1958:
454:
of inversion. A subset of a group that is closed under multiplication and inversion is also closed under the nullary operation (that is, it contains the identity) if and only if it is non-empty. So, a non-empty subset of a group that is closed under multiplication and inversion is a group that is
235:
The concepts of closed sets and closure are often extended to any property of subsets that are stable under intersection; that is, every intersection of subsets that have the property has also the property. For example, in
1686:
A closure on the subsets of a given set may be defined either by a closure operator or by a set of closed sets that is stable under intersection and includes the given set. These two definitions are equivalent.
1483:
1562:
774:
266:
1728:
1419:
202:
619:
380:. It follows that, in a specific example, when closeness is proved, there is no need to check the axioms for proving that a substructure is a structure of the same type.
1334:
678:
544:
1597:
928:
811:
339:
1292:
1645:
707:
1826:
1791:
1360:
1027:
878:
992:
960:
843:
1862:
1757:
578:
376:, including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as
2040:
2071:
485:
Similar examples can be given for almost every algebraic structures, with, sometimes some specific terminology. For example, in a
57:
of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the
720:
1172:
of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language.
1831:
Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator
1872:
87:
of a subset under some operations is the smallest superset that is closed under these operations. It is often called the
2030:
478:
of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of
2119:
1430:
365:
35:
1500:
2124:
1663:
621:
Many properties or operations on relations can be used to define closures. Some of the most common ones follow:
341:
in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely
1666:, axioms K2, K3, K4' correspond to the above defining properties. An example not operating on subsets is the
1891:
304:
54:
1228:
In the preceding sections, closures are considered for subsets of a given set. The subsets of a set form a
1260:
342:
316:
312:
1927:
1899:
1229:
1191:
431:
225:
1957:
239:
1918:
is an example, where there are an infinity of input elements and the result is not always defined. If
1058:
2114:
1923:
1911:
1659:
1184:
1127:
1120:
895:
490:
427:
346:
300:
1875:, if one replace "closed sets" by "closed elements" and "intersection" by "greatest lower bound".
1371:
181:
1963:
1697:
1195:
1034:
778:
625:
583:
479:
31:
1304:
645:
520:
459:. The subgroup generated by a single element, that is, the closure of this element, is called a
1567:
907:
790:
321:
158:
The main property of closed sets, which results immediately from the definition, is that every
2087:
2067:
2036:
1907:
1265:
1180:
1101:
1094:
901:
885:
710:
112:
46:
1612:
683:
2059:
1895:
1796:
1766:
1667:
1339:
1223:
1131:
1112:
997:
848:
486:
439:
403:. In the context of algebraic structures, this closure is generally called the substructure
350:
159:
80:
17:
965:
933:
816:
1838:
1733:
1206:
1165:
1116:
1105:
784:
506:
494:
451:
443:
354:
280:
2032:
Matrix
Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory
557:
1980:
1648:
1169:
1090:
709:
As every intersection of reflexive relations is reflexive, this defines a closure. The
467:
420:
270:
221:
58:
2090:
2005:
2108:
1233:
1210:
276:
474:(under vector-space operations, that is, addition and scalar multiplication) is the
1176:
547:
471:
460:
30:
This article is about closures in general. For the specific use in topology, see
1142:
475:
92:
1953:
1935:
1365:
1150:
2047:...convex hull of S, denoted by coS, is the smallest convex set containing S.
2095:
1424:
1203:
1138:
1042:
456:
72:
of operations if it is closed under each of the operations individually.
1240:
allow generalizing the concept of closure to any partially ordered set.
1071:
447:
2066:. Colloquium Publications. Vol. 25. Am. Math. Soc. p. 111.
1158:
1049:
of a relation is the smallest preorder containing it. Similarly, the
1045:
is a relation that is reflective and transitive. It follows that the
308:
279:, is the set of the common zeros of a family of polynomials, and the
42:
1871:
This equivalence remains true for partially ordered sets with the
162:
of closed sets is a closed set. It follows that for every subset
115:
equipped with one or several methods for producing elements of
1187:
elements is the smallest normal subgroup containing the set.
204:(it is the intersection of all closed subsets that contain
1694:
implies that an intersection of closed sets is closed: if
135:
under these methods, if, when all input elements are in
450:
operation that results in the identity element and the
1942:
is another example where the result may be not unique.
1700:
1690:
Indeed, the defining properties of a closure operator
287:
of points is the smallest algebraic set that contains
61:
are closed under addition, but not under subtraction:
1841:
1799:
1769:
1736:
1615:
1570:
1503:
1433:
1374:
1342:
1307:
1268:
1000:
968:
936:
910:
851:
819:
793:
723:
686:
648:
586:
560:
523:
324:
242:
184:
65:
is not a natural number, although both 1 and 2 are.
372:is a subset that is closed under all operations of
1864:is the intersection of the closed sets containing
1856:
1820:
1785:
1751:
1722:
1639:
1591:
1556:
1477:
1413:
1354:
1328:
1286:
1037:of a relation is its closure under this operation.
1021:
986:
954:
922:
872:
837:
805:
768:
701:
672:
613:
572:
538:
333:
260:
196:
1674:to the smallest integer that is not smaller than
68:Similarly, a subset is said to be closed under a
1033:if it is closed under this operation, and the
884:if it is closed under this operation, and the
360:In this context, given an algebraic structure
1478:{\displaystyle x\leq y\implies C(x)\leq C(y)}
8:
1557:{\displaystyle x\leq C(y)\iff C(x)\leq C(y)}
1198:, the closure of a collection of subsets of
760:
730:
769:{\displaystyle R\cup \{(x,x)\mid x\in A\}.}
2035:. Princeton University Press. p. 25.
1526:
1522:
1447:
1443:
1840:
1798:
1774:
1768:
1735:
1714:
1699:
1614:
1569:
1502:
1432:
1373:
1341:
1306:
1267:
999:
967:
935:
909:
850:
818:
792:
722:
685:
647:
585:
559:
522:
470:, the closure of a non-empty subset of a
446:. Here, the auxiliary operations are the
323:
249:
245:
244:
241:
183:
1730:is an intersection of closed sets, then
489:, the closure of a single element under
139:, then all possible results are also in
1971:
1884:
399:that is closed under all operations of
34:. For the use in computer science, see
1609:if it is its own closure, that is, if
1051:reflexive transitive symmetric closure
1651:it is the closure of some element of
1647:By idempotency, an element is closed
7:
1247:whose partial order is denoted with
170:, there is a smallest closed subset
1828:by definition of the intersection.
892:is its closure under this relation.
143:. Sometimes, one may also say that
325:
25:
1902:are examples of such methods. If
442:, such that every element has an
349:for details. A set with a single
261:{\displaystyle \mathbb {C} ^{n},}
27:Operation on the subsets of a set
1956:
1682:Closure operator vs. closed sets
395:is the smallest substructure of
1670:, which maps every real number
900:Transitivity is defined by the
79:of a subset is the result of a
1851:
1845:
1809:
1803:
1746:
1740:
1631:
1625:
1551:
1545:
1536:
1530:
1523:
1519:
1513:
1489:Equivalently, a function from
1472:
1466:
1457:
1451:
1444:
1408:
1402:
1393:
1390:
1384:
1378:
1323:
1317:
1278:
1183:or normal closure of a set of
1057:of a relation is the smallest
1013:
1001:
981:
969:
949:
937:
864:
852:
832:
820:
745:
733:
661:
649:
599:
587:
1:
2029:Bernstein, Dennis S. (2005).
1873:greatest-lower-bound property
1664:Kuratowski's characterization
208:). Depending on the context,
1723:{\textstyle X=\bigcap X_{i}}
1414:{\displaystyle C(C(x))=C(x)}
1213:generated by the collection.
1047:reflexive transitive closure
197:{\displaystyle Y\subseteq X}
18:Reflexive transitive closure
1078:is the largest superset of
614:{\displaystyle (x,y)\in R.}
513:can be defined as a subset
482:of elements of the subset.
353:that is closed is called a
83:applied to the subset. The
2141:
1763:and be contained in every
1329:{\displaystyle x\leq C(x)}
1221:
1149:of points is the smallest
1082:that has the same rank as
673:{\displaystyle (x,x)\in R}
539:{\displaystyle A\times A,}
387:of an algebraic structure
315:. Some axioms may contain
36:closure (computer science)
29:
1592:{\displaystyle x,y\in S.}
1497:is a closure operator if
923:{\displaystyle A\times A}
806:{\displaystyle A\times A}
334:{\displaystyle \exists ;}
1287:{\displaystyle C:S\to S}
902:partial binary operation
1640:{\displaystyle x=C(x).}
1074:theory, the closure of
702:{\displaystyle x\in A.}
493:operations is called a
317:existential quantifiers
303:is a set equipped with
295:In algebraic structures
119:from other elements of
1858:
1822:
1821:{\displaystyle C(X)=X}
1787:
1786:{\displaystyle X_{i}.}
1753:
1724:
1641:
1593:
1558:
1479:
1415:
1356:
1355:{\displaystyle x\in S}
1330:
1288:
1023:
1022:{\displaystyle (x,z).}
988:
956:
924:
874:
873:{\displaystyle (y,x).}
839:
807:
770:
703:
674:
615:
574:
540:
423:of the substructure.
343:universally quantified
335:
311:. These axioms may be
262:
198:
2010:mathworld.wolfram.com
1985:mathworld.wolfram.com
1938:with coefficients in
1900:multivariate function
1859:
1823:
1788:
1754:
1725:
1642:
1594:
1559:
1480:
1416:
1357:
1331:
1289:
1230:partially ordered set
1192:mathematical analysis
1024:
989:
987:{\displaystyle (y,z)}
957:
955:{\displaystyle (x,y)}
925:
875:
840:
838:{\displaystyle (x,y)}
808:
771:
704:
675:
616:
580:is commonly used for
575:
541:
432:associative operation
336:
263:
199:
1981:"Transitive Closure"
1857:{\displaystyle C(X)}
1839:
1797:
1767:
1752:{\displaystyle C(X)}
1734:
1698:
1613:
1568:
1501:
1431:
1372:
1340:
1305:
1266:
1059:equivalence relation
998:
966:
934:
908:
849:
817:
791:
721:
684:
646:
584:
558:
521:
415:, and one says that
322:
240:
182:
2091:"Algebraic Closure"
2006:"Algebraic Closure"
2004:Weisstein, Eric W.
1979:Weisstein, Eric W.
1912:limit of a sequence
1660:topological closure
1128:radical of an ideal
1055:equivalence closure
573:{\displaystyle xRy}
480:linear combinations
347:Algebraic structure
301:algebraic structure
275:, also known as an
2088:Weisstein, Eric W.
1964:Mathematics portal
1854:
1818:
1783:
1749:
1720:
1658:An example is the
1637:
1589:
1554:
1475:
1411:
1352:
1326:
1284:
1196:probability theory
1091:transitive closure
1061:that contains it.
1035:transitive closure
1019:
984:
952:
920:
870:
835:
803:
766:
699:
670:
611:
570:
536:
331:
307:that satisfy some
258:
194:
41:In mathematics, a
32:Closure (topology)
2120:Closure operators
2060:Birkhoff, Garrett
2042:978-0-691-11802-4
1908:topological space
1238:Closure operators
1181:conjugate closure
1123:that contains it.
1102:algebraic closure
886:symmetric closure
711:reflexive closure
430:is a set with an
391:, the closure of
16:(Redirected from
2132:
2125:Abstract algebra
2101:
2100:
2078:
2077:
2056:
2050:
2049:
2026:
2020:
2019:
2017:
2016:
2001:
1995:
1994:
1992:
1991:
1976:
1966:
1961:
1960:
1943:
1941:
1933:
1921:
1917:
1905:
1889:
1867:
1863:
1861:
1860:
1855:
1834:
1827:
1825:
1824:
1819:
1792:
1790:
1789:
1784:
1779:
1778:
1762:
1758:
1756:
1755:
1750:
1729:
1727:
1726:
1721:
1719:
1718:
1693:
1677:
1673:
1668:ceiling function
1654:
1646:
1644:
1643:
1638:
1604:
1598:
1596:
1595:
1590:
1563:
1561:
1560:
1555:
1496:
1492:
1484:
1482:
1481:
1476:
1420:
1418:
1417:
1412:
1361:
1359:
1358:
1353:
1335:
1333:
1332:
1327:
1293:
1291:
1290:
1285:
1258:
1253:closure operator
1250:
1246:
1224:Closure operator
1218:Closure operator
1166:formal languages
1132:commutative ring
1113:integral closure
1028:
1026:
1025:
1020:
993:
991:
990:
985:
961:
959:
958:
953:
929:
927:
926:
921:
891:
879:
877:
876:
871:
844:
842:
841:
836:
812:
810:
809:
804:
783:Symmetry is the
775:
773:
772:
767:
716:
708:
706:
705:
700:
679:
677:
676:
671:
637:
633:
620:
618:
617:
612:
579:
577:
576:
571:
553:
545:
543:
542:
537:
516:
512:
501:Binary relations
487:commutative ring
440:identity element
418:
414:
402:
398:
394:
390:
386:
379:
375:
371:
363:
351:binary operation
340:
338:
337:
332:
290:
286:
267:
265:
264:
259:
254:
253:
248:
231:
219:
211:
207:
203:
201:
200:
195:
177:
173:
169:
165:
153:
152:
151:closure property
146:
142:
138:
130:
126:
122:
118:
110:
81:closure operator
64:
21:
2140:
2139:
2135:
2134:
2133:
2131:
2130:
2129:
2105:
2104:
2086:
2085:
2082:
2081:
2074:
2058:
2057:
2053:
2043:
2028:
2027:
2023:
2014:
2012:
2003:
2002:
1998:
1989:
1987:
1978:
1977:
1973:
1962:
1955:
1952:
1947:
1946:
1939:
1931:
1919:
1915:
1914:of elements of
1903:
1890:
1886:
1881:
1865:
1837:
1836:
1832:
1795:
1794:
1770:
1765:
1764:
1760:
1732:
1731:
1710:
1696:
1695:
1691:
1684:
1675:
1671:
1652:
1611:
1610:
1602:
1566:
1565:
1499:
1498:
1494:
1490:
1429:
1428:
1370:
1369:
1338:
1337:
1303:
1302:
1264:
1263:
1256:
1248:
1244:
1226:
1220:
1117:integral domain
1067:
996:
995:
964:
963:
932:
931:
906:
905:
889:
847:
846:
815:
814:
789:
788:
785:unary operation
719:
718:
714:
682:
681:
644:
643:
635:
631:
582:
581:
556:
555:
554:. The notation
551:
550:of elements of
546:the set of the
519:
518:
514:
510:
507:binary relation
503:
495:principal ideal
452:unary operation
444:inverse element
434:, often called
426:For example, a
416:
412:
400:
396:
392:
388:
384:
383:Given a subset
377:
373:
369:
361:
320:
319:
297:
288:
284:
281:Zariski closure
243:
238:
237:
229:
217:
209:
205:
180:
179:
175:
171:
167:
163:
150:
149:
144:
140:
136:
128:
124:
120:
116:
108:
105:
62:
59:natural numbers
39:
28:
23:
22:
15:
12:
11:
5:
2138:
2136:
2128:
2127:
2122:
2117:
2107:
2106:
2103:
2102:
2080:
2079:
2072:
2064:Lattice Theory
2051:
2041:
2021:
1996:
1970:
1969:
1968:
1967:
1951:
1948:
1945:
1944:
1883:
1882:
1880:
1877:
1853:
1850:
1847:
1844:
1817:
1814:
1811:
1808:
1805:
1802:
1782:
1777:
1773:
1748:
1745:
1742:
1739:
1717:
1713:
1709:
1706:
1703:
1683:
1680:
1649:if and only if
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1601:An element of
1588:
1585:
1582:
1579:
1576:
1573:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1525:
1521:
1518:
1515:
1512:
1509:
1506:
1487:
1486:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1446:
1442:
1439:
1436:
1422:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1363:
1351:
1348:
1345:
1325:
1322:
1319:
1316:
1313:
1310:
1283:
1280:
1277:
1274:
1271:
1243:Given a poset
1222:Main article:
1219:
1216:
1215:
1214:
1209:is called the
1207:set operations
1204:countably many
1188:
1173:
1170:Kleene closure
1162:
1135:
1124:
1109:
1098:
1087:
1066:
1065:Other examples
1063:
1039:
1038:
1029:A relation is
1018:
1015:
1012:
1009:
1006:
1003:
983:
980:
977:
974:
971:
951:
948:
945:
942:
939:
919:
916:
913:
898:
893:
888:of a relation
880:A relation is
869:
866:
863:
860:
857:
854:
834:
831:
828:
825:
822:
802:
799:
796:
781:
776:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
713:of a relation
698:
695:
692:
689:
669:
666:
663:
660:
657:
654:
651:
628:
610:
607:
604:
601:
598:
595:
592:
589:
569:
566:
563:
535:
532:
529:
526:
502:
499:
468:linear algebra
436:multiplication
421:generating set
345:formulas. See
330:
327:
296:
293:
271:Zariski-closed
257:
252:
247:
212:is called the
193:
190:
187:
131:is said to be
104:
101:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2137:
2126:
2123:
2121:
2118:
2116:
2113:
2112:
2110:
2098:
2097:
2092:
2089:
2084:
2083:
2075:
2073:9780821889534
2069:
2065:
2061:
2055:
2052:
2048:
2044:
2038:
2034:
2033:
2025:
2022:
2011:
2007:
2000:
1997:
1986:
1982:
1975:
1972:
1965:
1959:
1954:
1949:
1937:
1929:
1925:
1913:
1909:
1901:
1897:
1893:
1888:
1885:
1878:
1876:
1874:
1869:
1848:
1842:
1829:
1815:
1812:
1806:
1800:
1793:This implies
1780:
1775:
1771:
1759:must contain
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2115:Set theory
2109:Categories
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1990:2020-07-25
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1936:polynomial
1892:Operations
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1366:idempotent
1299:increasing
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1031:transitive
930:that maps
813:that maps
680:for every
438:, with an
313:identities
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