20:
2169:
2097:
1289:: map each antichain to its upper closure (see below); conversely, map each upper set to the set of its minimal elements. This correspondence does not hold for more general partial orders; for example the sets of
1830:
2281:. As a result, the upper closure of a set is equal to the intersection of all upper sets containing it, and similarly for lower sets. (Indeed, this is a general phenomenon of closure operators. For example, the
1639:
1209:
2672:
2325:
is usually identified with the set of all smaller ordinal numbers. Thus each ordinal number forms a lower set in the class of all ordinal numbers, which are totally ordered by set inclusion.
2244:
1382:
1336:
2206:
1241:
1700:
2028:
1729:
1058:
1756:
1512:
99:
1998:
1880:
1541:
1929:
1568:
1021:
1856:
647:
464:
187:
1448:
968:
847:
76:
2397:
389:
341:
271:
161:
2475:
2102:
2426:
761:
732:
706:
578:
549:
523:
677:
494:
233:
49:
1968:
1479:
1264:
1101:
925:
2446:
2365:
2271:
2033:
1900:
1667:
1413:
1167:
1143:
1123:
1078:
995:
945:
902:
867:
419:
1761:
1573:
2642:
3360:
1173:
3343:
2873:
2709:
2654:
2597:
2564:
3190:
2517:
2250:. The upper closure and lower closure of a set are, respectively, the smallest upper set and lower set containing it.
780:
are sometimes used as synonyms for lower set. This choice of terminology fails to reflect the notion of an ideal of a
3326:
3185:
2531:
2211:
3180:
1341:
1295:
2816:
2274:
2898:
2176:
3217:
3137:
1214:
3002:
2931:
2811:
2905:
2893:
2856:
2831:
2806:
2760:
2729:
2334:
3202:
2836:
2826:
2702:
2523:
1672:
1282:
2003:
1704:
1034:
3175:
2841:
2298:
2278:
1734:
1484:
797:
81:
1973:
1861:
1516:
3107:
2734:
2559:. Cambridge studies in advanced mathematics. Vol. 1. Cambridge University Press. p. 100.
1905:
1546:
1000:
1837:
626:
443:
166:
3355:
3338:
808:
1418:
950:
817:
58:
3267:
2883:
2370:
870:
584:
362:
324:
254:
134:
2589:
2451:
3393:
3245:
3080:
3071:
2940:
2821:
2775:
2739:
2695:
2402:
737:
711:
682:
554:
528:
499:
129:
2581:
656:
473:
212:
3333:
3292:
3282:
3272:
3017:
2980:
2970:
2950:
2935:
775:
8:
3260:
3171:
3117:
3076:
3066:
2955:
2888:
2851:
2338:
2306:
2302:
2282:
34:
1950:
1461:
1246:
1083:
907:
3372:
3299:
3152:
3061:
3051:
2992:
2910:
2846:
2431:
2350:
2310:
2256:
1885:
1652:
1398:
1152:
1128:
1108:
1063:
980:
930:
887:
852:
801:
404:
3212:
2633:
3309:
3287:
3147:
3132:
3112:
2915:
2660:
2650:
2593:
2582:
2560:
2537:
2527:
2613:
3122:
2975:
2628:
2513:
2341:) - a set-family that is downwards-closed with respect to the containment relation.
874:
3304:
3087:
2965:
2960:
2945:
2861:
2770:
2755:
2294:
2164:{\displaystyle A^{\downarrow X}=A^{\downarrow }=\bigcup _{a\in A}\downarrow \!a.}
1146:
781:
3222:
3207:
3197:
3056:
3034:
3012:
2322:
1285:, antichains and upper sets are in one-to-one correspondence via the following
392:
3387:
3321:
3277:
3255:
3127:
2997:
2985:
2790:
2664:
2253:
The upper and lower closures, when viewed as functions from the power set of
24:
3142:
3024:
3007:
2925:
2765:
2718:
1271:
2092:{\displaystyle A^{\uparrow X}=A^{\uparrow }=\bigcup _{a\in A}\uparrow \!a}
3348:
3041:
2920:
2785:
2344:
2290:
2286:
1290:
1275:
105:
3316:
3250:
3091:
1286:
3240:
3046:
1825:{\displaystyle x^{\downarrow X}=\;\downarrow \!x=\{l\in X:l\leq x\}.}
3162:
3029:
2780:
2541:
2511:
784:
because a lower set of a lattice is not necessarily a sublattice.
28:
2492:
2490:
1882:
are, respectively, the smallest upper and lower sets containing
1634:{\displaystyle x^{\uparrow X}=\;\uparrow \!x=\{u\in X:x\leq u\}}
2687:
1204:{\displaystyle \downarrow Y=\downarrow \operatorname {Max} (Y)}
19:
2487:
2313:
is the intersection of all ideals containing it; and so on.)
2305:
is the intersection of all subgroups containing it; the
2246:
where upper sets and lower sets of this form are called
793:
Every partially ordered set is an upper set of itself.
16:
Subset of a preorder that contains all larger elements
2454:
2434:
2405:
2373:
2353:
2259:
2214:
2179:
2105:
2036:
2006:
1976:
1953:
1908:
1888:
1864:
1840:
1764:
1737:
1707:
1675:
1655:
1576:
1549:
1519:
1487:
1464:
1421:
1401:
1344:
1298:
1249:
1217:
1176:
1155:
1131:
1111:
1086:
1066:
1037:
1003:
983:
953:
933:
910:
890:
855:
820:
740:
714:
685:
659:
649:
that is "closed under going down", in the sense that
629:
557:
531:
502:
476:
446:
407:
365:
327:
257:
215:
169:
137:
84:
61:
37:
2584:
Inverse semigroups: the theory of partial symmetries
971:
2649:. New Jersey: World Scientific Publishing Company.
1243:denotes the set containing the maximal elements of
466:that is "closed under going up", in the sense that
2469:
2440:
2420:
2391:
2359:
2265:
2238:
2200:
2163:
2091:
2022:
1992:
1962:
1923:
1894:
1874:
1850:
1824:
1750:
1723:
1694:
1661:
1633:
1562:
1535:
1506:
1473:
1442:
1407:
1390:
1376:
1330:
1258:
1235:
1203:
1161:
1145:is equal to the smallest lower set containing all
1137:
1117:
1095:
1072:
1052:
1015:
989:
962:
939:
919:
896:
861:
841:
804:of any family of upper sets is again an upper set.
755:
726:
700:
671:
641:
572:
543:
517:
488:
458:
413:
383:
335:
265:
227:
181:
155:
93:
70:
43:
2154:
2085:
1868:
1844:
1785:
1741:
1597:
1553:
78:colored green. The white sets form the lower set
3385:
1902:as an element. More generally, given a subset
811:of any upper set is a lower set, and vice versa.
2293:of a set of vectors is the intersection of all
2614:"Domain representations of topological spaces"
2239:{\displaystyle \downarrow x=\downarrow \{x\},}
2703:
2641:
2496:
2230:
2224:
2195:
2189:
1816:
1792:
1628:
1604:
1377:{\displaystyle \{x\in \mathbb {R} :x>1\}}
1371:
1345:
1331:{\displaystyle \{x\in \mathbb {R} :x>0\}}
1325:
1299:
1047:
1041:
3361:Positive cone of a partially ordered group
2710:
2696:
2611:
1781:
1593:
977:Dually, the smallest lower set containing
2632:
2201:{\displaystyle \uparrow x=\uparrow \{x\}}
1355:
1309:
332:
328:
305:) is defined similarly as being a subset
262:
258:
3344:Positive cone of an ordered vector space
18:
2554:
1384:are both mapped to the empty antichain.
1236:{\displaystyle \operatorname {Max} (Y)}
3386:
2579:
243:. In other words, this means that any
2691:
2507:
2505:
2285:of a set is the intersection of all
2647:Convergence Foundations Of Topology
313:with the property that any element
13:
2871:Properties & Types (
2519:Introduction to Lattices and Order
2316:
1281:For partial orders satisfying the
1125:of a finite partially ordered set
927:the smallest upper set containing
347:is necessarily also an element of
277:is necessarily also an element of
14:
3405:
3327:Positive cone of an ordered field
2502:
1695:{\displaystyle x^{\downarrow X},}
997:is denoted using a down arrow as
3181:Ordered topological vector space
2717:
2399:that contains for every element
2023:{\displaystyle A^{\downarrow X}}
1724:{\displaystyle x^{\downarrow },}
1053:{\displaystyle \downarrow \{x\}}
947:is denoted using an up arrow as
189:with the following property: if
1751:{\displaystyle \downarrow \!x,}
1507:{\displaystyle x^{\uparrow X},}
1391:Upper closure and lower closure
972:upper closure and lower closure
94:{\displaystyle \downarrow 105.}
2573:
2548:
2386:
2374:
2299:subgroup generated by a subset
2277:since they satisfy all of the
2215:
2180:
2151:
2127:
2111:
2082:
2058:
2042:
2012:
1993:{\displaystyle A^{\uparrow X}}
1982:
1875:{\displaystyle \downarrow \!x}
1865:
1841:
1782:
1770:
1738:
1713:
1681:
1594:
1582:
1550:
1536:{\displaystyle x^{\uparrow },}
1525:
1493:
1434:
1422:
1230:
1224:
1198:
1192:
1177:
1038:
1004:
954:
833:
821:
814:Given a partially ordered set
378:
366:
150:
138:
85:
62:
1:
3138:Series-parallel partial order
2634:10.1016/s0304-3975(99)00045-6
2480:
1924:{\displaystyle A\subseteq X,}
1563:{\displaystyle \uparrow \!x,}
1016:{\displaystyle \downarrow Y.}
787:
354:
2817:Cantor's isomorphism theorem
2674:The low separation axioms (T
2621:Theoretical Computer Science
2588:. World Scientific. p.
1851:{\displaystyle \uparrow \!x}
849:the family of upper sets of
642:{\displaystyle L\subseteq X}
459:{\displaystyle U\subseteq X}
182:{\displaystyle S\subseteq X}
7:
2857:Szpilrajn extension theorem
2832:Hausdorff maximal principle
2807:Boolean prime ideal theorem
2645:; Mynard, Frédéric (2016).
2367:of a partially ordered set
2335:Abstract simplicial complex
2328:
2309:generated by a subset of a
2273:to itself, are examples of
1415:of a partially ordered set
904:of a partially ordered set
10:
3410:
3203:Topological vector lattice
2524:Cambridge University Press
1443:{\displaystyle (X,\leq ),}
1283:descending chain condition
963:{\displaystyle \uparrow Y}
884:Given an arbitrary subset
842:{\displaystyle (X,\leq ),}
71:{\displaystyle \uparrow 2}
51:, ordered by the relation
3233:
3161:
3100:
2870:
2799:
2748:
2725:
2557:Enumerative combinatorics
2497:Dolecki & Mynard 2016
2392:{\displaystyle (X,\leq )}
2279:Kuratowski closure axioms
384:{\displaystyle (X,\leq )}
336:{\displaystyle \,\leq \,}
266:{\displaystyle \,\geq \,}
156:{\displaystyle (X,\leq )}
2812:Cantor–Bernstein theorem
2470:{\displaystyle x\leq y.}
3356:Partially ordered group
3176:Specialization preorder
2671:Hoffman, K. H. (2001),
2421:{\displaystyle x\in X,}
1274:lower set is called an
756:{\displaystyle x\in L.}
727:{\displaystyle x\leq l}
701:{\displaystyle x\in X,}
573:{\displaystyle x\in U.}
544:{\displaystyle u\leq x}
518:{\displaystyle x\in X,}
2842:Kruskal's tree theorem
2837:Knaster–Tarski theorem
2827:Dushnik–Miller theorem
2555:Stanley, R.P. (2002).
2471:
2442:
2422:
2393:
2361:
2267:
2240:
2202:
2165:
2093:
2024:
1994:
1964:
1925:
1896:
1876:
1852:
1826:
1752:
1725:
1696:
1663:
1635:
1564:
1537:
1508:
1475:
1444:
1409:
1378:
1332:
1260:
1237:
1205:
1163:
1139:
1119:
1097:
1074:
1054:
1027:A lower set is called
1017:
991:
964:
941:
921:
898:
863:
843:
757:
728:
702:
673:
672:{\displaystyle l\in L}
643:
574:
545:
519:
490:
489:{\displaystyle u\in U}
460:
415:
385:
337:
267:
229:
228:{\displaystyle s<x}
183:
157:
101:
95:
72:
45:
2580:Lawson, M.V. (1998).
2472:
2443:
2423:
2394:
2362:
2268:
2241:
2203:
2166:
2094:
2025:
1995:
1965:
1926:
1897:
1877:
1853:
1827:
1753:
1726:
1697:
1664:
1636:
1565:
1538:
1509:
1476:
1445:
1410:
1379:
1333:
1261:
1238:
1206:
1164:
1140:
1120:
1098:
1075:
1055:
1031:if it is of the form
1018:
992:
965:
942:
922:
899:
864:
844:
758:
729:
703:
674:
644:
623:), which is a subset
575:
546:
520:
491:
461:
416:
386:
338:
268:
230:
184:
158:
130:partially ordered set
96:
73:
55:, with the upper set
46:
22:
3334:Ordered vector space
2514:Hilary Ann Priestley
2452:
2432:
2403:
2371:
2351:
2257:
2212:
2177:
2103:
2034:
2004:
1974:
1951:
1906:
1886:
1862:
1838:
1762:
1735:
1705:
1673:
1653:
1574:
1547:
1517:
1485:
1462:
1419:
1399:
1342:
1296:
1247:
1215:
1174:
1153:
1129:
1109:
1084:
1064:
1035:
1001:
981:
951:
931:
908:
888:
853:
818:
738:
712:
683:
657:
627:
555:
529:
500:
474:
444:
405:
363:
325:
255:
213:
167:
135:
82:
59:
35:
3172:Alexandrov topology
3118:Lexicographic order
3077:Well-quasi-ordering
2612:Blanck, J. (2000).
2526:. pp. 20, 44.
2339:Independence system
2297:containing it; the
2289:containing it; the
2283:topological closure
596:downward closed set
343:to some element of
287:downward closed set
273:to some element of
44:{\displaystyle 210}
3153:Transitive closure
3113:Converse/Transpose
2822:Dilworth's theorem
2467:
2438:
2418:
2389:
2357:
2263:
2236:
2198:
2161:
2150:
2089:
2081:
2030:respectively, as
2020:
1990:
1963:{\displaystyle A,}
1960:
1921:
1892:
1872:
1848:
1822:
1748:
1721:
1692:
1659:
1631:
1560:
1533:
1504:
1474:{\displaystyle x,}
1471:
1440:
1405:
1374:
1328:
1259:{\displaystyle Y.}
1256:
1233:
1201:
1159:
1135:
1115:
1096:{\displaystyle X.}
1093:
1070:
1050:
1013:
987:
960:
937:
920:{\displaystyle X,}
917:
894:
859:
839:
753:
724:
698:
669:
639:
570:
541:
515:
486:
456:
411:
381:
333:
263:
225:
179:
153:
102:
91:
68:
41:
3381:
3380:
3339:Partially ordered
3148:Symmetric closure
3133:Reflexive closure
2876:
2656:978-981-4571-52-4
2599:978-981-02-3316-7
2566:978-0-521-66351-9
2499:, pp. 27–29.
2441:{\displaystyle y}
2360:{\displaystyle U}
2275:closure operators
2266:{\displaystyle X}
2135:
2066:
1895:{\displaystyle x}
1662:{\displaystyle x}
1408:{\displaystyle x}
1395:Given an element
1162:{\displaystyle Y}
1138:{\displaystyle X}
1118:{\displaystyle Y}
1080:is an element of
1073:{\displaystyle x}
990:{\displaystyle Y}
940:{\displaystyle Y}
897:{\displaystyle Y}
879:upper set lattice
869:ordered with the
862:{\displaystyle X}
424:upward closed set
414:{\displaystyle X}
114:upward closed set
3401:
3123:Linear extension
2872:
2852:Mirsky's theorem
2712:
2705:
2698:
2689:
2688:
2668:
2638:
2636:
2627:(1–2): 229–255.
2618:
2604:
2603:
2587:
2577:
2571:
2570:
2552:
2546:
2545:
2522:(2nd ed.).
2512:Brian A. Davey;
2509:
2500:
2494:
2476:
2474:
2473:
2468:
2447:
2445:
2444:
2439:
2427:
2425:
2424:
2419:
2398:
2396:
2395:
2390:
2366:
2364:
2363:
2358:
2272:
2270:
2269:
2264:
2245:
2243:
2242:
2237:
2207:
2205:
2204:
2199:
2170:
2168:
2167:
2162:
2149:
2131:
2130:
2118:
2117:
2098:
2096:
2095:
2090:
2080:
2062:
2061:
2049:
2048:
2029:
2027:
2026:
2021:
2019:
2018:
1999:
1997:
1996:
1991:
1989:
1988:
1969:
1967:
1966:
1961:
1945:downward closure
1930:
1928:
1927:
1922:
1901:
1899:
1898:
1893:
1881:
1879:
1878:
1873:
1857:
1855:
1854:
1849:
1831:
1829:
1828:
1823:
1777:
1776:
1757:
1755:
1754:
1749:
1730:
1728:
1727:
1722:
1717:
1716:
1701:
1699:
1698:
1693:
1688:
1687:
1668:
1666:
1665:
1660:
1647:downward closure
1640:
1638:
1637:
1632:
1589:
1588:
1569:
1567:
1566:
1561:
1542:
1540:
1539:
1534:
1529:
1528:
1513:
1511:
1510:
1505:
1500:
1499:
1480:
1478:
1477:
1472:
1449:
1447:
1446:
1441:
1414:
1412:
1411:
1406:
1383:
1381:
1380:
1375:
1358:
1337:
1335:
1334:
1329:
1312:
1265:
1263:
1262:
1257:
1242:
1240:
1239:
1234:
1210:
1208:
1207:
1202:
1168:
1166:
1165:
1160:
1147:maximal elements
1144:
1142:
1141:
1136:
1124:
1122:
1121:
1116:
1105:Every lower set
1102:
1100:
1099:
1094:
1079:
1077:
1076:
1071:
1059:
1057:
1056:
1051:
1022:
1020:
1019:
1014:
996:
994:
993:
988:
969:
967:
966:
961:
946:
944:
943:
938:
926:
924:
923:
918:
903:
901:
900:
895:
875:complete lattice
868:
866:
865:
860:
848:
846:
845:
840:
762:
760:
759:
754:
733:
731:
730:
725:
707:
705:
704:
699:
678:
676:
675:
670:
648:
646:
645:
640:
579:
577:
576:
571:
550:
548:
547:
542:
524:
522:
521:
516:
495:
493:
492:
487:
465:
463:
462:
457:
421:(also called an
420:
418:
417:
412:
390:
388:
387:
382:
342:
340:
339:
334:
272:
270:
269:
264:
234:
232:
231:
226:
188:
186:
185:
180:
162:
160:
159:
154:
112:(also called an
100:
98:
97:
92:
77:
75:
74:
69:
50:
48:
47:
42:
3409:
3408:
3404:
3403:
3402:
3400:
3399:
3398:
3384:
3383:
3382:
3377:
3373:Young's lattice
3229:
3157:
3096:
2946:Heyting algebra
2894:Boolean algebra
2866:
2847:Laver's theorem
2795:
2761:Boolean algebra
2756:Binary relation
2744:
2721:
2716:
2681:
2677:
2657:
2643:Dolecki, Szymon
2616:
2608:
2607:
2600:
2578:
2574:
2567:
2553:
2549:
2534:
2510:
2503:
2495:
2488:
2483:
2453:
2450:
2449:
2433:
2430:
2429:
2404:
2401:
2400:
2372:
2369:
2368:
2352:
2349:
2348:
2331:
2319:
2317:Ordinal numbers
2258:
2255:
2254:
2213:
2210:
2209:
2178:
2175:
2174:
2139:
2126:
2122:
2110:
2106:
2104:
2101:
2100:
2070:
2057:
2053:
2041:
2037:
2035:
2032:
2031:
2011:
2007:
2005:
2002:
2001:
1981:
1977:
1975:
1972:
1971:
1952:
1949:
1948:
1907:
1904:
1903:
1887:
1884:
1883:
1863:
1860:
1859:
1839:
1836:
1835:
1769:
1765:
1763:
1760:
1759:
1736:
1733:
1732:
1712:
1708:
1706:
1703:
1702:
1680:
1676:
1674:
1671:
1670:
1654:
1651:
1650:
1581:
1577:
1575:
1572:
1571:
1548:
1545:
1544:
1524:
1520:
1518:
1515:
1514:
1492:
1488:
1486:
1483:
1482:
1463:
1460:
1459:
1420:
1417:
1416:
1400:
1397:
1396:
1393:
1354:
1343:
1340:
1339:
1308:
1297:
1294:
1293:
1248:
1245:
1244:
1216:
1213:
1212:
1175:
1172:
1171:
1154:
1151:
1150:
1130:
1127:
1126:
1110:
1107:
1106:
1085:
1082:
1081:
1065:
1062:
1061:
1036:
1033:
1032:
1002:
999:
998:
982:
979:
978:
952:
949:
948:
932:
929:
928:
909:
906:
905:
889:
886:
885:
854:
851:
850:
819:
816:
815:
790:
739:
736:
735:
713:
710:
709:
684:
681:
680:
658:
655:
654:
628:
625:
624:
614:initial segment
593:(also called a
556:
553:
552:
530:
527:
526:
501:
498:
497:
475:
472:
471:
445:
442:
441:
406:
403:
402:
364:
361:
360:
357:
326:
323:
322:
299:initial segment
285:(also called a
256:
253:
252:
214:
211:
210:
205:is larger than
168:
165:
164:
136:
133:
132:
83:
80:
79:
60:
57:
56:
36:
33:
32:
17:
12:
11:
5:
3407:
3397:
3396:
3379:
3378:
3376:
3375:
3370:
3365:
3364:
3363:
3353:
3352:
3351:
3346:
3341:
3331:
3330:
3329:
3319:
3314:
3313:
3312:
3307:
3300:Order morphism
3297:
3296:
3295:
3285:
3280:
3275:
3270:
3265:
3264:
3263:
3253:
3248:
3243:
3237:
3235:
3231:
3230:
3228:
3227:
3226:
3225:
3220:
3218:Locally convex
3215:
3210:
3200:
3198:Order topology
3195:
3194:
3193:
3191:Order topology
3188:
3178:
3168:
3166:
3159:
3158:
3156:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3104:
3102:
3098:
3097:
3095:
3094:
3084:
3074:
3069:
3064:
3059:
3054:
3049:
3044:
3039:
3038:
3037:
3027:
3022:
3021:
3020:
3015:
3010:
3005:
3003:Chain-complete
2995:
2990:
2989:
2988:
2983:
2978:
2973:
2968:
2958:
2953:
2948:
2943:
2938:
2928:
2923:
2918:
2913:
2908:
2903:
2902:
2901:
2891:
2886:
2880:
2878:
2868:
2867:
2865:
2864:
2859:
2854:
2849:
2844:
2839:
2834:
2829:
2824:
2819:
2814:
2809:
2803:
2801:
2797:
2796:
2794:
2793:
2788:
2783:
2778:
2773:
2768:
2763:
2758:
2752:
2750:
2746:
2745:
2743:
2742:
2737:
2732:
2726:
2723:
2722:
2715:
2714:
2707:
2700:
2692:
2686:
2685:
2679:
2675:
2669:
2655:
2639:
2606:
2605:
2598:
2572:
2565:
2547:
2532:
2501:
2485:
2484:
2482:
2479:
2478:
2477:
2466:
2463:
2460:
2457:
2437:
2417:
2414:
2411:
2408:
2388:
2385:
2382:
2379:
2376:
2356:
2342:
2337:(also called:
2330:
2327:
2323:ordinal number
2318:
2315:
2262:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2197:
2194:
2191:
2188:
2185:
2182:
2160:
2157:
2153:
2148:
2145:
2142:
2138:
2134:
2129:
2125:
2121:
2116:
2113:
2109:
2088:
2084:
2079:
2076:
2073:
2069:
2065:
2060:
2056:
2052:
2047:
2044:
2040:
2017:
2014:
2010:
1987:
1984:
1980:
1959:
1956:
1937:upward closure
1920:
1917:
1914:
1911:
1891:
1871:
1867:
1847:
1843:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1784:
1780:
1775:
1772:
1768:
1758:is defined by
1747:
1744:
1740:
1720:
1715:
1711:
1691:
1686:
1683:
1679:
1658:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1596:
1592:
1587:
1584:
1580:
1570:is defined by
1559:
1556:
1552:
1532:
1527:
1523:
1503:
1498:
1495:
1491:
1470:
1467:
1456:upward closure
1439:
1436:
1433:
1430:
1427:
1424:
1404:
1392:
1389:
1386:
1385:
1373:
1370:
1367:
1364:
1361:
1357:
1353:
1350:
1347:
1327:
1324:
1321:
1318:
1315:
1311:
1307:
1304:
1301:
1279:
1268:
1267:
1266:
1255:
1252:
1232:
1229:
1226:
1223:
1220:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1158:
1134:
1114:
1103:
1092:
1089:
1069:
1049:
1046:
1043:
1040:
1025:
1024:
1023:
1012:
1009:
1006:
986:
959:
956:
936:
916:
913:
893:
882:
873:relation is a
858:
838:
835:
832:
829:
826:
823:
812:
805:
794:
789:
786:
778:
770:
764:
763:
752:
749:
746:
743:
723:
720:
717:
697:
694:
691:
688:
668:
665:
662:
638:
635:
632:
621:
615:
609:
608:decreasing set
603:
597:
591:
581:
580:
569:
566:
563:
560:
540:
537:
534:
514:
511:
508:
505:
485:
482:
479:
455:
452:
449:
440:) is a subset
437:
431:
425:
410:
399:
393:preordered set
380:
377:
374:
371:
368:
356:
353:
331:
295:decreasing set
261:
224:
221:
218:
178:
175:
172:
152:
149:
146:
143:
140:
90:
87:
67:
64:
40:
15:
9:
6:
4:
3:
2:
3406:
3395:
3392:
3391:
3389:
3374:
3371:
3369:
3366:
3362:
3359:
3358:
3357:
3354:
3350:
3347:
3345:
3342:
3340:
3337:
3336:
3335:
3332:
3328:
3325:
3324:
3323:
3322:Ordered field
3320:
3318:
3315:
3311:
3308:
3306:
3303:
3302:
3301:
3298:
3294:
3291:
3290:
3289:
3286:
3284:
3281:
3279:
3278:Hasse diagram
3276:
3274:
3271:
3269:
3266:
3262:
3259:
3258:
3257:
3256:Comparability
3254:
3252:
3249:
3247:
3244:
3242:
3239:
3238:
3236:
3232:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3205:
3204:
3201:
3199:
3196:
3192:
3189:
3187:
3184:
3183:
3182:
3179:
3177:
3173:
3170:
3169:
3167:
3164:
3160:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3128:Product order
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3105:
3103:
3101:Constructions
3099:
3093:
3089:
3085:
3082:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3036:
3033:
3032:
3031:
3028:
3026:
3023:
3019:
3016:
3014:
3011:
3009:
3006:
3004:
3001:
3000:
2999:
2998:Partial order
2996:
2994:
2991:
2987:
2986:Join and meet
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2963:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2933:
2929:
2927:
2924:
2922:
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2900:
2897:
2896:
2895:
2892:
2890:
2887:
2885:
2884:Antisymmetric
2882:
2881:
2879:
2875:
2869:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2804:
2802:
2798:
2792:
2791:Weak ordering
2789:
2787:
2784:
2782:
2779:
2777:
2776:Partial order
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2753:
2751:
2747:
2741:
2738:
2736:
2733:
2731:
2728:
2727:
2724:
2720:
2713:
2708:
2706:
2701:
2699:
2694:
2693:
2690:
2684:
2683:
2670:
2666:
2662:
2658:
2652:
2648:
2644:
2640:
2635:
2630:
2626:
2622:
2615:
2610:
2609:
2601:
2595:
2591:
2586:
2585:
2576:
2568:
2562:
2558:
2551:
2543:
2539:
2535:
2533:0-521-78451-4
2529:
2525:
2521:
2520:
2515:
2508:
2506:
2498:
2493:
2491:
2486:
2464:
2461:
2458:
2455:
2435:
2428:some element
2415:
2412:
2409:
2406:
2383:
2380:
2377:
2354:
2346:
2343:
2340:
2336:
2333:
2332:
2326:
2324:
2314:
2312:
2308:
2304:
2300:
2296:
2292:
2288:
2284:
2280:
2276:
2260:
2251:
2249:
2233:
2227:
2221:
2218:
2192:
2186:
2183:
2173:In this way,
2171:
2158:
2155:
2146:
2143:
2140:
2136:
2132:
2123:
2119:
2114:
2107:
2086:
2077:
2074:
2071:
2067:
2063:
2054:
2050:
2045:
2038:
2015:
2008:
1985:
1978:
1957:
1954:
1946:
1942:
1938:
1934:
1918:
1915:
1912:
1909:
1889:
1869:
1845:
1832:
1819:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1789:
1786:
1778:
1773:
1766:
1745:
1742:
1718:
1709:
1689:
1684:
1677:
1669:, denoted by
1656:
1648:
1644:
1643:lower closure
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1601:
1598:
1590:
1585:
1578:
1557:
1554:
1530:
1521:
1501:
1496:
1489:
1468:
1465:
1457:
1453:
1452:upper closure
1437:
1431:
1428:
1425:
1402:
1388:
1368:
1365:
1362:
1359:
1351:
1348:
1322:
1319:
1316:
1313:
1305:
1302:
1292:
1288:
1284:
1280:
1277:
1273:
1269:
1253:
1250:
1227:
1221:
1218:
1195:
1189:
1186:
1183:
1180:
1170:
1169:
1156:
1148:
1132:
1112:
1104:
1090:
1087:
1067:
1044:
1030:
1026:
1010:
1007:
984:
976:
975:
973:
957:
934:
914:
911:
891:
883:
880:
876:
872:
856:
836:
830:
827:
824:
813:
810:
806:
803:
799:
795:
792:
791:
785:
783:
779:
777:
774:
771:
768:
750:
747:
744:
741:
721:
718:
715:
695:
692:
689:
686:
666:
663:
660:
652:
651:
650:
636:
633:
630:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
567:
564:
561:
558:
538:
535:
532:
512:
509:
506:
503:
483:
480:
477:
469:
468:
467:
453:
450:
447:
439:
435:
432:
429:
426:
423:
408:
400:
397:
394:
375:
372:
369:
352:
350:
346:
329:
320:
316:
312:
308:
304:
300:
296:
292:
288:
284:
280:
276:
259:
250:
246:
242:
238:
222:
219:
216:
209:(that is, if
208:
204:
200:
196:
192:
176:
173:
170:
147:
144:
141:
131:
127:
123:
119:
115:
111:
107:
88:
65:
54:
53:is divisor of
38:
30:
26:
25:Hasse diagram
21:
3394:Order theory
3367:
3165:& Orders
3143:Star product
3072:Well-founded
3025:Prefix order
2981:Distributive
2971:Complemented
2941:Foundational
2906:Completeness
2862:Zorn's lemma
2766:Cyclic order
2749:Key concepts
2719:Order theory
2673:
2646:
2624:
2620:
2583:
2575:
2556:
2550:
2518:
2320:
2252:
2247:
2172:
1944:
1940:
1936:
1932:
1833:
1646:
1642:
1455:
1451:
1394:
1387:
1291:real numbers
1028:
878:
798:intersection
773:
767:
765:
618:
612:
606:
600:
594:
588:
587:notion is a
582:
434:
428:
422:
396:
358:
348:
344:
318:
314:
310:
306:
302:
298:
294:
290:
286:
282:
281:. The term
278:
274:
248:
244:
240:
236:
206:
202:
198:
194:
190:
163:is a subset
125:
121:
117:
113:
109:
103:
52:
3349:Riesz space
3310:Isomorphism
3186:Normal cone
3108:Composition
3042:Semilattice
2951:Homogeneous
2936:Equivalence
2786:Total order
2347:– a subset
2345:Cofinal set
2287:closed sets
1970:denoted by
1931:define the
1481:denoted by
1276:order ideal
769:order ideal
247:element of
106:mathematics
3317:Order type
3251:Cofinality
3092:Well-order
3067:Transitive
2956:Idempotent
2889:Asymmetric
2542:2001043910
2481:References
2448:such that
2222:=↓
2187:=↑
1641:while the
1287:bijections
1184:=↓
809:complement
788:Properties
766:The terms
620:semi-ideal
355:Definition
303:semi-ideal
3368:Upper set
3305:Embedding
3241:Antichain
3062:Tolerance
3052:Symmetric
3047:Semiorder
2993:Reflexive
2911:Connected
2665:945169917
2459:≤
2410:∈
2384:≤
2295:subspaces
2248:principal
2216:↓
2181:↑
2152:↓
2144:∈
2137:⋃
2128:↓
2112:↓
2083:↑
2075:∈
2068:⋃
2059:↑
2043:↑
2013:↓
1983:↑
1913:⊆
1866:↓
1842:↑
1834:The sets
1811:≤
1799:∈
1783:↓
1771:↓
1739:↓
1714:↓
1682:↓
1623:≤
1611:∈
1595:↑
1583:↑
1551:↑
1526:↑
1494:↑
1432:≤
1352:∈
1306:∈
1222:
1190:
1178:↓
1039:↓
1029:principal
1005:↓
955:↑
871:inclusion
831:≤
745:∈
719:≤
690:∈
664:∈
634:⊆
590:lower set
562:∈
536:≤
507:∈
481:∈
451:⊆
398:upper set
376:≤
330:≤
283:lower set
260:≥
174:⊆
148:≤
110:upper set
86:↓
63:↑
3388:Category
3163:Topology
3030:Preorder
3013:Eulerian
2976:Complete
2926:Directed
2916:Covering
2781:Preorder
2740:Category
2735:Glossary
2678:) and (T
2516:(2002).
2329:See also
1939:and the
1272:directed
800:and the
679:and all
653:for all
602:down set
496:and all
470:for all
433:, or an
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