31:
2180:
2108:
1300:: 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
1841:
2292:. 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
1650:
1220:
2683:
2336:
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.
2255:
1393:
1347:
2217:
1252:
1711:
2039:
1740:
1069:
1767:
1523:
110:
2009:
1891:
1552:
1940:
1579:
1032:
1867:
658:
475:
198:
1459:
979:
858:
87:
2408:
400:
352:
282:
172:
2486:
2113:
2437:
772:
743:
717:
589:
560:
534:
688:
505:
244:
60:
1979:
1490:
1275:
1112:
936:
2457:
2376:
2282:
2044:
1911:
1678:
1424:
1178:
1154:
1134:
1089:
1006:
956:
913:
878:
430:
1772:
1584:
2653:
3371:
1184:
3354:
2884:
2720:
2665:
2608:
2575:
3201:
2528:
2261:. The upper closure and lower closure of a set are, respectively, the smallest upper set and lower set containing it.
791:
are sometimes used as synonyms for lower set. This choice of terminology fails to reflect the notion of an ideal of a
3337:
3196:
2542:
2222:
3191:
1352:
1306:
2827:
2285:
2909:
2187:
3228:
3148:
1225:
3013:
2942:
2822:
2916:
2904:
2867:
2842:
2817:
2771:
2740:
2345:
3213:
2847:
2837:
2713:
2534:
1683:
1293:
2014:
1715:
1045:
3186:
2852:
2309:
2289:
1745:
1495:
808:
92:
17:
1984:
1872:
1527:
3118:
2745:
2570:. Cambridge studies in advanced mathematics. Vol. 1. Cambridge University Press. p. 100.
1916:
1557:
1011:
1848:
637:
454:
177:
3366:
3349:
819:
1429:
961:
828:
69:
3278:
2894:
2381:
881:
595:
373:
335:
265:
145:
2600:
2462:
3404:
3256:
3091:
3082:
2951:
2832:
2786:
2750:
2706:
2413:
748:
722:
693:
565:
539:
510:
140:
2592:
667:
484:
223:
3344:
3303:
3293:
3283:
3028:
2991:
2981:
2961:
2946:
786:
8:
3271:
3182:
3128:
3087:
3077:
2966:
2899:
2862:
2349:
2317:
2313:
2293:
45:
1961:
1472:
1257:
1094:
918:
3383:
3310:
3163:
3072:
3062:
3003:
2921:
2857:
2442:
2361:
2321:
2267:
1896:
1663:
1409:
1163:
1139:
1119:
1074:
991:
941:
898:
863:
812:
415:
3223:
2644:
3320:
3298:
3158:
3143:
3123:
2926:
2671:
2661:
2604:
2593:
2571:
2548:
2538:
2624:
3133:
2986:
2639:
2524:
2352:) - a set-family that is downwards-closed with respect to the containment relation.
885:
3315:
3098:
2976:
2971:
2956:
2872:
2781:
2766:
2305:
2175:{\displaystyle A^{\downarrow X}=A^{\downarrow }=\bigcup _{a\in A}\downarrow \!a.}
1157:
792:
3233:
3218:
3208:
3067:
3045:
3023:
2333:
1296:, antichains and upper sets are in one-to-one correspondence via the following
403:
3398:
3332:
3288:
3266:
3138:
3008:
2996:
2801:
2675:
2264:
The upper and lower closures, when viewed as functions from the power set of
35:
3153:
3035:
3018:
2936:
2776:
2729:
1282:
2103:{\displaystyle A^{\uparrow X}=A^{\uparrow }=\bigcup _{a\in A}\uparrow \!a}
3359:
3052:
2931:
2796:
2355:
2301:
2297:
1301:
1286:
116:
3327:
3261:
3102:
1297:
3251:
3057:
1836:{\displaystyle x^{\downarrow X}=\;\downarrow \!x=\{l\in X:l\leq x\}.}
3173:
3040:
2791:
2552:
2522:
795:
because a lower set of a lattice is not necessarily a sublattice.
39:
2503:
2501:
1893:
are, respectively, the smallest upper and lower sets containing
1645:{\displaystyle x^{\uparrow X}=\;\uparrow \!x=\{u\in X:x\leq u\}}
2698:
1215:{\displaystyle \downarrow Y=\downarrow \operatorname {Max} (Y)}
30:
2498:
2324:
is the intersection of all ideals containing it; and so on.)
2316:
is the intersection of all subgroups containing it; the
2257:
where upper sets and lower sets of this form are called
804:
Every partially ordered set is an upper set of itself.
27:
Subset of a preorder that contains all larger elements
2465:
2445:
2416:
2384:
2364:
2270:
2225:
2190:
2116:
2047:
2017:
1987:
1964:
1919:
1899:
1875:
1851:
1775:
1748:
1718:
1686:
1666:
1587:
1560:
1530:
1498:
1475:
1432:
1412:
1355:
1309:
1260:
1228:
1187:
1166:
1142:
1122:
1097:
1077:
1048:
1014:
994:
964:
944:
921:
901:
866:
831:
751:
725:
696:
670:
660:
that is "closed under going down", in the sense that
640:
568:
542:
513:
487:
457:
418:
376:
338:
268:
226:
180:
148:
95:
72:
48:
2595:
Inverse semigroups: the theory of partial symmetries
982:
2660:. New Jersey: World Scientific Publishing Company.
1254:denotes the set containing the maximal elements of
477:that is "closed under going up", in the sense that
2480:
2451:
2431:
2402:
2370:
2276:
2249:
2211:
2174:
2102:
2033:
2003:
1973:
1934:
1905:
1885:
1861:
1835:
1761:
1734:
1705:
1672:
1644:
1573:
1546:
1517:
1484:
1453:
1418:
1401:
1387:
1341:
1269:
1246:
1214:
1172:
1156:is equal to the smallest lower set containing all
1148:
1128:
1106:
1083:
1063:
1026:
1000:
973:
950:
930:
907:
872:
852:
815:of any family of upper sets is again an upper set.
766:
737:
711:
682:
652:
583:
554:
528:
499:
469:
424:
394:
346:
276:
238:
192:
166:
104:
81:
54:
2165:
2096:
1879:
1855:
1796:
1752:
1608:
1564:
89:colored green. The white sets form the lower set
3396:
1913:as an element. More generally, given a subset
822:of any upper set is a lower set, and vice versa.
2304:of a set of vectors is the intersection of all
2625:"Domain representations of topological spaces"
2250:{\displaystyle \downarrow x=\downarrow \{x\},}
2714:
2652:
2507:
2241:
2235:
2206:
2200:
1827:
1803:
1639:
1615:
1388:{\displaystyle \{x\in \mathbb {R} :x>1\}}
1382:
1356:
1342:{\displaystyle \{x\in \mathbb {R} :x>0\}}
1336:
1310:
1058:
1052:
3372:Positive cone of a partially ordered group
2721:
2707:
2622:
1792:
1604:
988:Dually, the smallest lower set containing
2643:
2212:{\displaystyle \uparrow x=\uparrow \{x\}}
1366:
1320:
343:
339:
316:) is defined similarly as being a subset
273:
269:
3355:Positive cone of an ordered vector space
29:
2565:
1395:are both mapped to the empty antichain.
1247:{\displaystyle \operatorname {Max} (Y)}
14:
3397:
2590:
254:. In other words, this means that any
2702:
2518:
2516:
2296:of a set is the intersection of all
2658:Convergence Foundations Of Topology
324:with the property that any element
24:
2882:Properties & Types (
2530:Introduction to Lattices and Order
2327:
1292:For partial orders satisfying the
1136:of a finite partially ordered set
938:the smallest upper set containing
358:is necessarily also an element of
288:is necessarily also an element of
25:
3416:
3338:Positive cone of an ordered field
2513:
1706:{\displaystyle x^{\downarrow X},}
1008:is denoted using a down arrow as
3192:Ordered topological vector space
2728:
2410:that contains for every element
2034:{\displaystyle A^{\downarrow X}}
1735:{\displaystyle x^{\downarrow },}
1064:{\displaystyle \downarrow \{x\}}
958:is denoted using an up arrow as
200:with the following property: if
1762:{\displaystyle \downarrow \!x,}
1518:{\displaystyle x^{\uparrow X},}
1402:Upper closure and lower closure
983:upper closure and lower closure
105:{\displaystyle \downarrow 105.}
2584:
2559:
2397:
2385:
2310:subgroup generated by a subset
2288:since they satisfy all of the
2226:
2191:
2162:
2138:
2122:
2093:
2069:
2053:
2023:
2004:{\displaystyle A^{\uparrow X}}
1993:
1886:{\displaystyle \downarrow \!x}
1876:
1852:
1793:
1781:
1749:
1724:
1692:
1605:
1593:
1561:
1547:{\displaystyle x^{\uparrow },}
1536:
1504:
1445:
1433:
1241:
1235:
1209:
1203:
1188:
1049:
1015:
965:
844:
832:
825:Given a partially ordered set
389:
377:
161:
149:
96:
73:
13:
1:
3149:Series-parallel partial order
2645:10.1016/s0304-3975(99)00045-6
2491:
1935:{\displaystyle A\subseteq X,}
1574:{\displaystyle \uparrow \!x,}
1027:{\displaystyle \downarrow Y.}
798:
365:
2828:Cantor's isomorphism theorem
2685:The low separation axioms (T
2632:Theoretical Computer Science
2599:. World Scientific. p.
1862:{\displaystyle \uparrow \!x}
860:the family of upper sets of
653:{\displaystyle L\subseteq X}
470:{\displaystyle U\subseteq X}
193:{\displaystyle S\subseteq X}
7:
2868:Szpilrajn extension theorem
2843:Hausdorff maximal principle
2818:Boolean prime ideal theorem
2656:; Mynard, Frédéric (2016).
2378:of a partially ordered set
2346:Abstract simplicial complex
2339:
2320:generated by a subset of a
2284:to itself, are examples of
1426:of a partially ordered set
915:of a partially ordered set
10:
3421:
3214:Topological vector lattice
2535:Cambridge University Press
1454:{\displaystyle (X,\leq ),}
1294:descending chain condition
974:{\displaystyle \uparrow Y}
895:Given an arbitrary subset
853:{\displaystyle (X,\leq ),}
82:{\displaystyle \uparrow 2}
62:, ordered by the relation
3244:
3172:
3111:
2881:
2810:
2759:
2736:
2568:Enumerative combinatorics
2508:Dolecki & Mynard 2016
2403:{\displaystyle (X,\leq )}
2290:Kuratowski closure axioms
395:{\displaystyle (X,\leq )}
347:{\displaystyle \,\leq \,}
277:{\displaystyle \,\geq \,}
167:{\displaystyle (X,\leq )}
2823:Cantor–Bernstein theorem
2481:{\displaystyle x\leq y.}
3367:Partially ordered group
3187:Specialization preorder
2682:Hoffman, K. H. (2001),
2432:{\displaystyle x\in X,}
1285:lower set is called an
767:{\displaystyle x\in L.}
738:{\displaystyle x\leq l}
712:{\displaystyle x\in X,}
584:{\displaystyle x\in U.}
555:{\displaystyle u\leq x}
529:{\displaystyle x\in X,}
2853:Kruskal's tree theorem
2848:Knaster–Tarski theorem
2838:Dushnik–Miller theorem
2566:Stanley, R.P. (2002).
2482:
2453:
2433:
2404:
2372:
2278:
2251:
2213:
2176:
2104:
2035:
2005:
1975:
1936:
1907:
1887:
1863:
1837:
1763:
1736:
1707:
1674:
1646:
1575:
1548:
1519:
1486:
1455:
1420:
1389:
1343:
1271:
1248:
1216:
1174:
1150:
1130:
1108:
1085:
1065:
1038:A lower set is called
1028:
1002:
975:
952:
932:
909:
874:
854:
768:
739:
713:
684:
683:{\displaystyle l\in L}
654:
585:
556:
530:
501:
500:{\displaystyle u\in U}
471:
426:
396:
348:
278:
240:
239:{\displaystyle s<x}
194:
168:
112:
106:
83:
56:
2591:Lawson, M.V. (1998).
2483:
2454:
2434:
2405:
2373:
2279:
2252:
2214:
2177:
2105:
2036:
2006:
1976:
1937:
1908:
1888:
1864:
1838:
1764:
1737:
1708:
1675:
1647:
1576:
1549:
1520:
1487:
1456:
1421:
1390:
1344:
1272:
1249:
1217:
1175:
1151:
1131:
1109:
1086:
1066:
1042:if it is of the form
1029:
1003:
976:
953:
933:
910:
875:
855:
769:
740:
714:
685:
655:
634:), which is a subset
586:
557:
531:
502:
472:
427:
397:
349:
279:
241:
195:
169:
141:partially ordered set
107:
84:
66:, with the upper set
57:
33:
3345:Ordered vector space
2525:Hilary Ann Priestley
2463:
2443:
2414:
2382:
2362:
2268:
2223:
2188:
2114:
2045:
2015:
1985:
1962:
1917:
1897:
1873:
1849:
1773:
1746:
1716:
1684:
1664:
1585:
1558:
1528:
1496:
1473:
1430:
1410:
1353:
1307:
1258:
1226:
1185:
1164:
1140:
1120:
1095:
1075:
1046:
1012:
992:
962:
942:
919:
899:
864:
829:
749:
723:
694:
668:
638:
566:
540:
511:
485:
455:
416:
374:
336:
266:
224:
178:
146:
93:
70:
46:
3183:Alexandrov topology
3129:Lexicographic order
3088:Well-quasi-ordering
2623:Blanck, J. (2000).
2537:. pp. 20, 44.
2350:Independence system
2308:containing it; the
2300:containing it; the
2294:topological closure
607:downward closed set
354:to some element of
298:downward closed set
284:to some element of
55:{\displaystyle 210}
3164:Transitive closure
3124:Converse/Transpose
2833:Dilworth's theorem
2478:
2449:
2429:
2400:
2368:
2274:
2247:
2209:
2172:
2161:
2100:
2092:
2041:respectively, as
2031:
2001:
1974:{\displaystyle A,}
1971:
1932:
1903:
1883:
1859:
1833:
1759:
1732:
1703:
1670:
1642:
1571:
1544:
1515:
1485:{\displaystyle x,}
1482:
1451:
1416:
1385:
1339:
1270:{\displaystyle Y.}
1267:
1244:
1212:
1170:
1146:
1126:
1107:{\displaystyle X.}
1104:
1081:
1061:
1024:
998:
971:
948:
931:{\displaystyle X,}
928:
905:
870:
850:
764:
735:
709:
680:
650:
581:
552:
526:
497:
467:
422:
392:
344:
274:
236:
190:
164:
113:
102:
79:
52:
3392:
3391:
3350:Partially ordered
3159:Symmetric closure
3144:Reflexive closure
2887:
2667:978-981-4571-52-4
2610:978-981-02-3316-7
2577:978-0-521-66351-9
2510:, pp. 27–29.
2452:{\displaystyle y}
2371:{\displaystyle U}
2286:closure operators
2277:{\displaystyle X}
2146:
2077:
1906:{\displaystyle x}
1673:{\displaystyle x}
1419:{\displaystyle x}
1406:Given an element
1173:{\displaystyle Y}
1149:{\displaystyle X}
1129:{\displaystyle Y}
1091:is an element of
1084:{\displaystyle x}
1001:{\displaystyle Y}
951:{\displaystyle Y}
908:{\displaystyle Y}
890:upper set lattice
880:ordered with the
873:{\displaystyle X}
435:upward closed set
425:{\displaystyle X}
125:upward closed set
16:(Redirected from
3412:
3134:Linear extension
2883:
2863:Mirsky's theorem
2723:
2716:
2709:
2700:
2699:
2679:
2649:
2647:
2638:(1–2): 229–255.
2629:
2615:
2614:
2598:
2588:
2582:
2581:
2563:
2557:
2556:
2533:(2nd ed.).
2523:Brian A. Davey;
2520:
2511:
2505:
2487:
2485:
2484:
2479:
2458:
2456:
2455:
2450:
2438:
2436:
2435:
2430:
2409:
2407:
2406:
2401:
2377:
2375:
2374:
2369:
2283:
2281:
2280:
2275:
2256:
2254:
2253:
2248:
2218:
2216:
2215:
2210:
2181:
2179:
2178:
2173:
2160:
2142:
2141:
2129:
2128:
2109:
2107:
2106:
2101:
2091:
2073:
2072:
2060:
2059:
2040:
2038:
2037:
2032:
2030:
2029:
2010:
2008:
2007:
2002:
2000:
1999:
1980:
1978:
1977:
1972:
1956:downward closure
1941:
1939:
1938:
1933:
1912:
1910:
1909:
1904:
1892:
1890:
1889:
1884:
1868:
1866:
1865:
1860:
1842:
1840:
1839:
1834:
1788:
1787:
1768:
1766:
1765:
1760:
1741:
1739:
1738:
1733:
1728:
1727:
1712:
1710:
1709:
1704:
1699:
1698:
1679:
1677:
1676:
1671:
1658:downward closure
1651:
1649:
1648:
1643:
1600:
1599:
1580:
1578:
1577:
1572:
1553:
1551:
1550:
1545:
1540:
1539:
1524:
1522:
1521:
1516:
1511:
1510:
1491:
1489:
1488:
1483:
1460:
1458:
1457:
1452:
1425:
1423:
1422:
1417:
1394:
1392:
1391:
1386:
1369:
1348:
1346:
1345:
1340:
1323:
1276:
1274:
1273:
1268:
1253:
1251:
1250:
1245:
1221:
1219:
1218:
1213:
1179:
1177:
1176:
1171:
1158:maximal elements
1155:
1153:
1152:
1147:
1135:
1133:
1132:
1127:
1116:Every lower set
1113:
1111:
1110:
1105:
1090:
1088:
1087:
1082:
1070:
1068:
1067:
1062:
1033:
1031:
1030:
1025:
1007:
1005:
1004:
999:
980:
978:
977:
972:
957:
955:
954:
949:
937:
935:
934:
929:
914:
912:
911:
906:
886:complete lattice
879:
877:
876:
871:
859:
857:
856:
851:
773:
771:
770:
765:
744:
742:
741:
736:
718:
716:
715:
710:
689:
687:
686:
681:
659:
657:
656:
651:
590:
588:
587:
582:
561:
559:
558:
553:
535:
533:
532:
527:
506:
504:
503:
498:
476:
474:
473:
468:
432:(also called an
431:
429:
428:
423:
401:
399:
398:
393:
353:
351:
350:
345:
283:
281:
280:
275:
245:
243:
242:
237:
199:
197:
196:
191:
173:
171:
170:
165:
123:(also called an
111:
109:
108:
103:
88:
86:
85:
80:
61:
59:
58:
53:
21:
3420:
3419:
3415:
3414:
3413:
3411:
3410:
3409:
3395:
3394:
3393:
3388:
3384:Young's lattice
3240:
3168:
3107:
2957:Heyting algebra
2905:Boolean algebra
2877:
2858:Laver's theorem
2806:
2772:Boolean algebra
2767:Binary relation
2755:
2732:
2727:
2692:
2688:
2668:
2654:Dolecki, Szymon
2627:
2619:
2618:
2611:
2589:
2585:
2578:
2564:
2560:
2545:
2521:
2514:
2506:
2499:
2494:
2464:
2461:
2460:
2444:
2441:
2440:
2415:
2412:
2411:
2383:
2380:
2379:
2363:
2360:
2359:
2342:
2330:
2328:Ordinal numbers
2269:
2266:
2265:
2224:
2221:
2220:
2189:
2186:
2185:
2150:
2137:
2133:
2121:
2117:
2115:
2112:
2111:
2081:
2068:
2064:
2052:
2048:
2046:
2043:
2042:
2022:
2018:
2016:
2013:
2012:
1992:
1988:
1986:
1983:
1982:
1963:
1960:
1959:
1918:
1915:
1914:
1898:
1895:
1894:
1874:
1871:
1870:
1850:
1847:
1846:
1780:
1776:
1774:
1771:
1770:
1747:
1744:
1743:
1723:
1719:
1717:
1714:
1713:
1691:
1687:
1685:
1682:
1681:
1665:
1662:
1661:
1592:
1588:
1586:
1583:
1582:
1559:
1556:
1555:
1535:
1531:
1529:
1526:
1525:
1503:
1499:
1497:
1494:
1493:
1474:
1471:
1470:
1431:
1428:
1427:
1411:
1408:
1407:
1404:
1365:
1354:
1351:
1350:
1319:
1308:
1305:
1304:
1259:
1256:
1255:
1227:
1224:
1223:
1186:
1183:
1182:
1165:
1162:
1161:
1141:
1138:
1137:
1121:
1118:
1117:
1096:
1093:
1092:
1076:
1073:
1072:
1047:
1044:
1043:
1013:
1010:
1009:
993:
990:
989:
963:
960:
959:
943:
940:
939:
920:
917:
916:
900:
897:
896:
865:
862:
861:
830:
827:
826:
801:
750:
747:
746:
724:
721:
720:
695:
692:
691:
669:
666:
665:
639:
636:
635:
625:initial segment
604:(also called a
567:
564:
563:
541:
538:
537:
512:
509:
508:
486:
483:
482:
456:
453:
452:
417:
414:
413:
375:
372:
371:
368:
337:
334:
333:
310:initial segment
296:(also called a
267:
264:
263:
225:
222:
221:
216:is larger than
179:
176:
175:
147:
144:
143:
94:
91:
90:
71:
68:
67:
47:
44:
43:
28:
23:
22:
15:
12:
11:
5:
3418:
3408:
3407:
3390:
3389:
3387:
3386:
3381:
3376:
3375:
3374:
3364:
3363:
3362:
3357:
3352:
3342:
3341:
3340:
3330:
3325:
3324:
3323:
3318:
3311:Order morphism
3308:
3307:
3306:
3296:
3291:
3286:
3281:
3276:
3275:
3274:
3264:
3259:
3254:
3248:
3246:
3242:
3241:
3239:
3238:
3237:
3236:
3231:
3229:Locally convex
3226:
3221:
3211:
3209:Order topology
3206:
3205:
3204:
3202:Order topology
3199:
3189:
3179:
3177:
3170:
3169:
3167:
3166:
3161:
3156:
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3115:
3113:
3109:
3108:
3106:
3105:
3095:
3085:
3080:
3075:
3070:
3065:
3060:
3055:
3050:
3049:
3048:
3038:
3033:
3032:
3031:
3026:
3021:
3016:
3014:Chain-complete
3006:
3001:
3000:
2999:
2994:
2989:
2984:
2979:
2969:
2964:
2959:
2954:
2949:
2939:
2934:
2929:
2924:
2919:
2914:
2913:
2912:
2902:
2897:
2891:
2889:
2879:
2878:
2876:
2875:
2870:
2865:
2860:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2814:
2812:
2808:
2807:
2805:
2804:
2799:
2794:
2789:
2784:
2779:
2774:
2769:
2763:
2761:
2757:
2756:
2754:
2753:
2748:
2743:
2737:
2734:
2733:
2726:
2725:
2718:
2711:
2703:
2697:
2696:
2690:
2686:
2680:
2666:
2650:
2617:
2616:
2609:
2583:
2576:
2558:
2543:
2512:
2496:
2495:
2493:
2490:
2489:
2488:
2477:
2474:
2471:
2468:
2448:
2428:
2425:
2422:
2419:
2399:
2396:
2393:
2390:
2387:
2367:
2353:
2348:(also called:
2341:
2338:
2334:ordinal number
2329:
2326:
2273:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2208:
2205:
2202:
2199:
2196:
2193:
2171:
2168:
2164:
2159:
2156:
2153:
2149:
2145:
2140:
2136:
2132:
2127:
2124:
2120:
2099:
2095:
2090:
2087:
2084:
2080:
2076:
2071:
2067:
2063:
2058:
2055:
2051:
2028:
2025:
2021:
1998:
1995:
1991:
1970:
1967:
1948:upward closure
1931:
1928:
1925:
1922:
1902:
1882:
1878:
1858:
1854:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1795:
1791:
1786:
1783:
1779:
1769:is defined by
1758:
1755:
1751:
1731:
1726:
1722:
1702:
1697:
1694:
1690:
1669:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1607:
1603:
1598:
1595:
1591:
1581:is defined by
1570:
1567:
1563:
1543:
1538:
1534:
1514:
1509:
1506:
1502:
1481:
1478:
1467:upward closure
1450:
1447:
1444:
1441:
1438:
1435:
1415:
1403:
1400:
1397:
1396:
1384:
1381:
1378:
1375:
1372:
1368:
1364:
1361:
1358:
1338:
1335:
1332:
1329:
1326:
1322:
1318:
1315:
1312:
1290:
1279:
1278:
1277:
1266:
1263:
1243:
1240:
1237:
1234:
1231:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1169:
1145:
1125:
1114:
1103:
1100:
1080:
1060:
1057:
1054:
1051:
1036:
1035:
1034:
1023:
1020:
1017:
997:
970:
967:
947:
927:
924:
904:
893:
884:relation is a
869:
849:
846:
843:
840:
837:
834:
823:
816:
805:
800:
797:
789:
781:
775:
774:
763:
760:
757:
754:
734:
731:
728:
708:
705:
702:
699:
679:
676:
673:
649:
646:
643:
632:
626:
620:
619:decreasing set
614:
608:
602:
592:
591:
580:
577:
574:
571:
551:
548:
545:
525:
522:
519:
516:
496:
493:
490:
466:
463:
460:
451:) is a subset
448:
442:
436:
421:
410:
404:preordered set
391:
388:
385:
382:
379:
367:
364:
342:
306:decreasing set
272:
235:
232:
229:
189:
186:
183:
163:
160:
157:
154:
151:
101:
98:
78:
75:
51:
26:
9:
6:
4:
3:
2:
3417:
3406:
3403:
3402:
3400:
3385:
3382:
3380:
3377:
3373:
3370:
3369:
3368:
3365:
3361:
3358:
3356:
3353:
3351:
3348:
3347:
3346:
3343:
3339:
3336:
3335:
3334:
3333:Ordered field
3331:
3329:
3326:
3322:
3319:
3317:
3314:
3313:
3312:
3309:
3305:
3302:
3301:
3300:
3297:
3295:
3292:
3290:
3289:Hasse diagram
3287:
3285:
3282:
3280:
3277:
3273:
3270:
3269:
3268:
3267:Comparability
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3249:
3247:
3243:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3216:
3215:
3212:
3210:
3207:
3203:
3200:
3198:
3195:
3194:
3193:
3190:
3188:
3184:
3181:
3180:
3178:
3175:
3171:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3139:Product order
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3116:
3114:
3112:Constructions
3110:
3104:
3100:
3096:
3093:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3047:
3044:
3043:
3042:
3039:
3037:
3034:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3011:
3010:
3009:Partial order
3007:
3005:
3002:
2998:
2997:Join and meet
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2974:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2944:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2911:
2908:
2907:
2906:
2903:
2901:
2898:
2896:
2895:Antisymmetric
2893:
2892:
2890:
2886:
2880:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2815:
2813:
2809:
2803:
2802:Weak ordering
2800:
2798:
2795:
2793:
2790:
2788:
2787:Partial order
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2768:
2765:
2764:
2762:
2758:
2752:
2749:
2747:
2744:
2742:
2739:
2738:
2735:
2731:
2724:
2719:
2717:
2712:
2710:
2705:
2704:
2701:
2695:
2694:
2681:
2677:
2673:
2669:
2663:
2659:
2655:
2651:
2646:
2641:
2637:
2633:
2626:
2621:
2620:
2612:
2606:
2602:
2597:
2596:
2587:
2579:
2573:
2569:
2562:
2554:
2550:
2546:
2544:0-521-78451-4
2540:
2536:
2532:
2531:
2526:
2519:
2517:
2509:
2504:
2502:
2497:
2475:
2472:
2469:
2466:
2446:
2439:some element
2426:
2423:
2420:
2417:
2394:
2391:
2388:
2365:
2357:
2354:
2351:
2347:
2344:
2343:
2337:
2335:
2325:
2323:
2319:
2315:
2311:
2307:
2303:
2299:
2295:
2291:
2287:
2271:
2262:
2260:
2244:
2238:
2232:
2229:
2203:
2197:
2194:
2184:In this way,
2182:
2169:
2166:
2157:
2154:
2151:
2147:
2143:
2134:
2130:
2125:
2118:
2097:
2088:
2085:
2082:
2078:
2074:
2065:
2061:
2056:
2049:
2026:
2019:
1996:
1989:
1968:
1965:
1957:
1953:
1949:
1945:
1929:
1926:
1923:
1920:
1900:
1880:
1856:
1843:
1830:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1800:
1797:
1789:
1784:
1777:
1756:
1753:
1729:
1720:
1700:
1695:
1688:
1680:, denoted by
1667:
1659:
1655:
1654:lower closure
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1612:
1609:
1601:
1596:
1589:
1568:
1565:
1541:
1532:
1512:
1507:
1500:
1479:
1476:
1468:
1464:
1463:upper closure
1448:
1442:
1439:
1436:
1413:
1399:
1379:
1376:
1373:
1370:
1362:
1359:
1333:
1330:
1327:
1324:
1316:
1313:
1303:
1299:
1295:
1291:
1288:
1284:
1280:
1264:
1261:
1238:
1232:
1229:
1206:
1200:
1197:
1194:
1191:
1181:
1180:
1167:
1159:
1143:
1123:
1115:
1101:
1098:
1078:
1055:
1041:
1037:
1021:
1018:
995:
987:
986:
984:
968:
945:
925:
922:
902:
894:
891:
887:
883:
867:
847:
841:
838:
835:
824:
821:
817:
814:
810:
806:
803:
802:
796:
794:
790:
788:
785:
782:
779:
761:
758:
755:
752:
732:
729:
726:
706:
703:
700:
697:
677:
674:
671:
663:
662:
661:
647:
644:
641:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
578:
575:
572:
569:
549:
546:
543:
523:
520:
517:
514:
494:
491:
488:
480:
479:
478:
464:
461:
458:
450:
446:
443:
440:
437:
434:
419:
411:
408:
405:
386:
383:
380:
363:
361:
357:
340:
331:
327:
323:
319:
315:
311:
307:
303:
299:
295:
291:
287:
270:
261:
257:
253:
249:
233:
230:
227:
220:(that is, if
219:
215:
211:
207:
203:
187:
184:
181:
158:
155:
152:
142:
138:
134:
130:
126:
122:
118:
99:
76:
65:
64:is divisor of
49:
41:
37:
36:Hasse diagram
32:
19:
3405:Order theory
3378:
3176:& Orders
3154:Star product
3083:Well-founded
3036:Prefix order
2992:Distributive
2982:Complemented
2952:Foundational
2917:Completeness
2873:Zorn's lemma
2777:Cyclic order
2760:Key concepts
2730:Order theory
2684:
2657:
2635:
2631:
2594:
2586:
2567:
2561:
2529:
2331:
2263:
2258:
2183:
1955:
1951:
1947:
1943:
1844:
1657:
1653:
1466:
1462:
1405:
1398:
1302:real numbers
1039:
889:
809:intersection
784:
778:
776:
629:
623:
617:
611:
605:
599:
598:notion is a
593:
445:
439:
433:
407:
369:
359:
355:
329:
325:
321:
317:
313:
309:
305:
301:
297:
293:
292:. The term
289:
285:
259:
255:
251:
247:
217:
213:
209:
205:
201:
174:is a subset
136:
132:
128:
124:
120:
114:
63:
3360:Riesz space
3321:Isomorphism
3197:Normal cone
3119:Composition
3053:Semilattice
2962:Homogeneous
2947:Equivalence
2797:Total order
2358:– a subset
2356:Cofinal set
2298:closed sets
1981:denoted by
1942:define the
1492:denoted by
1287:order ideal
780:order ideal
258:element of
117:mathematics
3328:Order type
3262:Cofinality
3103:Well-order
3078:Transitive
2967:Idempotent
2900:Asymmetric
2553:2001043910
2492:References
2459:such that
2233:=↓
2198:=↑
1652:while the
1298:bijections
1195:=↓
820:complement
799:Properties
777:The terms
631:semi-ideal
366:Definition
314:semi-ideal
3379:Upper set
3316:Embedding
3252:Antichain
3073:Tolerance
3063:Symmetric
3058:Semiorder
3004:Reflexive
2922:Connected
2676:945169917
2470:≤
2421:∈
2395:≤
2306:subspaces
2259:principal
2227:↓
2192:↑
2163:↓
2155:∈
2148:⋃
2139:↓
2123:↓
2094:↑
2086:∈
2079:⋃
2070:↑
2054:↑
2024:↓
1994:↑
1924:⊆
1877:↓
1853:↑
1845:The sets
1822:≤
1810:∈
1794:↓
1782:↓
1750:↓
1725:↓
1693:↓
1634:≤
1622:∈
1606:↑
1594:↑
1562:↑
1537:↑
1505:↑
1443:≤
1363:∈
1317:∈
1233:
1201:
1189:↓
1050:↓
1040:principal
1016:↓
966:↑
882:inclusion
842:≤
756:∈
730:≤
701:∈
675:∈
645:⊆
601:lower set
573:∈
547:≤
518:∈
492:∈
462:⊆
409:upper set
387:≤
341:≤
294:lower set
271:≥
185:⊆
159:≤
121:upper set
97:↓
74:↑
18:Lower set
3399:Category
3174:Topology
3041:Preorder
3024:Eulerian
2987:Complete
2937:Directed
2927:Covering
2792:Preorder
2751:Category
2746:Glossary
2689:) and (T
2527:(2002).
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690:and all
664:for all
613:down set
507:and all
481:for all
444:, or an
332:that is
302:down set
262:that is
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131:, or an
40:divisors
3279:Duality
3257:Cofinal
3245:Related
3224:Fréchet
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2977:Bounded
2972:Lattice
2945:)
2943:Partial
2811:Results
2782:Lattice
793:lattice
447:isotone
208:and if
139:) of a
135:set in
133:isotone
38:of the
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1071:where
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2628:(PDF)
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