4154:
4506:
compare to division and produce quotients. Three quotients are exhibited here: left residual, right residual, and symmetric quotient. The left residual of two relations is defined presuming that they have the same domain (source), and the right residual presumes the same codomain (range, target). The
2252:
of two logical matrices will be 1, then, only if the row and column multiplied have a corresponding 1. Thus the logical matrix of a composition of relations can be found by computing the matrix product of the matrices representing the factors of the composition. "Matrices constitute a method for
3644:
4035:
4758:
477:
5455:. The usual composition of two binary relations as defined here can be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle component. For example, in the query language SQL there is the operation
3336:
2184:. The entries of these matrices are either zero or one, depending on whether the relation represented is false or true for the row and column corresponding to compared objects. Working with such matrices involves the Boolean arithmetic with
235:
4390:
4241:
3922:
3511:
5392:
4576:
4644:
1871:
348:
1467:
4449:
4314:
4651:
2881:
4149:{\displaystyle QR\subseteq S\quad {\text{ is equivalent to }}\quad Q^{\textsf {T}}{\bar {S}}\subseteq {\bar {R}}\quad {\text{ is equivalent to }}\quad {\bar {S}}R^{\textsf {T}}\subseteq {\bar {Q}}.}
1576:
3241:
3026:
2925:
1249:
5114:
3971:
1648:
2727:
4825:
2973:
517:
3734:
hence any two languages share a nation where they both are spoken (in fact: Switzerland). Vice versa, the question whether two given nations share a language can be answered using
2302:
850:
172:
4170:
3771:
2653:
1329:
1011:
317:
285:
3705:
5193:
3409:
2780:
5254:
5152:
3506:
3444:
2685:
2481:
2429:
2397:
1762:
1680:
5222:
4923:
4022:
3370:
1122:
1076:
4955:
4857:
5858:
5307:
4886:
4790:
3861:
623:
2246:
1351:
1271:
1208:
561:
947:
920:
774:
699:
661:
3474:
5443:
5059:
5027:
1180:
1043:
4156:
Verbally, one equivalence can be obtained from another: select the first or second factor and transpose it; then complement the other two relations and permute them.
2559:
4995:
2530:
1602:
1297:
1148:
971:
587:
2214:
2052:
1956:
1795:
4319:
890:
3125:
2585:
343:
167:
141:
115:
3801:
3732:
3667:
3172:
3079:
3056:
2075:
1979:
949:
explicitly when necessary, depending whether the left or the right relation is the first one applied. A further variation encountered in computer science is the
5414:
5302:
5282:
4491:
4471:
4263:
3993:
3825:
3232:
3212:
3192:
3145:
3099:
2800:
2605:
2501:
2449:
2362:
2342:
2322:
2161:
2137:
2117:
2097:
2021:
2001:
1925:
1905:
1518:
1490:
1398:
1378:
3870:
6595:
4268:
4517:
6578:
2809:
6108:
3639:{\displaystyle RR^{\textsf {T}}={\begin{pmatrix}1&0&0&1\\0&1&0&1\\0&0&1&1\\1&1&1&1\end{pmatrix}}.}
5944:
4583:
1800:
973:
is used to denote the traditional (right) composition, while left composition is denoted by a fat semicolon. The unicode symbols are ⨾ and ⨟.
5888:
5630:
5533:
5505:
6425:
1407:
4753:{\displaystyle \operatorname {syq} (E,F)\mathrel {:=} {\overline {E^{\textsf {T}}{\bar {F}}}}\cap {\overline {{\bar {E}}^{\textsf {T}}F}}}
4395:
1088:, the morphisms are binary relations and the composition of morphisms is exactly composition of relations as defined above. The category
5677:
6561:
6420:
5916:
5703:
5568:
5653:
4160:
6415:
6638:
6051:
472:{\displaystyle R;S=\{(x,z)\in X\times Z:{\text{ there exists }}y\in Y{\text{ such that }}(x,y)\in R{\text{ and }}(y,z)\in S\}.}
6133:
6452:
6372:
5498:
6046:
1523:
6237:
6166:
5876:
6140:
6128:
6091:
6066:
6041:
5995:
5964:
6071:
6061:
6633:
6437:
5937:
5526:
2978:
2886:
1213:
5064:
3931:
1607:
6410:
6076:
5778:
5447:
4960:
2694:
1874:
6628:
5969:
4795:
2937:
785:
6590:
6573:
3864:
484:
2272:
790:
6502:
6118:
5812:
De Morgan indicated contraries by lower case, conversion as M, and inclusion with )), so his notation was
3832:
3737:
2614:
1493:
1401:
1357:
1302:
1046:
984:
290:
258:
77:
3674:
3331:{\displaystyle {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\1&1&1\end{pmatrix}}.}
6480:
6315:
6306:
6175:
6010:
5974:
5930:
3925:
1080:
61:
6056:
5157:
3379:
2732:
892:
is commonly used in algebra to signify multiplication, so too, it can signify relative multiplication.
5227:
5119:
3479:
3417:
2658:
2454:
2402:
2370:
1702:
1656:
856:
the text sequence from the operation sequence. The small circle was used in the introductory pages of
6568:
6527:
6517:
6507:
6252:
6215:
6205:
6185:
6170:
5198:
4891:
3998:
3346:
1696:
1497:
1097:
1051:
730:
73:
5782:
5673:
4928:
4830:
6495:
6406:
6352:
6311:
6301:
6190:
6123:
6086:
5815:
5801:
5468:
4862:
4766:
3837:
3447:
2803:
2608:
1470:
1354:
592:
87:
indicates a compound relation: for a person to be an uncle, he must be the brother of a parent. In
2219:
1334:
1254:
1191:
522:
6534:
6387:
6296:
6286:
6227:
6145:
5473:
5452:
4164:
3708:
2688:
240:
6607:
6447:
6081:
925:
898:
747:
666:
628:
5737:
5733:
3453:
6544:
6522:
6382:
6367:
6347:
6150:
5912:
5908:
5884:
5774:
5699:
5661:
5626:
5564:
5529:
5501:
5419:
5032:
5000:
4025:
3373:
3341:
3148:
2365:
2168:
1767:
1153:
1085:
1016:
738:
230:{\displaystyle U=BP\quad {\text{ is equivalent to: }}\quad xByPz{\text{ if and only if }}xUz.}
5620:
4502:
Just as composition of relations is a type of multiplication resulting in a product, so some
4385:{\displaystyle RX\subseteq S{\text{ implies }}R^{\textsf {T}}{\bar {S}}\subseteq {\bar {X}},}
4236:{\displaystyle LM\subseteq N{\text{ implies }}{\bar {N}}M^{\textsf {T}}\subseteq {\bar {L}}.}
2535:
6357:
6210:
4974:
2509:
1881:
1581:
1276:
1186:
1127:
1090:
956:
566:
2187:
2030:
1934:
1773:
6539:
6322:
6200:
6195:
6180:
6005:
5990:
5868:
5696:
5657:
5584:
5561:
4503:
3828:
3804:
2164:
866:
734:
722:
88:
35:
6096:
4159:
Though this transformation of an inclusion of a composition of relations was detailed by
3917:{\displaystyle A\subseteq B{\text{ implies }}B^{\complement }\subseteq A^{\complement }.}
3104:
2564:
322:
146:
120:
94:
5650:
3783:
3714:
3649:
3238:, assuming rows (top to bottom) and columns (left to right) are ordered alphabetically:
3154:
3061:
3038:
2057:
1961:
6457:
6442:
6432:
6291:
6269:
6247:
5880:
5603:
5592:
5399:
5287:
5267:
4476:
4456:
4248:
3978:
3810:
3412:
3235:
3217:
3197:
3177:
3130:
3084:
2785:
2590:
2486:
2434:
2347:
2327:
2307:
2249:
2181:
2146:
2122:
2102:
2082:
2006:
1986:
1910:
1890:
1503:
1475:
1383:
1363:
895:
Further with the circle notation, subscripts may be used. Some authors prefer to write
718:
714:
5387:{\displaystyle c\,(<)\,d~\mathrel {:=} ~c;a^{\textsf {T}}\cap \ d;b^{\textsf {T}}.}
6622:
6556:
6512:
6490:
6362:
6232:
6220:
6025:
5688:
5638:
2258:
861:
777:
5607:
17:
6377:
6259:
6242:
6160:
6000:
5953:
5794:
2928:
1651:
733:
with the notation for function composition used (mostly by computer scientists) in
5750:
4571:{\displaystyle A\backslash B\mathrel {:=} {\overline {A^{\textsf {T}}{\bar {B}}}}}
76:
is the special case of composition of relations where all relations involved are
6583:
6276:
6155:
6020:
1683:
31:
3508:
contains a 1 at every position, while the reversed matrix product computes as:
6551:
6485:
6326:
5456:
4245:
With Schröder rules and complementation one can solve for an unknown relation
2024:
950:
6602:
6475:
6281:
5905:
Monoids, Acts and
Categories with Applications to Wreath Products and Graphs
2257:
the conclusions traditionally drawn by means of hypothetical syllogisms and
1928:
781:
710:
244:
247:
has been subsumed by relational logical expressions and their composition.
6397:
6264:
6015:
5797:
4507:
symmetric quotient presumes two relations share a domain and a codomain.
5544:
A. De Morgan (1860) "On the
Syllogism: IV and on the Logic of Relations"
4639:{\displaystyle D/C\mathrel {:=} {\overline {{\bar {D}}C^{\textsf {T}}}}}
1866:{\displaystyle (R\,;S)^{\textsf {T}}=S^{\textsf {T}}\,;R^{\textsf {T}}.}
4167:
first articulated the transformation as
Theorem K in 1860. He wrote
2140:
1462:{\displaystyle {\mathsf {Rel}}({\mathsf {Set}})\cong {\mathsf {Rel}}}
860:
until it was dropped in favor of juxtaposition (no infix notation).
784:
of relations. However, the small circle is widely used to represent
776:
has been used for the infix notation of composition of relations by
737:, as well as the notation for dynamic conjunction within linguistic
5777:(December 1948) "Matrix development of the calculus of relations",
5396:
Another form of composition of relations, which applies to general
5195:
understood as relations, meaning that there are converse relations
4444:{\displaystyle X\subseteq {\overline {R^{\textsf {T}}{\bar {S}}}},}
4309:{\displaystyle RX\subseteq S\quad {\text{and}}\quad XR\subseteq S.}
3646:
This matrix is symmetric, and represents a homogeneous relation on
84:
5493:
Bjarni JĂłnssen (1984) "Maximal
Algebras of Binary Relations", in
3928:
it is common to represent the complement of a set by an overbar:
2975:
is used to distinguish relations of Ferrer's type, which satisfy
2054:
is surjective, which conversely implies only the surjectivity of
5926:
5922:
1958:
is injective, which conversely implies only the injectivity of
1873:
This property makes the set of all binary relations on a set a
5525:, Encyclopedia of Mathematics and its Applications, vol. 132,
1884:(that is, functional relations) is again a (partial) function.
2876:{\displaystyle R^{\textsf {T}}R\supseteq I=\{(x,x):x\in B\}.}
2139:) together with (left or right) relation composition forms a
5476: – Human contact that exists because of a mutual friend
4808:
4524:
4925:
and the right residual is the greatest relation satisfying
4827:
Thus the left residual is the greatest relation satisfying
3538:
3250:
2344:
are distinct sets. Then using composition of relation
725:
has renewed the use of the semicolon, particularly in
5818:
5422:
5402:
5310:
5290:
5270:
5230:
5201:
5160:
5122:
5067:
5035:
5003:
4977:
4931:
4894:
4865:
4833:
4798:
4769:
4654:
4586:
4520:
4479:
4459:
4398:
4322:
4271:
4251:
4173:
4038:
4001:
3981:
3934:
3873:
3840:
3813:
3786:
3740:
3717:
3677:
3652:
3514:
3482:
3456:
3420:
3382:
3349:
3244:
3220:
3200:
3180:
3157:
3133:
3107:
3087:
3064:
3041:
2981:
2940:
2889:
2812:
2788:
2735:
2697:
2661:
2617:
2593:
2567:
2538:
2512:
2489:
2457:
2437:
2405:
2373:
2350:
2330:
2310:
2275:
2222:
2190:
2149:
2125:
2105:
2085:
2060:
2033:
2009:
1989:
1964:
1937:
1913:
1893:
1803:
1776:
1705:
1659:
1610:
1584:
1526:
1506:
1478:
1410:
1386:
1366:
1337:
1305:
1279:
1257:
1216:
1194:
1156:
1130:
1100:
1054:
1019:
987:
959:
928:
901:
869:
793:
750:
669:
631:
595:
569:
525:
487:
351:
325:
293:
261:
175:
149:
123:
117:) is the composition of relations "is a brother of" (
97:
6468:
6396:
6335:
6105:
6034:
5983:
1571:{\displaystyle {\mathsf {Rel}}({\mathsf {Mat}}(k))}
5852:
5639:http://www.cs.man.ac.uk/~pt/Practical_Foundations/
5437:
5408:
5386:
5296:
5276:
5248:
5216:
5187:
5146:
5108:
5053:
5021:
4989:
4949:
4917:
4880:
4851:
4819:
4784:
4752:
4638:
4570:
4485:
4465:
4443:
4384:
4308:
4257:
4235:
4148:
4016:
3987:
3965:
3916:
3855:
3819:
3795:
3765:
3726:
3699:
3661:
3638:
3500:
3468:
3438:
3403:
3364:
3330:
3226:
3206:
3186:
3166:
3139:
3119:
3093:
3073:
3050:
3020:
2967:
2919:
2875:
2794:
2774:
2721:
2679:
2647:
2599:
2579:
2553:
2524:
2495:
2475:
2443:
2423:
2391:
2356:
2336:
2316:
2296:
2240:
2208:
2155:
2131:
2111:
2091:
2069:
2046:
2015:
1995:
1973:
1950:
1919:
1899:
1865:
1789:
1756:
1674:
1642:
1596:
1570:
1512:
1484:
1461:
1392:
1372:
1345:
1323:
1291:
1265:
1243:
1202:
1174:
1142:
1116:
1070:
1037:
1005:
965:
941:
914:
884:
844:
768:
693:
655:
617:
581:
555:
511:
471:
337:
311:
279:
229:
161:
135:
109:
5907:, De Gruyter Expositions in Mathematics vol. 29,
319:are two binary relations, then their composition
5637:A free HTML version of the book is available at
3081:{ French, German, Italian } with the relation
5938:
5558:Augustus De Morgan and the Logic of Relations
4959:One can practice the logic of residuals with
3021:{\displaystyle R{\bar {R}}^{\textsf {T}}R=R.}
2920:{\displaystyle R\subseteq RR^{\textsf {T}}R.}
1650:. The category of linear relations over the
1244:{\displaystyle {\mathsf {Rel}}(\mathbb {X} )}
8:
5109:{\displaystyle c\,(<)\,d:H\to A\times B.}
3966:{\displaystyle {\bar {A}}=A^{\complement }.}
3058:{ France, Germany, Italy, Switzerland } and
2867:
2837:
2766:
2736:
1643:{\displaystyle R\subseteq k^{n}\oplus k^{m}}
463:
364:
2722:{\displaystyle I\subseteq RR^{\textsf {T}}}
2180:Finite binary relations are represented by
717:for composition of relations dates back to
6596:Positive cone of a partially ordered group
5945:
5931:
5923:
5625:. Cambridge University Press. p. 24.
4997:has been introduced to fuse two relations
1094:of sets and functions is a subcategory of
5903:M. Kilp, U. Knauer, A.V. Mikhalev (2000)
5826:
5817:
5421:
5401:
5375:
5374:
5373:
5351:
5350:
5349:
5331:
5324:
5314:
5309:
5289:
5269:
5237:
5236:
5235:
5229:
5208:
5207:
5206:
5200:
5159:
5121:
5081:
5071:
5066:
5034:
5002:
4976:
4930:
4904:
4893:
4864:
4832:
4820:{\displaystyle X\subseteq A\backslash B.}
4797:
4768:
4735:
4734:
4733:
4722:
4721:
4717:
4697:
4696:
4690:
4689:
4688:
4681:
4676:
4653:
4624:
4623:
4622:
4607:
4606:
4603:
4598:
4590:
4585:
4551:
4550:
4544:
4543:
4542:
4535:
4530:
4519:
4478:
4458:
4421:
4420:
4414:
4413:
4412:
4405:
4397:
4368:
4367:
4353:
4352:
4346:
4345:
4344:
4335:
4321:
4285:
4270:
4250:
4219:
4218:
4209:
4208:
4207:
4192:
4191:
4186:
4172:
4132:
4131:
4122:
4121:
4120:
4105:
4104:
4098:
4086:
4085:
4071:
4070:
4064:
4063:
4062:
4052:
4037:
4008:
4007:
4006:
4000:
3980:
3954:
3936:
3935:
3933:
3905:
3892:
3883:
3872:
3839:
3812:
3785:
3754:
3753:
3752:
3744:
3739:
3716:
3690:
3684:
3683:
3682:
3676:
3651:
3533:
3524:
3523:
3522:
3513:
3489:
3488:
3487:
3481:
3455:
3427:
3426:
3425:
3419:
3389:
3388:
3387:
3381:
3356:
3355:
3354:
3348:
3245:
3243:
3219:
3199:
3179:
3156:
3132:
3106:
3086:
3063:
3040:
3000:
2999:
2998:
2987:
2986:
2980:
2968:{\displaystyle {\bar {R}}^{\textsf {T}}R}
2956:
2955:
2954:
2943:
2942:
2939:
2905:
2904:
2903:
2888:
2819:
2818:
2817:
2811:
2787:
2734:
2713:
2712:
2711:
2696:
2671:
2670:
2669:
2660:
2636:
2635:
2634:
2616:
2592:
2566:
2537:
2511:
2488:
2464:
2463:
2462:
2456:
2436:
2415:
2414:
2413:
2404:
2380:
2379:
2378:
2372:
2349:
2329:
2309:
2274:
2221:
2189:
2148:
2124:
2104:
2084:
2059:
2037:
2032:
2008:
1988:
1963:
1941:
1936:
1912:
1892:
1854:
1853:
1852:
1844:
1838:
1837:
1836:
1823:
1822:
1821:
1810:
1802:
1780:
1775:
1704:
1666:
1662:
1661:
1658:
1634:
1621:
1609:
1583:
1544:
1543:
1528:
1527:
1525:
1505:
1496:), the category of relations internal to
1477:
1447:
1446:
1428:
1427:
1412:
1411:
1409:
1385:
1365:
1339:
1338:
1336:
1304:
1278:
1259:
1258:
1256:
1234:
1233:
1218:
1217:
1215:
1196:
1195:
1193:
1155:
1129:
1102:
1101:
1099:
1056:
1055:
1053:
1018:
986:
958:
933:
927:
906:
900:
868:
792:
749:
668:
630:
611:
607:
603:
599:
594:
568:
524:
486:
437:
411:
397:
350:
324:
292:
260:
210:
189:
174:
148:
122:
96:
64:, the composition of relations is called
6579:Positive cone of an ordered vector space
5798:The Origins of the Calculus of Relations
5649:Michael Barr & Charles Wells (1998)
5116:The construction depends on projections
42:is the forming of a new binary relation
5651:Category Theory for Computer Scientists
5486:
1400:. Categories of internal relations are
243:, the traditional form of reasoning by
91:it is said that the relation of Uncle (
5580:
5578:
5576:
5517:
5515:
5513:
1682:is isomorphic to the phase-free qubit
1551:
1548:
1545:
1535:
1532:
1529:
1454:
1451:
1448:
1435:
1432:
1429:
1419:
1416:
1413:
1225:
1222:
1219:
1109:
1106:
1103:
1063:
1060:
1057:
512:{\displaystyle R;S\subseteq X\times Z}
5552:
5550:
2297:{\displaystyle R\subseteq A\times B;}
2143:with zero, where the identity map on
2079:The set of binary relations on a set
1210:, its category of internal relations
845:{\displaystyle g(f(x))=(g\circ f)(x)}
7:
5622:Practical Foundations of Mathematics
3766:{\displaystyle R\,;R^{\textsf {T}}.}
2927:The opposite inclusion occurs for a
2648:{\displaystyle x,xRR^{\textsf {T}}x}
1324:{\displaystyle R\subseteq X\times Y}
1006:{\displaystyle R\subseteq X\times Y}
312:{\displaystyle S\subseteq Y\times Z}
280:{\displaystyle R\subseteq X\times Y}
5678:Stanford Encyclopedia of Philosophy
3700:{\displaystyle R^{\textsf {T}}\,;R}
563:if and only if there is an element
6106:Properties & Types (
2269:Consider a heterogeneous relation
25:
6562:Positive cone of an ordered field
5714:Kilp, Knauer & Mikhalev, p. 7
5188:{\displaystyle b:A\times B\to B,}
4967:Join: another form of composition
3446:when summation is implemented by
3404:{\displaystyle R^{\textsf {T}};R}
2775:{\displaystyle \{(x,x):x\in A\}.}
2729:where I is the identity relation
519:is defined by the rule that says
6416:Ordered topological vector space
5693:Fundamentals of Semigroup Theory
5249:{\displaystyle b^{\textsf {T}}.}
5147:{\displaystyle a:A\times B\to A}
4265:in relation inclusions such as
3501:{\displaystyle R^{\textsf {T}}R}
3439:{\displaystyle R^{\textsf {T}}R}
2680:{\displaystyle RR^{\textsf {T}}}
2476:{\displaystyle R^{\textsf {T}}R}
2424:{\displaystyle RR^{\textsf {T}}}
2399:there are homogeneous relations
2392:{\displaystyle R^{\textsf {T}},}
2176:Composition in terms of matrices
1757:{\displaystyle R;(S;T)=(R;S);T.}
1675:{\displaystyle \mathbb {F} _{2}}
52:from two given binary relations
5217:{\displaystyle a^{\textsf {T}}}
4918:{\displaystyle Y\subseteq D/C,}
4316:For instance, by Schröder rule
4290:
4284:
4103:
4097:
4057:
4051:
4017:{\displaystyle S^{\textsf {T}}}
3376:, and the relation composition
3365:{\displaystyle R^{\textsf {T}}}
1117:{\displaystyle {\mathsf {Rel}}}
1071:{\displaystyle {\mathsf {Rel}}}
194:
188:
5838:
5835:
5695:, page 16, LMS Monograph #12,
5672:Rick Nouwen and others (2016)
5608:Algebra und Logik der Relative
5587:& Thomas Ströhlein (1993)
5321:
5315:
5176:
5138:
5091:
5078:
5072:
5045:
5013:
4984:
4978:
4950:{\displaystyle YC\subseteq D.}
4852:{\displaystyle AX\subseteq B.}
4727:
4702:
4673:
4661:
4612:
4556:
4426:
4373:
4358:
4224:
4197:
4137:
4110:
4091:
4076:
4032:. Then the Schröder rules are
3941:
3847:
3841:
2992:
2948:
2852:
2840:
2751:
2739:
1818:
1804:
1742:
1730:
1724:
1712:
1588:
1565:
1562:
1556:
1540:
1440:
1424:
1283:
1238:
1230:
1166:
1134:
1029:
879:
870:
839:
833:
830:
818:
812:
809:
803:
797:
763:
751:
682:
670:
644:
632:
538:
526:
454:
442:
428:
416:
379:
367:
1:
6373:Series-parallel partial order
5853:{\displaystyle nM^{-1}))\ l.}
5499:American Mathematical Society
5495:Contributions to Group Theory
4881:{\displaystyle YC\subseteq D}
4785:{\displaystyle AX\subseteq B}
3856:{\displaystyle (\subseteq ).}
729:(2011). The use of semicolon
618:{\displaystyle x\,R\,y\,S\,z}
191: is equivalent to:
68:, and its result is called a
6052:Cantor's isomorphism theorem
5877:Lecture Notes in Mathematics
5723:ISO/IEC 13568:2002(E), p. 23
4745:
4709:
4631:
4563:
4433:
4100: is equivalent to
4054: is equivalent to
2241:{\displaystyle 1\times 1=1.}
1695:Composition of relations is
1346:{\displaystyle \mathbb {X} }
1266:{\displaystyle \mathbb {X} }
1203:{\displaystyle \mathbb {X} }
977:Mathematical generalizations
556:{\displaystyle (x,z)\in R;S}
6092:Szpilrajn extension theorem
6067:Hausdorff maximal principle
6042:Boolean prime ideal theorem
5871:and Michael Winter (2018):
2167:, and the empty set is the
6655:
6438:Topological vector lattice
5527:Cambridge University Press
4392:and complementation gives
3995:is a binary relation, let
1273:, but now the morphisms
942:{\displaystyle \circ _{r}}
915:{\displaystyle \circ _{l}}
769:{\displaystyle (R\circ S)}
694:{\displaystyle (y,z)\in S}
656:{\displaystyle (x,y)\in R}
212: if and only if
5960:
5779:Journal of Symbolic Logic
5556:Daniel D. Merrill (1990)
4859:Similarly, the inclusion
3469:{\displaystyle 3\times 3}
2099:(that is, relations from
1875:semigroup with involution
1251:has the same objects as
780:in his books considering
6047:Cantor–Bernstein theorem
5438:{\displaystyle n\geq 2,}
5054:{\displaystyle d:H\to B}
5022:{\displaystyle c:H\to A}
4763:Using Schröder's rules,
3450:. It turns out that the
3234:can be represented by a
1299:are given by subobjects
1175:{\displaystyle f:X\to Y}
1038:{\displaystyle R:X\to Y}
786:composition of functions
399: there exists
143:) and "is a parent of" (
40:composition of relations
6591:Partially ordered group
6411:Specialization preorder
5521:Gunther Schmidt (2011)
2554:{\displaystyle y\in B,}
2265:Heterogeneous relations
1353:. Formally, these are
66:relative multiplication
6639:Mathematical relations
6077:Kruskal's tree theorem
6072:Knaster–Tarski theorem
6062:Dushnik–Miller theorem
5854:
5523:Relational Mathematics
5439:
5410:
5388:
5298:
5278:
5250:
5218:
5189:
5148:
5110:
5055:
5023:
4991:
4990:{\displaystyle (<)}
4951:
4919:
4882:
4853:
4821:
4786:
4754:
4640:
4572:
4487:
4467:
4445:
4386:
4310:
4259:
4237:
4150:
4018:
3989:
3967:
3918:
3857:
3821:
3803:the collection of all
3797:
3767:
3728:
3701:
3663:
3640:
3502:
3470:
3440:
3405:
3366:
3332:
3228:
3208:
3188:
3168:
3141:
3121:
3095:
3075:
3052:
3022:
2969:
2921:
2877:
2796:
2776:
2723:
2681:
2649:
2601:
2581:
2555:
2526:
2525:{\displaystyle x\in A}
2497:
2477:
2445:
2425:
2393:
2358:
2338:
2318:
2298:
2242:
2210:
2157:
2133:
2113:
2093:
2071:
2048:
2017:
1997:
1975:
1952:
1921:
1901:
1867:
1791:
1758:
1676:
1644:
1598:
1597:{\displaystyle n\to m}
1572:
1514:
1494:principal ideal domain
1486:
1463:
1394:
1374:
1347:
1325:
1293:
1292:{\displaystyle X\to Y}
1267:
1245:
1204:
1176:
1144:
1143:{\displaystyle X\to Y}
1118:
1072:
1039:
1007:
967:
966:{\displaystyle \circ }
943:
916:
886:
846:
770:
727:Relational Mathematics
695:
657:
619:
583:
582:{\displaystyle y\in Y}
557:
513:
473:
339:
313:
281:
231:
163:
137:
111:
27:Mathematical operation
5855:
5785:, quote from page 203
5440:
5416:-place relations for
5411:
5389:
5299:
5279:
5251:
5219:
5190:
5149:
5111:
5056:
5024:
4992:
4952:
4920:
4883:
4854:
4822:
4787:
4755:
4641:
4573:
4488:
4468:
4446:
4387:
4311:
4260:
4238:
4151:
4019:
3990:
3968:
3926:calculus of relations
3919:
3858:
3822:
3798:
3768:
3729:
3702:
3664:
3641:
3503:
3471:
3441:
3406:
3367:
3333:
3229:
3209:
3189:
3169:
3142:
3122:
3096:
3076:
3053:
3023:
2970:
2922:
2878:
2797:
2777:
2724:
2682:
2650:
2609:(left-)total relation
2602:
2582:
2556:
2527:
2498:
2478:
2446:
2426:
2394:
2359:
2339:
2319:
2299:
2243:
2211:
2209:{\displaystyle 1+1=1}
2158:
2134:
2114:
2094:
2072:
2049:
2047:{\displaystyle R\,;S}
2018:
1998:
1976:
1953:
1951:{\displaystyle R\,;S}
1922:
1902:
1868:
1792:
1790:{\displaystyle R\,;S}
1759:
1677:
1645:
1599:
1573:
1515:
1492:(or more generally a
1487:
1464:
1395:
1375:
1348:
1326:
1294:
1268:
1246:
1205:
1177:
1145:
1119:
1073:
1040:
1008:
968:
944:
917:
887:
847:
771:
721:'s textbook of 1895.
705:Notational variations
696:
658:
620:
584:
558:
514:
474:
413: such that
340:
314:
282:
232:
164:
138:
112:
62:calculus of relations
6569:Ordered vector space
5816:
5751:"internal relations"
5619:Paul Taylor (1999).
5589:Relations and Graphs
5497:, K.I. Appel editor
5420:
5400:
5308:
5288:
5268:
5228:
5199:
5158:
5120:
5065:
5033:
5001:
4975:
4929:
4892:
4863:
4831:
4796:
4767:
4652:
4648:Symmetric quotient:
4584:
4518:
4477:
4457:
4451:which is called the
4396:
4320:
4269:
4249:
4171:
4036:
3999:
3979:
3932:
3871:
3867:reverses inclusion:
3838:
3811:
3784:
3738:
3715:
3675:
3650:
3512:
3480:
3454:
3418:
3380:
3347:
3242:
3218:
3198:
3178:
3155:
3131:
3105:
3085:
3062:
3039:
2979:
2938:
2887:
2810:
2786:
2733:
2695:
2659:
2615:
2591:
2565:
2536:
2510:
2487:
2455:
2435:
2403:
2371:
2348:
2328:
2308:
2273:
2220:
2188:
2147:
2123:
2103:
2083:
2058:
2031:
2007:
1987:
1962:
1935:
1911:
1891:
1801:
1774:
1703:
1657:
1608:
1582:
1524:
1504:
1476:
1408:
1384:
1364:
1335:
1303:
1277:
1255:
1214:
1192:
1154:
1128:
1098:
1052:
1017:
985:
957:
926:
899:
885:{\displaystyle (RS)}
867:
858:Graphs and Relations
791:
748:
667:
629:
593:
567:
523:
485:
349:
323:
291:
259:
173:
147:
121:
95:
74:Function composition
18:Relation composition
6407:Alexandrov topology
6353:Lexicographic order
6312:Well-quasi-ordering
5873:Relational Topology
5802:Stanford University
5469:Demonic composition
4337: implies
4188: implies
3885: implies
3448:logical disjunction
3411:corresponds to the
3372:corresponds to the
3120:{\displaystyle aRb}
2804:surjective relation
2580:{\displaystyle xRy}
1882:(partial) functions
1880:The composition of
338:{\displaystyle R;S}
162:{\displaystyle yPz}
136:{\displaystyle xBy}
110:{\displaystyle xUz}
6388:Transitive closure
6348:Converse/Transpose
6057:Dilworth's theorem
5850:
5740:on FileFormat.info
5656:2016-03-04 at the
5474:Friend of a friend
5453:relational algebra
5435:
5406:
5384:
5294:
5274:
5246:
5214:
5185:
5144:
5106:
5051:
5019:
4987:
4947:
4915:
4878:
4849:
4817:
4782:
4750:
4636:
4568:
4483:
4463:
4441:
4382:
4306:
4255:
4233:
4165:Augustus De Morgan
4146:
4028:, also called the
4014:
3985:
3963:
3914:
3853:
3817:
3796:{\displaystyle V,}
3793:
3763:
3727:{\displaystyle B,}
3724:
3709:universal relation
3697:
3662:{\displaystyle A.}
3659:
3636:
3627:
3498:
3466:
3436:
3401:
3362:
3328:
3319:
3224:
3204:
3184:
3167:{\displaystyle a.}
3164:
3137:
3117:
3091:
3074:{\displaystyle B=}
3071:
3051:{\displaystyle A=}
3048:
3018:
2965:
2917:
2873:
2792:
2772:
2719:
2689:reflexive relation
2677:
2645:
2597:
2577:
2551:
2532:there exists some
2522:
2493:
2473:
2441:
2421:
2389:
2354:
2334:
2314:
2294:
2238:
2206:
2153:
2129:
2109:
2089:
2070:{\displaystyle S.}
2067:
2044:
2013:
1993:
1974:{\displaystyle R.}
1971:
1948:
1917:
1897:
1863:
1787:
1754:
1672:
1640:
1594:
1568:
1510:
1482:
1459:
1390:
1370:
1343:
1321:
1289:
1263:
1241:
1200:
1172:
1140:
1114:
1068:
1035:
1003:
963:
939:
912:
882:
842:
766:
691:
653:
615:
579:
553:
509:
469:
335:
309:
277:
241:Augustus De Morgan
227:
159:
133:
107:
6634:Binary operations
6616:
6615:
6574:Partially ordered
6383:Symmetric closure
6368:Reflexive closure
6111:
5909:Walter de Gruyter
5889:978-3-319-74451-3
5843:
5775:Irving Copilowish
5674:Dynamic Semantics
5662:McGill University
5632:978-0-521-63107-5
5534:978-0-521-76268-7
5506:978-0-8218-5035-0
5409:{\displaystyle n}
5377:
5362:
5353:
5338:
5330:
5297:{\displaystyle d}
5277:{\displaystyle c}
5239:
5210:
4888:is equivalent to
4792:is equivalent to
4748:
4737:
4730:
4712:
4705:
4692:
4634:
4626:
4615:
4566:
4559:
4546:
4486:{\displaystyle R}
4466:{\displaystyle S}
4453:left residual of
4436:
4429:
4416:
4376:
4361:
4348:
4338:
4288:
4258:{\displaystyle X}
4227:
4211:
4200:
4189:
4140:
4124:
4113:
4101:
4094:
4079:
4066:
4055:
4026:converse relation
4010:
3988:{\displaystyle S}
3944:
3886:
3820:{\displaystyle V}
3756:
3686:
3671:Correspondingly,
3526:
3491:
3429:
3391:
3374:transposed matrix
3358:
3342:converse relation
3227:{\displaystyle R}
3207:{\displaystyle B}
3187:{\displaystyle A}
3149:national language
3140:{\displaystyle b}
3094:{\displaystyle R}
3002:
2995:
2958:
2951:
2907:
2821:
2795:{\displaystyle R}
2715:
2673:
2638:
2600:{\displaystyle R}
2496:{\displaystyle B}
2466:
2444:{\displaystyle A}
2417:
2382:
2357:{\displaystyle R}
2337:{\displaystyle B}
2317:{\displaystyle A}
2156:{\displaystyle X}
2132:{\displaystyle X}
2112:{\displaystyle X}
2092:{\displaystyle X}
2016:{\displaystyle S}
1996:{\displaystyle R}
1920:{\displaystyle S}
1900:{\displaystyle R}
1856:
1840:
1825:
1768:converse relation
1604:linear subspaces
1513:{\displaystyle k}
1485:{\displaystyle k}
1404:. In particular
1393:{\displaystyle Y}
1373:{\displaystyle X}
1124:where the maps
981:Binary relations
739:dynamic semantics
440:
414:
400:
213:
192:
16:(Redirected from
6646:
6358:Linear extension
6107:
6087:Mirsky's theorem
5947:
5940:
5933:
5924:
5891:
5866:
5860:
5859:
5857:
5856:
5851:
5841:
5834:
5833:
5810:
5804:
5792:
5786:
5772:
5766:
5765:
5763:
5761:
5747:
5741:
5730:
5724:
5721:
5715:
5712:
5706:
5686:
5680:
5670:
5664:
5647:
5641:
5636:
5616:
5610:
5601:
5595:
5582:
5571:
5554:
5545:
5542:
5536:
5519:
5508:
5491:
5444:
5442:
5441:
5436:
5415:
5413:
5412:
5407:
5393:
5391:
5390:
5385:
5380:
5379:
5378:
5360:
5356:
5355:
5354:
5336:
5335:
5328:
5303:
5301:
5300:
5295:
5283:
5281:
5280:
5275:
5262:
5261:
5255:
5253:
5252:
5247:
5242:
5241:
5240:
5223:
5221:
5220:
5215:
5213:
5212:
5211:
5194:
5192:
5191:
5186:
5153:
5151:
5150:
5145:
5115:
5113:
5112:
5107:
5060:
5058:
5057:
5052:
5028:
5026:
5025:
5020:
4996:
4994:
4993:
4988:
4971:A fork operator
4956:
4954:
4953:
4948:
4924:
4922:
4921:
4916:
4908:
4887:
4885:
4884:
4879:
4858:
4856:
4855:
4850:
4826:
4824:
4823:
4818:
4791:
4789:
4788:
4783:
4759:
4757:
4756:
4751:
4749:
4744:
4740:
4739:
4738:
4732:
4731:
4723:
4718:
4713:
4708:
4707:
4706:
4698:
4695:
4694:
4693:
4682:
4680:
4645:
4643:
4642:
4637:
4635:
4630:
4629:
4628:
4627:
4617:
4616:
4608:
4604:
4602:
4594:
4580:Right residual:
4577:
4575:
4574:
4569:
4567:
4562:
4561:
4560:
4552:
4549:
4548:
4547:
4536:
4534:
4492:
4490:
4489:
4484:
4472:
4470:
4469:
4464:
4450:
4448:
4447:
4442:
4437:
4432:
4431:
4430:
4422:
4419:
4418:
4417:
4406:
4391:
4389:
4388:
4383:
4378:
4377:
4369:
4363:
4362:
4354:
4351:
4350:
4349:
4339:
4336:
4315:
4313:
4312:
4307:
4289:
4286:
4264:
4262:
4261:
4256:
4242:
4240:
4239:
4234:
4229:
4228:
4220:
4214:
4213:
4212:
4202:
4201:
4193:
4190:
4187:
4155:
4153:
4152:
4147:
4142:
4141:
4133:
4127:
4126:
4125:
4115:
4114:
4106:
4102:
4099:
4096:
4095:
4087:
4081:
4080:
4072:
4069:
4068:
4067:
4056:
4053:
4023:
4021:
4020:
4015:
4013:
4012:
4011:
3994:
3992:
3991:
3986:
3972:
3970:
3969:
3964:
3959:
3958:
3946:
3945:
3937:
3923:
3921:
3920:
3915:
3910:
3909:
3897:
3896:
3887:
3884:
3862:
3860:
3859:
3854:
3826:
3824:
3823:
3818:
3805:binary relations
3802:
3800:
3799:
3794:
3780:For a given set
3772:
3770:
3769:
3764:
3759:
3758:
3757:
3733:
3731:
3730:
3725:
3706:
3704:
3703:
3698:
3689:
3688:
3687:
3668:
3666:
3665:
3660:
3645:
3643:
3642:
3637:
3632:
3631:
3529:
3528:
3527:
3507:
3505:
3504:
3499:
3494:
3493:
3492:
3475:
3473:
3472:
3467:
3445:
3443:
3442:
3437:
3432:
3431:
3430:
3410:
3408:
3407:
3402:
3394:
3393:
3392:
3371:
3369:
3368:
3363:
3361:
3360:
3359:
3337:
3335:
3334:
3329:
3324:
3323:
3233:
3231:
3230:
3225:
3213:
3211:
3210:
3205:
3193:
3191:
3190:
3185:
3173:
3171:
3170:
3165:
3146:
3144:
3143:
3138:
3126:
3124:
3123:
3118:
3100:
3098:
3097:
3092:
3080:
3078:
3077:
3072:
3057:
3055:
3054:
3049:
3027:
3025:
3024:
3019:
3005:
3004:
3003:
2997:
2996:
2988:
2974:
2972:
2971:
2966:
2961:
2960:
2959:
2953:
2952:
2944:
2934:The composition
2926:
2924:
2923:
2918:
2910:
2909:
2908:
2882:
2880:
2879:
2874:
2824:
2823:
2822:
2801:
2799:
2798:
2793:
2781:
2779:
2778:
2773:
2728:
2726:
2725:
2720:
2718:
2717:
2716:
2686:
2684:
2683:
2678:
2676:
2675:
2674:
2654:
2652:
2651:
2646:
2641:
2640:
2639:
2611:), then for all
2606:
2604:
2603:
2598:
2586:
2584:
2583:
2578:
2560:
2558:
2557:
2552:
2531:
2529:
2528:
2523:
2502:
2500:
2499:
2494:
2482:
2480:
2479:
2474:
2469:
2468:
2467:
2450:
2448:
2447:
2442:
2430:
2428:
2427:
2422:
2420:
2419:
2418:
2398:
2396:
2395:
2390:
2385:
2384:
2383:
2363:
2361:
2360:
2355:
2343:
2341:
2340:
2335:
2323:
2321:
2320:
2315:
2303:
2301:
2300:
2295:
2248:An entry in the
2247:
2245:
2244:
2239:
2215:
2213:
2212:
2207:
2182:logical matrices
2162:
2160:
2159:
2154:
2138:
2136:
2135:
2130:
2118:
2116:
2115:
2110:
2098:
2096:
2095:
2090:
2076:
2074:
2073:
2068:
2053:
2051:
2050:
2045:
2022:
2020:
2019:
2014:
2002:
2000:
1999:
1994:
1980:
1978:
1977:
1972:
1957:
1955:
1954:
1949:
1926:
1924:
1923:
1918:
1906:
1904:
1903:
1898:
1872:
1870:
1869:
1864:
1859:
1858:
1857:
1843:
1842:
1841:
1828:
1827:
1826:
1796:
1794:
1793:
1788:
1763:
1761:
1760:
1755:
1686:modulo scalars.
1681:
1679:
1678:
1673:
1671:
1670:
1665:
1649:
1647:
1646:
1641:
1639:
1638:
1626:
1625:
1603:
1601:
1600:
1595:
1577:
1575:
1574:
1569:
1555:
1554:
1539:
1538:
1519:
1517:
1516:
1511:
1491:
1489:
1488:
1483:
1468:
1466:
1465:
1460:
1458:
1457:
1439:
1438:
1423:
1422:
1399:
1397:
1396:
1391:
1379:
1377:
1376:
1371:
1352:
1350:
1349:
1344:
1342:
1330:
1328:
1327:
1322:
1298:
1296:
1295:
1290:
1272:
1270:
1269:
1264:
1262:
1250:
1248:
1247:
1242:
1237:
1229:
1228:
1209:
1207:
1206:
1201:
1199:
1187:regular category
1181:
1179:
1178:
1173:
1149:
1147:
1146:
1141:
1123:
1121:
1120:
1115:
1113:
1112:
1084:the objects are
1077:
1075:
1074:
1069:
1067:
1066:
1044:
1042:
1041:
1036:
1012:
1010:
1009:
1004:
972:
970:
969:
964:
948:
946:
945:
940:
938:
937:
921:
919:
918:
913:
911:
910:
891:
889:
888:
883:
851:
849:
848:
843:
775:
773:
772:
767:
700:
698:
697:
692:
662:
660:
659:
654:
624:
622:
621:
616:
588:
586:
585:
580:
562:
560:
559:
554:
518:
516:
515:
510:
481:In other words,
478:
476:
475:
470:
441:
438:
415:
412:
401:
398:
345:is the relation
344:
342:
341:
336:
318:
316:
315:
310:
286:
284:
283:
278:
236:
234:
233:
228:
214:
211:
193:
190:
168:
166:
165:
160:
142:
140:
139:
134:
116:
114:
113:
108:
70:relative product
51:
36:binary relations
21:
6654:
6653:
6649:
6648:
6647:
6645:
6644:
6643:
6629:Algebraic logic
6619:
6618:
6617:
6612:
6608:Young's lattice
6464:
6392:
6331:
6181:Heyting algebra
6129:Boolean algebra
6101:
6082:Laver's theorem
6030:
5996:Boolean algebra
5991:Binary relation
5979:
5956:
5951:
5900:
5895:
5894:
5869:Gunther Schmidt
5867:
5863:
5822:
5814:
5813:
5811:
5807:
5793:
5789:
5781:13(4): 193–203
5773:
5769:
5759:
5757:
5749:
5748:
5744:
5731:
5727:
5722:
5718:
5713:
5709:
5697:Clarendon Press
5687:
5683:
5671:
5667:
5660:, page 6, from
5658:Wayback Machine
5648:
5644:
5633:
5618:
5617:
5613:
5602:
5598:
5585:Gunther Schmidt
5583:
5574:
5562:Kluwer Academic
5555:
5548:
5543:
5539:
5520:
5511:
5492:
5488:
5483:
5465:
5418:
5417:
5398:
5397:
5369:
5345:
5306:
5305:
5286:
5285:
5266:
5265:
5259:
5258:
5231:
5226:
5225:
5202:
5197:
5196:
5156:
5155:
5118:
5117:
5063:
5062:
5031:
5030:
4999:
4998:
4973:
4972:
4969:
4927:
4926:
4890:
4889:
4861:
4860:
4829:
4828:
4794:
4793:
4765:
4764:
4720:
4719:
4684:
4683:
4650:
4649:
4618:
4605:
4582:
4581:
4538:
4537:
4516:
4515:
4514:Left residual:
4500:
4475:
4474:
4455:
4454:
4408:
4407:
4394:
4393:
4340:
4318:
4317:
4267:
4266:
4247:
4246:
4203:
4169:
4168:
4116:
4058:
4034:
4033:
4002:
3997:
3996:
3977:
3976:
3950:
3930:
3929:
3901:
3888:
3869:
3868:
3865:complementation
3836:
3835:
3829:Boolean lattice
3809:
3808:
3782:
3781:
3778:
3748:
3736:
3735:
3713:
3712:
3678:
3673:
3672:
3648:
3647:
3626:
3625:
3620:
3615:
3610:
3604:
3603:
3598:
3593:
3588:
3582:
3581:
3576:
3571:
3566:
3560:
3559:
3554:
3549:
3544:
3534:
3518:
3510:
3509:
3483:
3478:
3477:
3452:
3451:
3421:
3416:
3415:
3383:
3378:
3377:
3350:
3345:
3344:
3318:
3317:
3312:
3307:
3301:
3300:
3295:
3290:
3284:
3283:
3278:
3273:
3267:
3266:
3261:
3256:
3246:
3240:
3239:
3216:
3215:
3196:
3195:
3176:
3175:
3153:
3152:
3129:
3128:
3103:
3102:
3083:
3082:
3060:
3059:
3037:
3036:
3033:
2985:
2977:
2976:
2941:
2936:
2935:
2899:
2885:
2884:
2813:
2808:
2807:
2784:
2783:
2731:
2730:
2707:
2693:
2692:
2665:
2657:
2656:
2630:
2613:
2612:
2589:
2588:
2563:
2562:
2534:
2533:
2508:
2507:
2485:
2484:
2458:
2453:
2452:
2433:
2432:
2409:
2401:
2400:
2374:
2369:
2368:
2346:
2345:
2326:
2325:
2306:
2305:
2304:that is, where
2271:
2270:
2267:
2218:
2217:
2186:
2185:
2178:
2165:neutral element
2145:
2144:
2121:
2120:
2101:
2100:
2081:
2080:
2056:
2055:
2029:
2028:
2005:
2004:
1985:
1984:
1960:
1959:
1933:
1932:
1909:
1908:
1889:
1888:
1848:
1832:
1817:
1799:
1798:
1772:
1771:
1701:
1700:
1692:
1660:
1655:
1654:
1630:
1617:
1606:
1605:
1580:
1579:
1522:
1521:
1502:
1501:
1474:
1473:
1406:
1405:
1382:
1381:
1362:
1361:
1333:
1332:
1301:
1300:
1275:
1274:
1253:
1252:
1212:
1211:
1190:
1189:
1152:
1151:
1126:
1125:
1096:
1095:
1050:
1049:
1015:
1014:
983:
982:
979:
955:
954:
929:
924:
923:
902:
897:
896:
865:
864:
789:
788:
746:
745:
744:A small circle
735:category theory
723:Gunther Schmidt
707:
665:
664:
627:
626:
591:
590:
565:
564:
521:
520:
483:
482:
439: and
347:
346:
321:
320:
289:
288:
257:
256:
253:
239:Beginning with
171:
170:
145:
144:
119:
118:
93:
92:
89:algebraic logic
43:
28:
23:
22:
15:
12:
11:
5:
6652:
6650:
6642:
6641:
6636:
6631:
6621:
6620:
6614:
6613:
6611:
6610:
6605:
6600:
6599:
6598:
6588:
6587:
6586:
6581:
6576:
6566:
6565:
6564:
6554:
6549:
6548:
6547:
6542:
6535:Order morphism
6532:
6531:
6530:
6520:
6515:
6510:
6505:
6500:
6499:
6498:
6488:
6483:
6478:
6472:
6470:
6466:
6465:
6463:
6462:
6461:
6460:
6455:
6453:Locally convex
6450:
6445:
6435:
6433:Order topology
6430:
6429:
6428:
6426:Order topology
6423:
6413:
6403:
6401:
6394:
6393:
6391:
6390:
6385:
6380:
6375:
6370:
6365:
6360:
6355:
6350:
6345:
6339:
6337:
6333:
6332:
6330:
6329:
6319:
6309:
6304:
6299:
6294:
6289:
6284:
6279:
6274:
6273:
6272:
6262:
6257:
6256:
6255:
6250:
6245:
6240:
6238:Chain-complete
6230:
6225:
6224:
6223:
6218:
6213:
6208:
6203:
6193:
6188:
6183:
6178:
6173:
6163:
6158:
6153:
6148:
6143:
6138:
6137:
6136:
6126:
6121:
6115:
6113:
6103:
6102:
6100:
6099:
6094:
6089:
6084:
6079:
6074:
6069:
6064:
6059:
6054:
6049:
6044:
6038:
6036:
6032:
6031:
6029:
6028:
6023:
6018:
6013:
6008:
6003:
5998:
5993:
5987:
5985:
5981:
5980:
5978:
5977:
5972:
5967:
5961:
5958:
5957:
5952:
5950:
5949:
5942:
5935:
5927:
5921:
5920:
5899:
5896:
5893:
5892:
5881:Springer books
5861:
5849:
5846:
5840:
5837:
5832:
5829:
5825:
5821:
5805:
5787:
5767:
5742:
5725:
5716:
5707:
5689:John M. Howie
5681:
5665:
5642:
5631:
5611:
5604:Ernst Schroder
5596:
5593:Springer books
5572:
5546:
5537:
5509:
5485:
5484:
5482:
5479:
5478:
5477:
5471:
5464:
5461:
5434:
5431:
5428:
5425:
5405:
5383:
5372:
5368:
5365:
5359:
5348:
5344:
5341:
5334:
5327:
5323:
5320:
5317:
5313:
5293:
5273:
5245:
5234:
5205:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5080:
5077:
5074:
5070:
5050:
5047:
5044:
5041:
5038:
5018:
5015:
5012:
5009:
5006:
4986:
4983:
4980:
4968:
4965:
4946:
4943:
4940:
4937:
4934:
4914:
4911:
4907:
4903:
4900:
4897:
4877:
4874:
4871:
4868:
4848:
4845:
4842:
4839:
4836:
4816:
4813:
4810:
4807:
4804:
4801:
4781:
4778:
4775:
4772:
4761:
4760:
4747:
4743:
4729:
4726:
4716:
4711:
4704:
4701:
4687:
4679:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4646:
4633:
4621:
4614:
4611:
4601:
4597:
4593:
4589:
4578:
4565:
4558:
4555:
4541:
4533:
4529:
4526:
4523:
4499:
4496:
4482:
4462:
4440:
4435:
4428:
4425:
4411:
4404:
4401:
4381:
4375:
4372:
4366:
4360:
4357:
4343:
4334:
4331:
4328:
4325:
4305:
4302:
4299:
4296:
4293:
4283:
4280:
4277:
4274:
4254:
4232:
4226:
4223:
4217:
4206:
4199:
4196:
4185:
4182:
4179:
4176:
4161:Ernst Schröder
4145:
4139:
4136:
4130:
4119:
4112:
4109:
4093:
4090:
4084:
4078:
4075:
4061:
4050:
4047:
4044:
4041:
4024:represent the
4005:
3984:
3962:
3957:
3953:
3949:
3943:
3940:
3913:
3908:
3904:
3900:
3895:
3891:
3882:
3879:
3876:
3852:
3849:
3846:
3843:
3816:
3792:
3789:
3777:
3776:Schröder rules
3774:
3762:
3751:
3747:
3743:
3723:
3720:
3696:
3693:
3681:
3658:
3655:
3635:
3630:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3605:
3602:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3583:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3561:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3539:
3537:
3532:
3521:
3517:
3497:
3486:
3465:
3462:
3459:
3435:
3424:
3413:matrix product
3400:
3397:
3386:
3353:
3327:
3322:
3316:
3313:
3311:
3308:
3306:
3303:
3302:
3299:
3296:
3294:
3291:
3289:
3286:
3285:
3282:
3279:
3277:
3274:
3272:
3269:
3268:
3265:
3262:
3260:
3257:
3255:
3252:
3251:
3249:
3236:logical matrix
3223:
3203:
3183:
3163:
3160:
3136:
3116:
3113:
3110:
3090:
3070:
3067:
3047:
3044:
3032:
3029:
3017:
3014:
3011:
3008:
2994:
2991:
2984:
2964:
2950:
2947:
2916:
2913:
2902:
2898:
2895:
2892:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2816:
2791:
2782:Similarly, if
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2710:
2706:
2703:
2700:
2668:
2664:
2644:
2633:
2629:
2626:
2623:
2620:
2596:
2576:
2573:
2570:
2550:
2547:
2544:
2541:
2521:
2518:
2515:
2492:
2472:
2461:
2440:
2412:
2408:
2388:
2377:
2353:
2333:
2313:
2293:
2290:
2287:
2284:
2281:
2278:
2266:
2263:
2250:matrix product
2237:
2234:
2231:
2228:
2225:
2205:
2202:
2199:
2196:
2193:
2177:
2174:
2173:
2172:
2152:
2128:
2108:
2088:
2077:
2066:
2063:
2043:
2040:
2036:
2012:
1992:
1981:
1970:
1967:
1947:
1944:
1940:
1916:
1896:
1885:
1878:
1862:
1851:
1847:
1835:
1831:
1820:
1816:
1813:
1809:
1806:
1786:
1783:
1779:
1764:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1691:
1688:
1669:
1664:
1637:
1633:
1629:
1624:
1620:
1616:
1613:
1593:
1590:
1587:
1578:has morphisms
1567:
1564:
1561:
1558:
1553:
1550:
1547:
1542:
1537:
1534:
1531:
1509:
1481:
1456:
1453:
1450:
1445:
1442:
1437:
1434:
1431:
1426:
1421:
1418:
1415:
1389:
1369:
1341:
1320:
1317:
1314:
1311:
1308:
1288:
1285:
1282:
1261:
1240:
1236:
1232:
1227:
1224:
1221:
1198:
1171:
1168:
1165:
1162:
1159:
1150:are functions
1139:
1136:
1133:
1111:
1108:
1105:
1065:
1062:
1059:
1034:
1031:
1028:
1025:
1022:
1013:are morphisms
1002:
999:
996:
993:
990:
978:
975:
962:
936:
932:
909:
905:
881:
878:
875:
872:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
765:
762:
759:
756:
753:
719:Ernst Schroder
715:infix notation
706:
703:
690:
687:
684:
681:
678:
675:
672:
652:
649:
646:
643:
640:
637:
634:
614:
610:
606:
602:
598:
578:
575:
572:
552:
549:
546:
543:
540:
537:
534:
531:
528:
508:
505:
502:
499:
496:
493:
490:
468:
465:
462:
459:
456:
453:
450:
447:
444:
436:
433:
430:
427:
424:
421:
418:
410:
407:
404:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
334:
331:
328:
308:
305:
302:
299:
296:
276:
273:
270:
267:
264:
252:
249:
226:
223:
220:
217:
209:
206:
203:
200:
197:
187:
184:
181:
178:
158:
155:
152:
132:
129:
126:
106:
103:
100:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6651:
6640:
6637:
6635:
6632:
6630:
6627:
6626:
6624:
6609:
6606:
6604:
6601:
6597:
6594:
6593:
6592:
6589:
6585:
6582:
6580:
6577:
6575:
6572:
6571:
6570:
6567:
6563:
6560:
6559:
6558:
6557:Ordered field
6555:
6553:
6550:
6546:
6543:
6541:
6538:
6537:
6536:
6533:
6529:
6526:
6525:
6524:
6521:
6519:
6516:
6514:
6513:Hasse diagram
6511:
6509:
6506:
6504:
6501:
6497:
6494:
6493:
6492:
6491:Comparability
6489:
6487:
6484:
6482:
6479:
6477:
6474:
6473:
6471:
6467:
6459:
6456:
6454:
6451:
6449:
6446:
6444:
6441:
6440:
6439:
6436:
6434:
6431:
6427:
6424:
6422:
6419:
6418:
6417:
6414:
6412:
6408:
6405:
6404:
6402:
6399:
6395:
6389:
6386:
6384:
6381:
6379:
6376:
6374:
6371:
6369:
6366:
6364:
6363:Product order
6361:
6359:
6356:
6354:
6351:
6349:
6346:
6344:
6341:
6340:
6338:
6336:Constructions
6334:
6328:
6324:
6320:
6317:
6313:
6310:
6308:
6305:
6303:
6300:
6298:
6295:
6293:
6290:
6288:
6285:
6283:
6280:
6278:
6275:
6271:
6268:
6267:
6266:
6263:
6261:
6258:
6254:
6251:
6249:
6246:
6244:
6241:
6239:
6236:
6235:
6234:
6233:Partial order
6231:
6229:
6226:
6222:
6221:Join and meet
6219:
6217:
6214:
6212:
6209:
6207:
6204:
6202:
6199:
6198:
6197:
6194:
6192:
6189:
6187:
6184:
6182:
6179:
6177:
6174:
6172:
6168:
6164:
6162:
6159:
6157:
6154:
6152:
6149:
6147:
6144:
6142:
6139:
6135:
6132:
6131:
6130:
6127:
6125:
6122:
6120:
6119:Antisymmetric
6117:
6116:
6114:
6110:
6104:
6098:
6095:
6093:
6090:
6088:
6085:
6083:
6080:
6078:
6075:
6073:
6070:
6068:
6065:
6063:
6060:
6058:
6055:
6053:
6050:
6048:
6045:
6043:
6040:
6039:
6037:
6033:
6027:
6026:Weak ordering
6024:
6022:
6019:
6017:
6014:
6012:
6011:Partial order
6009:
6007:
6004:
6002:
5999:
5997:
5994:
5992:
5989:
5988:
5986:
5982:
5976:
5973:
5971:
5968:
5966:
5963:
5962:
5959:
5955:
5948:
5943:
5941:
5936:
5934:
5929:
5928:
5925:
5918:
5917:3-11-015248-7
5914:
5910:
5906:
5902:
5901:
5897:
5890:
5886:
5882:
5878:
5874:
5870:
5865:
5862:
5847:
5844:
5830:
5827:
5823:
5819:
5809:
5806:
5803:
5799:
5796:
5791:
5788:
5784:
5780:
5776:
5771:
5768:
5756:
5752:
5746:
5743:
5739:
5735:
5729:
5726:
5720:
5717:
5711:
5708:
5705:
5704:0-19-851194-9
5701:
5698:
5694:
5690:
5685:
5682:
5679:
5675:
5669:
5666:
5663:
5659:
5655:
5652:
5646:
5643:
5640:
5634:
5628:
5624:
5623:
5615:
5612:
5609:
5605:
5600:
5597:
5594:
5590:
5586:
5581:
5579:
5577:
5573:
5570:
5569:9789400920477
5566:
5563:
5559:
5553:
5551:
5547:
5541:
5538:
5535:
5531:
5528:
5524:
5518:
5516:
5514:
5510:
5507:
5503:
5500:
5496:
5490:
5487:
5480:
5475:
5472:
5470:
5467:
5466:
5462:
5460:
5458:
5454:
5451:operation of
5450:
5449:
5432:
5429:
5426:
5423:
5403:
5394:
5381:
5370:
5366:
5363:
5357:
5346:
5342:
5339:
5332:
5325:
5318:
5311:
5291:
5271:
5263:
5243:
5232:
5203:
5182:
5179:
5173:
5170:
5167:
5164:
5161:
5141:
5135:
5132:
5129:
5126:
5123:
5103:
5100:
5097:
5094:
5088:
5085:
5082:
5075:
5068:
5048:
5042:
5039:
5036:
5016:
5010:
5007:
5004:
4981:
4966:
4964:
4962:
4957:
4944:
4941:
4938:
4935:
4932:
4912:
4909:
4905:
4901:
4898:
4895:
4875:
4872:
4869:
4866:
4846:
4843:
4840:
4837:
4834:
4814:
4811:
4805:
4802:
4799:
4779:
4776:
4773:
4770:
4741:
4724:
4714:
4699:
4685:
4677:
4670:
4667:
4664:
4658:
4655:
4647:
4619:
4609:
4599:
4595:
4591:
4587:
4579:
4553:
4539:
4531:
4527:
4521:
4513:
4512:
4511:
4510:Definitions:
4508:
4505:
4497:
4495:
4493:
4480:
4460:
4438:
4423:
4409:
4402:
4399:
4379:
4370:
4364:
4355:
4341:
4332:
4329:
4326:
4323:
4303:
4300:
4297:
4294:
4291:
4281:
4278:
4275:
4272:
4252:
4243:
4230:
4221:
4215:
4204:
4194:
4183:
4180:
4177:
4174:
4166:
4162:
4157:
4143:
4134:
4128:
4117:
4107:
4088:
4082:
4073:
4059:
4048:
4045:
4042:
4039:
4031:
4027:
4003:
3982:
3973:
3960:
3955:
3951:
3947:
3938:
3927:
3911:
3906:
3902:
3898:
3893:
3889:
3880:
3877:
3874:
3866:
3850:
3844:
3834:
3830:
3814:
3806:
3790:
3787:
3775:
3773:
3760:
3749:
3745:
3741:
3721:
3718:
3710:
3694:
3691:
3679:
3669:
3656:
3653:
3633:
3628:
3622:
3617:
3612:
3607:
3600:
3595:
3590:
3585:
3578:
3573:
3568:
3563:
3556:
3551:
3546:
3541:
3535:
3530:
3519:
3515:
3495:
3484:
3463:
3460:
3457:
3449:
3433:
3422:
3414:
3398:
3395:
3384:
3375:
3351:
3343:
3338:
3325:
3320:
3314:
3309:
3304:
3297:
3292:
3287:
3280:
3275:
3270:
3263:
3258:
3253:
3247:
3237:
3221:
3201:
3181:
3161:
3158:
3150:
3134:
3114:
3111:
3108:
3088:
3068:
3065:
3045:
3042:
3030:
3028:
3015:
3012:
3009:
3006:
2989:
2982:
2962:
2945:
2932:
2930:
2914:
2911:
2900:
2896:
2893:
2890:
2883:In this case
2870:
2864:
2861:
2858:
2855:
2849:
2846:
2843:
2834:
2831:
2828:
2825:
2814:
2805:
2789:
2769:
2763:
2760:
2757:
2754:
2748:
2745:
2742:
2708:
2704:
2701:
2698:
2690:
2666:
2662:
2642:
2631:
2627:
2624:
2621:
2618:
2610:
2594:
2574:
2571:
2568:
2548:
2545:
2542:
2539:
2519:
2516:
2513:
2504:
2490:
2470:
2459:
2438:
2410:
2406:
2386:
2375:
2367:
2351:
2331:
2311:
2291:
2288:
2285:
2282:
2279:
2276:
2264:
2262:
2260:
2256:
2251:
2235:
2232:
2229:
2226:
2223:
2203:
2200:
2197:
2194:
2191:
2183:
2175:
2170:
2166:
2150:
2142:
2126:
2106:
2086:
2078:
2064:
2061:
2041:
2038:
2034:
2026:
2010:
1990:
1982:
1968:
1965:
1945:
1942:
1938:
1930:
1914:
1894:
1886:
1883:
1879:
1876:
1860:
1849:
1845:
1833:
1829:
1814:
1811:
1807:
1784:
1781:
1777:
1769:
1765:
1751:
1748:
1745:
1739:
1736:
1733:
1727:
1721:
1718:
1715:
1709:
1706:
1698:
1694:
1693:
1689:
1687:
1685:
1667:
1653:
1635:
1631:
1627:
1622:
1618:
1614:
1611:
1591:
1585:
1559:
1507:
1499:
1495:
1479:
1472:
1443:
1403:
1387:
1367:
1359:
1356:
1355:jointly monic
1318:
1315:
1312:
1309:
1306:
1286:
1280:
1188:
1183:
1169:
1163:
1160:
1157:
1137:
1131:
1093:
1092:
1087:
1083:
1082:
1048:
1032:
1026:
1023:
1020:
1000:
997:
994:
991:
988:
976:
974:
960:
952:
934:
930:
907:
903:
893:
876:
873:
863:
862:Juxtaposition
859:
855:
836:
827:
824:
821:
815:
806:
800:
794:
787:
783:
779:
778:John M. Howie
760:
757:
754:
742:
740:
736:
732:
728:
724:
720:
716:
712:
704:
702:
688:
685:
679:
676:
673:
650:
647:
641:
638:
635:
612:
608:
604:
600:
596:
576:
573:
570:
550:
547:
544:
541:
535:
532:
529:
506:
503:
500:
497:
494:
491:
488:
479:
466:
460:
457:
451:
448:
445:
434:
431:
425:
422:
419:
408:
405:
402:
394:
391:
388:
385:
382:
376:
373:
370:
361:
358:
355:
352:
332:
329:
326:
306:
303:
300:
297:
294:
274:
271:
268:
265:
262:
250:
248:
246:
242:
237:
224:
221:
218:
215:
207:
204:
201:
198:
195:
185:
182:
179:
176:
156:
153:
150:
130:
127:
124:
104:
101:
98:
90:
86:
81:
79:
75:
71:
67:
63:
59:
55:
50:
46:
41:
37:
33:
19:
6400:& Orders
6378:Star product
6342:
6307:Well-founded
6260:Prefix order
6216:Distributive
6206:Complemented
6176:Foundational
6141:Completeness
6097:Zorn's lemma
6001:Cyclic order
5984:Key concepts
5954:Order theory
5904:
5872:
5864:
5808:
5795:Vaughn Pratt
5790:
5770:
5760:26 September
5758:. Retrieved
5754:
5745:
5728:
5719:
5710:
5692:
5684:
5668:
5645:
5621:
5614:
5599:
5588:
5560:, page 121,
5557:
5540:
5522:
5494:
5489:
5446:
5395:
5304:is given by
5257:
4970:
4958:
4762:
4509:
4501:
4452:
4244:
4158:
4029:
3974:
3863:Recall that
3779:
3670:
3339:
3034:
2933:
2929:difunctional
2505:
2268:
2254:
2179:
2169:zero element
1652:finite field
1184:
1089:
1079:
980:
894:
857:
853:
743:
726:
708:
480:
254:
238:
82:
69:
65:
57:
53:
48:
44:
39:
29:
6584:Riesz space
6545:Isomorphism
6421:Normal cone
6343:Composition
6277:Semilattice
6186:Homogeneous
6171:Equivalence
6021:Total order
5879:vol. 2208,
5875:, page 26,
5676:§2.2, from
3831:ordered by
3214:is finite,
3174:Since both
2506:If for all
1697:associative
1684:ZX-calculus
625:(that is,
32:mathematics
6623:Categories
6552:Order type
6486:Cofinality
6327:Well-order
6302:Transitive
6191:Idempotent
6124:Asymmetric
5898:References
5783:Jstor link
5457:Join (SQL)
4504:operations
4163:, in fact
2931:relation.
2587:(that is,
2561:such that
2025:surjective
1690:Properties
1469:. Given a
1402:allegories
951:Z notation
782:semigroups
589:such that
251:Definition
6603:Upper set
6540:Embedding
6476:Antichain
6297:Tolerance
6287:Symmetric
6282:Semiorder
6228:Reflexive
6146:Connected
5828:−
5427:≥
5358:∩
5256:Then the
5177:→
5171:×
5139:→
5133:×
5098:×
5092:→
5046:→
5014:→
4939:⊆
4899:⊆
4873:⊆
4841:⊆
4809:∖
4803:⊆
4777:⊆
4746:¯
4728:¯
4715:∩
4710:¯
4703:¯
4659:
4632:¯
4613:¯
4564:¯
4557:¯
4525:∖
4498:Quotients
4434:¯
4427:¯
4403:⊆
4374:¯
4365:⊆
4359:¯
4330:⊆
4298:⊆
4279:⊆
4225:¯
4216:⊆
4198:¯
4181:⊆
4138:¯
4129:⊆
4111:¯
4092:¯
4083:⊆
4077:¯
4046:⊆
4030:transpose
3956:∁
3942:¯
3907:∁
3899:⊆
3894:∁
3878:⊆
3845:⊆
3833:inclusion
3461:×
3101:given by
2993:¯
2949:¯
2894:⊆
2862:∈
2829:⊇
2761:∈
2702:⊆
2543:∈
2517:∈
2364:with its
2286:×
2280:⊆
2255:computing
2227:×
1929:injective
1628:⊕
1615:⊆
1589:→
1444:≅
1316:×
1310:⊆
1284:→
1167:→
1135:→
1030:→
998:×
992:⊆
961:∘
931:∘
904:∘
825:∘
758:∘
731:coincides
711:semicolon
686:∈
648:∈
574:∈
542:∈
504:×
498:⊆
458:∈
432:∈
406:∈
389:×
383:∈
304:×
298:⊆
272:×
266:⊆
245:syllogism
83:The word
78:functions
60:. In the
6398:Topology
6265:Preorder
6248:Eulerian
6211:Complete
6161:Directed
6151:Covering
6016:Preorder
5975:Category
5970:Glossary
5654:Archived
5463:See also
3827:forms a
2655:so that
2366:converse
1498:matrices
1360:between
1185:Given a
1047:category
854:reverses
6503:Duality
6481:Cofinal
6469:Related
6448:Fréchet
6325:)
6201:Bounded
6196:Lattice
6169:)
6167:Partial
6035:Results
6006:Lattice
5800:, from
5691:(1995)
5606:(1895)
5445:is the
3924:In the
3707:is the
3476:matrix
3031:Example
2259:sorites
2163:is the
2027:, then
1931:, then
1045:in the
30:In the
6528:Subnet
6508:Filter
6458:Normed
6443:Banach
6409:&
6316:Better
6253:Strict
6243:Graded
6134:topics
5965:Topics
5915:
5887:
5842:
5738:U+2A1F
5734:U+2A3E
5702:
5629:
5567:
5532:
5504:
5361:
5337:
5329:
4961:Sudoku
2806:then
2451:) and
2141:monoid
852:which
713:as an
38:, the
6518:Ideal
6496:Graph
6292:Total
6270:Total
6156:Dense
5481:Notes
5061:into
3147:is a
3127:when
2802:is a
2687:is a
2607:is a
1500:over
1471:field
1358:spans
1078:. In
85:uncle
6109:list
5913:ISBN
5885:ISBN
5762:2023
5755:nlab
5736:and
5732:See
5700:ISBN
5627:ISBN
5565:ISBN
5530:ISBN
5502:ISBN
5448:join
5319:<
5284:and
5260:fork
5224:and
5154:and
5076:<
5029:and
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3340:The
3194:and
3035:Let
2483:(on
2431:(on
2324:and
2216:and
2023:are
2003:and
1927:are
1907:and
1766:The
1380:and
1086:sets
922:and
709:The
663:and
287:and
56:and
6523:Net
6323:Pre
5264:of
4656:syq
4473:by
4287:and
3975:If
3807:on
3711:on
3151:of
2691:or
2503:).
2261:."
2119:to
1983:If
1887:If
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1770:of
1331:in
1091:Set
1081:Rel
701:).
255:If
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6625::
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5753:.
5591:,
5575:^
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5512:^
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