1404:
531:
501:
337:
307:
277:
153:
1445:
1409:
43:
1335:
1694:
1315:
The correspondence between
Priestley spaces and pairwise Stone spaces is functorial and yields an isomorphism between the category
961:
1718:
1474:
Cignoli, R.; Lafalce, S.; Petrovich, A. (September 1991). "Remarks on
Priestley duality for distributive lattices".
1678:
506:
476:
312:
282:
252:
128:
1574:
1109:
51:
1654:
1552:
Priestley, H. A. (1970). "Representation of distributive lattices by means of ordered Stone spaces".
1435:
1430:
1413:
1232:
1117:
39:
1087:
1324:
1194:
1610:
38:
who introduced and investigated them. Priestley spaces play a fundamental role in the study of
1713:
1690:
419:
95:
31:
1682:
1639:
1597:
1589:
1561:
1485:
965:
415:
35:
1476:
210:
1096:; that is, the topology generated by the subbasis consisting of compact open subsets of
1425:
1182:
984:
1644:
1627:
1707:
1440:
119:
88:
28:
1412:
is that each of these categories is dually equivalent to the category of bounded
1170:
634:
571:
is compact, there exists a finite intersection of these clopen neighborhoods of
20:
1593:
1601:
1166:
533:. By the Priestley separation axiom, there exists a clopen up-set or a clopen
156:
1165:
In fact, this correspondence between
Priestley spaces and spectral spaces is
1575:"Ordered topological spaces and the representation of distributive lattices"
1399:{\displaystyle \mathbf {Spec} \cong \mathbf {Pries} \cong \mathbf {PStone} }
1565:
1686:
1611:"On H. Priestley's dual of the category of bounded distributive lattices"
833:
534:
1489:
54:
of
Priestley spaces and the category of bounded distributive lattices.
1655:"Bitopological duality for distributive lattices and Heyting algebras"
640:
Some further useful properties of
Priestley spaces are listed below.
159:
1653:
Bezhanishvili, G.; Bezhanishvili, N.; Gabelaia, D.; Kurz, A (2010).
975:
denote the category of
Priestley spaces and Priestley morphisms.
587:. This finite intersection is the desired clopen neighborhood
1319:
of
Priestley spaces and Priestley morphisms and the category
637:; that is, it is a compact Hausdorff zero-dimensional space.
34:
with special properties. Priestley spaces are named after
1677:. New Mathematical Monographs. Vol. 35. Cambridge:
1330:
Thus, one has the following isomorphisms of categories:
1673:
Dickmann, Max; Schwartz, Niels; Tressl, Marcus (2019).
510:
480:
316:
286:
256:
132:
1338:
509:
479:
315:
285:
255:
131:
549:. The intersection of these clopen neighborhoods of
1398:
525:
495:
331:
301:
271:
147:
457:. To see this, one proceeds as follows. For each
16:Ordered topological space with special properties
1017:denote the collection of all open down-sets of
1529:Cornish, (1975). Bezhanishvili et al. (2010).
1520:Cornish, (1975). Bezhanishvili et al. (2010).
1193:Priestley spaces are also closely related to
1005:denote the collection of all open up-sets of
526:{\displaystyle \scriptstyle y\,\not \leq \,x}
496:{\displaystyle \scriptstyle x\,\not \leq \,y}
332:{\displaystyle \scriptstyle x\,\not \leq \,y}
309:. Assuming, without loss of generality, that
302:{\displaystyle \scriptstyle y\,\not \leq \,x}
272:{\displaystyle \scriptstyle x\,\not \leq \,y}
148:{\displaystyle \scriptstyle x\,\not \leq \,y}
8:
1628:"Prime ideal structure in commutative rings"
103:, satisfying the following two conditions:
1662:Mathematical Structures in Computer Science
826:(d) Clopen up-sets and clopen down-sets of
811:is an intersection of clopen down-sets of
1643:
1376:
1356:
1339:
1337:
609:It follows that for each Priestley space
518:
514:
508:
488:
484:
478:
324:
320:
314:
294:
290:
284:
264:
260:
254:
140:
136:
130:
1446:Duality theory for distributive lattices
1410:duality theory for distributive lattices
983:Priestley spaces are closely related to
799:is an intersection of clopen up-sets of
1457:
7:
1408:One of the main consequences of the
1189:Connection with bitopological spaces
887:, then there exists a clopen up-set
857:(e) For each pair of closed subsets
1068:Conversely, given a spectral space
778:is a union of clopen down-sets of
195:. (This condition is known as the
14:
1645:10.1090/S0002-9947-1969-0251026-X
1392:
1389:
1386:
1383:
1380:
1377:
1369:
1366:
1363:
1360:
1357:
1349:
1346:
1343:
1340:
1257:is a pairwise Stone space, then
1041:is a Priestley space, then both
766:is a union of clopen up-sets of
339:, (ii) provides a clopen up-set
1297:is the specialization order of
979:Connection with spectral spaces
445:contains a clopen neighborhood
204:Properties of Priestley spaces
1:
1323:of pairwise Stone spaces and
414:Each Priestley space is also
393:are disjoint open subsets of
1538:Bezhanishvili et al. (2010).
1269:is a Priestley space, where
805:and each closed down-set of
42:. In particular, there is a
1511:Bezhanishvili et al. (2010)
1215:is a Priestley space, then
936:to another Priestley space
661:(a) For each closed subset
249:is a partial order, either
213:. Indeed, given two points
1735:
1679:Cambridge University Press
1150:is a spectral space, then
793:(c) Each closed up-set of
772:and each open down-set of
197:Priestley separation axiom
1573:Priestley, H. A. (1972).
68:ordered topological space
1594:10.1112/plms/s3-24.3.507
987:. For a Priestley space
760:(b) Each open up-set of
621:, the topological space
209:Each Priestley space is
1609:Cornish, W. H. (1975).
1181:of spectral spaces and
924:from a Priestley space
1632:Trans. Amer. Math. Soc
1582:Proc. London Math. Soc
1554:Bull. London Math. Soc
1400:
1162:is a Priestley space.
745:are closed subsets of
655:be a Priestley space.
527:
497:
333:
303:
273:
155:, then there exists a
149:
1687:10.1017/9781316543870
1626:Hochster, M. (1969).
1414:distributive lattices
1401:
1065:are spectral spaces.
528:
498:
433:of a Priestley space
334:
304:
274:
223:of a Priestley space
150:
40:distributive lattices
1566:10.1112/blms/2.2.186
1436:Distributive lattice
1431:Pairwise Stone space
1336:
1233:pairwise Stone space
1195:bitopological spaces
1118:specialization order
725: :
689: :
507:
477:
313:
283:
253:
129:
1719:Topological spaces
1602:10338.dmlcz/134149
1490:10.1007/BF00383451
1464:Priestley, (1970).
1396:
1325:bi-continuous maps
922:Priestley morphism
523:
522:
493:
492:
329:
328:
299:
298:
269:
268:
145:
144:
1696:978-1-107-14672-3
1235:. Conversely, if
1177:and the category
1011:. Similarly, let
739:
721:
703:
685:
464:
420:open neighborhood
358:
180:
48:Priestley duality
32:topological space
1726:
1700:
1669:
1659:
1649:
1647:
1622:
1605:
1579:
1569:
1539:
1536:
1530:
1527:
1521:
1518:
1512:
1509:
1503:
1502:Cornish, (1975).
1500:
1494:
1493:
1471:
1465:
1462:
1405:
1403:
1402:
1397:
1395:
1372:
1352:
1311:
1296:
1292:
1283:
1274:
1268:
1256:
1230:
1214:
1161:
1149:
1131:
1115:
1107:
1095:
1085:
1079:
1064:
1052:
1040:
1022:
1016:
1010:
1004:
998:
966:order-preserving
959:
947:
935:
913:
902:
892:
886:
874:
868:
862:
847:
831:
816:
810:
804:
798:
783:
777:
771:
765:
750:
744:
737:
719:
708:
701:
683:
672:
666:
654:
632:
620:
604:
598:
592:
586:
576:
570:
565:. Therefore, as
564:
554:
548:
542:
532:
530:
529:
524:
502:
500:
499:
494:
472:
462:
456:
450:
444:
432:
426:
418:; that is, each
416:zero-dimensional
410:
404:
398:
392:
378:
372:
362:
356:
350:
344:
338:
336:
335:
330:
308:
306:
305:
300:
278:
276:
275:
270:
248:
244:
234:
222:
194:
184:
178:
172:
166:
154:
152:
151:
146:
117:
102:
93:
87:equipped with a
86:
80:
36:Hilary Priestley
1734:
1733:
1729:
1728:
1727:
1725:
1724:
1723:
1704:
1703:
1697:
1675:Spectral Spaces
1672:
1657:
1652:
1625:
1608:
1577:
1572:
1551:
1548:
1543:
1542:
1537:
1533:
1528:
1524:
1519:
1515:
1510:
1506:
1501:
1497:
1473:
1472:
1468:
1463:
1459:
1454:
1422:
1334:
1333:
1309:
1298:
1294:
1291:
1285:
1282:
1276:
1275:is the join of
1270:
1258:
1254:
1247:
1236:
1216:
1204:
1191:
1151:
1139:
1121:
1113:
1097:
1091:
1081:
1069:
1054:
1042:
1030:
1018:
1012:
1006:
1000:
988:
985:spectral spaces
981:
949:
937:
925:
904:
894:
888:
876:
870:
864:
858:
837:
827:
812:
806:
800:
794:
779:
773:
767:
761:
746:
710:
674:
668:
662:
644:
622:
610:
600:
594:
588:
578:
572:
566:
556:
550:
544:
538:
505:
504:
475:
474:
458:
452:
446:
434:
428:
422:
406:
400:
394:
380:
374:
364:
352:
346:
340:
311:
310:
281:
280:
251:
250:
246:
236:
224:
214:
206:
186:
174:
168:
162:
127:
126:
107:
98:
91:
82:
70:
64:Priestley space
60:
50:") between the
25:Priestley space
17:
12:
11:
5:
1732:
1730:
1722:
1721:
1716:
1706:
1705:
1702:
1701:
1695:
1670:
1650:
1623:
1621:(27): 329–332.
1606:
1588:(3): 507–530.
1570:
1560:(2): 186–190.
1547:
1544:
1541:
1540:
1531:
1522:
1513:
1504:
1495:
1484:(3): 299–315.
1466:
1456:
1455:
1453:
1450:
1449:
1448:
1443:
1438:
1433:
1428:
1426:Spectral space
1421:
1418:
1394:
1391:
1388:
1385:
1382:
1379:
1375:
1371:
1368:
1365:
1362:
1359:
1355:
1351:
1348:
1345:
1342:
1307:
1289:
1280:
1252:
1245:
1190:
1187:
1169:and yields an
1088:patch topology
980:
977:
918:
917:
916:
915:
852:
851:
850:
849:
821:
820:
819:
818:
788:
787:
786:
785:
755:
754:
753:
752:
607:
606:
555:does not meet
521:
517:
513:
491:
487:
483:
412:
327:
323:
319:
297:
293:
289:
267:
263:
259:
205:
202:
201:
200:
143:
139:
135:
123:
59:
56:
15:
13:
10:
9:
6:
4:
3:
2:
1731:
1720:
1717:
1715:
1712:
1711:
1709:
1698:
1692:
1688:
1684:
1680:
1676:
1671:
1667:
1663:
1656:
1651:
1646:
1641:
1637:
1633:
1629:
1624:
1620:
1616:
1612:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1576:
1571:
1567:
1563:
1559:
1555:
1550:
1549:
1545:
1535:
1532:
1526:
1523:
1517:
1514:
1508:
1505:
1499:
1496:
1491:
1487:
1483:
1479:
1478:
1470:
1467:
1461:
1458:
1451:
1447:
1444:
1442:
1441:Stone duality
1439:
1437:
1434:
1432:
1429:
1427:
1424:
1423:
1419:
1417:
1415:
1411:
1406:
1373:
1353:
1331:
1328:
1326:
1322:
1318:
1313:
1306:
1302:
1288:
1279:
1273:
1266:
1262:
1251:
1244:
1240:
1234:
1228:
1224:
1220:
1212:
1208:
1202:
1198:
1196:
1188:
1186:
1184:
1183:spectral maps
1180:
1176:
1172:
1168:
1163:
1159:
1155:
1147:
1143:
1137:
1133:
1129:
1125:
1119:
1111:
1105:
1101:
1094:
1089:
1084:
1077:
1073:
1066:
1062:
1058:
1050:
1046:
1038:
1034:
1028:
1024:
1021:
1015:
1009:
1003:
996:
992:
986:
978:
976:
974:
969:
967:
963:
957:
953:
945:
941:
933:
929:
923:
911:
907:
901:
897:
891:
884:
880:
873:
867:
861:
856:
855:
854:
853:
845:
841:
835:
830:
825:
824:
823:
822:
815:
809:
803:
797:
792:
791:
790:
789:
782:
776:
770:
764:
759:
758:
757:
756:
749:
742:
736:
732:
728:
724:
718:
714:
706:
700:
696:
692:
688:
682:
678:
671:
665:
660:
659:
658:
657:
656:
652:
648:
641:
638:
636:
630:
626:
618:
614:
603:
599:contained in
597:
591:
585:
581:
575:
569:
563:
559:
553:
547:
541:
536:
519:
515:
511:
489:
485:
481:
471:
467:
461:
455:
449:
442:
438:
431:
425:
421:
417:
413:
409:
403:
397:
391:
387:
383:
377:
373:. Therefore,
371:
367:
361:
355:
349:
343:
325:
321:
317:
295:
291:
287:
265:
261:
257:
243:
239:
232:
228:
221:
217:
212:
208:
207:
203:
198:
193:
189:
183:
177:
171:
165:
161:
158:
141:
137:
133:
124:
121:
115:
111:
106:
105:
104:
101:
97:
90:
89:partial order
85:
81:, i.e. a set
78:
74:
69:
65:
57:
55:
53:
49:
45:
41:
37:
33:
30:
26:
22:
1674:
1665:
1661:
1635:
1631:
1618:
1614:
1585:
1581:
1557:
1553:
1534:
1525:
1516:
1507:
1498:
1481:
1475:
1469:
1460:
1407:
1332:
1329:
1320:
1316:
1314:
1304:
1300:
1286:
1277:
1271:
1264:
1260:
1249:
1242:
1238:
1226:
1222:
1218:
1210:
1206:
1200:
1199:
1192:
1178:
1174:
1164:
1157:
1153:
1145:
1141:
1135:
1134:
1127:
1123:
1103:
1099:
1092:
1082:
1075:
1071:
1067:
1060:
1056:
1048:
1044:
1036:
1032:
1026:
1025:
1019:
1013:
1007:
1001:
994:
990:
982:
972:
970:
955:
951:
943:
939:
931:
927:
921:
919:
909:
905:
899:
895:
889:
882:
878:
871:
865:
859:
843:
839:
828:
813:
807:
801:
795:
780:
774:
768:
762:
747:
740:
734:
730:
726:
722:
716:
712:
704:
698:
694:
690:
686:
680:
676:
669:
663:
650:
646:
642:
639:
628:
624:
616:
612:
608:
601:
595:
589:
583:
579:
573:
567:
561:
557:
551:
545:
543:and missing
539:
469:
465:
459:
453:
447:
440:
436:
429:
423:
407:
401:
395:
389:
385:
381:
375:
369:
365:
359:
353:
347:
341:
241:
237:
230:
226:
219:
215:
196:
191:
187:
181:
175:
169:
163:
113:
109:
99:
83:
76:
72:
67:
63:
61:
47:
24:
18:
1615:Mat. Vesnik
1171:isomorphism
1116:denote the
1112:. Let also
1110:complements
1086:denote the
950:f :
635:Stone space
537:containing
427:of a point
399:separating
21:mathematics
1708:Categories
1546:References
1167:functorial
1108:and their
962:continuous
893:such that
351:such that
245:, then as
173:such that
58:Definition
1638:: 43–60.
1374:≅
1354:≅
960:which is
948:is a map
733:for some
697:for some
473:, either
211:Hausdorff
1714:Topology
1420:See also
1201:Theorem:
1173:between
1136:Theorem:
1027:Theorem:
834:subbasis
577:missing
535:down-set
516:≰
486:≰
322:≰
292:≰
262:≰
138:≰
96:topology
52:category
832:form a
673:, both
120:compact
44:duality
29:ordered
1693:
1321:PStone
1080:, let
999:, let
160:up-set
157:clopen
94:and a
66:is an
27:is an
1658:(PDF)
1578:(PDF)
1477:Order
1452:Notes
1317:Pries
1231:is a
1175:Pries
973:Pries
946:′,≤′)
875:, if
633:is a
235:, if
1691:ISBN
1293:and
1284:and
1179:Spec
1053:and
971:Let
964:and
903:and
863:and
836:for
715:= {
709:and
643:Let
405:and
379:and
363:and
185:and
23:, a
1683:doi
1640:doi
1636:142
1598:hdl
1590:doi
1562:doi
1486:doi
1267:,≤)
1213:,≤)
1203:If
1160:,≤)
1138:If
1120:of
1090:on
1039:,≤)
1029:If
997:,≤)
934:,≤)
912:= ∅
885:= ∅
881:∩ ↓
869:of
679:= {
667:of
653:,≤)
619:,≤)
593:of
503:or
451:of
443:,≤)
345:of
279:or
233:,≤)
167:of
125:If
118:is
79:,≤)
19:In
1710::
1689:.
1681:.
1666:20
1664:.
1660:.
1634:.
1630:.
1619:12
1617:.
1613:.
1596:.
1586:24
1584:.
1580:.
1556:.
1480:.
1416:.
1327:.
1312:.
1197:.
1185:.
1132:.
1023:.
968:.
954:→
942:′,
920:A
908:∩
898:⊆
743:}
729:≤
711:↓
707:}
693:≤
675:↑
582:−
560:−
468:−
388:−
384:=
368:∉
240:≠
199:.)
190:∉
62:A
46:("
1699:.
1685::
1668:.
1648:.
1642::
1604:.
1600::
1592::
1568:.
1564::
1558:2
1492:.
1488::
1482:8
1393:e
1390:n
1387:o
1384:t
1381:S
1378:P
1370:s
1367:e
1364:i
1361:r
1358:P
1350:c
1347:e
1344:p
1341:S
1310:)
1308:1
1305:τ
1303:,
1301:X
1299:(
1295:≤
1290:2
1287:τ
1281:1
1278:τ
1272:τ
1265:τ
1263:,
1261:X
1259:(
1255:)
1253:2
1250:τ
1248:,
1246:1
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1239:X
1237:(
1229:)
1227:τ
1225:,
1223:τ
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1219:X
1217:(
1211:τ
1209:,
1207:X
1205:(
1158:τ
1156:,
1154:X
1152:(
1148:)
1146:τ
1144:,
1142:X
1140:(
1130:)
1128:τ
1126:,
1124:X
1122:(
1114:≤
1106:)
1104:τ
1102:,
1100:X
1098:(
1093:X
1083:τ
1078:)
1076:τ
1074:,
1072:X
1070:(
1063:)
1061:τ
1059:,
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1055:(
1051:)
1049:τ
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1043:(
1037:τ
1035:,
1033:X
1031:(
1020:X
1014:τ
1008:X
1002:τ
995:τ
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991:X
989:(
958:′
956:X
952:X
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938:(
932:τ
930:,
928:X
926:(
914:.
910:G
906:U
900:U
896:F
890:U
883:G
879:F
877:↑
872:X
866:G
860:F
848:.
846:)
844:τ
842:,
840:X
838:(
829:X
817:.
814:X
808:X
802:X
796:X
784:.
781:X
775:X
769:X
763:X
751:.
748:X
741:F
738:∈
735:y
731:y
727:x
723:X
720:∈
717:x
713:F
705:F
702:∈
699:y
695:x
691:y
687:X
684:∈
681:x
677:F
670:X
664:F
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645:(
631:)
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605:.
602:U
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568:X
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552:x
546:y
540:x
520:x
512:y
490:y
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430:x
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411:.
408:y
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116:)
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