904:
778:
1374:
2607:
652:
2065:
1306:
348:
789:
663:
504:
2657:
45:
is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if
1878:
1226:
2403:
2502:
1971:
222:
189:
119:
1758:
1660:
2447:
1311:
534:
405:
2345:
1427:(the unary operation of taking the converse relation) commutes with other binary operations of union and intersection. Conversion also commutes with unary operation of
2677:
1929:
1835:
2270:
2189:
2155:
2121:
2541:
593:
251:
2299:
1794:
1723:
1692:
1625:
1594:
1193:
428:
162:
2236:
2212:
2087:
1560:
1536:
1516:
1254:
1170:
1126:
1102:
1082:
1054:
1034:
1010:
990:
970:
950:
930:
139:
83:
63:
1404:
Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations as sets), and actually an involutive
2516:, the converse may be composed with the original relation. For example, the subset relation composed with its converse is always the universal relation:
1976:
1263:
3053:
263:
435:
578:) commutes with the order-related operations of the calculus of relations, that is it commutes with union, intersection, and complement.
3863:
2615:
3846:
3376:
899:{\displaystyle {\begin{pmatrix}1&0&0&0\\1&1&0&0\\1&0&1&0\\1&1&0&1\end{pmatrix}}.}
773:{\displaystyle {\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}}.}
3212:
3182:
3153:
3126:
3029:
2977:
2950:
2925:
2887:
3693:
3002:
3829:
3688:
3093:
3074:
2840:
2807:
3683:
1840:
1198:
3319:
2350:
1896:
if and only if its converse relation is a function, in which case the converse relation is the inverse function.
2452:
3401:
1934:
194:
3720:
3640:
2520:∀A ∀B ∅ ⊂ A ∩B ⇔ A ⊃ ∅ ⊂ B ⇔ A ⊃ ⊂ B. Similarly,
2769:
167:
3505:
3434:
3314:
2856:
92:
3901:
3408:
3396:
3359:
3334:
3309:
3263:
3232:
3068:
Guide to
Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications
3705:
3339:
3329:
3205:
3024:. Rosen, Kenneth H., Shier, Douglas R., Goddard, Wayne. (Second ed.). Boca Raton, FL. p. 43.
2879:
1420:
1152:), the converse relation does not satisfy the definition of an inverse from group theory, that is, if
3896:
3678:
3344:
1483:
1257:
551:
3610:
3237:
2513:
1467:
1369:{\displaystyle (L\circ R)^{\operatorname {T} }=R^{\operatorname {T} }\circ L^{\operatorname {T} }.}
1149:
547:
3174:
542:. Although many functions do not have an inverse, every relation does have a unique converse. The
3858:
3841:
3169:
1728:
1630:
2410:
509:
383:
3770:
3386:
2781:
2524:
2306:
1889:
1451:
1409:
1379:
2967:
2917:
2662:
2602:{\displaystyle A\ni z\in B\Leftrightarrow z\in A\cap B\Leftrightarrow A\cap B\neq \emptyset .}
1902:
647:{\displaystyle {\leq ^{\operatorname {T} }}={\geq },\quad {<^{\operatorname {T} }}={>}.}
3748:
3583:
3574:
3443:
3324:
3278:
3242:
3198:
1807:
1383:
559:
27:
2909:
2245:
2164:
2130:
2096:
1428:
3836:
3795:
3785:
3775:
3520:
3483:
3473:
3453:
3438:
1487:
1443:
254:
227:
8:
3763:
3674:
3620:
3579:
3569:
3458:
3391:
3354:
1459:
1455:
2281:
1776:
1705:
1674:
1607:
1576:
1175:
410:
144:
3875:
3802:
3655:
3564:
3554:
3495:
3413:
3349:
3047:
2221:
2197:
2072:
1545:
1521:
1501:
1463:
1447:
1439:
1435:
and infima. Conversion is also compatible with the ordering of relations by inclusion.
1394:
1239:
1155:
1111:
1087:
1067:
1039:
1019:
995:
975:
955:
935:
915:
124:
68:
48:
20:
3715:
2812:
3812:
3790:
3650:
3635:
3418:
3178:
3149:
3122:
3089:
3070:
3035:
3025:
2998:
2973:
2946:
2921:
2910:
2883:
2687:
1479:
1233:
86:
3625:
3478:
2836:
2821:
2274:
2158:
1893:
1145:
1013:
783:
539:
2682:
The compositions are used to classify relations according to type: for a relation
3807:
3590:
3468:
3463:
3448:
3364:
3273:
3258:
3141:
2873:
2869:
2787:
1390:
567:
555:
543:
42:
3114:
1378:
Since one may generally consider relations between different sets (which form a
3725:
3710:
3700:
3559:
3537:
3515:
3164:
3110:
3085:
2532:
2192:
657:
587:
354:
3890:
3824:
3780:
3758:
3630:
3500:
3488:
3293:
3039:
2817:
2813:
Algebra der Logik (Exakte Logik) Dritter Band, Algebra und Logik der
Relative
2060:{\displaystyle \operatorname {graph} \,f^{-1}=\{(y,x)\in Y\times X:y=f(x)\}.}
1471:
3645:
3527:
3510:
3428:
3268:
3221:
2124:
1398:
1141:
3088:& Andre Scedrov (1990) Categories, Allegories, page 79, North Holland
3019:
1301:{\displaystyle \left(L^{\operatorname {T} }\right)^{\operatorname {T} }=L}
3851:
3544:
3423:
3288:
2852:
1475:
343:{\displaystyle L^{\operatorname {T} }=\{(y,x)\in Y\times X:(x,y)\in L\}.}
34:
1256:
in general. The converse relation does satisfy the (weaker) axioms of a
3819:
3753:
3594:
2940:
2215:
590:, the converse is the naively expected "opposite" order, for examples,
2527:, A ∪ B ⊂ U ⇔ A ⊂ U ⊃ B ⇔ A ⊂ ⊃ B.
1393:(aka category with involution). A relation equal to its converse is a
499:{\displaystyle L^{\operatorname {C} },L^{-1},{\breve {L}},L^{\circ },}
3870:
3743:
3549:
2239:
2090:
1389:), in this context the converse relation conforms to the axioms of a
358:
2069:
This is not necessarily a function: One necessary condition is that
1796:
all right and left inverses coincide; this unique set is called its
3665:
3532:
3283:
1432:
1405:
2912:
Relations and Graphs: Discrete
Mathematics for Computer Scientists
2652:{\displaystyle A\ni \in B\Leftrightarrow A\cap B\neq \emptyset .}
1105:
909:
554:
on the binary relations on a set, or, more generally, induces a
3190:
2907:
1137:
1057:
2943:
Nearrings: Some
Developments Linked to Semigroups and Groups
3121:. Springer Science & Business Media. pp. 135–146.
2992:
1061:
361:
of the original, the converse relation is also called the
1128:" is its own converse, since it is a symmetric relation.
357:, and the logical matrix of the converse relation is the
3148:, Springer Lecture Notes in Mathematics #2208, page 8,
798:
672:
16:
Reversal of the order of elements of a binary relation
2784: – Term in the mathematical area of order theory
2665:
2618:
2544:
2455:
2413:
2353:
2309:
2284:
2248:
2224:
2200:
2167:
2133:
2099:
2075:
1979:
1937:
1905:
1843:
1810:
1779:
1731:
1708:
1677:
1633:
1610:
1579:
1548:
1524:
1504:
1314:
1266:
1242:
1201:
1178:
1158:
1114:
1090:
1070:
1042:
1022:
998:
978:
958:
938:
918:
792:
666:
596:
512:
438:
413:
386:
266:
230:
197:
170:
147:
127:
95:
71:
51:
3119:
Relational
Methods for Computer Science Applications
546:
that maps a relation to the converse relation is an
1769:
if it is both right-invertible and left-invertible.
3021:Handbook of discrete and combinatorial mathematics
2997:. World Scientific Publishing Company. p. 9.
2965:
2941:Celestina Cotti Ferrero; Giovanni Ferrero (2002).
2671:
2651:
2601:
2496:
2441:
2397:
2339:
2293:
2264:
2230:
2206:
2183:
2149:
2115:
2081:
2059:
1965:
1923:
1883:
1872:
1829:
1788:
1752:
1717:
1686:
1654:
1619:
1588:
1554:
1530:
1518:represents the identity relation, then a relation
1510:
1368:
1300:
1248:
1220:
1187:
1164:
1120:
1096:
1076:
1048:
1028:
1004:
984:
964:
944:
924:
898:
772:
646:
528:
498:
432:Other notations for the converse relation include
422:
399:
342:
245:
216:
183:
156:
133:
113:
77:
57:
782:Then the converse relation is represented by its
3888:
2709:. When the identity relation on the domain of
3109:
1397:; in the language of dagger categories, it is
3206:
3105:
3103:
3101:
2504:which is not a function, being multi-valued.
1873:{\displaystyle R^{-1}=R^{\operatorname {T} }}
1221:{\displaystyle L\circ L^{\operatorname {T} }}
26:For inverse relationships in statistics, see
2764:is an equivalence relation on the domain of
2507:
2051:
2000:
334:
280:
2972:. Cambridge University Press. p. 173.
2398:{\displaystyle f^{-1}(x)={\frac {x}{2}}-1.}
538:The notation is analogous with that for an
3864:Positive cone of a partially ordered group
3213:
3199:
3098:
3052:: CS1 maint: location missing publisher (
2908:Gunther Schmidt; Thomas Ströhlein (1993).
2790: – Directed graph with reversed edges
2497:{\displaystyle g^{-1}(x)=\pm {\sqrt {x}},}
2945:. Kluwer Academic Publishers. p. 3.
2903:
2901:
2899:
2729:is both univalent and total then it is a
1983:
1966:{\displaystyle f^{-1}\subseteq Y\times X}
353:Since a relation may be represented by a
3847:Positive cone of an ordered vector space
3060:
586:For the usual (maybe strict or partial)
570:, taking the converse (sometimes called
217:{\displaystyle yL^{\operatorname {T} }x}
2916:. Springer Berlin Heidelberg. pp.
2868:
1773:For an invertible homogeneous relation
3889:
3163:
2993:Shlomo Sternberg; Lynn Loomis (2014).
2986:
2969:How to Prove It: A Structured Approach
2896:
2770:Transitive relation#Related properties
184:{\displaystyle L^{\operatorname {T} }}
3194:
3017:
19:For functions decreasing as 1/x, see
3113:(2001). "Relations Old and New". In
2857:A Survey of Symbolic Logic, page 273
1899:The converse relation of a function
114:{\displaystyle L\subseteq X\times Y}
2127:. This condition is sufficient for
656:A relation may be represented by a
550:, so it induces the structure of a
13:
3374:Properties & Types (
2643:
2593:
1865:
1358:
1345:
1332:
1287:
1277:
1213:
627:
603:
444:
272:
206:
176:
14:
3913:
3830:Positive cone of an ordered field
1382:rather than a monoid, namely the
377:of the original relation, or the
3684:Ordered topological vector space
3220:
191:is the relation defined so that
3135:
3079:
1884:Converse relation of a function
620:
563:
3011:
2959:
2934:
2862:
2846:
2843:, page 97 via Internet Archive
2830:
2801:
2628:
2578:
2560:
2475:
2469:
2423:
2417:
2373:
2367:
2319:
2313:
2048:
2042:
2015:
2003:
1915:
1328:
1315:
373:of the original relation, the
365:. It has also been called the
325:
313:
295:
283:
1:
3641:Series-parallel partial order
2794:
1416:is also an ordered category.
1408:. Similarly, the category of
1131:
3320:Cantor's isomorphism theorem
3144:& Michael Winter (2018)
1671:if there exists a relation
1172:is an arbitrary relation on
7:
3360:Szpilrajn extension theorem
3335:Hausdorff maximal principle
3310:Boolean prime ideal theorem
2966:Daniel J. Velleman (2006).
2775:
2679:is the universal relation.
2535:relation and its converse.
2218:. In that case, meaning if
2191:then is a (total) function
1753:{\displaystyle Y\circ R=I.}
1655:{\displaystyle R\circ X=I.}
1573:if there exists a relation
1493:
581:
10:
3918:
3706:Topological vector lattice
3018:Rosen, Kenneth H. (2017).
2880:Cambridge University Press
2442:{\displaystyle g(x)=x^{2}}
2303:For example, the function
529:{\displaystyle L^{\vee }.}
400:{\displaystyle L^{\circ }}
25:
18:
3736:
3664:
3603:
3373:
3302:
3251:
3228:
3117:; Andrzej Szalas (eds.).
2841:Principles of Mathematics
2659:The opposite composition
2508:Composition with relation
2449:has the inverse relation
2347:has the inverse function
2340:{\displaystyle f(x)=2x+2}
1258:semigroup with involution
552:semigroup with involution
3315:Cantor–Bernstein theorem
2672:{\displaystyle \in \ni }
2514:composition of relations
1924:{\displaystyle f:X\to Y}
1150:composition of relations
3859:Partially ordered group
3679:Specialization preorder
3066:Gerard O'Regan (2016):
2161:, and it is clear that
1830:{\displaystyle R^{-1}.}
1490:, its converse is too.
1431:as well as with taking
1410:heterogeneous relations
1148:on relations being the
3345:Kruskal's tree theorem
3340:Knaster–Tarski theorem
3330:Dushnik–Miller theorem
2875:Relational Mathematics
2782:Duality (order theory)
2749:is total, Q is termed
2673:
2653:
2603:
2498:
2443:
2407:However, the function
2399:
2341:
2295:
2266:
2265:{\displaystyle f^{-1}}
2232:
2208:
2185:
2184:{\displaystyle f^{-1}}
2151:
2150:{\displaystyle f^{-1}}
2117:
2116:{\displaystyle f^{-1}}
2083:
2061:
1967:
1925:
1874:
1831:
1790:
1754:
1719:
1688:
1656:
1621:
1590:
1556:
1532:
1512:
1370:
1302:
1250:
1222:
1189:
1166:
1122:
1098:
1078:
1050:
1030:
1006:
986:
966:
946:
926:
912:relations are named: "
900:
774:
648:
530:
500:
424:
401:
344:
247:
218:
185:
158:
135:
115:
79:
59:
2674:
2654:
2604:
2499:
2444:
2400:
2342:
2296:
2267:
2233:
2209:
2186:
2152:
2118:
2084:
2062:
1968:
1926:
1875:
1832:
1804:and it is denoted by
1791:
1755:
1720:
1689:
1657:
1622:
1591:
1557:
1533:
1513:
1486:(weak order), or an
1421:calculus of relations
1384:category of relations
1371:
1303:
1251:
1223:
1190:
1167:
1123:
1099:
1079:
1051:
1031:
1007:
987:
967:
947:
927:
901:
775:
649:
560:category of relations
531:
501:
425:
402:
345:
248:
219:
186:
159:
136:
116:
80:
60:
28:negative relationship
3837:Ordered vector space
3173:, Springer, p.
2859:via Internet Archive
2663:
2616:
2542:
2453:
2411:
2351:
2307:
2282:
2246:
2222:
2198:
2165:
2131:
2097:
2073:
1977:
1935:
1903:
1841:
1808:
1777:
1729:
1706:
1675:
1631:
1608:
1577:
1546:
1522:
1502:
1488:equivalence relation
1312:
1264:
1240:
1199:
1176:
1156:
1112:
1088:
1068:
1040:
1020:
996:
976:
956:
936:
916:
790:
664:
594:
510:
436:
411:
384:
264:
255:set-builder notation
246:{\displaystyle xLy.}
228:
195:
168:
145:
125:
93:
69:
49:
3675:Alexandrov topology
3621:Lexicographic order
3580:Well-quasi-ordering
3146:Relational Topology
2760:is univalent, then
2737:is univalent, then
1144:on a set (with the
121:is a relation from
3902:Mathematical logic
3656:Transitive closure
3616:Converse/Transpose
3325:Dilworth's theorem
2669:
2667:∈ ∋
2649:
2623:∋ ∈
2599:
2531:Now consider the
2494:
2439:
2395:
2337:
2294:{\displaystyle f.}
2291:
2272:may be called the
2262:
2228:
2204:
2181:
2147:
2113:
2079:
2057:
1963:
1921:
1870:
1827:
1789:{\displaystyle R,}
1786:
1750:
1718:{\displaystyle R,}
1715:
1687:{\displaystyle Y,}
1684:
1652:
1620:{\displaystyle R,}
1617:
1589:{\displaystyle X,}
1586:
1552:
1528:
1508:
1395:symmetric relation
1366:
1298:
1246:
1218:
1188:{\displaystyle X,}
1185:
1162:
1118:
1094:
1074:
1046:
1026:
1002:
982:
962:
942:
922:
896:
887:
770:
761:
644:
526:
496:
423:{\displaystyle L.}
420:
397:
363:transpose relation
340:
243:
214:
181:
157:{\displaystyle Y,}
154:
131:
111:
75:
55:
21:inverse proportion
3884:
3883:
3842:Partially ordered
3651:Symmetric closure
3636:Reflexive closure
3379:
3184:978-0-387-90092-6
3154:978-3-319-74450-6
3128:978-3-7908-1365-4
3031:978-1-315-15648-4
2995:Advanced Calculus
2979:978-1-139-45097-3
2952:978-1-4613-0267-4
2927:978-3-642-77970-1
2889:978-0-521-76268-7
2688:identity relation
2489:
2387:
2231:{\displaystyle f}
2207:{\displaystyle f}
2082:{\displaystyle f}
1555:{\displaystyle R}
1531:{\displaystyle R}
1511:{\displaystyle I}
1480:strict weak order
1438:If a relation is
1249:{\displaystyle X}
1234:identity relation
1165:{\displaystyle L}
1121:{\displaystyle B}
1097:{\displaystyle A}
1084:". The relation "
1077:{\displaystyle A}
1049:{\displaystyle B}
1029:{\displaystyle B}
1005:{\displaystyle A}
985:{\displaystyle A}
965:{\displaystyle B}
945:{\displaystyle B}
925:{\displaystyle A}
477:
134:{\displaystyle X}
78:{\displaystyle Y}
58:{\displaystyle X}
3909:
3897:Binary relations
3626:Linear extension
3375:
3355:Mirsky's theorem
3215:
3208:
3201:
3192:
3191:
3187:
3170:Naive Set Theory
3156:
3139:
3133:
3132:
3107:
3096:
3083:
3077:
3064:
3058:
3057:
3051:
3043:
3015:
3009:
3008:
2990:
2984:
2983:
2963:
2957:
2956:
2938:
2932:
2931:
2915:
2905:
2894:
2893:
2870:Schmidt, Gunther
2866:
2860:
2850:
2844:
2837:Bertrand Russell
2834:
2828:
2822:Internet Archive
2805:
2713:is contained in
2690:on the range of
2678:
2676:
2675:
2670:
2658:
2656:
2655:
2650:
2608:
2606:
2605:
2600:
2503:
2501:
2500:
2495:
2490:
2485:
2468:
2467:
2448:
2446:
2445:
2440:
2438:
2437:
2404:
2402:
2401:
2396:
2388:
2380:
2366:
2365:
2346:
2344:
2343:
2338:
2300:
2298:
2297:
2292:
2275:inverse function
2271:
2269:
2268:
2263:
2261:
2260:
2237:
2235:
2234:
2229:
2213:
2211:
2210:
2205:
2190:
2188:
2187:
2182:
2180:
2179:
2159:partial function
2156:
2154:
2153:
2148:
2146:
2145:
2122:
2120:
2119:
2114:
2112:
2111:
2088:
2086:
2085:
2080:
2066:
2064:
2063:
2058:
1996:
1995:
1972:
1970:
1969:
1964:
1950:
1949:
1931:is the relation
1930:
1928:
1927:
1922:
1879:
1877:
1876:
1871:
1869:
1868:
1856:
1855:
1836:
1834:
1833:
1828:
1823:
1822:
1802:
1801:
1795:
1793:
1792:
1787:
1766:
1765:
1759:
1757:
1756:
1751:
1724:
1722:
1721:
1716:
1700:
1699:
1693:
1691:
1690:
1685:
1668:
1667:
1661:
1659:
1658:
1653:
1626:
1624:
1623:
1618:
1602:
1601:
1595:
1593:
1592:
1587:
1570:
1569:
1568:right-invertible
1561:
1559:
1558:
1553:
1537:
1535:
1534:
1529:
1517:
1515:
1514:
1509:
1375:
1373:
1372:
1367:
1362:
1361:
1349:
1348:
1336:
1335:
1307:
1305:
1304:
1299:
1291:
1290:
1285:
1281:
1280:
1255:
1253:
1252:
1247:
1227:
1225:
1224:
1219:
1217:
1216:
1194:
1192:
1191:
1186:
1171:
1169:
1168:
1163:
1146:binary operation
1127:
1125:
1124:
1119:
1103:
1101:
1100:
1095:
1083:
1081:
1080:
1075:
1055:
1053:
1052:
1047:
1036:" has converse "
1035:
1033:
1032:
1027:
1011:
1009:
1008:
1003:
991:
989:
988:
983:
971:
969:
968:
963:
952:" has converse "
951:
949:
948:
943:
931:
929:
928:
923:
908:The converse of
905:
903:
902:
897:
892:
891:
784:transpose matrix
779:
777:
776:
771:
766:
765:
653:
651:
650:
645:
640:
632:
631:
630:
616:
608:
607:
606:
540:inverse function
535:
533:
532:
527:
522:
521:
505:
503:
502:
497:
492:
491:
479:
478:
470:
464:
463:
448:
447:
429:
427:
426:
421:
407:of the relation
406:
404:
403:
398:
396:
395:
349:
347:
346:
341:
276:
275:
252:
250:
249:
244:
223:
221:
220:
215:
210:
209:
190:
188:
187:
182:
180:
179:
163:
161:
160:
155:
140:
138:
137:
132:
120:
118:
117:
112:
84:
82:
81:
76:
64:
62:
61:
56:
3917:
3916:
3912:
3911:
3910:
3908:
3907:
3906:
3887:
3886:
3885:
3880:
3876:Young's lattice
3732:
3660:
3599:
3449:Heyting algebra
3397:Boolean algebra
3369:
3350:Laver's theorem
3298:
3264:Boolean algebra
3259:Binary relation
3247:
3224:
3219:
3185:
3165:Halmos, Paul R.
3160:
3159:
3142:Gunther Schmidt
3140:
3136:
3129:
3108:
3099:
3084:
3080:
3065:
3061:
3045:
3044:
3032:
3016:
3012:
3005:
2991:
2987:
2980:
2964:
2960:
2953:
2939:
2935:
2928:
2906:
2897:
2890:
2867:
2863:
2851:
2847:
2835:
2831:
2806:
2802:
2797:
2788:Transpose graph
2778:
2664:
2661:
2660:
2617:
2614:
2613:
2543:
2540:
2539:
2510:
2484:
2460:
2456:
2454:
2451:
2450:
2433:
2429:
2412:
2409:
2408:
2379:
2358:
2354:
2352:
2349:
2348:
2308:
2305:
2304:
2283:
2280:
2279:
2253:
2249:
2247:
2244:
2243:
2223:
2220:
2219:
2199:
2196:
2195:
2172:
2168:
2166:
2163:
2162:
2138:
2134:
2132:
2129:
2128:
2104:
2100:
2098:
2095:
2094:
2074:
2071:
2070:
1988:
1984:
1978:
1975:
1974:
1973:defined by the
1942:
1938:
1936:
1933:
1932:
1904:
1901:
1900:
1886:
1864:
1860:
1848:
1844:
1842:
1839:
1838:
1815:
1811:
1809:
1806:
1805:
1799:
1798:
1778:
1775:
1774:
1763:
1762:
1730:
1727:
1726:
1725:that satisfies
1707:
1704:
1703:
1697:
1696:
1676:
1673:
1672:
1666:left-invertible
1665:
1664:
1632:
1629:
1628:
1627:that satisfies
1609:
1606:
1605:
1599:
1598:
1578:
1575:
1574:
1567:
1566:
1547:
1544:
1543:
1523:
1520:
1519:
1503:
1500:
1499:
1496:
1429:complementation
1391:dagger category
1357:
1353:
1344:
1340:
1331:
1327:
1313:
1310:
1309:
1286:
1276:
1272:
1268:
1267:
1265:
1262:
1261:
1241:
1238:
1237:
1212:
1208:
1200:
1197:
1196:
1177:
1174:
1173:
1157:
1154:
1153:
1134:
1113:
1110:
1109:
1089:
1086:
1085:
1069:
1066:
1065:
1041:
1038:
1037:
1021:
1018:
1017:
1014:nephew or niece
997:
994:
993:
977:
974:
973:
972:is a parent of
957:
954:
953:
937:
934:
933:
917:
914:
913:
886:
885:
880:
875:
870:
864:
863:
858:
853:
848:
842:
841:
836:
831:
826:
820:
819:
814:
809:
804:
794:
793:
791:
788:
787:
760:
759:
754:
749:
744:
738:
737:
732:
727:
722:
716:
715:
710:
705:
700:
694:
693:
688:
683:
678:
668:
667:
665:
662:
661:
636:
626:
622:
621:
612:
602:
598:
597:
595:
592:
591:
588:order relations
584:
568:unary operation
556:dagger category
544:unary operation
517:
513:
511:
508:
507:
487:
483:
469:
468:
456:
452:
443:
439:
437:
434:
433:
412:
409:
408:
391:
387:
385:
382:
381:
271:
267:
265:
262:
261:
229:
226:
225:
224:if and only if
205:
201:
196:
193:
192:
175:
171:
169:
166:
165:
146:
143:
142:
126:
123:
122:
94:
91:
90:
70:
67:
66:
50:
47:
46:
43:binary relation
31:
24:
17:
12:
11:
5:
3915:
3905:
3904:
3899:
3882:
3881:
3879:
3878:
3873:
3868:
3867:
3866:
3856:
3855:
3854:
3849:
3844:
3834:
3833:
3832:
3822:
3817:
3816:
3815:
3810:
3803:Order morphism
3800:
3799:
3798:
3788:
3783:
3778:
3773:
3768:
3767:
3766:
3756:
3751:
3746:
3740:
3738:
3734:
3733:
3731:
3730:
3729:
3728:
3723:
3721:Locally convex
3718:
3713:
3703:
3701:Order topology
3698:
3697:
3696:
3694:Order topology
3691:
3681:
3671:
3669:
3662:
3661:
3659:
3658:
3653:
3648:
3643:
3638:
3633:
3628:
3623:
3618:
3613:
3607:
3605:
3601:
3600:
3598:
3597:
3587:
3577:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3541:
3540:
3530:
3525:
3524:
3523:
3518:
3513:
3508:
3506:Chain-complete
3498:
3493:
3492:
3491:
3486:
3481:
3476:
3471:
3461:
3456:
3451:
3446:
3441:
3431:
3426:
3421:
3416:
3411:
3406:
3405:
3404:
3394:
3389:
3383:
3381:
3371:
3370:
3368:
3367:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3322:
3317:
3312:
3306:
3304:
3300:
3299:
3297:
3296:
3291:
3286:
3281:
3276:
3271:
3266:
3261:
3255:
3253:
3249:
3248:
3246:
3245:
3240:
3235:
3229:
3226:
3225:
3218:
3217:
3210:
3203:
3195:
3189:
3188:
3183:
3158:
3157:
3134:
3127:
3111:Joachim Lambek
3097:
3086:Peter J. Freyd
3078:
3059:
3030:
3010:
3004:978-9814583930
3003:
2985:
2978:
2958:
2951:
2933:
2926:
2895:
2888:
2882:. p. 39.
2861:
2845:
2829:
2808:Ernst Schröder
2799:
2798:
2796:
2793:
2792:
2791:
2785:
2777:
2774:
2668:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2610:
2609:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2533:set membership
2529:
2528:
2521:
2509:
2506:
2493:
2488:
2483:
2480:
2477:
2474:
2471:
2466:
2463:
2459:
2436:
2432:
2428:
2425:
2422:
2419:
2416:
2394:
2391:
2386:
2383:
2378:
2375:
2372:
2369:
2364:
2361:
2357:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2290:
2287:
2259:
2256:
2252:
2227:
2203:
2193:if and only if
2178:
2175:
2171:
2144:
2141:
2137:
2110:
2107:
2103:
2093:, since else
2078:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1994:
1991:
1987:
1982:
1962:
1959:
1956:
1953:
1948:
1945:
1941:
1920:
1917:
1914:
1911:
1908:
1885:
1882:
1867:
1863:
1859:
1854:
1851:
1847:
1837:In this case,
1826:
1821:
1818:
1814:
1785:
1782:
1771:
1770:
1767:
1760:
1749:
1746:
1743:
1740:
1737:
1734:
1714:
1711:
1683:
1680:
1669:
1662:
1651:
1648:
1645:
1642:
1639:
1636:
1616:
1613:
1585:
1582:
1571:
1551:
1527:
1507:
1495:
1492:
1484:total preorder
1426:
1365:
1360:
1356:
1352:
1347:
1343:
1339:
1334:
1330:
1326:
1323:
1320:
1317:
1297:
1294:
1289:
1284:
1279:
1275:
1271:
1245:
1231:
1215:
1211:
1207:
1204:
1184:
1181:
1161:
1133:
1130:
1117:
1093:
1073:
1045:
1025:
1001:
981:
961:
941:
932:is a child of
921:
895:
890:
884:
881:
879:
876:
874:
871:
869:
866:
865:
862:
859:
857:
854:
852:
849:
847:
844:
843:
840:
837:
835:
832:
830:
827:
825:
822:
821:
818:
815:
813:
810:
808:
805:
803:
800:
799:
797:
769:
764:
758:
755:
753:
750:
748:
745:
743:
740:
739:
736:
733:
731:
728:
726:
723:
721:
718:
717:
714:
711:
709:
706:
704:
701:
699:
696:
695:
692:
689:
687:
684:
682:
679:
677:
674:
673:
671:
658:logical matrix
643:
639:
635:
629:
625:
619:
615:
611:
605:
601:
583:
580:
564:detailed below
525:
520:
516:
495:
490:
486:
482:
476:
473:
467:
462:
459:
455:
451:
446:
442:
419:
416:
394:
390:
355:logical matrix
351:
350:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
274:
270:
242:
239:
236:
233:
213:
208:
204:
200:
178:
174:
153:
150:
130:
110:
107:
104:
101:
98:
74:
54:
15:
9:
6:
4:
3:
2:
3914:
3903:
3900:
3898:
3895:
3894:
3892:
3877:
3874:
3872:
3869:
3865:
3862:
3861:
3860:
3857:
3853:
3850:
3848:
3845:
3843:
3840:
3839:
3838:
3835:
3831:
3828:
3827:
3826:
3825:Ordered field
3823:
3821:
3818:
3814:
3811:
3809:
3806:
3805:
3804:
3801:
3797:
3794:
3793:
3792:
3789:
3787:
3784:
3782:
3781:Hasse diagram
3779:
3777:
3774:
3772:
3769:
3765:
3762:
3761:
3760:
3759:Comparability
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3741:
3739:
3735:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3708:
3707:
3704:
3702:
3699:
3695:
3692:
3690:
3687:
3686:
3685:
3682:
3680:
3676:
3673:
3672:
3670:
3667:
3663:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3631:Product order
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3608:
3606:
3604:Constructions
3602:
3596:
3592:
3588:
3585:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3539:
3536:
3535:
3534:
3531:
3529:
3526:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3503:
3502:
3501:Partial order
3499:
3497:
3494:
3490:
3489:Join and meet
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3466:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3436:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3407:
3403:
3400:
3399:
3398:
3395:
3393:
3390:
3388:
3387:Antisymmetric
3385:
3384:
3382:
3378:
3372:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3307:
3305:
3301:
3295:
3294:Weak ordering
3292:
3290:
3287:
3285:
3282:
3280:
3279:Partial order
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3256:
3254:
3250:
3244:
3241:
3239:
3236:
3234:
3231:
3230:
3227:
3223:
3216:
3211:
3209:
3204:
3202:
3197:
3196:
3193:
3186:
3180:
3176:
3172:
3171:
3166:
3162:
3161:
3155:
3151:
3147:
3143:
3138:
3130:
3124:
3120:
3116:
3112:
3106:
3104:
3102:
3095:
3094:0-444-70368-3
3091:
3087:
3082:
3076:
3075:9783319445618
3072:
3069:
3063:
3055:
3049:
3041:
3037:
3033:
3027:
3023:
3022:
3014:
3006:
3000:
2996:
2989:
2981:
2975:
2971:
2970:
2962:
2954:
2948:
2944:
2937:
2929:
2923:
2919:
2914:
2913:
2904:
2902:
2900:
2891:
2885:
2881:
2878:. Cambridge:
2877:
2876:
2871:
2865:
2858:
2854:
2849:
2842:
2838:
2833:
2827:
2823:
2819:
2818:B. G. Teubner
2815:
2814:
2809:
2804:
2800:
2789:
2786:
2783:
2780:
2779:
2773:
2771:
2767:
2763:
2759:
2754:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2697:
2693:
2689:
2685:
2680:
2666:
2646:
2640:
2637:
2634:
2631:
2625:
2622:
2619:
2596:
2590:
2587:
2584:
2581:
2575:
2572:
2569:
2566:
2563:
2557:
2554:
2551:
2548:
2545:
2538:
2537:
2536:
2534:
2526:
2522:
2519:
2518:
2517:
2515:
2505:
2491:
2486:
2481:
2478:
2472:
2464:
2461:
2457:
2434:
2430:
2426:
2420:
2414:
2405:
2392:
2389:
2384:
2381:
2376:
2370:
2362:
2359:
2355:
2334:
2331:
2328:
2325:
2322:
2316:
2310:
2301:
2288:
2285:
2277:
2276:
2257:
2254:
2250:
2241:
2225:
2217:
2201:
2194:
2176:
2173:
2169:
2160:
2142:
2139:
2135:
2126:
2108:
2105:
2101:
2092:
2076:
2067:
2054:
2045:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2012:
2009:
2006:
1997:
1992:
1989:
1985:
1980:
1960:
1957:
1954:
1951:
1946:
1943:
1939:
1918:
1912:
1909:
1906:
1897:
1895:
1891:
1881:
1861:
1857:
1852:
1849:
1845:
1824:
1819:
1816:
1812:
1803:
1783:
1780:
1768:
1761:
1747:
1744:
1741:
1738:
1735:
1732:
1712:
1709:
1701:
1681:
1678:
1670:
1663:
1649:
1646:
1643:
1640:
1637:
1634:
1614:
1611:
1603:
1600:right inverse
1583:
1580:
1572:
1565:
1564:
1563:
1549:
1541:
1525:
1505:
1491:
1489:
1485:
1481:
1477:
1473:
1472:partial order
1469:
1465:
1461:
1457:
1453:
1452:antisymmetric
1449:
1445:
1441:
1436:
1434:
1430:
1424:
1422:
1417:
1415:
1411:
1407:
1402:
1400:
1396:
1392:
1388:
1385:
1381:
1376:
1363:
1354:
1350:
1341:
1337:
1324:
1321:
1318:
1295:
1292:
1282:
1273:
1269:
1259:
1243:
1235:
1229:
1209:
1205:
1202:
1182:
1179:
1159:
1151:
1147:
1143:
1142:endorelations
1139:
1129:
1115:
1107:
1091:
1071:
1063:
1059:
1043:
1023:
1015:
999:
979:
959:
939:
919:
911:
906:
893:
888:
882:
877:
872:
867:
860:
855:
850:
845:
838:
833:
828:
823:
816:
811:
806:
801:
795:
785:
780:
767:
762:
756:
751:
746:
741:
734:
729:
724:
719:
712:
707:
702:
697:
690:
685:
680:
675:
669:
659:
654:
641:
637:
633:
623:
617:
613:
609:
599:
589:
579:
577:
576:transposition
573:
569:
565:
561:
557:
553:
549:
545:
541:
536:
523:
518:
514:
493:
488:
484:
480:
474:
471:
465:
460:
457:
453:
449:
440:
430:
417:
414:
392:
388:
380:
376:
372:
368:
364:
360:
356:
337:
331:
328:
322:
319:
316:
310:
307:
304:
301:
298:
292:
289:
286:
277:
268:
260:
259:
258:
256:
240:
237:
234:
231:
211:
202:
198:
172:
151:
148:
128:
108:
105:
102:
99:
96:
88:
72:
52:
44:
40:
36:
29:
22:
3668:& Orders
3646:Star product
3615:
3575:Well-founded
3528:Prefix order
3484:Distributive
3474:Complemented
3444:Foundational
3409:Completeness
3365:Zorn's lemma
3269:Cyclic order
3252:Key concepts
3222:Order theory
3168:
3145:
3137:
3118:
3115:Ewa Orłowska
3081:
3067:
3062:
3020:
3013:
2994:
2988:
2968:
2961:
2942:
2936:
2911:
2874:
2864:
2848:
2832:
2825:
2811:
2803:
2765:
2761:
2757:
2755:
2750:
2746:
2742:
2738:
2734:
2730:
2726:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2695:
2691:
2683:
2681:
2611:
2530:
2511:
2406:
2302:
2273:
2125:multi-valued
2068:
1898:
1887:
1797:
1772:
1698:left inverse
1695:
1597:
1542:as follows:
1539:
1538:may have an
1497:
1468:trichotomous
1437:
1418:
1413:
1403:
1399:self-adjoint
1386:
1377:
1135:
907:
781:
655:
585:
575:
571:
537:
431:
378:
374:
370:
366:
362:
352:
38:
32:
3852:Riesz space
3813:Isomorphism
3689:Normal cone
3611:Composition
3545:Semilattice
3454:Homogeneous
3439:Equivalence
3289:Total order
2853:C. I. Lewis
2816:, Leibzig:
2686:, when the
1476:total order
1444:irreflexive
35:mathematics
3891:Categories
3820:Order type
3754:Cofinality
3595:Well-order
3570:Transitive
3459:Idempotent
3392:Asymmetric
2826:Konversion
2810:, (1895),
2795:References
2751:surjective
2741:is termed
2721:is called
2705:is called
2216:surjective
1894:invertible
1764:invertible
1694:called a
1562:is called
1460:transitive
1456:asymmetric
1425:conversion
1232:equal the
1140:of binary
1132:Properties
572:conversion
548:involution
379:reciprocal
3871:Upper set
3808:Embedding
3744:Antichain
3565:Tolerance
3555:Symmetric
3550:Semiorder
3496:Reflexive
3414:Connected
3048:cite book
3040:994604351
2743:injective
2707:univalent
2694:contains
2644:∅
2641:≠
2635:∩
2629:⇔
2594:∅
2591:≠
2585:∩
2579:⇔
2573:∩
2567:∈
2561:⇔
2555:∈
2549:∋
2523:For U =
2482:±
2462:−
2390:−
2360:−
2255:−
2240:bijective
2174:−
2140:−
2106:−
2091:injective
2025:×
2019:∈
1990:−
1958:×
1952:⊆
1944:−
1916:→
1850:−
1817:−
1736:∘
1638:∘
1596:called a
1464:connected
1448:symmetric
1440:reflexive
1351:∘
1322:∘
1206:∘
614:≥
600:≤
519:∨
489:∘
475:˘
458:−
393:∘
359:transpose
329:∈
305:×
299:∈
106:×
100:⊆
3666:Topology
3533:Preorder
3516:Eulerian
3479:Complete
3429:Directed
3419:Covering
3284:Preorder
3243:Category
3238:Glossary
3167:(1974),
2872:(2010).
2824:Seite 3
2776:See also
2731:function
2525:universe
2157:being a
1890:function
1494:Inverses
1406:quantale
1380:category
660:such as
582:Examples
367:opposite
39:converse
3771:Duality
3749:Cofinal
3737:Related
3716:Fréchet
3593:)
3469:Bounded
3464:Lattice
3437:)
3435:Partial
3303:Results
3274:Lattice
2855:(1918)
2839:(1903)
2745:. When
2733:. When
2725:. When
2717:, then
2701:, then
1880:holds.
1800:inverse
1540:inverse
1433:suprema
1419:In the
1136:In the
1106:sibling
910:kinship
566:. As a
558:on the
375:inverse
3796:Subnet
3776:Filter
3726:Normed
3711:Banach
3677:&
3584:Better
3521:Strict
3511:Graded
3402:topics
3233:Topics
3181:
3152:
3125:
3092:
3073:
3038:
3028:
3001:
2976:
2949:
2924:
2886:
2768:, see
2512:Using
1138:monoid
1056:is an
37:, the
3786:Ideal
3764:Graph
3560:Total
3538:Total
3424:Dense
2920:–10.
2723:total
2612:Thus
1981:graph
1228:does
1195:then
1104:is a
1058:uncle
1012:is a
992:". "
164:then
41:of a
3377:list
3179:ISBN
3150:ISBN
3123:ISBN
3090:ISBN
3071:ISBN
3054:link
3036:OCLC
3026:ISBN
2999:ISBN
2974:ISBN
2947:ISBN
2922:ISBN
2884:ISBN
2820:via
1470:, a
1308:and
1062:aunt
638:>
624:<
371:dual
89:and
87:sets
85:are
65:and
3791:Net
3591:Pre
2756:If
2715:Q Q
2278:of
2238:is
2214:is
2123:is
2089:be
1892:is
1702:of
1604:of
1498:If
1414:Rel
1387:Rel
1236:on
1230:not
1108:of
1064:of
1060:or
1016:of
574:or
562:as
506:or
369:or
257:,
253:In
141:to
33:In
3893::
3177:,
3175:40
3100:^
3050:}}
3046:{{
3034:.
2898:^
2772:.
2762:QQ
2753:.
2393:1.
2242:,
1888:A
1482:,
1478:,
1474:,
1466:,
1462:,
1458:,
1454:,
1450:,
1446:,
1442:,
1423:,
1412:,
1401:.
1260::
786::
3589:(
3586:)
3582:(
3433:(
3380:)
3214:e
3207:t
3200:v
3131:.
3056:)
3042:.
3007:.
2982:.
2955:.
2930:.
2918:9
2892:.
2766:Q
2758:Q
2747:Q
2739:Q
2735:Q
2727:Q
2719:Q
2711:Q
2703:Q
2699:Q
2696:Q
2692:Q
2684:Q
2647:.
2638:B
2632:A
2626:B
2620:A
2597:.
2588:B
2582:A
2576:B
2570:A
2564:z
2558:B
2552:z
2546:A
2492:,
2487:x
2479:=
2476:)
2473:x
2470:(
2465:1
2458:g
2435:2
2431:x
2427:=
2424:)
2421:x
2418:(
2415:g
2385:2
2382:x
2377:=
2374:)
2371:x
2368:(
2363:1
2356:f
2335:2
2332:+
2329:x
2326:2
2323:=
2320:)
2317:x
2314:(
2311:f
2289:.
2286:f
2258:1
2251:f
2226:f
2202:f
2177:1
2170:f
2143:1
2136:f
2109:1
2102:f
2077:f
2055:.
2052:}
2049:)
2046:x
2043:(
2040:f
2037:=
2034:y
2031::
2028:X
2022:Y
2016:)
2013:x
2010:,
2007:y
2004:(
2001:{
1998:=
1993:1
1986:f
1961:X
1955:Y
1947:1
1940:f
1919:Y
1913:X
1910::
1907:f
1866:T
1862:R
1858:=
1853:1
1846:R
1825:.
1820:1
1813:R
1784:,
1781:R
1748:.
1745:I
1742:=
1739:R
1733:Y
1713:,
1710:R
1682:,
1679:Y
1650:.
1647:I
1644:=
1641:X
1635:R
1615:,
1612:R
1584:,
1581:X
1550:R
1526:R
1506:I
1364:.
1359:T
1355:L
1346:T
1342:R
1338:=
1333:T
1329:)
1325:R
1319:L
1316:(
1296:L
1293:=
1288:T
1283:)
1278:T
1274:L
1270:(
1244:X
1214:T
1210:L
1203:L
1183:,
1180:X
1160:L
1116:B
1092:A
1072:A
1044:B
1024:B
1000:A
980:A
960:B
940:B
920:A
894:.
889:)
883:1
878:0
873:1
868:1
861:0
856:1
851:0
846:1
839:0
834:0
829:1
824:1
817:0
812:0
807:0
802:1
796:(
768:.
763:)
757:1
752:0
747:0
742:0
735:0
730:1
725:0
720:0
713:1
708:0
703:1
698:0
691:1
686:1
681:1
676:1
670:(
642:.
634:=
628:T
618:,
610:=
604:T
524:.
515:L
494:,
485:L
481:,
472:L
466:,
461:1
454:L
450:,
445:C
441:L
418:.
415:L
389:L
338:.
335:}
332:L
326:)
323:y
320:,
317:x
314:(
311::
308:X
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293:x
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287:y
284:(
281:{
278:=
273:T
269:L
241:.
238:y
235:L
232:x
212:x
207:T
203:L
199:y
177:T
173:L
152:,
149:Y
129:X
109:Y
103:X
97:L
73:Y
53:X
30:.
23:.
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