
9686:
9685:
9684:
9681:
9677:
9674:
9673:
9672:
9671:Ordered field
9669:
9667:
9664:
9660:
9657:
9655:
9652:
9651:
9650:
9647:
9643:
9640:
9639:
9638:
9635:
9633:
9630:
9628:
9627:Hasse diagram
9625:
9623:
9620:
9618:
9615:
9611:
9608:
9607:
9606:
9605:Comparability
9603:
9601:
9598:
9596:
9593:
9591:
9588:
9587:
9585:
9581:
9573:
9570:
9568:
9565:
9563:
9560:
9558:
9555:
9554:
9553:
9550:
9548:
9545:
9541:
9538:
9536:
9533:
9532:
9531:
9528:
9526:
9522:
9519:
9518:
9516:
9513:
9509:
9503:
9500:
9498:
9495:
9493:
9490:
9488:
9485:
9483:
9480:
9478:
9477:Product order
9475:
9473:
9470:
9468:
9465:
9463:
9460:
9458:
9455:
9454:
9452:
9450:Constructions
9448:
9442:
9438:
9434:
9431:
9427:
9424:
9422:
9419:
9417:
9414:
9412:
9409:
9407:
9404:
9402:
9399:
9397:
9394:
9392:
9389:
9385:
9382:
9381:
9380:
9377:
9375:
9372:
9368:
9365:
9363:
9360:
9358:
9355:
9353:
9350:
9349:
9348:
9347:Partial order
9345:
9343:
9340:
9336:
9335:Join and meet
9333:
9331:
9328:
9326:
9323:
9321:
9318:
9316:
9313:
9312:
9311:
9308:
9306:
9303:
9301:
9298:
9296:
9293:
9291:
9288:
9286:
9282:
9278:
9276:
9273:
9271:
9268:
9266:
9263:
9261:
9258:
9256:
9253:
9249:
9246:
9245:
9244:
9241:
9239:
9236:
9234:
9233:Antisymmetric
9231:
9230:
9228:
9224:
9218:
9212:
9209:
9207:
9204:
9202:
9199:
9197:
9194:
9192:
9189:
9187:
9184:
9182:
9179:
9177:
9174:
9172:
9169:
9167:
9164:
9162:
9159:
9157:
9154:
9153:
9151:
9147:
9141:
9140:Weak ordering
9138:
9136:
9133:
9131:
9128:
9126:
9125:Partial order
9123:
9121:
9118:
9116:
9113:
9111:
9108:
9106:
9103:
9102:
9100:
9096:
9090:
9087:
9085:
9082:
9080:
9077:
9076:
9073:
9069:
9062:
9057:
9055:
9050:
9048:
9043:
9042:
9039:
9032:
9030:0-8176-4128-9
9026:
9022:
9017:
9015:
9011:
9007:
9006:
8993:
8986:
8957:
8949:
8945:
8940:
8935:
8931:
8927:
8923:
8916:
8908:
8906:0-262-16209-1
8902:
8898:
8897:
8892:
8886:
8878:
8874:
8870:
8866:
8862:
8858:
8850:
8846:
8837:
8834:
8832:
8829:
8827:
8824:
8822:
8819:
8816:
8813:
8811:
8807:
8804:
8802:
8798:
8795:
8793:
8792:antisymmetric
8789:
8788:Partial order
8786:
8785:
8779:
8762:
8759:
8756:
8735:
8732:
8729:
8717:
8704:
8701:
8698:
8695:
8675:
8672:
8669:
8666:
8663:
8660:
8653:also written
8640:
8637:
8634:
8631:
8611:
8608:
8605:
8597:
8578:
8575:
8572:
8549:
8540:
8523:
8520:
8517:
8506:
8502:
8486:
8483:
8480:
8477:
8474:
8471:
8464:also written
8451:
8448:
8445:
8442:
8422:
8419:
8416:
8408:
8389:
8386:
8383:
8373:
8357:
8354:
8351:
8348:
8326:1 + 1 + 1 + 1
8323:
8308:
8305:
8302:
8299:
8296:
8284:
8269:
8266:
8263:
8260:
8257:
8241:
8238:
8237:
8224:
8221:
8218:
8215:
8207:
8197:
8182:
8179:
8176:
8173:
8170:
8158:
8155:
8154:
8141:
8138:
8135:
8132:
8124:
8123:
8121:
8117:
8115:
8109:
8105:
8101:
8091:
8088:
8086:
8083:
8081:
8078:
8076:
8073:
8071:
8068:
8066:
8063:
8061:
8058:
8056:
8053:
8051:
8048:
8046:
8043:
8042:
8037:
8033:
8029:
8022:
8013:
8010:
8007:
8003:
7999:
7995:
7991:
7984:
7975:
7973:
7971:
7968:
7965:
7963:
7960:
7958:
7955:
7954:
7950:
7947:
7944:
7941:
7938:
7933:
7928:
7923:
7918:
7915:
7914:
7910:
7907:
7904:
7901:
7898:
7895:
7892:
7889:
7886:
7883:
7882:
7878:
7875:
7872:
7869:
7866:
7863:
7860:
7857:
7854:
7851:
7850:
7846:
7843:
7840:
7837:
7834:
7831:
7828:
7825:
7822:
7819:
7818:
7814:
7811:
7808:
7805:
7802:
7799:
7796:
7793:
7790:
7787:
7786:
7783:
7780:
7778:
7775:
7773:
7770:
7768:
7767:Partial order
7765:
7763:
7760:
7758:
7755:
7753:
7750:
7748:
7745:
7743:
7740:
7737:
7736:
7732:
7718:
7714:
7710:
7706:
7702:
7699:
7695:
7691:
7688:
7684:
7681:
7677:
7676:
7675:
7667:
7665:
7661:
7656:
7643:
7640:
7637:
7634:
7631:
7628:
7608:
7605:
7602:
7582:
7579:
7576:
7568:
7563:
7561:
7545:
7542:
7539:
7536:
7516:
7513:
7510:
7502:
7497:
7495:
7494:partial order
7492:then it is a
7479:
7476:
7473:
7470:
7450:
7447:
7444:
7424:
7421:
7418:
7410:
7409:antisymmetric
7400:
7398:
7382:
7362:
7359:
7351:
7330:
7308:
7305:
7297:
7273:
7264:
7263:left residual
7237:
7232:
7221:
7214:
7211:
7205:
7185:
7182:
7173:
7172:
7168:
7155:
7152:
7132:
7124:
7108:
7088:
7083:
7079:
7075:
7072:
7069:
7049:
7044:
7041:
7037:
7028:
7024:
7008:
6988:
6968:
6957:Constructions
6951:
6948:
6947:
6946:
6944:
6941:Example of a
6936:
6932:
6928:
6925:
6921:
6917:
6913:
6909:
6905:
6889:
6886:
6883:
6875:
6872:
6856:
6850:
6844:
6841:
6835:
6829:
6809:
6806:
6803:
6795:
6794:
6790:
6786:
6782:
6778:
6774:
6770:
6769:
6765:
6761:
6757:
6753:
6749:
6733:
6730:
6727:
6719:
6715:
6714:
6713:
6691:
6685:
6679:
6673:
6670:
6662:
6658:
6642:
6639:
6631:
6627:
6623:
6619:
6615:
6611:
6607:
6606:
6597:
6593:
6590:
6587:
6583:
6579:
6575:
6559:
6556:
6553:
6546:, defined by
6545:
6541:
6537:
6534:
6530:
6527:
6524:
6521:
6518:
6515:
6512:
6508:
6504:
6501:
6498:
6469:
6466:
6458:
6455:
6454:
6453:
6442:
6438:
6435:
6431:
6415:
6412:
6409:
6399:
6395:
6389:
6385:
6381:
6377:
6361:
6358:
6355:
6347:
6343:
6339:
6338:
6327:
6314:
6311:
6308:
6305:
6285:
6282:
6279:
6259:
6256:
6253:
6232:
6227:
6222:
6201:
6198:
6195:
6192:
6172:
6169:
6166:
6154:
6150:
6146:
6130:
6109:
6087:
6065:
6044:
6041:
6038:
6030:
6027:
6024:
6004:
6001:
5998:
5978:
5975:
5972:
5947:
5944:
5941:
5933:
5917:
5895:
5874:
5871:
5850:
5830:
5827:
5824:
5816:
5813:
5810:
5790:
5787:
5784:
5776:
5775:
5774:
5759:
5737:
5734:
5712:
5690:
5668:
5665:
5650:
5637:
5634:
5631:
5628:
5620:
5617:
5614:
5594:
5591:
5588:
5568:
5548:
5527:
5505:
5484:
5481:
5478:
5470:
5467:
5464:
5444:
5441:
5438:
5428:
5413:
5391:
5380:
5366:
5355:
5342:
5339:
5336:
5326:
5323:
5320:
5310:
5307:
5304:
5283:
5262:
5259:
5256:
5236:
5233:
5230:
5209:
5200:
5199:antisymmetric
5183:
5175:the preorder
5172:
5168:
5166:
5149:
5141:The relation
5128:
5125:
5122:
5119:
5109:
5106:
5103:
5093:
5090:
5087:
5067:
5064:
5061:
5058:
5050:
5047:
5044:
5024:
5021:
5018:
4997:
4976:
4973:
4970:
4950:
4947:
4944:
4924:
4921:
4918:
4897:
4889:Any preorder
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4827:
4824:
4821:
4813:
4810:
4807:
4799:
4796:
4793:
4785:
4782:
4767:
4764:
4761:
4758:
4748:
4734:
4731:
4728:
4707:
4704:
4701:
4698:
4678:
4675:
4672:
4663:
4661:
4660:Irreflexivity
4658:
4657:
4656:
4642:
4621:
4612:
4609:
4594:
4586:
4582:
4581:irreflexivity
4572:
4570:
4553:
4546:
4543:
4539:
4535:
4531:
4510:
4507:
4504:
4501:
4481:
4475:
4455:
4449:
4428:
4425:
4417:
4401:
4398:
4395:
4392:
4389:
4369:
4366:
4363:
4360:
4357:
4349:
4327:
4324:
4300:
4289:
4248:
4242:
4239:
4216:
4210:
4207:
4187:
4184:
4178:
4155:
4143:
4119:
4094:
4070:
4066:
4062:
4042:
4034:
4014:
4011:
4008:
3988:
3968:
3965:
3962:
3942:
3936:
3913:
3910:
3890:
3887:
3842:
3839:
3831:
3811:
3808:
3805:
3797:
3759:
3753:
3745:
3739:
3719:
3716:
3713:
3694:
3691:
3688:
3685:
3682:
3679:
3659:
3656:
3650:
3630:
3624:
3604:
3598:
3578:
3572:
3552:
3522:
3521:
3504:
3501:
3498:
3478:
3475:
3469:
3449:
3443:
3423:
3420:
3400:
3397:
3394:
3374:
3366:
3362:
3358:
3342:
3334:
3330:
3326:
3325:formal theory
3310:
3300:
3298:
3296:
3280:
3260:
3257:
3248:
3233:
3230:
3206:
3180:
3157:
3154:
3151:
3148:
3125:
3119:
3113:
3090:
3086:
3082:
3062:
3042:
3022:
3002:
2999:
2996:
2973:
2967:
2964:
2956:
2940:
2920:
2916:
2912:
2892:
2887:
2884:
2880:
2859:
2856:
2847:
2831:
2828:
2824:
2820:
2811:
2796:
2774:
2752:
2731:
2728:
2725:
2722:
2712:
2709:
2706:
2696:
2693:
2690:
2670:
2649:
2641:
2625:
2604:
2589:
2587:
2583:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2523:
2520:
2517:
2509:
2506:
2503:
2495:
2492:
2489:
2481:
2478:
2463:
2460:
2457:
2454:
2434:
2431:
2428:
2420:
2417:
2416:
2415:
2413:
2409:
2405:
2402:
2399:
2396:
2380:
2359:
2356:
2352:
2346:
2343:
2340:
2317:
2314:
2311:
2291:
2288:
2285:
2264:
2243:
2240:
2233:
2216:
2201:
2199:
2193:
2192:
2188:
2183:
2182:
2178:
2173:
2172:
2156:
2153:
2150:
2147:
2126:
2104:
2094:
2092:
2088:
2083:
2081:
2065:
2045:
2037:
2033:
2028:
2026:
2022:
2006:
1986:
1983:
1963:
1960:
1940:
1932:
1928:
1924:
1919:
1917:
1916:antisymmetric
1913:
1910:
1904:
1902:
1898:
1894:
1890:
1886:
1882:
1878:
1870:
1869:partial order
1866:
1862:
1858:
1854:
1851:
1847:
1844:
1840:
1836:
1835:Hasse diagram
1832:
1822:
1807:
1804:
1801:
1798:
1778:
1775:
1772:
1752:
1749:
1746:
1726:
1723:
1720:
1717:
1714:
1711:
1703:
1687:
1680:
1648:
1647:
1624:
1621:
1618:
1603:
1600:
1597:
1586:
1572:
1569:
1566:
1554:
1540:
1537:
1534:
1527:
1500:
1497:
1494:
1483:
1456:
1453:
1450:
1439:
1412:
1398:
1380:
1377:
1374:
1364:
1361:
1358:
1348:
1341:
1338:
1327:
1309:
1302:
1299:
1289:
1286:
1283:
1273:
1270:
1267:
1256:
1238:
1235:
1232:
1218:
1215:
1212:
1200:
1186:
1180:
1177:
1157:
1154:
1151:
1143:
1142:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1103:Antisymmetric
1101:
1099:
1096:
1094:
1093:
1082:
1072:
1067:
1062:
1057:
1052:
1042:
1032:
1027:
1025:
1022:
1021:
1010:
1000:
995:
990:
985:
980:
975:
965:
960:
958:
955:
954:
943:
933:
928:
923:
918:
913:
908:
898:
893:
891:
888:
887:
881:
876:
866:
856:
851:
846:
841:
831:
826:
824:
821:
820:
814:
809:
799:
794:
784:
779:
774:
764:
759:
757:
754:
753:
747:
742:
732:
722:
712:
707:
702:
692:
687:
685:
682:
681:
675:
670:
660:
655:
650:
640:
630:
620:
615:
613:
612:Well-ordering
610:
609:
603:
598:
588:
583:
578:
568:
563:
558:
553:
550:
546:
545:
539:
534:
524:
519:
514:
504:
494:
489:
484:
482:
479:
478:
472:
467:
457:
452:
447:
442:
432:
422:
417:
415:
412:
411:
405:
400:
390:
385:
380:
375:
365:
360:
355:
353:
350:
349:
343:
338:
328:
323:
318:
313:
308:
298:
293:
291:
290:Partial order
288:
287:
281:
276:
266:
261:
256:
251:
246:
241:
236:
233:
226:
225:
219:
214:
204:
199:
194:
189:
184:
179:
169:
166:
162:
161:
158:
153:
151:
149:
147:
145:
142:
140:
138:
136:
135:
132:
129:
127:
124:
122:
119:
117:
114:
112:
109:
107:
104:
102:
99:
97:
96:Antisymmetric
94:
92:
89:
87:
86:
83:
82:
76:
71:
69:
64:
62:
57:
56:
53:
49:
45:
44:
37:
30:
26:
22:
9748:Order theory
9514:& Orders
9492:Star product
9421:Well-founded
9378:
9374:Prefix order
9330:Distributive
9320:Complemented
9290:Foundational
9255:Completeness
9211:Zorn's lemma
9129:
9115:Cyclic order
9098:Key concepts
9068:Order theory
9020:
8991:
8985:
8956:
8932:(1): 23–41.
8929:
8925:
8915:
8895:
8885:
8860:
8856:
8849:
8821:Directed set
8718:
8595:
8541:
8504:
8500:
8406:
8340:
8118:
8107:
8103:
8099:
8096:
8035:
8031:
8027:
8020:
8008:
8001:
7997:
7993:
7989:
7982:
7956:
7761:
7730:
7717:independence
7673:
7657:
7564:
7498:
7406:
7352:relation of
7348:denotes the
7294:denotes the
7174:
7170:
7169:
6960:
6940:
6911:
6907:
6870:
6755:
6752:neighborhood
6747:
6711:
6621:
6617:
6596:substitution
6585:
6577:
6451:
6397:
6393:
6387:
6383:
6379:
6375:
6342:reachability
6335:Graph theory
6158:
6017:as "neither
5656:
5431:defined as:
4888:
4784:Transitivity
4608:
4578:
4348:directed set
3520:modus ponens
3518:
3462:and also as
3299:
3249:
3055:-cycle with
2812:
2595:
2585:
2581:
2579:
2480:Transitivity
2401:
2395:
2394:is called a
2207:
2197:
2190:
2186:
2180:
2176:
2170:
2095:
2084:
2029:
1920:
1908:
1888:
1884:
1881:order theory
1874:
1864:
1860:
1849:
1842:
1838:
1676:
1113:Well-founded
231:(Quasiorder)
228:
106:Well-founded
36:strict order
9698:Riesz space
9659:Isomorphism
9535:Normal cone
9457:Composition
9391:Semilattice
9300:Homogeneous
9285:Equivalence
9135:Total order
8863:: 219–233,
8815:Total order
8598:satisfying
8409:satisfying
7777:Total order
7713:consistency
7698:modal logic
7411:, that is,
6924:permutation
6781:upper bound
6628:. Here the
6441:graph-minor
4350:because if
4083:induced by
3926:denoted by
3355:could be a
2683:such that
2419:Reflexivity
1927:polynomials
1903:. The name
1877:mathematics
1841:defined by
1133:Irreflexive
414:Total order
126:Irreflexive
9737:Categories
9666:Order type
9600:Cofinality
9441:Well-order
9416:Transitive
9305:Idempotent
9238:Asymmetric
9003:References
8250:), giving
8112:refers to
8097:Note that
7747:Transitive
7729:Number of
7709:set theory
7350:complement
6950:Preference
6916:surjection
5405:(that is,
4027:such that
2412:transitive
2403:quasiorder
2204:Definition
2189:, or that
2080:asymmetric
1901:transitive
1889:quasiorder
1704:: for all
1702:transitive
1138:Asymmetric
131:Asymmetric
48:Transitive
9717:Upper set
9654:Embedding
9590:Antichain
9411:Tolerance
9401:Symmetric
9396:Semiorder
9342:Reflexive
9260:Connected
8968:∖
8801:symmetric
8481:≲
8475:≲
8446:≲
8420:≲
8352:≲
8321:preorders
8300:×
8289:, giving
8287:2 + 1 + 1
8282:preorders
8261:×
8246:and 3 of
8200:1 + 1 + 1
8195:preorders
8174:×
8163:, giving
7757:Symmetric
7752:Reflexive
7711:to prove
7638:∈
7606:≲
7580:≲
7540:≲
7514:≲
7501:symmetric
7448:≲
7422:≲
7383:∘
7334:¯
7247:¯
7241:¯
7233:∘
7209:∖
6981:on a set
6931:embedding
6904:injection
6887:≲
6842:≲
6807:≲
6731:≲
6683:→
6656:relation.
6557:≲
6517:Subtyping
6478:→
6413:≲
6359:≲
6257:≠
6233:≤
6223:≲
6196:≲
6170:≤
6149:connected
6110:∼
6088:∼
6066:≲
6002:∼
5945:≲
5918:∼
5896:≲
5788:≤
5713:∼
5592:≤
5569:≤
5549:≲
5506:≲
5482:≠
5468:≲
5356:≲
5340:≠
5324:≲
5234:∼
5210:∼
5184:≲
5123:∼
5091:≲
5062:∼
5048:≲
4998:∼
4974:≲
4948:≲
4898:≲
4863:∈
4762:∈
4720:that is,
4702:∈
4550:⇐
4544:∼
4505:∈
4479:⇐
4453:⇐
4426:∧
4399:∧
4367:∈
4331:⇐
4298:⇒
4273:⇐
4243:∈
4211:∈
4182:⇐
4150:⇐
4117:⇐
4092:⇐
4071:∼
4039:⟺
4012:∈
3966:∈
3888:∼
3865:⟺
3836:⟺
3809:∼
3781:⇐
3757:⇐
3743:⇐
3717:∼
3689:∈
3654:⇐
3628:⇐
3602:⇐
3576:⇐
3532:⇐
3502:∈
3473:⇐
3447:⇒
3398:∈
3329:sentences
3231:∼
3152:≲
3120:≤
3091:∼
3035:is in an
2968:∈
2921:∼
2857:∼
2829:∼
2797:≲
2775:≲
2753:∼
2726:≲
2710:≲
2694:∼
2650:∼
2605:≲
2559:∈
2521:≲
2507:≲
2493:≲
2458:∈
2432:≲
2408:reflexive
2406:if it is
2381:≲
2357:≲
2353:∈
2315:≲
2289:×
2265:≲
2217:≲
2151:≲
2127:≤
2105:≲
2036:symmetric
1984:−
1961:−
1897:reflexive
1855:4 on the
1615:not
1607:⇒
1563:not
1498:∧
1454:∨
1352:⇒
1342:≠
1297:⇒
1226:⇒
1184:∅
1181:≠
1128:Reflexive
1123:Has meets
1118:Has joins
1108:Connected
1098:Symmetric
229:Preorder
156:reflexive
121:Reflexive
116:Has meets
111:Has joins
101:Connected
91:Symmetric
25:pre-order
9512:Topology
9379:Preorder
9362:Eulerian
9325:Complete
9275:Directed
9265:Covering
9130:Preorder
9089:Category
9084:Glossary
8948:14389185
8893:(2002).
8782:See also
8372:interval
8337:Interval
7762:Preorder
7719:results.
7621:for all
7529:implies
7463:implies
7395:denotes
7265:, where
6785:topology
6777:directed
6614:morphism
6610:category
6330:Examples
5607:implies
4963:and not
4840:for all
4751:for all
4691:for all
3565:because
2536:for all
2447:for all
2397:preorder
2184:precedes
2179:or that
1999:divides
1953:divides
1905:preorder
1895:that is
1885:preorder
1672:✗
1659:✗
1069:✗
1064:✗
1059:✗
1054:✗
1029:✗
997:✗
992:✗
987:✗
982:✗
977:✗
962:✗
930:✗
925:✗
920:✗
915:✗
910:✗
895:✗
883:✗
878:✗
853:✗
848:✗
843:✗
828:✗
816:✗
811:✗
796:✗
781:✗
776:✗
761:✗
749:✗
744:✗
709:✗
704:✗
689:✗
677:✗
672:✗
657:✗
652:✗
617:✗
605:✗
600:✗
585:✗
580:✗
565:✗
560:✗
555:✗
541:✗
536:✗
521:✗
516:✗
491:✗
486:✗
474:✗
469:✗
454:✗
449:✗
444:✗
419:✗
407:✗
402:✗
387:✗
382:✗
377:✗
362:✗
357:✗
345:✗
340:✗
325:✗
320:✗
315:✗
310:✗
295:✗
283:✗
278:✗
263:✗
258:✗
253:✗
248:✗
243:✗
238:✗
221:✗
216:✗
201:✗
196:✗
191:✗
186:✗
181:✗
9617:Duality
9595:Cofinal
9583:Related
9562:Fréchet
9439:)
9315:Bounded
9310:Lattice
9283:)
9281:Partial
9149:Results
9120:Lattice
8877:1127325
8090:A000110
8085:A000142
8080:A000670
8075:A001035
8070:A000798
8065:A006125
8060:A053763
8055:A006905
8050:A002416
7705:forcing
6630:objects
6574:subterm
6143:" is a
5934:Define
5777:Define
4382:and if
3301:Example
2194:reduces
2023:" and "
684:Lattice
9642:Subnet
9622:Filter
9572:Normed
9557:Banach
9523:&
9430:Better
9367:Strict
9357:Graded
9248:topics
9079:Topics
9027:
9012:
8946:
8903:
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7920:65,536
7375:while
6869:where
6716:Every
4494:where
3359:(like
3303:: Let
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1909:almost
1509:exists
1465:exists
1421:exists
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9632:Ideal
9610:Graph
9406:Total
9384:Total
9270:Dense
8944:S2CID
8842:Notes
8810:total
8719:Also
8248:2 + 2
8244:3 + 1
8161:2 + 1
7935:1,024
7930:4,096
7925:3,994
7664:class
7662:is a
7567:total
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6906:from
6775:is a
6708:Other
6580:is a
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6544:terms
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6272:then
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