36:
552:
The above definitions of "ideal" and "order ideal" are the standard ones, but there is some confusion in terminology. Sometimes the words and definitions such as "ideal", "order ideal", "
1213:
573:. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper. For lattices, prime ideals can be characterized as follows:
716:
564:
An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order. Such ideals are called
514:
662:
610:
316:
161:
435:
362:
412:
826:, but this terminology is often reserved for Boolean algebras, where a maximal filter (ideal) is a filter (ideal) that contains exactly one of the elements {
2429:
2412:
1942:
1778:
1200:. It is strictly weaker than the axiom of choice and it turns out that nothing more is needed for many order-theoretic applications of ideals.
2259:
57:
1568:
764:
The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within ZF (
2395:
2254:
1597:
1578:
1553:
1266:
in the ideal completion, so the original poset can be recovered as the sub-poset consisting of compact elements. Furthermore, every
79:
2249:
115:, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and
765:
1885:
1255:
1967:
1394:
2462:
2286:
2206:
1687:
1637:
1880:
2071:
2000:
1974:
1962:
1925:
1900:
1875:
1829:
1798:
1338:
1292:
1287:, where the name was derived from the ring ideals of abstract algebra. He adopted this terminology because, using the
1284:
1236:
1197:
1193:
773:
1905:
1895:
1216:, the maximal ideals (or, equivalently via the negation map, ultrafilters) are used to obtain the set of points of a
50:
44:
2271:
1771:
2244:
1910:
1288:
1232:
61:
2176:
1803:
819:, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general.
2424:
2407:
2336:
1952:
391:
278:
While this is the most general way to define an ideal for arbitrary posets, it was originally defined for
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2314:
2149:
2140:
2009:
1844:
1808:
1764:
1615:
104:
1890:
1385:. Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press. p.
1240:
1192:
for every disjoint filter–ideal-pair can be shown. In the special case that the considered order is a
2402:
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2341:
2086:
2049:
2039:
2019:
2004:
1696:
1646:
1315:
816:
438:
2329:
2240:
2186:
2145:
2135:
2024:
1957:
1920:
1321:
1592:, Cambridge Studies in Advanced Mathematics, vol. 59, Cambridge University Press, Cambridge,
1208:
The construction of ideals and filters is an important tool in many applications of order theory.
695:
496:
2368:
2221:
2130:
2120:
2061:
1979:
1493:
1478:
1327:
2441:
2281:
1915:
1685:
Stone, M. H. (1935), "Subsumption of the Theory of
Boolean Algebras under the Theory of Rings",
641:
583:
289:
134:
2378:
2356:
2216:
2201:
2181:
1984:
1724:
1674:
1593:
1574:
1549:
1457:
1390:
1386:
1380:
1362:
1280:
1217:
761:. So this is just a specific prime ideal that extends the above conditions to infinite meets.
417:
341:
2191:
2044:
1746:
1714:
1704:
1664:
1654:
1564:
722:
112:
1607:
1379:
Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M. W.; Scott, D. S. (2003).
397:
2373:
2156:
2034:
2029:
2014:
1839:
1824:
1603:
1333:
1267:
1263:
1185:
769:
279:
1930:
1700:
1650:
383:
that satisfies the above conditions 1 and 2. In other words, an order ideal is simply a
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2276:
2266:
2125:
2103:
2081:
1737:
1719:
1669:
1451:
319:
116:
17:
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2390:
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2196:
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812:
is maximal if it is proper and there is no proper filter that is a strict superset.
2211:
2093:
2076:
1994:
1834:
1787:
1750:
1543:
1296:
271:
96:
972:
by taking the downward closure of the set of all binary joins of this form, i.e.
394:
notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging
2417:
2110:
1989:
1854:
1303:
1252:
1225:
823:
553:
452:
448:
444:
93:
1330: – Non-empty family of sets that is closed under finite unions and subsets
857:
There is another interesting notion of maximality of ideals: Consider an ideal
282:
only. In this case, the following equivalent definition can be given: a subset
2385:
2319:
2160:
1221:
108:
2436:
2309:
2115:
1324: – Additive subgroup of a mathematical ring that absorbs multiplication
1270:
can be reconstructed as the ideal completion of its set of compact elements.
384:
220:
175:
1728:
1678:
1635:
Stone, M. H. (1934), "Boolean
Algebras and Their Application to Topology",
1709:
1659:
107:(poset). Although this term historically was derived from the notion of a
2231:
2098:
1849:
1479:"On lattices and their ideal lattices, and posets and their ideal posets"
1410:
1408:
1406:
1621:
776:, which are necessary for many applications that require prime ideals.
738:
323:
387:. Similarly, an ideal can also be defined as a "directed lower set".
692:
It is easily checked that this is indeed equivalent to stating that
1341: – Ideals in a Boolean algebra can be extended to prime ideals
1318: – In mathematics, a special subset of a partially ordered set
1180:
However, in general it is not clear whether there exists any ideal
1498:
1144:
contradicts the disjointness of the two sets. Hence all elements
455:
are different generalizations of the notion of a lattice ideal.
1760:
1756:
29:
1542:
Burris, Stanley N.; Sankappanavar, Hanamantagouda P. (1981).
893:
is always a prime ideal. A proof of this statement follows.
718:
is a filter (which is then also prime, in the dual sense).
1251:
ordered by subset inclusion. This construction yields the
1735:
Frink, Orrin (1954), "Ideals In
Partially Ordered Sets",
1160:. Consequently one can apply the above construction with
1132:
too. On the other hand, this finite join of elements of
904:
is maximal with respect to disjointness from the filter
838:
of the
Boolean algebra. In Boolean algebras, the terms
737:
with the additional property that, whenever the meet (
1184:
that is maximal in this sense. Yet, if we assume the
698:
644:
586:
499:
420:
400:
344:
322:
it is a lower set that is closed under finite joins (
292:
137:
493:
of the ideal in this situation. The principal ideal
2302:
2230:
2169:
1939:
1868:
1817:
1414:
1214:
Stone's representation theorem for
Boolean algebras
912:is not prime, i.e. there exists a pair of elements
733:is meaningful. It is defined to be a proper ideal
710:
656:
604:
508:
429:
406:
356:
310:
155:
1168:to obtain an ideal that is strictly greater than
469:The smallest ideal that contains a given element
889:. In the case of distributive lattices such an
27:Nonempty, upper-bounded, downward-closed subset
1426:
1026:For the other case, assume that there is some
881:that is maximal among all ideals that contain
556:", or "partial order ideal" mean one another.
1772:
1438:
1262:. An ideal is principal if and only if it is
8:
1573:(2nd ed.). Cambridge University Press.
1235:to turn posets into posets with additional
2430:Positive cone of a partially ordered group
1779:
1765:
1757:
1718:
1708:
1668:
1658:
1497:
1302:Generalization to any posets was done by
1188:in our set theory, then the existence of
1015:. But this contradicts the maximality of
697:
643:
585:
498:
419:
399:
343:
291:
136:
80:Learn how and when to remove this message
2413:Positive cone of an ordered vector space
43:This article includes a list of general
1351:
702:
326:); that is, it is nonempty and for all
1365:: "A directed lower subset of a poset
1358:
1299:, the two notions do indeed coincide.
772:). This issue is discussed in various
1614:Frenchman, Zack; Hart, James (2020),
1526:
1515:
1511:
1140:, such that the assumed existence of
822:Maximal filters are sometimes called
379:is defined to be a subset of a poset
7:
1590:Practical foundations of mathematics
462:if it is not equal to the whole set
167:, if the following conditions hold:
908:. Suppose for a contradiction that
1940:Properties & Types (
1570:Introduction to Lattices and Order
49:it lacks sufficient corresponding
25:
2396:Positive cone of an ordered field
1007:is indeed an ideal disjoint from
946:. Consider the case that for all
612:is a prime ideal, if and only if
458:An ideal or filter is said to be
2250:Ordered topological vector space
1228:to the original Boolean algebra.
877:. We are interested in an ideal
796:if it is proper and there is no
34:
1617:An Introduction to Order Theory
1415:Burris & Sankappanavar 1981
1382:Continuous Lattices and Domains
1011:which is strictly greater than
1751:10.1080/00029890.1954.11988449
599:
587:
500:
305:
293:
150:
138:
1:
2207:Series-parallel partial order
1688:Proc. Natl. Acad. Sci. U.S.A.
1638:Proc. Natl. Acad. Sci. U.S.A.
1545:A Course in Universal Algebra
1289:isomorphism of the categories
1239:properties. For example, the
1196:, this theorem is called the
1019:and thus the assumption that
1003:. It is readily checked that
968:. One can construct an ideal
804:that is a strict superset of
1886:Cantor's isomorphism theorem
1247:is the set of all ideals of
1104:. But then their meet is in
711:{\displaystyle P\setminus I}
509:{\displaystyle \downarrow p}
1926:Szpilrajn extension theorem
1901:Hausdorff maximal principle
1876:Boolean prime ideal theorem
1339:Boolean prime ideal theorem
1198:Boolean prime ideal theorem
1176:. This finishes the proof.
766:Zermelo–Fraenkel set theory
131:of a partially ordered set
2484:
2272:Topological vector lattice
1477:George M. Bergman (2008),
1427:Davey & Priestley 2002
1279:Ideals were introduced by
1172:while being disjoint from
846:coincide, as do the terms
1794:
1439:Frenchman & Hart 2020
1243:of a given partial order
657:{\displaystyle x\wedge y}
605:{\displaystyle (P,\leq )}
311:{\displaystyle (P,\leq )}
156:{\displaystyle (P,\leq )}
103:is a special subset of a
1881:Cantor–Bernstein theorem
1231:Order theory knows many
1108:and, by distributivity,
741:) of some arbitrary set
725:the further notion of a
430:{\displaystyle \wedge ,}
238:, there is some element
2425:Partially ordered group
2245:Specialization preorder
357:{\displaystyle x\vee y}
64:more precise citations.
1911:Kruskal's tree theorem
1906:Knaster–Tarski theorem
1896:Dushnik–Miller theorem
885:and are disjoint from
729:completely prime ideal
712:
658:
606:
510:
431:
408:
358:
312:
157:
18:Ideal (lattice theory)
1710:10.1073/pnas.21.2.103
1660:10.1073/pnas.20.3.197
1588:Taylor, Paul (1999),
1565:Priestley, Hilary Ann
1233:completion procedures
1048:. Now if any element
808:. Likewise, a filter
713:
659:
619:is a proper ideal of
607:
548:Terminology confusion
511:
432:
409:
407:{\displaystyle \vee }
359:
313:
158:
105:partially ordered set
2463:Ideals (ring theory)
2403:Ordered vector space
1316:Filter (mathematics)
834:}, for each element
817:distributive lattice
774:prime ideal theorems
696:
642:
584:
497:
418:
398:
342:
290:
135:
2241:Alexandrov topology
2187:Lexicographic order
2146:Well-quasi-ordering
1701:1935PNAS...21..103S
1651:1934PNAS...20..197S
1548:. Springer-Verlag.
1453:Partial Order Ideal
1369:is called an ideal"
1322:Ideal (ring theory)
375:A weaker notion of
2222:Transitive closure
2182:Converse/Transpose
1891:Dilworth's theorem
1429:, pp. 20, 44.
1328:Ideal (set theory)
898:
815:When a poset is a
753:, some element of
708:
654:
602:
506:
453:Doyle pseudoideals
427:
404:
354:
308:
153:
2450:
2449:
2408:Partially ordered
2217:Symmetric closure
2202:Reflexive closure
1945:
1563:Davey, Brian A.;
1458:Wolfram MathWorld
1281:Marshall H. Stone
1218:topological space
1152:have a join with
1070:, one finds that
900:Assume the ideal
896:
626:for all elements
520:is thus given by
489:principal element
90:
89:
82:
16:(Redirected from
2475:
2192:Linear extension
1941:
1921:Mirsky's theorem
1781:
1774:
1767:
1758:
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1731:
1722:
1712:
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1624:
1610:
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1524:
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1502:
1501:
1486:Tbilisi Math. J.
1483:
1474:
1468:
1467:
1466:
1465:
1448:
1442:
1441:, pp. 2, 7.
1436:
1430:
1424:
1418:
1412:
1401:
1400:
1376:
1370:
1356:
1293:Boolean algebras
1285:Boolean algebras
1241:ideal completion
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746:
736:
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730:
723:complete lattice
717:
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516:for a principal
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485:is said to be a
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113:abstract algebra
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
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21:
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2442:Young's lattice
2298:
2226:
2165:
2015:Heyting algebra
1963:Boolean algebra
1935:
1916:Laver's theorem
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1830:Boolean algebra
1825:Binary relation
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1334:Semigroup ideal
1312:
1277:
1206:
1194:Boolean algebra
1186:axiom of choice
1178:
1156:that is not in
1109:
1086:
1071:
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1035:
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770:axiom of choice
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477:principal ideal
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56:Please help to
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5:
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2369:Order morphism
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2289:
2287:Locally convex
2284:
2279:
2269:
2267:Order topology
2264:
2263:
2262:
2260:Order topology
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2247:
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2113:
2108:
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2106:
2096:
2091:
2090:
2089:
2084:
2079:
2074:
2072:Chain-complete
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1814:
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1801:
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1755:
1754:
1745:(4): 223–234,
1738:Am. Math. Mon.
1732:
1695:(2): 103–105,
1682:
1645:(3): 197–202,
1630:
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1611:
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1585:
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1560:
1554:
1537:
1534:
1531:
1530:
1519:
1504:
1469:
1443:
1431:
1419:
1402:
1395:
1371:
1350:
1349:
1347:
1344:
1343:
1342:
1336:
1331:
1325:
1319:
1311:
1308:
1276:
1273:
1272:
1271:
1268:algebraic dcpo
1229:
1205:
1202:
1136:is clearly in
1023:is not prime.
895:
852:maximal filter
795:
781:
780:Maximal ideals
778:
707:
704:
701:
690:
689:
653:
650:
647:
624:
601:
598:
595:
592:
589:
561:
558:
549:
546:
505:
502:
492:
480:
426:
423:
403:
353:
350:
347:
338:, the element
320:if and only if
307:
304:
301:
298:
295:
276:
275:
224:
179:
152:
149:
146:
143:
140:
124:
121:
117:lattice theory
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2480:
2469:
2466:
2464:
2461:
2460:
2458:
2443:
2440:
2438:
2435:
2431:
2428:
2427:
2426:
2423:
2419:
2416:
2414:
2411:
2409:
2406:
2405:
2404:
2401:
2397:
2394:
2393:
2392:
2391:Ordered field
2389:
2387:
2384:
2380:
2377:
2375:
2372:
2371:
2370:
2367:
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2360:
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2350:
2348:
2347:Hasse diagram
2345:
2343:
2340:
2338:
2335:
2331:
2328:
2327:
2326:
2325:Comparability
2323:
2321:
2318:
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2256:
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2229:
2223:
2220:
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2213:
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2203:
2200:
2198:
2197:Product order
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2174:
2172:
2170:Constructions
2168:
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2129:
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2124:
2122:
2119:
2117:
2114:
2112:
2109:
2105:
2102:
2101:
2100:
2097:
2095:
2092:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2069:
2068:
2067:Partial order
2065:
2063:
2060:
2056:
2055:Join and meet
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2032:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2002:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1969:
1966:
1965:
1964:
1961:
1959:
1956:
1954:
1953:Antisymmetric
1951:
1950:
1948:
1944:
1938:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1873:
1871:
1867:
1861:
1860:Weak ordering
1858:
1856:
1853:
1851:
1848:
1846:
1845:Partial order
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1822:
1820:
1816:
1810:
1807:
1805:
1802:
1800:
1797:
1796:
1793:
1789:
1782:
1777:
1775:
1770:
1768:
1763:
1762:
1759:
1752:
1748:
1744:
1740:
1739:
1733:
1730:
1726:
1721:
1716:
1711:
1706:
1702:
1698:
1694:
1690:
1689:
1683:
1680:
1676:
1671:
1666:
1661:
1656:
1652:
1648:
1644:
1640:
1639:
1633:
1632:
1629:About history
1628:
1623:
1619:
1618:
1612:
1609:
1605:
1601:
1599:0-521-63107-6
1595:
1591:
1586:
1582:
1580:0-521-78451-4
1576:
1572:
1571:
1566:
1561:
1557:
1555:3-540-90578-2
1551:
1547:
1546:
1540:
1539:
1535:
1528:
1523:
1520:
1517:
1513:
1508:
1505:
1500:
1495:
1491:
1487:
1480:
1473:
1470:
1459:
1455:
1454:
1447:
1444:
1440:
1435:
1432:
1428:
1423:
1420:
1416:
1411:
1409:
1407:
1403:
1398:
1392:
1388:
1384:
1383:
1375:
1372:
1368:
1364:
1360:
1359:Taylor (1999)
1355:
1352:
1345:
1340:
1337:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1313:
1309:
1307:
1305:
1300:
1298:
1297:Boolean rings
1294:
1290:
1286:
1282:
1274:
1269:
1265:
1261:
1258:generated by
1257:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1227:
1223:
1219:
1215:
1211:
1210:
1209:
1203:
1201:
1199:
1195:
1191:
1187:
1183:
1177:
1175:
1171:
1167:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1135:
1131:
1125:
1121:
1117:
1113:
1107:
1103:
1098:
1094:
1090:
1083:
1079:
1075:
1069:
1064:
1060:
1056:is such that
1055:
1051:
1047:
1042:
1038:
1033:
1029:
1024:
1022:
1018:
1014:
1010:
1006:
1000:
996:
992:
988:
984:
980:
976:
971:
967:
962:
958:
953:
949:
945:
941:
937:
933:
928:
924:
919:
915:
911:
907:
903:
894:
892:
888:
880:
876:
872:
864:
861:and a filter
855:
853:
849:
845:
844:maximal ideal
841:
837:
833:
829:
825:
820:
818:
813:
811:
803:
799:
792:maximal ideal
789:
779:
777:
775:
771:
767:
762:
756:
751:
745:
740:
732:
724:
719:
705:
699:
686:
682:
676:
672:
668:implies that
651:
648:
645:
637:
633:
629:
625:
622:
615:
614:
613:
596:
593:
590:
580:of a lattice
574:
572:
559:
557:
555:
547:
545:
541:
537:
533:
529:
525:
519:
503:
486:
484:
474:
472:
467:
465:
461:
456:
454:
450:
446:
442:
440:
424:
421:
401:
393:
388:
386:
378:
373:
367:
351:
348:
345:
333:
329:
325:
321:
302:
299:
296:
286:of a lattice
281:
273:
264:
260:
254:
250:
241:
233:
229:
225:
222:
210:
207:implies that
205:
201:
196:
192:
184:
180:
177:
170:
169:
168:
166:
147:
144:
141:
122:
120:
118:
114:
110:
106:
102:
98:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
2468:Order theory
2351:
2234:& Orders
2212:Star product
2141:Well-founded
2094:Prefix order
2050:Distributive
2040:Complemented
2010:Foundational
1975:Completeness
1931:Zorn's lemma
1835:Cyclic order
1818:Key concepts
1788:Order theory
1742:
1736:
1692:
1686:
1642:
1636:
1616:
1589:
1569:
1544:
1527:Frink (1954)
1522:
1516:Stone (1935)
1512:Stone (1934)
1507:
1489:
1485:
1472:
1462:, retrieved
1452:
1446:
1434:
1422:
1381:
1374:
1366:
1354:
1301:
1278:
1259:
1248:
1244:
1237:completeness
1207:
1204:Applications
1189:
1181:
1179:
1173:
1169:
1165:
1164:in place of
1161:
1157:
1153:
1149:
1145:
1141:
1137:
1133:
1129:
1123:
1119:
1115:
1111:
1105:
1101:
1100:are both in
1096:
1092:
1088:
1081:
1077:
1073:
1067:
1062:
1058:
1053:
1049:
1045:
1040:
1036:
1031:
1027:
1025:
1020:
1016:
1012:
1008:
1004:
998:
994:
990:
986:
982:
978:
974:
969:
965:
960:
956:
951:
947:
943:
939:
935:
934:but neither
931:
926:
922:
917:
913:
909:
905:
901:
899:
890:
886:
878:
874:
862:
856:
851:
848:prime filter
847:
843:
839:
835:
831:
827:
824:ultrafilters
821:
814:
809:
801:
797:
783:
768:without the
763:
754:
749:
743:
726:
720:
691:
684:
680:
674:
670:
635:
631:
627:
620:
575:
565:
563:
560:Prime ideals
551:
539:
535:
531:
527:
523:
517:
482:
470:
468:
463:
459:
457:
449:pseudoideals
445:Frink ideals
443:
389:
376:
374:
365:
331:
327:
318:is an ideal
277:
272:directed set
262:
258:
252:
248:
246:, such that
239:
231:
227:
208:
203:
199:
194:
190:
182:
164:
126:
100:
97:order theory
94:mathematical
91:
76:
67:
48:
2418:Riesz space
2379:Isomorphism
2255:Normal cone
2177:Composition
2111:Semilattice
2020:Homogeneous
2005:Equivalence
1855:Total order
1417:, Def. 8.2.
1222:clopen sets
1118:) ∨ (
840:prime ideal
757:is also in
568:prime ideal
554:Frink ideal
377:order ideal
368:is also in
123:Definitions
62:introducing
2457:Categories
2386:Order type
2320:Cofinality
2161:Well-order
2136:Transitive
2025:Idempotent
1958:Asymmetric
1536:References
1492:: 89–103,
1464:2023-02-26
1396:0521803381
1283:first for
1226:isomorphic
1095:) ∨
1080:) ∨
964:is not in
920:such that
865:such that
226:for every
181:for every
109:ring ideal
45:references
2437:Upper set
2374:Embedding
2310:Antichain
2131:Tolerance
2121:Symmetric
2116:Semiorder
2062:Reflexive
1980:Connected
1499:0801.0751
993:for some
784:An ideal
703:∖
649:∧
597:≤
576:A subset
501:↓
422:∧
402:∨
385:lower set
349:∨
303:≤
221:lower set
176:non-empty
148:≤
127:A subset
70:June 2017
2232:Topology
2099:Preorder
2082:Eulerian
2045:Complete
1995:Directed
1985:Covering
1850:Preorder
1809:Category
1804:Glossary
1729:16587931
1679:16587875
1567:(2002).
1310:See also
1220:, whose
1122:∧
1114:∨
1091:∨
1076:∨
1061:∨
1039:∨
997:∈
989:∨
959:∨
925:∧
871:disjoint
683:∈
673:∈
530:∈
280:lattices
2337:Duality
2315:Cofinal
2303:Related
2282:Fréchet
2159:)
2035:Bounded
2030:Lattice
2003:)
2001:Partial
1869:Results
1840:Lattice
1720:1076539
1697:Bibcode
1670:1076376
1647:Bibcode
1608:1694820
1295:and of
1275:History
1264:compact
942:are in
739:infimum
324:suprema
266: (
215: (
58:improve
2362:Subnet
2342:Filter
2292:Normed
2277:Banach
2243:&
2150:Better
2087:Strict
2077:Graded
1968:topics
1799:Topics
1727:
1717:
1677:
1667:
1606:
1596:
1577:
1552:
1460:, 2002
1393:
1363:p. 141
1128:is in
1066:is in
800:ideal
798:proper
747:is in
721:For a
460:proper
439:filter
211:is in
163:is an
47:, but
2352:Ideal
2330:Graph
2126:Total
2104:Total
1990:Dense
1494:arXiv
1482:(PDF)
1346:Notes
1304:Frink
1034:with
897:Proof
873:from
788:is a
623:, and
473:is a
437:is a
414:with
270:is a
219:is a
165:ideal
101:ideal
99:, an
1943:list
1725:PMID
1675:PMID
1594:ISBN
1575:ISBN
1550:ISBN
1514:and
1391:ISBN
1256:dcpo
1253:free
1224:are
1085:and
977:= {
938:nor
916:and
850:and
842:and
630:and
481:and
451:and
392:dual
390:The
256:and
189:and
174:is
2357:Net
2157:Pre
1747:doi
1715:PMC
1705:doi
1665:PMC
1655:doi
1622:AMS
1291:of
1212:In
1148:of
1052:in
1044:in
1030:in
950:in
930:in
869:is
830:, ¬
678:or
664:in
634:of
526:= {
364:of
334:in
242:in
234:in
193:in
185:in
111:of
92:In
2459::
1743:61
1741:,
1723:,
1713:,
1703:,
1693:21
1691:,
1673:,
1663:,
1653:,
1643:20
1641:,
1620:,
1604:MR
1602:,
1488:,
1484:,
1456:,
1405:^
1389:.
1361:,
1306:.
985:≤
981:|
954:,
854:.
638:,
544:.
538:≤
534:|
522:↓
466:.
447:,
441:.
372:.
330:,
274:).
261:≤
251:≤
230:,
223:),
202:≤
197:,
119:.
2155:(
2152:)
2148:(
1999:(
1946:)
1780:e
1773:t
1766:v
1749::
1707::
1699::
1657::
1649::
1583:.
1558:.
1496::
1490:1
1399:.
1387:3
1367:X
1260:P
1249:P
1245:P
1190:M
1182:M
1174:F
1170:M
1166:a
1162:b
1158:F
1154:b
1150:M
1146:n
1142:n
1138:M
1134:M
1130:F
1126:)
1124:b
1120:a
1116:n
1112:m
1110:(
1106:F
1102:F
1097:a
1093:n
1089:m
1087:(
1082:b
1078:n
1074:m
1072:(
1068:F
1063:b
1059:n
1054:M
1050:n
1046:F
1041:a
1037:m
1032:M
1028:m
1021:M
1017:M
1013:M
1009:F
1005:N
1001:}
999:M
995:m
991:a
987:m
983:x
979:x
975:N
970:N
966:F
961:a
957:m
952:M
948:m
944:M
940:b
936:a
932:M
927:b
923:a
918:b
914:a
910:M
906:F
902:M
891:M
887:F
883:I
879:M
875:F
867:I
863:F
859:I
836:a
832:a
828:a
810:F
806:I
802:J
786:I
759:I
755:A
750:I
744:A
735:I
706:I
700:P
688:.
685:I
681:y
675:I
671:x
666:I
652:y
646:x
636:P
632:y
628:x
621:P
617:I
600:)
594:,
591:P
588:(
578:I
571:s
542:}
540:p
536:x
532:P
528:x
524:p
518:p
504:p
483:p
471:p
464:P
425:,
381:P
370:I
366:P
352:y
346:x
336:I
332:y
328:x
306:)
300:,
297:P
294:(
284:I
268:I
263:z
259:y
253:z
249:x
244:I
240:z
236:I
232:y
228:x
217:I
213:I
209:y
204:x
200:y
195:P
191:y
187:I
183:x
178:,
172:I
151:)
145:,
142:P
139:(
129:I
83:)
77:(
72:)
68:(
54:.
20:)
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