1698:: if the union of a family of finite sets is given the empty partial order, and this is extended to a total order, the extension defines a choice from each finite set, its minimum element in the total order. Although finite choice is a weak version of the axiom of choice, it is independent of
1712:, can be combined to prove the full axiom of choice. With these assumptions, one can choose an element from any given set by extending its empty partial order, finding a cofinal well-order, and choosing the minimum element from that well-ordering.
813:
partial order cannot be extended, by definition, so it follows from this step that a maximal partial order must be a total order. In the second step, Zorn's lemma is applied to find a maximal partial order that extends any given partial order.
1247:
1577:
1548:
1151:
1058:
809:
The theorem is proved in two steps. First, one shows that, if a partial order does not compare some two elements, it can be extended to an order with a superset of comparable pairs. A
1283:
950:
446:
335:
551:
2801:
1641:
1383:
1102:
1029:
952:
This produces a relation that is still reflexive, antisymmetric and transitive and that strictly contains the original one. It follows that if the partial orders extending
402:
1359:
1321:
201:
759:
586:
507:
291:
1797:
1829:
1519:
1487:
1455:
1215:
1183:
140:
230:
1855:
1762:
907:
881:
724:
698:
472:
361:
262:
672:
163:
1684:
1664:
1617:
1597:
1423:
1403:
1122:
1078:
997:
970:
855:
835:
799:
779:
649:
629:
609:
108:
88:
2758:
1007:
has a maximal element. A chain in this poset is a set of relations in which, for every two relations, one extends the other. An upper bound for a chain
2741:
2271:
2107:
2010:
2588:
588:
A partial order is, by definition, a reflexive, transitive and antisymmetric relation. A total order is a partial order that is connex.
1705:
The
Szpilrajn extension theorem together with another consequence of the axiom of choice, the principle that every total order has a
2724:
2583:
1975:
1931:
2578:
1699:
2214:
2296:
1694:
Some form of the axiom of choice is necessary in proving the
Szpilrajn extension theorem. The extension theorem implies the
1962:, Mathematical Surveys and Monographs, vol. 59, Providence, Rhode Island: American Mathematical Society, p. 299,
2615:
2535:
2209:
2400:
2329:
1220:
1553:
1524:
1127:
1034:
972:
are themselves partially ordered by extension, then any maximal element of this extension order must be a total order.
2303:
2291:
2229:
2204:
2158:
2127:
2234:
2224:
2063:
Cato, Susumu (August 2011), "Szpilrajn, Arrow and
Suzumura: concise proofs of extension theorems and an extension",
1918:, Studies in the History of Mathematics and Physical Sciences, vol. 8, New York: Springer-Verlag, p. 222,
883:
to the relation, which may result in a non-transitive relation, and then restoring transitivity by adding all pairs
2600:
2100:
2573:
2239:
2791:
2505:
2132:
999:, ordered by extension, has a maximal element. The existence of such a maximal element is proved by applying
2753:
2736:
781:
that is reflexive, transitive and antisymmetric (that is, a partial order) is contained in another relation
1252:
912:
2665:
2281:
1695:
411:
405:
300:
516:
2796:
2643:
2478:
2469:
2338:
2173:
2137:
2093:
1706:
976:
51:
in such a way that every pair becomes comparable. The theorem is one of many examples of the use of the
2219:
2731:
2690:
2680:
2670:
2415:
2378:
2368:
2348:
2333:
1622:
1364:
1083:
1010:
2658:
2569:
2515:
2474:
2464:
2353:
2286:
2249:
294:
2697:
2550:
2459:
2449:
2390:
2308:
2042:
366:
235:
2770:
2610:
2244:
1734:
Suzumura proved that a binary relation can be extended to a total preorder if and only if it is
1326:
1288:
168:
729:
556:
477:
2707:
2685:
2545:
2530:
2510:
2313:
2006:
1971:
1927:
1913:
267:
2520:
2373:
2072:
2034:
1963:
1919:
1893:
1874:
1767:
48:
32:
1985:
1941:
1802:
1492:
1460:
1428:
1188:
1156:
113:
2702:
2485:
2363:
2358:
2343:
2168:
2153:
1981:
1937:
810:
510:
206:
68:
52:
2259:
1834:
1741:
1000:
886:
860:
703:
677:
451:
340:
241:
56:
654:
145:
2620:
2605:
2595:
2454:
2432:
2410:
1878:
1669:
1649:
1602:
1582:
1408:
1388:
1107:
1063:
982:
955:
840:
820:
784:
764:
634:
614:
594:
93:
73:
1686:. By the first step this maximal element must be a total order, completing the proof.
2785:
2719:
2675:
2653:
2525:
2395:
2383:
2188:
2076:
1998:
1955:
1728:
1720:
44:
36:
1003:
to this poset. Zorn's lemma states that a partial order in which every chain has an
2540:
2422:
2405:
2323:
2163:
2116:
1646:
This argument shows that Zorn's lemma may be applied to the poset of extensions of
801:
which is reflexive, transitive, antisymmetric and connex (that is, a total order).
20:
2746:
2439:
2318:
2183:
1004:
40:
2714:
2648:
2489:
1923:
1709:
2765:
2638:
2444:
1898:
1731:(transitive and connex relation). This claim was later proved by Hansson.
2560:
2427:
2178:
1724:
2046:
817:
For the first step, suppose that a given partial order does not compare
2038:
1967:
2025:
Hansson, Bengt (1968), "Choice structures and preference relations",
1915:
Zermelo's Axiom of Choice: Its
Origins, Development, and Influence
2089:
2085:
2001:(2012), "IV.3: Quasi-orderings and compatible weak orderings",
1628:
1563:
1534:
1370:
1270:
1230:
1137:
1089:
1044:
1016:
1738:, which means that there is no cycle of elements such that
1727:(reflexive and transitive relation) can be extended to a
1031:
can be found as the union of the relations in the chain,
1557:
1528:
1224:
1131:
1038:
857:. Then the order is extended by first adding the pair
1837:
1805:
1770:
1744:
1672:
1652:
1625:
1605:
1585:
1556:
1527:
1521:, as does the union. Similarly, it can be shown that
1495:
1463:
1431:
1411:
1391:
1367:
1329:
1291:
1255:
1223:
1191:
1159:
1130:
1110:
1086:
1066:
1037:
1013:
985:
958:
915:
889:
863:
843:
823:
787:
767:
732:
706:
680:
657:
637:
617:
597:
559:
519:
480:
454:
414:
369:
343:
303:
270:
244:
209:
171:
148:
116:
96:
76:
47:
elements that leaves some pairs incomparable can be
2631:
2559:
2498:
2268:
2197:
2146:
2005:(3rd ed.), Yale University Press, p. 64,
43:. Intuitively, the theorem says that any method of
1849:
1823:
1791:
1756:
1678:
1658:
1635:
1611:
1591:
1571:
1542:
1513:
1481:
1449:
1417:
1397:
1377:
1353:
1315:
1277:
1242:{\displaystyle \textstyle \bigcup {\mathcal {C}},}
1241:
1209:
1177:
1145:
1116:
1096:
1072:
1052:
1023:
991:
964:
944:
901:
875:
849:
829:
793:
773:
753:
718:
692:
666:
643:
623:
603:
580:
545:
501:
466:
440:
396:
355:
329:
285:
256:
224:
195:
157:
134:
102:
82:
1572:{\displaystyle \textstyle \bigcup {\mathcal {C}}}
1543:{\displaystyle \textstyle \bigcup {\mathcal {C}}}
1146:{\displaystyle \textstyle \bigcup {\mathcal {C}}}
1053:{\displaystyle \textstyle \bigcup {\mathcal {C}}}
761:The extension theorem states that every relation
1799:and there is some pair of consecutive elements
59:to find a maximal set with certain properties.
1599:, so it belongs to the poset of extensions of
110:is formally defined as a set of ordered pairs
2101:
8:
1489:, and by its transitivity it also contains
2802:Theorems in the foundations of mathematics
2759:Positive cone of a partially ordered group
2108:
2094:
2086:
1897:
1836:
1804:
1769:
1743:
1671:
1651:
1627:
1626:
1624:
1604:
1584:
1562:
1561:
1555:
1533:
1532:
1526:
1494:
1462:
1430:
1410:
1390:
1369:
1368:
1366:
1328:
1290:
1269:
1268:
1254:
1229:
1228:
1222:
1190:
1158:
1136:
1135:
1129:
1109:
1088:
1087:
1085:
1065:
1043:
1042:
1036:
1015:
1014:
1012:
984:
957:
925:
914:
888:
862:
842:
822:
786:
766:
731:
705:
679:
656:
636:
616:
596:
558:
529:
518:
479:
453:
424:
413:
368:
342:
313:
302:
269:
243:
208:
170:
147:
115:
95:
75:
2742:Positive cone of an ordered vector space
1060:. This union is a relation that extends
1866:
1764:for every pair of consecutive elements
1425:must extend the other and contain both
1153:is a transitive relation. Suppose that
2058:
2056:
16:Mathematical result on order relations
1278:{\displaystyle S,T\in {\mathcal {C}}}
945:{\displaystyle qRx{\text{ and }}yRp.}
7:
1879:"Sur l'extension de l'ordre partiel"
1124:as a subset. Next, it is shown that
441:{\displaystyle xRy{\text{ and }}yRx}
330:{\displaystyle xRy{\text{ and }}yRz}
2003:Social Choice and Individual Values
1960:Consequences of the Axiom of Choice
546:{\displaystyle xRy{\text{ or }}yRx}
2269:Properties & Types (
14:
2725:Positive cone of an ordered field
611:is contained in another relation
2579:Ordered topological vector space
2077:10.1111/j.1467-999x.2011.04130.x
1818:
1806:
1783:
1771:
1666:, producing a maximal element
1636:{\displaystyle {\mathcal {C}}}
1508:
1496:
1476:
1464:
1444:
1432:
1378:{\displaystyle {\mathcal {C}}}
1342:
1330:
1304:
1292:
1204:
1192:
1172:
1160:
1097:{\displaystyle {\mathcal {C}}}
1024:{\displaystyle {\mathcal {C}}}
184:
172:
129:
117:
1:
2536:Series-parallel partial order
2215:Cantor's isomorphism theorem
1619:, and is an upper bound for
979:of partial orders extending
2255:Szpilrajn extension theorem
2230:Hausdorff maximal principle
2205:Boolean prime ideal theorem
1700:Zermelo–Fraenkel set theory
397:{\displaystyle x,y,z\in X;}
35:in 1930, states that every
25:Szpilrajn extension theorem
2818:
2601:Topological vector lattice
1912:Moore, Gregory H. (1982),
1354:{\displaystyle (y,z)\in T}
1316:{\displaystyle (x,y)\in S}
1104:is a partial order having
975:Next it is shown that the
631:when all ordered pairs in
196:{\displaystyle (x,y)\in R}
2123:
1924:10.1007/978-1-4613-9478-5
1080:, since every element of
754:{\displaystyle x,y\in X.}
581:{\displaystyle x,y\in X.}
502:{\displaystyle x,y\in X;}
63:Definitions and statement
29:order-extension principle
2210:Cantor–Bernstein theorem
1716:Other extension theorems
1550:is antisymmetric. Thus,
264:holds for every element
203:is often abbreviated as
2754:Partially ordered group
2574:Specialization preorder
1899:10.4064/fm-16-1-386-389
1886:Fundamenta Mathematicae
1831:in the cycle for which
286:{\displaystyle x\in X;}
2240:Kruskal's tree theorem
2235:Knaster–Tarski theorem
2225:Dushnik–Miller theorem
1851:
1825:
1793:
1792:{\displaystyle (x,y),}
1758:
1696:axiom of finite choice
1680:
1660:
1637:
1613:
1593:
1573:
1544:
1515:
1483:
1451:
1419:
1399:
1379:
1355:
1317:
1279:
1243:
1211:
1179:
1147:
1118:
1098:
1074:
1054:
1025:
993:
966:
946:
903:
877:
851:
831:
795:
775:
755:
720:
694:
668:
645:
625:
605:
582:
547:
503:
468:
442:
398:
357:
331:
287:
258:
226:
197:
159:
136:
104:
84:
1852:
1826:
1824:{\displaystyle (x,y)}
1794:
1759:
1681:
1661:
1638:
1614:
1594:
1574:
1545:
1516:
1514:{\displaystyle (x,z)}
1484:
1482:{\displaystyle (y,z)}
1452:
1450:{\displaystyle (x,y)}
1420:
1400:
1380:
1356:
1318:
1280:
1244:
1212:
1210:{\displaystyle (y,z)}
1180:
1178:{\displaystyle (x,y)}
1148:
1119:
1099:
1075:
1055:
1026:
994:
967:
947:
904:
878:
852:
832:
796:
776:
756:
721:
695:
669:
646:
626:
606:
583:
548:
504:
469:
443:
399:
358:
332:
288:
259:
227:
198:
160:
137:
135:{\displaystyle (x,y)}
105:
85:
2732:Ordered vector space
1958:(1998), "Note 121",
1835:
1803:
1768:
1742:
1670:
1650:
1623:
1603:
1583:
1554:
1525:
1493:
1461:
1429:
1409:
1389:
1365:
1327:
1289:
1253:
1249:so that there exist
1221:
1189:
1157:
1128:
1108:
1084:
1064:
1035:
1011:
983:
956:
913:
887:
861:
841:
821:
785:
765:
730:
704:
678:
655:
635:
615:
595:
557:
517:
478:
452:
412:
367:
341:
301:
268:
242:
225:{\displaystyle xRy.}
207:
169:
146:
114:
94:
74:
2570:Alexandrov topology
2516:Lexicographic order
2475:Well-quasi-ordering
1850:{\displaystyle yRx}
1757:{\displaystyle xRy}
1736:Suzumura-consistent
1579:is an extension of
1385:is a chain, one of
902:{\displaystyle qRp}
876:{\displaystyle xRy}
719:{\displaystyle xSy}
693:{\displaystyle xRy}
467:{\displaystyle x=y}
356:{\displaystyle xRz}
257:{\displaystyle xRx}
2551:Transitive closure
2511:Converse/Transpose
2220:Dilworth's theorem
2039:10.1007/BF00484979
1847:
1821:
1789:
1754:
1723:stated that every
1676:
1656:
1633:
1609:
1589:
1569:
1568:
1540:
1539:
1511:
1479:
1447:
1415:
1395:
1375:
1351:
1313:
1275:
1239:
1238:
1207:
1175:
1143:
1142:
1114:
1094:
1070:
1050:
1049:
1021:
989:
962:
942:
899:
873:
847:
827:
791:
771:
751:
716:
690:
667:{\displaystyle S;}
664:
641:
621:
601:
578:
543:
499:
464:
438:
394:
353:
327:
283:
254:
222:
193:
158:{\displaystyle X,}
155:
132:
100:
80:
39:is contained in a
2779:
2778:
2737:Partially ordered
2546:Symmetric closure
2531:Reflexive closure
2274:
2012:978-0-300-18698-7
1999:Arrow, Kenneth J.
1875:Szpilrajn, Edward
1679:{\displaystyle Q}
1659:{\displaystyle R}
1612:{\displaystyle R}
1592:{\displaystyle R}
1418:{\displaystyle T}
1398:{\displaystyle S}
1117:{\displaystyle R}
1073:{\displaystyle R}
992:{\displaystyle R}
965:{\displaystyle R}
928:
850:{\displaystyle y}
830:{\displaystyle x}
794:{\displaystyle S}
774:{\displaystyle R}
644:{\displaystyle R}
624:{\displaystyle S}
604:{\displaystyle R}
532:
427:
316:
103:{\displaystyle X}
83:{\displaystyle R}
27:(also called the
2809:
2521:Linear extension
2270:
2250:Mirsky's theorem
2110:
2103:
2096:
2087:
2080:
2079:
2060:
2051:
2049:
2022:
2016:
2015:
1995:
1989:
1988:
1968:10.1090/surv/059
1951:
1945:
1944:
1909:
1903:
1902:
1901:
1883:
1871:
1856:
1854:
1853:
1848:
1830:
1828:
1827:
1822:
1798:
1796:
1795:
1790:
1763:
1761:
1760:
1755:
1702:without choice.
1685:
1683:
1682:
1677:
1665:
1663:
1662:
1657:
1642:
1640:
1639:
1634:
1632:
1631:
1618:
1616:
1615:
1610:
1598:
1596:
1595:
1590:
1578:
1576:
1575:
1570:
1567:
1566:
1549:
1547:
1546:
1541:
1538:
1537:
1520:
1518:
1517:
1512:
1488:
1486:
1485:
1480:
1456:
1454:
1453:
1448:
1424:
1422:
1421:
1416:
1404:
1402:
1401:
1396:
1384:
1382:
1381:
1376:
1374:
1373:
1360:
1358:
1357:
1352:
1322:
1320:
1319:
1314:
1284:
1282:
1281:
1276:
1274:
1273:
1248:
1246:
1245:
1240:
1234:
1233:
1216:
1214:
1213:
1208:
1184:
1182:
1181:
1176:
1152:
1150:
1149:
1144:
1141:
1140:
1123:
1121:
1120:
1115:
1103:
1101:
1100:
1095:
1093:
1092:
1079:
1077:
1076:
1071:
1059:
1057:
1056:
1051:
1048:
1047:
1030:
1028:
1027:
1022:
1020:
1019:
998:
996:
995:
990:
971:
969:
968:
963:
951:
949:
948:
943:
929:
926:
908:
906:
905:
900:
882:
880:
879:
874:
856:
854:
853:
848:
836:
834:
833:
828:
800:
798:
797:
792:
780:
778:
777:
772:
760:
758:
757:
752:
725:
723:
722:
717:
699:
697:
696:
691:
673:
671:
670:
665:
650:
648:
647:
642:
630:
628:
627:
622:
610:
608:
607:
602:
587:
585:
584:
579:
552:
550:
549:
544:
533:
530:
508:
506:
505:
500:
473:
471:
470:
465:
447:
445:
444:
439:
428:
425:
403:
401:
400:
395:
362:
360:
359:
354:
336:
334:
333:
328:
317:
314:
292:
290:
289:
284:
263:
261:
260:
255:
231:
229:
228:
223:
202:
200:
199:
194:
164:
162:
161:
156:
141:
139:
138:
133:
109:
107:
106:
101:
89:
87:
86:
81:
33:Edward Szpilrajn
2817:
2816:
2812:
2811:
2810:
2808:
2807:
2806:
2792:Axiom of choice
2782:
2781:
2780:
2775:
2771:Young's lattice
2627:
2555:
2494:
2344:Heyting algebra
2292:Boolean algebra
2264:
2245:Laver's theorem
2193:
2159:Boolean algebra
2154:Binary relation
2142:
2119:
2114:
2084:
2083:
2062:
2061:
2054:
2024:
2023:
2019:
2013:
1997:
1996:
1992:
1978:
1953:
1952:
1948:
1934:
1911:
1910:
1906:
1881:
1873:
1872:
1868:
1863:
1857:does not hold.
1833:
1832:
1801:
1800:
1766:
1765:
1740:
1739:
1718:
1692:
1668:
1667:
1648:
1647:
1621:
1620:
1601:
1600:
1581:
1580:
1552:
1551:
1523:
1522:
1491:
1490:
1459:
1458:
1427:
1426:
1407:
1406:
1387:
1386:
1363:
1362:
1325:
1324:
1287:
1286:
1251:
1250:
1219:
1218:
1187:
1186:
1155:
1154:
1126:
1125:
1106:
1105:
1082:
1081:
1062:
1061:
1033:
1032:
1009:
1008:
981:
980:
954:
953:
927: and
911:
910:
885:
884:
859:
858:
839:
838:
819:
818:
807:
783:
782:
763:
762:
728:
727:
702:
701:
676:
675:
653:
652:
651:also appear in
633:
632:
613:
612:
593:
592:
555:
554:
515:
514:
511:connex relation
476:
475:
450:
449:
426: and
410:
409:
365:
364:
339:
338:
315: and
299:
298:
266:
265:
240:
239:
205:
204:
167:
166:
144:
143:
142:of elements of
112:
111:
92:
91:
72:
71:
69:binary relation
65:
55:in the form of
53:axiom of choice
17:
12:
11:
5:
2815:
2813:
2805:
2804:
2799:
2794:
2784:
2783:
2777:
2776:
2774:
2773:
2768:
2763:
2762:
2761:
2751:
2750:
2749:
2744:
2739:
2729:
2728:
2727:
2717:
2712:
2711:
2710:
2705:
2698:Order morphism
2695:
2694:
2693:
2683:
2678:
2673:
2668:
2663:
2662:
2661:
2651:
2646:
2641:
2635:
2633:
2629:
2628:
2626:
2625:
2624:
2623:
2618:
2616:Locally convex
2613:
2608:
2598:
2596:Order topology
2593:
2592:
2591:
2589:Order topology
2586:
2576:
2566:
2564:
2557:
2556:
2554:
2553:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2513:
2508:
2502:
2500:
2496:
2495:
2493:
2492:
2482:
2472:
2467:
2462:
2457:
2452:
2447:
2442:
2437:
2436:
2435:
2425:
2420:
2419:
2418:
2413:
2408:
2403:
2401:Chain-complete
2393:
2388:
2387:
2386:
2381:
2376:
2371:
2366:
2356:
2351:
2346:
2341:
2336:
2326:
2321:
2316:
2311:
2306:
2301:
2300:
2299:
2289:
2284:
2278:
2276:
2266:
2265:
2263:
2262:
2257:
2252:
2247:
2242:
2237:
2232:
2227:
2222:
2217:
2212:
2207:
2201:
2199:
2195:
2194:
2192:
2191:
2186:
2181:
2176:
2171:
2166:
2161:
2156:
2150:
2148:
2144:
2143:
2141:
2140:
2135:
2130:
2124:
2121:
2120:
2115:
2113:
2112:
2105:
2098:
2090:
2082:
2081:
2071:(2): 235–249,
2065:Metroeconomica
2052:
2033:(4): 443–458,
2017:
2011:
1990:
1976:
1956:Rubin, Jean E.
1954:Howard, Paul;
1946:
1932:
1904:
1865:
1864:
1862:
1859:
1846:
1843:
1840:
1820:
1817:
1814:
1811:
1808:
1788:
1785:
1782:
1779:
1776:
1773:
1753:
1750:
1747:
1737:
1729:total preorder
1717:
1714:
1691:
1688:
1675:
1655:
1630:
1608:
1588:
1565:
1560:
1536:
1531:
1510:
1507:
1504:
1501:
1498:
1478:
1475:
1472:
1469:
1466:
1446:
1443:
1440:
1437:
1434:
1414:
1394:
1372:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1272:
1267:
1264:
1261:
1258:
1237:
1232:
1227:
1206:
1203:
1200:
1197:
1194:
1174:
1171:
1168:
1165:
1162:
1139:
1134:
1113:
1091:
1069:
1046:
1041:
1018:
988:
961:
941:
938:
935:
932:
924:
921:
918:
898:
895:
892:
872:
869:
866:
846:
826:
806:
803:
790:
770:
750:
747:
744:
741:
738:
735:
715:
712:
709:
689:
686:
683:
663:
660:
640:
620:
600:
577:
574:
571:
568:
565:
562:
553:holds for all
542:
539:
536:
531: or
528:
525:
522:
498:
495:
492:
489:
486:
483:
463:
460:
457:
437:
434:
431:
423:
420:
417:
393:
390:
387:
384:
381:
378:
375:
372:
352:
349:
346:
326:
323:
320:
312:
309:
306:
282:
279:
276:
273:
253:
250:
247:
234:A relation is
221:
218:
215:
212:
192:
189:
186:
183:
180:
177:
174:
154:
151:
131:
128:
125:
122:
119:
99:
79:
64:
61:
15:
13:
10:
9:
6:
4:
3:
2:
2814:
2803:
2800:
2798:
2795:
2793:
2790:
2789:
2787:
2772:
2769:
2767:
2764:
2760:
2757:
2756:
2755:
2752:
2748:
2745:
2743:
2740:
2738:
2735:
2734:
2733:
2730:
2726:
2723:
2722:
2721:
2720:Ordered field
2718:
2716:
2713:
2709:
2706:
2704:
2701:
2700:
2699:
2696:
2692:
2689:
2688:
2687:
2684:
2682:
2679:
2677:
2676:Hasse diagram
2674:
2672:
2669:
2667:
2664:
2660:
2657:
2656:
2655:
2654:Comparability
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2636:
2634:
2630:
2622:
2619:
2617:
2614:
2612:
2609:
2607:
2604:
2603:
2602:
2599:
2597:
2594:
2590:
2587:
2585:
2582:
2581:
2580:
2577:
2575:
2571:
2568:
2567:
2565:
2562:
2558:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2526:Product order
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2503:
2501:
2499:Constructions
2497:
2491:
2487:
2483:
2480:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2434:
2431:
2430:
2429:
2426:
2424:
2421:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2398:
2397:
2396:Partial order
2394:
2392:
2389:
2385:
2384:Join and meet
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2361:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2331:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2298:
2295:
2294:
2293:
2290:
2288:
2285:
2283:
2282:Antisymmetric
2280:
2279:
2277:
2273:
2267:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2202:
2200:
2196:
2190:
2189:Weak ordering
2187:
2185:
2182:
2180:
2177:
2175:
2174:Partial order
2172:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2151:
2149:
2145:
2139:
2136:
2134:
2131:
2129:
2126:
2125:
2122:
2118:
2111:
2106:
2104:
2099:
2097:
2092:
2091:
2088:
2078:
2074:
2070:
2066:
2059:
2057:
2053:
2050:; see Lemma 3
2048:
2044:
2040:
2036:
2032:
2028:
2021:
2018:
2014:
2008:
2004:
2000:
1994:
1991:
1987:
1983:
1979:
1977:0-8218-0977-6
1973:
1969:
1965:
1961:
1957:
1950:
1947:
1943:
1939:
1935:
1933:0-387-90670-3
1929:
1925:
1921:
1917:
1916:
1908:
1905:
1900:
1895:
1891:
1888:(in French),
1887:
1880:
1876:
1870:
1867:
1860:
1858:
1844:
1841:
1838:
1815:
1812:
1809:
1786:
1780:
1777:
1774:
1751:
1748:
1745:
1735:
1732:
1730:
1726:
1722:
1715:
1713:
1711:
1708:
1703:
1701:
1697:
1689:
1687:
1673:
1653:
1644:
1606:
1586:
1558:
1529:
1505:
1502:
1499:
1473:
1470:
1467:
1441:
1438:
1435:
1412:
1392:
1348:
1345:
1339:
1336:
1333:
1310:
1307:
1301:
1298:
1295:
1265:
1262:
1259:
1256:
1235:
1225:
1201:
1198:
1195:
1169:
1166:
1163:
1132:
1111:
1067:
1039:
1006:
1002:
986:
978:
973:
959:
939:
936:
933:
930:
922:
919:
916:
896:
893:
890:
870:
867:
864:
844:
824:
815:
812:
804:
802:
788:
768:
748:
745:
742:
739:
736:
733:
713:
710:
707:
687:
684:
681:
661:
658:
638:
618:
598:
589:
575:
572:
569:
566:
563:
560:
540:
537:
534:
526:
523:
520:
512:
496:
493:
490:
487:
484:
481:
461:
458:
455:
435:
432:
429:
421:
418:
415:
407:
406:antisymmetric
391:
388:
385:
382:
379:
376:
373:
370:
350:
347:
344:
324:
321:
318:
310:
307:
304:
296:
280:
277:
274:
271:
251:
248:
245:
237:
232:
219:
216:
213:
210:
190:
187:
181:
178:
175:
152:
149:
126:
123:
120:
97:
77:
70:
62:
60:
58:
54:
50:
46:
42:
38:
37:partial order
34:
31:), proved by
30:
26:
22:
2797:Order theory
2563:& Orders
2541:Star product
2470:Well-founded
2423:Prefix order
2379:Distributive
2369:Complemented
2339:Foundational
2304:Completeness
2260:Zorn's lemma
2254:
2164:Cyclic order
2147:Key concepts
2117:Order theory
2068:
2064:
2030:
2026:
2020:
2002:
1993:
1959:
1949:
1914:
1907:
1889:
1885:
1869:
1733:
1719:
1704:
1693:
1645:
1001:Zorn's lemma
974:
816:
808:
590:
509:and it is a
233:
66:
57:Zorn's lemma
28:
24:
21:order theory
18:
2747:Riesz space
2708:Isomorphism
2584:Normal cone
2506:Composition
2440:Semilattice
2349:Homogeneous
2334:Equivalence
2184:Total order
1892:: 386–389,
1005:upper bound
591:A relation
41:total order
2786:Categories
2715:Order type
2649:Cofinality
2490:Well-order
2465:Transitive
2354:Idempotent
2287:Asymmetric
1861:References
1710:well-order
1361:. Because
1285:such that
909:such that
295:transitive
2766:Upper set
2703:Embedding
2639:Antichain
2460:Tolerance
2450:Symmetric
2445:Semiorder
2391:Reflexive
2309:Connected
1559:⋃
1530:⋃
1346:∈
1308:∈
1266:∈
1226:⋃
1133:⋃
1040:⋃
743:∈
570:∈
491:∈
386:∈
275:∈
236:reflexive
188:∈
90:on a set
45:comparing
2561:Topology
2428:Preorder
2411:Eulerian
2374:Complete
2324:Directed
2314:Covering
2179:Preorder
2138:Category
2133:Glossary
2047:20114617
2027:Synthese
1877:(1930),
1725:preorder
1690:Strength
726:for all
700:implies
674:that is,
474:for all
363:for all
49:extended
2666:Duality
2644:Cofinal
2632:Related
2611:Fréchet
2488:)
2364:Bounded
2359:Lattice
2332:)
2330:Partial
2198:Results
2169:Lattice
1986:1637107
1942:0679315
1707:cofinal
1217:are in
811:maximal
2691:Subnet
2671:Filter
2621:Normed
2606:Banach
2572:&
2479:Better
2416:Strict
2406:Graded
2297:topics
2128:Topics
2045:
2009:
1984:
1974:
1940:
1930:
448:imply
404:it is
337:imply
293:it is
23:, the
2681:Ideal
2659:Graph
2455:Total
2433:Total
2319:Dense
2043:JSTOR
1882:(PDF)
1721:Arrow
977:poset
805:Proof
2272:list
2007:ISBN
1972:ISBN
1928:ISBN
1457:and
1323:and
1185:and
837:and
165:and
2686:Net
2486:Pre
2073:doi
2035:doi
1964:doi
1920:doi
1894:doi
1405:or
513:if
408:if
297:if
238:if
19:In
2788::
2069:63
2067:,
2055:^
2041:,
2031:18
2029:,
1982:MR
1980:,
1970:,
1938:MR
1936:,
1926:,
1890:16
1884:,
1643:.
67:A
2484:(
2481:)
2477:(
2328:(
2275:)
2109:e
2102:t
2095:v
2075::
2037::
1966::
1922::
1896::
1845:x
1842:R
1839:y
1819:)
1816:y
1813:,
1810:x
1807:(
1787:,
1784:)
1781:y
1778:,
1775:x
1772:(
1752:y
1749:R
1746:x
1674:Q
1654:R
1629:C
1607:R
1587:R
1564:C
1535:C
1509:)
1506:z
1503:,
1500:x
1497:(
1477:)
1474:z
1471:,
1468:y
1465:(
1445:)
1442:y
1439:,
1436:x
1433:(
1413:T
1393:S
1371:C
1349:T
1343:)
1340:z
1337:,
1334:y
1331:(
1311:S
1305:)
1302:y
1299:,
1296:x
1293:(
1271:C
1263:T
1260:,
1257:S
1236:,
1231:C
1205:)
1202:z
1199:,
1196:y
1193:(
1173:)
1170:y
1167:,
1164:x
1161:(
1138:C
1112:R
1090:C
1068:R
1045:C
1017:C
987:R
960:R
940:.
937:p
934:R
931:y
923:x
920:R
917:q
897:p
894:R
891:q
871:y
868:R
865:x
845:y
825:x
789:S
769:R
749:.
746:X
740:y
737:,
734:x
714:y
711:S
708:x
688:y
685:R
682:x
662:;
659:S
639:R
619:S
599:R
576:.
573:X
567:y
564:,
561:x
541:x
538:R
535:y
527:y
524:R
521:x
497:;
494:X
488:y
485:,
482:x
462:y
459:=
456:x
436:x
433:R
430:y
422:y
419:R
416:x
392:;
389:X
383:z
380:,
377:y
374:,
371:x
351:z
348:R
345:x
325:z
322:R
319:y
311:y
308:R
305:x
281:;
278:X
272:x
252:x
249:R
246:x
220:.
217:y
214:R
211:x
191:R
185:)
182:y
179:,
176:x
173:(
153:,
150:X
130:)
127:y
124:,
121:x
118:(
98:X
78:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.