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was absent, it would be inserted for the reflexive closure. For example, if on the same set
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25:
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97:
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is already reflexive by itself, so it does not differ from its reflexive closure.
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951: – Smallest transitive relation containing a given binary relation
924:{\displaystyle S=R\cup \{(x,x):x\in X\}=\{(1,1),(2,2),(3,3),(4,4)\}.}
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Pages displaying wikidata descriptions as a fallback
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639:{\displaystyle R=\{(1,1),(2,2),(3,3),(4,4)\}}
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425:In plain English, the reflexive closure of
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1651:Positive cone of a partially ordered group
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1634:Positive cone of an ordered vector space
974:, Cambridge University Press, 1998, p. 8
416:{\displaystyle S=R\cup \{(x,x):x\in X\}}
781:{\displaystyle R=\{(1,1),(2,2),(4,4)\}}
942: – operation on binary relations
7:
1676:
1674:
1722:. You can help Knowledge (XXG) by
1687:
1161:Properties & Types (
14:
1784:Programming language theory stubs
1617:Positive cone of an ordered field
230:", then the reflexive closure of
164:is a set of distinct numbers and
1697:{\displaystyle \Gamma \!\vdash }
1471:Ordered topological vector space
669:However, if any of the pairs in
121:if it relates every element of
912:
900:
894:
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788:then the reflexive closure is
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1:
1428:Series-parallel partial order
542:{\displaystyle X=\{1,2,3,4\}}
1107:Cantor's isomorphism theorem
1712:programming language theory
1147:Szpilrajn extension theorem
1122:Hausdorff maximal principle
1097:Boolean prime ideal theorem
971:Term Rewriting and All That
1800:
1673:
1493:Topological vector lattice
1015:
1102:Cantor–Bernstein theorem
1646:Partially ordered group
1466:Specialization preorder
1718:-related article is a
1698:
1132:Kruskal's tree theorem
1127:Knaster–Tarski theorem
1117:Dushnik–Miller theorem
925:
782:
703:
683:
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640:
543:
486:
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301:The reflexive closure
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270:is less than or equal
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117:A relation is called
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42:
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1624:Ordered vector space
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54:
31:
1462:Alexandrov topology
1408:Lexicographic order
1367:Well-quasi-ordering
183:{\displaystyle xRy}
1694:
1443:Transitive closure
1403:Converse/Transpose
1112:Dilworth's theorem
949:Transitive closure
921:
778:
699:
679:
656:
646:then the relation
636:
539:
499:As an example, if
485:{\displaystyle X.}
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110:{\displaystyle R.}
107:
84:
72:reflexive relation
60:
37:
1779:Rewriting systems
1774:Closure operators
1731:
1730:
1671:
1670:
1629:Partially ordered
1438:Symmetric closure
1423:Reflexive closure
1166:
940:Symmetric closure
702:{\displaystyle X}
682:{\displaystyle R}
659:{\displaystyle R}
467:identity relation
458:{\displaystyle R}
438:{\displaystyle R}
354:{\displaystyle X}
334:{\displaystyle R}
314:{\displaystyle S}
285:{\displaystyle y}
263:{\displaystyle x}
250:is the relation "
243:{\displaystyle R}
223:{\displaystyle y}
203:{\displaystyle x}
157:{\displaystyle X}
134:{\displaystyle X}
87:{\displaystyle X}
63:{\displaystyle X}
40:{\displaystyle R}
22:reflexive closure
1791:
1769:Binary relations
1752:
1745:
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1413:Linear extension
1162:
1142:Mirsky's theorem
1002:
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445:is the union of
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144:For example, if
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70:is the smallest
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1663:Young's lattice
1519:
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1236:Heyting algebra
1184:Boolean algebra
1156:
1137:Laver's theorem
1085:
1051:Boolean algebra
1046:Binary relation
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26:binary relation
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1590:Order morphism
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1508:Locally convex
1505:
1500:
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1488:Order topology
1485:
1484:
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1481:Order topology
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1293:Chain-complete
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321:of a relation
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259:
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219:
199:
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176:
173:
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130:
120:
106:
103:
94:that contains
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59:
36:
13:
10:
9:
6:
4:
3:
2:
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1721:
1717:
1713:
1708:
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1647:
1644:
1640:
1637:
1635:
1632:
1630:
1627:
1626:
1625:
1622:
1618:
1615:
1614:
1613:
1612:Ordered field
1610:
1608:
1605:
1601:
1598:
1596:
1593:
1592:
1591:
1588:
1584:
1581:
1580:
1579:
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1574:
1571:
1569:
1568:Hasse diagram
1566:
1564:
1561:
1559:
1556:
1552:
1549:
1548:
1547:
1546:Comparability
1544:
1542:
1539:
1537:
1534:
1532:
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1528:
1526:
1522:
1514:
1511:
1509:
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1504:
1501:
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1489:
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1482:
1479:
1477:
1474:
1473:
1472:
1469:
1467:
1463:
1460:
1459:
1457:
1454:
1450:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1418:Product order
1416:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1395:
1393:
1391:Constructions
1389:
1383:
1379:
1375:
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1368:
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1360:
1358:
1355:
1353:
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1326:
1323:
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1321:
1318:
1316:
1313:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1291:
1290:
1289:
1288:Partial order
1286:
1284:
1281:
1277:
1276:Join and meet
1274:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1253:
1252:
1249:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1223:
1219:
1217:
1214:
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1209:
1207:
1204:
1202:
1199:
1197:
1194:
1190:
1187:
1186:
1185:
1182:
1180:
1177:
1175:
1174:Antisymmetric
1172:
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1159:
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1100:
1098:
1095:
1094:
1092:
1088:
1082:
1081:Weak ordering
1079:
1077:
1074:
1072:
1069:
1067:
1066:Partial order
1064:
1062:
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1057:
1054:
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1021:
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1010:
1003:
998:
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991:
989:
984:
983:
980:
973:
972:
967:
966:Tobias Nipkow
963:
960:
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955:
950:
947:
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579:
573:
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567:
558:
555:
533:
530:
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524:
521:
518:
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509:
506:
494:
492:
479:
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468:
452:
432:
423:
407:
404:
401:
398:
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389:
386:
377:
374:
371:
368:
348:
328:
308:
296:
294:
279:
257:
237:
217:
210:is less than
197:
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151:
142:
128:
118:
104:
101:
81:
73:
57:
50:
34:
27:
23:
19:
1724:expanding it
1709:
1455:& Orders
1433:Star product
1422:
1362:Well-founded
1315:Prefix order
1271:Distributive
1261:Complemented
1231:Foundational
1196:Completeness
1152:Zorn's lemma
1056:Cyclic order
1039:Key concepts
1009:Order theory
969:
962:Franz Baader
668:
498:
424:
361:is given by
300:
143:
141:to itself.
21:
15:
1716:type theory
1639:Riesz space
1600:Isomorphism
1476:Normal cone
1398:Composition
1332:Semilattice
1241:Homogeneous
1226:Equivalence
1076:Total order
18:mathematics
1763:Categories
1607:Order type
1541:Cofinality
1382:Well-order
1357:Transitive
1246:Idempotent
1179:Asymmetric
956:References
297:Definition
1692:⊢
1688:Γ
1658:Upper set
1595:Embedding
1531:Antichain
1352:Tolerance
1342:Symmetric
1337:Semiorder
1283:Reflexive
1201:Connected
832:∈
805:∪
465:with the
405:∈
378:∪
341:on a set
119:reflexive
1453:Topology
1320:Preorder
1303:Eulerian
1266:Complete
1216:Directed
1206:Covering
1071:Preorder
1030:Category
1025:Glossary
934:See also
1558:Duality
1536:Cofinal
1524:Related
1503:Fréchet
1380:)
1256:Bounded
1251:Lattice
1224:)
1222:Partial
1090:Results
1061:Lattice
495:Example
190:means "
1583:Subnet
1563:Filter
1513:Normed
1498:Banach
1464:&
1371:Better
1308:Strict
1298:Graded
1189:topics
1020:Topics
20:, the
1710:This
1573:Ideal
1551:Graph
1347:Total
1325:Total
1211:Dense
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