43:(posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that either of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are
1592:
when there exists an order isomorphism from one to the other. Identity functions, function inverses, and compositions of functions correspond, respectively, to the three defining characteristics of an
1078:
1044:
427:
1583:
1544:
1461:
1422:
1363:
1324:
983:
944:
901:
862:
819:
780:
741:
702:
220:
181:
138:
99:
1100:
376:
542:
614:
588:
1178:
1124:
640:
1505:
1485:
1383:
1285:
1261:
562:
498:
478:
458:
343:
323:
303:
283:
263:
243:
1213:
1795:
1215:: the closed interval has a least element, but the open interval does not, and order isomorphisms must preserve the existence of least elements.
2622:
2605:
2135:
1971:
1608:. Therefore, order isomorphism is an equivalence relation. The class of partially ordered sets can be partitioned by it into
1231:
2452:
1869:
2588:
2447:
1919:
1898:
1877:
1770:
2442:
2078:
1612:, families of partially ordered sets that are all isomorphic to each other. These equivalence classes are called
1219:
2160:
1940:
2479:
2399:
2264:
2193:
2073:
1049:
1015:
2167:
2155:
2118:
2093:
2068:
2022:
1991:
2464:
2098:
2088:
1964:
1893:, London Mathematical Society Student Texts, vol. 39, Cambridge University Press, pp. 38â39,
381:
2437:
2103:
1226:. Explicit order isomorphisms between the quadratic algebraic numbers, the rational numbers, and the
2369:
1996:
1549:
1510:
1427:
1388:
1329:
1290:
949:
910:
867:
828:
785:
746:
707:
668:
186:
147:
104:
65:
1083:
2617:
2600:
822:
348:
2529:
2145:
503:
1863:
2660:
2507:
2342:
2333:
2202:
2083:
2037:
2001:
1957:
1909:
1888:
593:
567:
60:
40:
1757:, Texts and Readings in Mathematics, vol. 12, Berlin: Springer-Verlag, pp. 77â86,
821:
must satisfy additional properties to be regarded as an isomorphism. For example, given two
2595:
2554:
2544:
2534:
2279:
2242:
2232:
2212:
2197:
1818:
1780:
1593:
1464:
1151:
1109:
8:
2522:
2433:
2379:
2338:
2328:
2217:
2150:
2113:
1625:
1605:
619:
2655:
2634:
2561:
2414:
2323:
2313:
2254:
2172:
2108:
1601:
1597:
1490:
1470:
1368:
1270:
1246:
547:
483:
463:
443:
328:
308:
288:
268:
248:
228:
2474:
1186:
2549:
2409:
2394:
2374:
2177:
1936:
1915:
1894:
1873:
1766:
1609:
1003:
986:
646:
48:
32:
1222:, every unbounded countable dense linear order is isomorphic to the ordering of the
2384:
2237:
1804:
1758:
1750:
1264:
1009:
1793:
Girgensohn, Roland (1996), "Constructing singular functions via Farey fractions",
2566:
2349:
2227:
2222:
2207:
2123:
2032:
2017:
1930:
1814:
1776:
1227:
1223:
1181:
990:
430:
44:
1762:
1628:, a permutation that is order-isomorphic to a subsequence of another permutation
645:
Yet another characterization of order isomorphisms is that they are exactly the
2484:
2469:
2459:
2318:
2296:
2274:
1180:(again, ordered numerically) does not have an order isomorphism to or from the
2649:
2583:
2539:
2517:
2389:
2259:
2247:
2052:
1146:
2404:
2286:
2269:
2187:
2027:
1980:
1809:
658:
24:
2610:
2303:
2182:
2047:
1103:
655:
An order isomorphism from a partially ordered set to itself is called an
36:
20:
2578:
2512:
2353:
1727:, p. 39., for a similar example with integers in place of real numbers.
1613:
437:
2629:
2502:
2308:
1935:, Dover Publications; Reprint edition (March 5, 2014), pp. 2â3,
649:
223:
1748:
2424:
2291:
2042:
1749:
Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.;
665:
When an additional algebraic structure is imposed on the posets
1949:
1865:
Proofs and
Fundamentals: A First Course in Abstract Mathematics
1710:
This definition is equivalent to the definition set forth in
1006:
on any partially ordered set is always an order automorphism.
436:
It is also possible to define an order isomorphism to be a
480:
and that it preserve orderings, are enough to ensure that
1552:
1513:
1493:
1473:
1430:
1391:
1371:
1332:
1293:
1273:
1249:
1189:
1154:
1112:
1086:
1052:
1018:
952:
913:
870:
831:
788:
749:
710:
671:
622:
616:, implying by the definition of a partial order that
596:
570:
550:
506:
486:
466:
446:
384:
351:
331:
311:
291:
271:
251:
231:
189:
150:
107:
68:
1577:
1538:
1499:
1479:
1455:
1416:
1377:
1357:
1318:
1279:
1255:
1207:
1172:
1118:
1094:
1072:
1038:
977:
938:
895:
856:
813:
774:
735:
696:
634:
608:
582:
556:
536:
492:
472:
452:
421:
370:
337:
317:
297:
277:
257:
237:
214:
175:
132:
93:
1796:Journal of Mathematical Analysis and Applications
2647:
1126:denotes the usual numerical comparison), since â
1672:it is a consequence of a different definition.
1965:
1872:(2nd ed.), Springer, pp. 276â277,
2623:Positive cone of a partially ordered group
1972:
1958:
1886:
1846:
1830:
1792:
1736:
1724:
1661:
1648:
1588:Two partially ordered sets are said to be
564:preserves the order) it would follow that
440:order-embedding. The two assumptions that
1908:Schröder, Bernd Siegfried Walter (2003),
1808:
1088:
1057:
1023:
2606:Positive cone of an ordered vector space
1907:
1890:Set Theory for the Working Mathematician
1834:
1698:
1686:
1669:
1263:is an order isomorphism, then so is its
985:is an order isomorphism that is also a
2648:
35:that constitutes a suitable notion of
1953:
1928:
1861:
1711:
1682:
1665:
1644:
1507:is itself an order isomorphism, from
16:Equivalence of partially ordered sets
1755:Notes on infinite permutation groups
1073:{\displaystyle (\mathbb {R} ,\geq )}
1039:{\displaystyle (\mathbb {R} ,\leq )}
989:, not merely a bijection that is an
1932:Partially Ordered Algebraic Systems
13:
2133:Properties & Types (
1870:Undergraduate Texts in Mathematics
1232:Minkowski's question-mark function
285:with the property that, for every
14:
2672:
2589:Positive cone of an ordered field
422:{\displaystyle f(x)\leq _{T}f(y)}
2443:Ordered topological vector space
1979:
1840:
1824:
1681:This is the definition used by
1660:This is the definition used by
1887:Ciesielski, Krzysztof (1997),
1786:
1742:
1730:
1717:
1704:
1692:
1675:
1654:
1638:
1572:
1553:
1533:
1514:
1450:
1431:
1411:
1392:
1352:
1333:
1313:
1294:
1238:
1202:
1190:
1167:
1155:
1067:
1053:
1033:
1019:
972:
953:
933:
914:
890:
871:
851:
832:
808:
789:
769:
750:
730:
711:
691:
672:
652:that have a monotone inverse.
531:
525:
516:
510:
416:
410:
394:
388:
209:
190:
170:
151:
127:
108:
88:
69:
1:
2400:Series-parallel partial order
1911:Ordered Sets: An Introduction
1855:
1578:{\displaystyle (U,\leq _{U})}
1539:{\displaystyle (S,\leq _{S})}
1456:{\displaystyle (U,\leq _{U})}
1417:{\displaystyle (T,\leq _{T})}
1385:is an order isomorphism from
1358:{\displaystyle (T,\leq _{T})}
1319:{\displaystyle (S,\leq _{S})}
1287:is an order isomorphism from
1012:is an order isomorphism from
978:{\displaystyle (H,\leq _{H})}
939:{\displaystyle (G,\leq _{G})}
896:{\displaystyle (H,\leq _{H})}
857:{\displaystyle (G,\leq _{G})}
814:{\displaystyle (T,\leq _{T})}
775:{\displaystyle (S,\leq _{S})}
736:{\displaystyle (T,\leq _{T})}
697:{\displaystyle (S,\leq _{S})}
544:then (by the assumption that
429:. That is, it is a bijective
215:{\displaystyle (T,\leq _{T})}
176:{\displaystyle (S,\leq _{S})}
133:{\displaystyle (T,\leq _{T})}
94:{\displaystyle (S,\leq _{S})}
54:
2079:Cantor's isomorphism theorem
1753:(1997), "Rational numbers",
1220:Cantor's isomorphism theorem
1095:{\displaystyle \mathbb {R} }
7:
2119:Szpilrajn extension theorem
2094:Hausdorff maximal principle
2069:Boolean prime ideal theorem
1763:10.1007/978-93-80250-91-5_9
1619:
996:
500:is also one-to-one, for if
371:{\displaystyle x\leq _{S}y}
10:
2677:
2465:Topological vector lattice
460:cover all the elements of
2495:
2423:
2362:
2132:
2061:
2010:
1987:
537:{\displaystyle f(x)=f(y)}
2074:CantorâBernstein theorem
1914:, Springer, p. 11,
1862:Bloch, Ethan D. (2011),
1632:
1230:numbers are provided by
905:isomorphism of po-groups
823:partially ordered groups
2618:Partially ordered group
2438:Specialization preorder
609:{\displaystyle y\leq x}
583:{\displaystyle x\leq y}
2104:Kruskal's tree theorem
2099:KnasterâTarski theorem
2089:DushnikâMiller theorem
1929:Fuchs, Laszlo (1963),
1810:10.1006/jmaa.1996.0370
1579:
1540:
1501:
1481:
1457:
1418:
1379:
1359:
1320:
1281:
1257:
1209:
1174:
1120:
1096:
1074:
1040:
979:
940:
897:
858:
815:
776:
737:
698:
636:
610:
584:
558:
538:
494:
474:
454:
423:
372:
339:
319:
299:
279:
259:
239:
216:
177:
134:
95:
41:partially ordered sets
1580:
1541:
1502:
1482:
1458:
1419:
1380:
1360:
1321:
1282:
1258:
1210:
1175:
1173:{\displaystyle (0,1)}
1121:
1119:{\displaystyle \leq }
1097:
1075:
1041:
980:
941:
898:
859:
816:
777:
738:
699:
637:
611:
585:
559:
539:
495:
475:
455:
424:
373:
340:
320:
300:
280:
260:
240:
217:
178:
135:
96:
31:is a special kind of
2596:Ordered vector space
1594:equivalence relation
1550:
1511:
1491:
1471:
1465:function composition
1428:
1389:
1369:
1330:
1291:
1271:
1247:
1187:
1152:
1110:
1084:
1050:
1016:
950:
911:
868:
829:
786:
747:
708:
669:
620:
594:
568:
548:
504:
484:
464:
444:
382:
349:
329:
309:
289:
269:
249:
229:
187:
148:
105:
66:
59:Formally, given two
2434:Alexandrov topology
2380:Lexicographic order
2339:Well-quasi-ordering
1739:, example 1, p. 39.
1626:Permutation pattern
1610:equivalence classes
635:{\displaystyle x=y}
2415:Transitive closure
2375:Converse/Transpose
2084:Dilworth's theorem
1575:
1536:
1497:
1477:
1453:
1414:
1375:
1355:
1316:
1277:
1253:
1205:
1170:
1116:
1092:
1070:
1036:
975:
936:
893:
854:
811:
772:
743:, a function from
733:
694:
632:
606:
580:
554:
534:
490:
470:
450:
419:
368:
335:
315:
295:
275:
255:
235:
224:bijective function
212:
173:
130:
91:
49:Galois connections
2643:
2642:
2601:Partially ordered
2410:Symmetric closure
2395:Reflexive closure
2138:
1847:Ciesielski (1997)
1831:Ciesielski (1997)
1751:Neumann, Peter M.
1737:Ciesielski (1997)
1725:Ciesielski (1997)
1723:See example 4 of
1662:Ciesielski (1997)
1649:Ciesielski (1997)
1500:{\displaystyle g}
1480:{\displaystyle f}
1378:{\displaystyle g}
1280:{\displaystyle f}
1256:{\displaystyle f}
1004:identity function
987:group isomorphism
557:{\displaystyle f}
493:{\displaystyle f}
473:{\displaystyle T}
453:{\displaystyle f}
338:{\displaystyle S}
318:{\displaystyle y}
298:{\displaystyle x}
278:{\displaystyle T}
258:{\displaystyle S}
238:{\displaystyle f}
142:order isomorphism
33:monotone function
29:order isomorphism
2668:
2385:Linear extension
2134:
2114:Mirsky's theorem
1974:
1967:
1960:
1951:
1950:
1945:
1924:
1903:
1882:
1850:
1844:
1838:
1828:
1822:
1821:
1812:
1790:
1784:
1783:
1746:
1740:
1734:
1728:
1721:
1715:
1708:
1702:
1696:
1690:
1679:
1673:
1658:
1652:
1642:
1590:order isomorphic
1584:
1582:
1581:
1576:
1571:
1570:
1545:
1543:
1542:
1537:
1532:
1531:
1506:
1504:
1503:
1498:
1486:
1484:
1483:
1478:
1462:
1460:
1459:
1454:
1449:
1448:
1423:
1421:
1420:
1415:
1410:
1409:
1384:
1382:
1381:
1376:
1364:
1362:
1361:
1356:
1351:
1350:
1325:
1323:
1322:
1317:
1312:
1311:
1286:
1284:
1283:
1278:
1265:inverse function
1262:
1260:
1259:
1254:
1224:rational numbers
1214:
1212:
1211:
1208:{\displaystyle }
1206:
1179:
1177:
1176:
1171:
1125:
1123:
1122:
1117:
1101:
1099:
1098:
1093:
1091:
1079:
1077:
1076:
1071:
1060:
1045:
1043:
1042:
1037:
1026:
984:
982:
981:
976:
971:
970:
945:
943:
942:
937:
932:
931:
902:
900:
899:
894:
889:
888:
863:
861:
860:
855:
850:
849:
820:
818:
817:
812:
807:
806:
781:
779:
778:
773:
768:
767:
742:
740:
739:
734:
729:
728:
703:
701:
700:
695:
690:
689:
641:
639:
638:
633:
615:
613:
612:
607:
589:
587:
586:
581:
563:
561:
560:
555:
543:
541:
540:
535:
499:
497:
496:
491:
479:
477:
476:
471:
459:
457:
456:
451:
428:
426:
425:
420:
406:
405:
377:
375:
374:
369:
364:
363:
344:
342:
341:
336:
324:
322:
321:
316:
304:
302:
301:
296:
284:
282:
281:
276:
264:
262:
261:
256:
244:
242:
241:
236:
221:
219:
218:
213:
208:
207:
182:
180:
179:
174:
169:
168:
139:
137:
136:
131:
126:
125:
100:
98:
97:
92:
87:
86:
45:order embeddings
2676:
2675:
2671:
2670:
2669:
2667:
2666:
2665:
2646:
2645:
2644:
2639:
2635:Young's lattice
2491:
2419:
2358:
2208:Heyting algebra
2156:Boolean algebra
2128:
2109:Laver's theorem
2057:
2023:Boolean algebra
2018:Binary relation
2006:
1983:
1978:
1943:
1922:
1901:
1880:
1858:
1853:
1845:
1841:
1835:Schröder (2003)
1829:
1825:
1791:
1787:
1773:
1747:
1743:
1735:
1731:
1722:
1718:
1709:
1705:
1699:Schröder (2003)
1697:
1693:
1687:Schröder (2003)
1680:
1676:
1670:Schröder (2003)
1659:
1655:
1643:
1639:
1635:
1622:
1566:
1562:
1551:
1548:
1547:
1527:
1523:
1512:
1509:
1508:
1492:
1489:
1488:
1472:
1469:
1468:
1444:
1440:
1429:
1426:
1425:
1405:
1401:
1390:
1387:
1386:
1370:
1367:
1366:
1346:
1342:
1331:
1328:
1327:
1307:
1303:
1292:
1289:
1288:
1272:
1269:
1268:
1248:
1245:
1244:
1241:
1228:dyadic rational
1188:
1185:
1184:
1182:closed interval
1153:
1150:
1149:
1134:if and only if
1111:
1108:
1107:
1087:
1085:
1082:
1081:
1056:
1051:
1048:
1047:
1022:
1017:
1014:
1013:
999:
991:order embedding
966:
962:
951:
948:
947:
927:
923:
912:
909:
908:
884:
880:
869:
866:
865:
845:
841:
830:
827:
826:
802:
798:
787:
784:
783:
763:
759:
748:
745:
744:
724:
720:
709:
706:
705:
685:
681:
670:
667:
666:
621:
618:
617:
595:
592:
591:
569:
566:
565:
549:
546:
545:
505:
502:
501:
485:
482:
481:
465:
462:
461:
445:
442:
441:
431:order-embedding
401:
397:
383:
380:
379:
378:if and only if
359:
355:
350:
347:
346:
330:
327:
326:
310:
307:
306:
290:
287:
286:
270:
267:
266:
250:
247:
246:
230:
227:
226:
203:
199:
188:
185:
184:
164:
160:
149:
146:
145:
121:
117:
106:
103:
102:
82:
78:
67:
64:
63:
57:
17:
12:
11:
5:
2674:
2664:
2663:
2658:
2641:
2640:
2638:
2637:
2632:
2627:
2626:
2625:
2615:
2614:
2613:
2608:
2603:
2593:
2592:
2591:
2581:
2576:
2575:
2574:
2569:
2562:Order morphism
2559:
2558:
2557:
2547:
2542:
2537:
2532:
2527:
2526:
2525:
2515:
2510:
2505:
2499:
2497:
2493:
2492:
2490:
2489:
2488:
2487:
2482:
2480:Locally convex
2477:
2472:
2462:
2460:Order topology
2457:
2456:
2455:
2453:Order topology
2450:
2440:
2430:
2428:
2421:
2420:
2418:
2417:
2412:
2407:
2402:
2397:
2392:
2387:
2382:
2377:
2372:
2366:
2364:
2360:
2359:
2357:
2356:
2346:
2336:
2331:
2326:
2321:
2316:
2311:
2306:
2301:
2300:
2299:
2289:
2284:
2283:
2282:
2277:
2272:
2267:
2265:Chain-complete
2257:
2252:
2251:
2250:
2245:
2240:
2235:
2230:
2220:
2215:
2210:
2205:
2200:
2190:
2185:
2180:
2175:
2170:
2165:
2164:
2163:
2153:
2148:
2142:
2140:
2130:
2129:
2127:
2126:
2121:
2116:
2111:
2106:
2101:
2096:
2091:
2086:
2081:
2076:
2071:
2065:
2063:
2059:
2058:
2056:
2055:
2050:
2045:
2040:
2035:
2030:
2025:
2020:
2014:
2012:
2008:
2007:
2005:
2004:
1999:
1994:
1988:
1985:
1984:
1977:
1976:
1969:
1962:
1954:
1948:
1947:
1941:
1926:
1920:
1905:
1899:
1884:
1878:
1857:
1854:
1852:
1851:
1839:
1823:
1803:(1): 127â141,
1785:
1771:
1741:
1729:
1716:
1703:
1691:
1674:
1653:
1636:
1634:
1631:
1630:
1629:
1621:
1618:
1574:
1569:
1565:
1561:
1558:
1555:
1535:
1530:
1526:
1522:
1519:
1516:
1496:
1476:
1452:
1447:
1443:
1439:
1436:
1433:
1413:
1408:
1404:
1400:
1397:
1394:
1374:
1354:
1349:
1345:
1341:
1338:
1335:
1315:
1310:
1306:
1302:
1299:
1296:
1276:
1252:
1240:
1237:
1236:
1235:
1216:
1204:
1201:
1198:
1195:
1192:
1169:
1166:
1163:
1160:
1157:
1143:
1115:
1102:is the set of
1090:
1069:
1066:
1063:
1059:
1055:
1035:
1032:
1029:
1025:
1021:
1007:
998:
995:
974:
969:
965:
961:
958:
955:
935:
930:
926:
922:
919:
916:
892:
887:
883:
879:
876:
873:
853:
848:
844:
840:
837:
834:
810:
805:
801:
797:
794:
791:
771:
766:
762:
758:
755:
752:
732:
727:
723:
719:
716:
713:
693:
688:
684:
680:
677:
674:
631:
628:
625:
605:
602:
599:
579:
576:
573:
553:
533:
530:
527:
524:
521:
518:
515:
512:
509:
489:
469:
449:
418:
415:
412:
409:
404:
400:
396:
393:
390:
387:
367:
362:
358:
354:
334:
314:
294:
274:
254:
234:
211:
206:
202:
198:
195:
192:
172:
167:
163:
159:
156:
153:
129:
124:
120:
116:
113:
110:
90:
85:
81:
77:
74:
71:
56:
53:
15:
9:
6:
4:
3:
2:
2673:
2662:
2659:
2657:
2654:
2653:
2651:
2636:
2633:
2631:
2628:
2624:
2621:
2620:
2619:
2616:
2612:
2609:
2607:
2604:
2602:
2599:
2598:
2597:
2594:
2590:
2587:
2586:
2585:
2584:Ordered field
2582:
2580:
2577:
2573:
2570:
2568:
2565:
2564:
2563:
2560:
2556:
2553:
2552:
2551:
2548:
2546:
2543:
2541:
2540:Hasse diagram
2538:
2536:
2533:
2531:
2528:
2524:
2521:
2520:
2519:
2518:Comparability
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2500:
2498:
2494:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2467:
2466:
2463:
2461:
2458:
2454:
2451:
2449:
2446:
2445:
2444:
2441:
2439:
2435:
2432:
2431:
2429:
2426:
2422:
2416:
2413:
2411:
2408:
2406:
2403:
2401:
2398:
2396:
2393:
2391:
2390:Product order
2388:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2367:
2365:
2363:Constructions
2361:
2355:
2351:
2347:
2344:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2298:
2295:
2294:
2293:
2290:
2288:
2285:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2262:
2261:
2260:Partial order
2258:
2256:
2253:
2249:
2248:Join and meet
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2225:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2195:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2162:
2159:
2158:
2157:
2154:
2152:
2149:
2147:
2146:Antisymmetric
2144:
2143:
2141:
2137:
2131:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2066:
2064:
2060:
2054:
2053:Weak ordering
2051:
2049:
2046:
2044:
2041:
2039:
2038:Partial order
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2019:
2016:
2015:
2013:
2009:
2003:
2000:
1998:
1995:
1993:
1990:
1989:
1986:
1982:
1975:
1970:
1968:
1963:
1961:
1956:
1955:
1952:
1944:
1938:
1934:
1933:
1927:
1923:
1921:9780817641283
1917:
1913:
1912:
1906:
1902:
1900:9780521594653
1896:
1892:
1891:
1885:
1881:
1879:9781441971265
1875:
1871:
1867:
1866:
1860:
1859:
1848:
1843:
1836:
1832:
1827:
1820:
1816:
1811:
1806:
1802:
1798:
1797:
1789:
1782:
1778:
1774:
1772:81-85931-13-5
1768:
1764:
1760:
1756:
1752:
1745:
1738:
1733:
1726:
1720:
1713:
1707:
1700:
1695:
1688:
1684:
1678:
1671:
1667:
1663:
1657:
1650:
1646:
1641:
1637:
1627:
1624:
1623:
1617:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1586:
1567:
1563:
1559:
1556:
1528:
1524:
1520:
1517:
1494:
1474:
1466:
1445:
1441:
1437:
1434:
1406:
1402:
1398:
1395:
1372:
1347:
1343:
1339:
1336:
1308:
1304:
1300:
1297:
1274:
1266:
1250:
1233:
1229:
1225:
1221:
1217:
1199:
1196:
1193:
1183:
1164:
1161:
1158:
1148:
1147:open interval
1144:
1141:
1137:
1133:
1129:
1113:
1105:
1064:
1061:
1030:
1027:
1011:
1008:
1005:
1001:
1000:
994:
992:
988:
967:
963:
959:
956:
928:
924:
920:
917:
906:
885:
881:
877:
874:
846:
842:
838:
835:
824:
803:
799:
795:
792:
764:
760:
756:
753:
725:
721:
717:
714:
686:
682:
678:
675:
663:
661:
660:
653:
651:
648:
643:
629:
626:
623:
603:
600:
597:
577:
574:
571:
551:
528:
522:
519:
513:
507:
487:
467:
447:
439:
434:
432:
413:
407:
402:
398:
391:
385:
365:
360:
356:
352:
332:
312:
292:
272:
252:
232:
225:
204:
200:
196:
193:
165:
161:
157:
154:
143:
122:
118:
114:
111:
83:
79:
75:
72:
62:
52:
50:
46:
42:
38:
34:
30:
26:
22:
2661:Order theory
2571:
2427:& Orders
2405:Star product
2334:Well-founded
2287:Prefix order
2243:Distributive
2233:Complemented
2203:Foundational
2168:Completeness
2124:Zorn's lemma
2028:Cyclic order
2011:Key concepts
1981:Order theory
1931:
1910:
1889:
1864:
1842:
1826:
1800:
1794:
1788:
1754:
1744:
1732:
1719:
1712:Fuchs (1963)
1706:
1694:
1683:Bloch (2011)
1677:
1666:Bloch (2011)
1656:
1645:Bloch (2011)
1640:
1606:transitivity
1589:
1587:
1242:
1139:
1135:
1131:
1127:
1104:real numbers
904:
825:(po-groups)
664:
659:automorphism
656:
654:
644:
435:
141:
58:
28:
25:order theory
21:mathematical
18:
2611:Riesz space
2572:Isomorphism
2448:Normal cone
2370:Composition
2304:Semilattice
2213:Homogeneous
2198:Equivalence
2048:Total order
1614:order types
1598:reflexivity
1463:, then the
1267:. Also, if
1239:Order types
37:isomorphism
2650:Categories
2579:Order type
2513:Cofinality
2354:Well-order
2329:Transitive
2218:Idempotent
2151:Asymmetric
1942:0486483878
1856:References
650:bijections
438:surjective
55:Definition
2656:Morphisms
2630:Upper set
2567:Embedding
2503:Antichain
2324:Tolerance
2314:Symmetric
2309:Semiorder
2255:Reflexive
2173:Connected
1564:≤
1525:≤
1442:≤
1403:≤
1344:≤
1305:≤
1114:≤
1065:≥
1031:≤
964:≤
925:≤
882:≤
843:≤
800:≤
761:≤
722:≤
683:≤
601:≤
575:≤
399:≤
357:≤
201:≤
162:≤
119:≤
80:≤
23:field of
2425:Topology
2292:Preorder
2275:Eulerian
2238:Complete
2188:Directed
2178:Covering
2043:Preorder
2002:Category
1997:Glossary
1701:, p. 13.
1620:See also
1602:symmetry
1010:Negation
997:Examples
647:monotone
2530:Duality
2508:Cofinal
2496:Related
2475:Fréchet
2352:)
2228:Bounded
2223:Lattice
2196:)
2194:Partial
2062:Results
2033:Lattice
1819:1412484
1781:1632579
1080:(where
19:In the
2555:Subnet
2535:Filter
2485:Normed
2470:Banach
2436:&
2343:Better
2280:Strict
2270:Graded
2161:topics
1992:Topics
1939:
1918:
1897:
1876:
1817:
1779:
1769:
1664:. For
1604:, and
657:order
61:posets
2545:Ideal
2523:Graph
2319:Total
2297:Total
2183:Dense
1633:Notes
907:from
903:, an
245:from
222:is a
144:from
140:, an
27:, an
2136:list
1937:ISBN
1916:ISBN
1895:ISBN
1874:ISBN
1767:ISBN
1685:and
1668:and
1487:and
1365:and
1145:The
1106:and
1002:The
864:and
704:and
590:and
305:and
101:and
47:and
39:for
2550:Net
2350:Pre
1805:doi
1801:203
1759:doi
1546:to
1467:of
1424:to
1326:to
1243:If
1218:By
1130:â„ â
1046:to
946:to
782:to
325:in
265:to
183:to
2652::
1868:,
1833:;
1815:MR
1813:,
1799:,
1777:MR
1775:,
1765:,
1647:;
1616:.
1600:,
1596::
1585:.
1138:â€
993:.
662:.
642:.
433:.
345:,
51:.
2348:(
2345:)
2341:(
2192:(
2139:)
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1966:t
1959:v
1946:.
1925:.
1904:.
1883:.
1849:.
1837:.
1807::
1761::
1714:.
1689:.
1651:.
1573:)
1568:U
1560:,
1557:U
1554:(
1534:)
1529:S
1521:,
1518:S
1515:(
1495:g
1475:f
1451:)
1446:U
1438:,
1435:U
1432:(
1412:)
1407:T
1399:,
1396:T
1393:(
1373:g
1353:)
1348:T
1340:,
1337:T
1334:(
1314:)
1309:S
1301:,
1298:S
1295:(
1275:f
1251:f
1234:.
1203:]
1200:1
1197:,
1194:0
1191:[
1168:)
1165:1
1162:,
1159:0
1156:(
1142:.
1140:y
1136:x
1132:y
1128:x
1089:R
1068:)
1062:,
1058:R
1054:(
1034:)
1028:,
1024:R
1020:(
973:)
968:H
960:,
957:H
954:(
934:)
929:G
921:,
918:G
915:(
891:)
886:H
878:,
875:H
872:(
852:)
847:G
839:,
836:G
833:(
809:)
804:T
796:,
793:T
790:(
770:)
765:S
757:,
754:S
751:(
731:)
726:T
718:,
715:T
712:(
692:)
687:S
679:,
676:S
673:(
630:y
627:=
624:x
604:x
598:y
578:y
572:x
552:f
532:)
529:y
526:(
523:f
520:=
517:)
514:x
511:(
508:f
488:f
468:T
448:f
417:)
414:y
411:(
408:f
403:T
395:)
392:x
389:(
386:f
366:y
361:S
353:x
333:S
313:y
293:x
273:T
253:S
233:f
210:)
205:T
197:,
194:T
191:(
171:)
166:S
158:,
155:S
152:(
128:)
123:T
115:,
112:T
109:(
89:)
84:S
76:,
73:S
70:(
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