4143:
4495:
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
2241:
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
3633:
4024:
4747:
466:
5444:. 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
3325:
2173:. 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
224:
4379:
4230:
3911:
3500:
5381:
4565:
4633:
1860:
337:
1456:
4438:
4303:
4640:
2870:
4138:{\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}}.}
1565:
3230:
3015:
2914:
1238:
5103:
3960:
1637:
2716:
4814:
2962:
506:
3723:
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
2291:
839:
161:
4159:
3760:
2642:
1318:
1000:
306:
274:
3694:
5182:
3398:
2769:
5243:
5141:
3495:
3433:
2674:
2470:
2418:
2386:
1751:
1669:
5211:
4912:
4011:
3359:
1111:
1065:
4944:
4846:
5847:
5296:
4875:
4779:
3850:
612:
2235:
1340:
1260:
1197:
550:
936:
909:
763:
688:
650:
3463:
5432:
5048:
5016:
1169:
1032:
4145:
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.
2548:
4984:
2519:
1591:
1286:
1137:
960:
576:
2203:
2041:
1945:
1784:
4308:
879:
3114:
2574:
332:
156:
130:
104:
3790:
3721:
3656:
3161:
3068:
3045:
2064:
1968:
938:
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
5403:
5291:
5271:
4480:
4460:
4252:
3982:
3814:
3221:
3201:
3181:
3134:
3088:
2789:
2594:
2490:
2438:
2351:
2331:
2311:
2150:
2126:
2106:
2086:
2010:
1990:
1914:
1894:
1507:
1479:
1387:
1367:
3859:
4257:
4506:
2798:
3628:{\displaystyle RR^{\textsf {T}}={\begin{pmatrix}1&0&0&1\\0&1&0&1\\0&0&1&1\\1&1&1&1\end{pmatrix}}.}
1343:
4572:
1789:
962:
is used to denote the traditional (right) composition, while left composition is denoted by a fat semicolon. The unicode symbols are ⨾ and ⨟.
1396:
4742:{\displaystyle \operatorname {syq} (E,F)\mathrel {:=} {\overline {E^{\textsf {T}}{\bar {F}}}}\cap {\overline {{\bar {E}}^{\textsf {T}}F}}}
4384:
1077:, the morphisms are binary relations and the composition of morphisms is exactly composition of relations as defined above. The category
6584:
6567:
5642:
6097:
5933:
461:{\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\}.}
5877:
5619:
5522:
5494:
4949:
6414:
5666:
1512:
6550:
6409:
5905:
5692:
5557:
4149:
6404:
2967:
2875:
1202:
5053:
3920:
1596:
6627:
6040:
2683:
719:
6122:
4784:
2926:
6441:
6361:
5487:
473:
6226:
6155:
6035:
5865:
2261:
779:
5801:
De Morgan indicated contraries by lower case, conversion as M, and inclusion with )), so his notation was
3726:
2603:
1291:
973:
279:
247:
6129:
6117:
6080:
6055:
6030:
5984:
5953:
3663:
3320:{\displaystyle {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\1&1&1\end{pmatrix}}.}
6622:
6426:
6060:
6050:
5926:
5515:
5146:
3368:
2721:
881:
is commonly used in algebra to signify multiplication, so too, it can signify relative multiplication.
5216:
5108:
3468:
3406:
2647:
2443:
2391:
2359:
1691:
1645:
845:
the text sequence from the operation sequence. The small circle was used in the introductory pages of
6399:
6065:
5767:
5436:
5187:
4880:
3987:
3335:
1863:
1086:
1040:
5771:
5662:
4917:
4819:
6617:
5958:
5804:
4851:
4755:
3826:
774:
581:
76:
indicates a compound relation: for a person to be an uncle, he must be the brother of a parent. In
2208:
1323:
1243:
1180:
511:
6579:
6562:
3853:
914:
887:
736:
655:
617:
6491:
6107:
5726:
5722:
3821:
3442:
1482:
1390:
1346:
1035:
66:
5408:
5021:
4989:
1142:
1005:
219:{\displaystyle U=BP\quad {\text{ is equivalent to: }}\quad xByPz{\text{ if and only if }}xUz.}
6469:
6304:
6295:
6164:
6045:
5999:
5963:
5919:
5609:
4491:
Just as composition of relations is a type of multiplication resulting in a product, so some
4374:{\displaystyle RX\subseteq S{\text{ implies }}R^{\textsf {T}}{\bar {S}}\subseteq {\bar {X}},}
4225:{\displaystyle LM\subseteq N{\text{ implies }}{\bar {N}}M^{\textsf {T}}\subseteq {\bar {L}}.}
3914:
2524:
1069:
50:
4963:
4492:
2498:
1570:
1265:
1116:
945:
555:
6557:
6516:
6506:
6496:
6241:
6204:
6194:
6174:
6159:
2176:
2019:
1923:
1762:
1685:
1486:
62:
855:
8:
6484:
6395:
6341:
6300:
6290:
6179:
6112:
6075:
5790:
5457:
4148:
Though this transformation of an inclusion of a composition of relations was detailed by
3906:{\displaystyle A\subseteq B{\text{ implies }}B^{\complement }\subseteq A^{\complement }.}
3436:
3093:
2792:
2597:
2553:
1459:
850:
311:
135:
109:
83:
5639:
3772:
3703:
3638:
3227:, assuming rows (top to bottom) and columns (left to right) are ordered alphabetically:
3143:
3050:
3027:
2046:
1950:
6596:
6523:
6376:
6285:
6275:
6216:
6134:
6070:
5462:
5441:
5388:
5276:
5256:
4465:
4445:
4237:
4153:
3967:
3799:
3697:
3206:
3186:
3166:
3119:
3073:
2774:
2677:
2579:
2475:
2423:
2336:
2316:
2296:
2135:
2111:
2091:
2071:
1995:
1975:
1899:
1879:
1492:
1464:
1372:
1352:
884:
Further with the circle notation, subscripts may be used. Some authors prefer to write
229:
6436:
5376:{\displaystyle c\,(<)\,d~\mathrel {:=} ~c;a^{\textsf {T}}\cap \ d;b^{\textsf {T}}.}
6533:
6511:
6371:
6356:
6336:
6139:
5901:
5897:
5873:
5763:
5688:
5650:
5627:
5615:
5553:
5518:
5490:
4014:
3362:
3330:
3137:
2354:
2157:
1756:
1074:
727:
5596:
6346:
6199:
1870:
1175:
1079:
722:
with the notation for function composition used (mostly by computer scientists) in
5739:
4560:{\displaystyle A\backslash B\mathrel {:=} {\overline {A^{\textsf {T}}{\bar {B}}}}}
6528:
6311:
6189:
6184:
6169:
6085:
5994:
5979:
5857:
5685:
5646:
5573:
5550:
3817:
3793:
2153:
723:
711:
77:
65:
is the special case of composition of relations where all relations involved are
24:
6446:
6431:
6421:
6280:
6258:
6236:
5869:
5592:
5581:
3497:
contains a 1 at every position, while the reversed matrix product computes as:
3401:
3224:
2238:
2170:
707:
703:
4234:
With Schröder rules and complementation one can solve for an unknown relation
6611:
6545:
6501:
6479:
6351:
6221:
6209:
6014:
5677:
2247:
766:
5894:
Monoids, Acts and
Categories with Applications to Wreath Products and Graphs
2246:
the conclusions traditionally drawn by means of hypothetical syllogisms and
6366:
6248:
6231:
6149:
5989:
5942:
5783:
2917:
1640:
236:
has been subsumed by relational logical expressions and their composition.
6572:
6265:
6144:
6009:
5786:
4496:
symmetric quotient presumes two relations share a domain and a codomain.
1672:
20:
5533:
A. De Morgan (1860) "On the
Syllogism: IV and on the Logic of Relations"
4628:{\displaystyle D/C\mathrel {:=} {\overline {{\bar {D}}C^{\textsf {T}}}}}
1855:{\displaystyle (R\,;S)^{\textsf {T}}=S^{\textsf {T}}\,;R^{\textsf {T}}.}
6540:
6474:
6315:
5445:
2013:
939:
6591:
6464:
6270:
4156:
first articulated the transformation as
Theorem K in 1860. He wrote
1917:
1451:{\displaystyle {\mathsf {Rel}}({\mathsf {Set}})\cong {\mathsf {Rel}}}
770:
699:
233:
849:
until it was dropped in favor of juxtaposition (no infix notation).
773:
of relations. However, the small circle is widely used to represent
765:
has been used for the infix notation of composition of relations by
726:, as well as the notation for dynamic conjunction within linguistic
6386:
6253:
6004:
5766:(December 1948) "Matrix development of the calculus of relations",
5385:
Another form of composition of relations, which applies to general
5184:
understood as relations, meaning that there are converse relations
4433:{\displaystyle X\subseteq {\overline {R^{\textsf {T}}{\bar {S}}}},}
4298:{\displaystyle RX\subseteq S\quad {\text{and}}\quad XR\subseteq S.}
3635:
This matrix is symmetric, and represents a homogeneous relation on
5482:
Bjarni JĂłnssen (1984) "Maximal
Algebras of Binary Relations", in
3917:
it is common to represent the complement of a set by an overbar:
2964:
is used to distinguish relations of Ferrer's type, which satisfy
2043:
is surjective, which conversely implies only the surjectivity of
5911:
2129:
1947:
is injective, which conversely implies only the injectivity of
1862:
This property makes the set of all binary relations on a set a
5514:, Encyclopedia of Mathematics and its Applications, vol. 132,
1873:(that is, functional relations) is again a (partial) function.
2865:{\displaystyle R^{\textsf {T}}R\supseteq I=\{(x,x):x\in B\}.}
2128:) together with (left or right) relation composition forms a
73:
5465: – Human contact that exists because of a mutual friend
4914:
and the right residual is the greatest relation satisfying
4816:
Thus the left residual is the greatest relation satisfying
4797:
4513:
3527:
3239:
2333:
are distinct sets. Then using composition of relation
714:
has renewed the use of the semicolon, particularly in
5807:
5411:
5391:
5299:
5279:
5259:
5219:
5190:
5149:
5111:
5056:
5024:
4992:
4966:
4920:
4883:
4854:
4822:
4787:
4758:
4643:
4575:
4509:
4468:
4448:
4387:
4311:
4260:
4240:
4162:
4027:
3990:
3970:
3923:
3862:
3829:
3802:
3775:
3729:
3706:
3666:
3641:
3503:
3471:
3445:
3409:
3371:
3338:
3233:
3209:
3189:
3169:
3146:
3122:
3096:
3076:
3053:
3030:
2970:
2929:
2878:
2801:
2777:
2724:
2686:
2650:
2606:
2582:
2556:
2527:
2501:
2478:
2446:
2426:
2394:
2362:
2339:
2319:
2299:
2264:
2211:
2179:
2138:
2114:
2094:
2074:
2049:
2022:
1998:
1978:
1953:
1926:
1902:
1882:
1792:
1765:
1694:
1648:
1599:
1573:
1515:
1495:
1467:
1399:
1375:
1355:
1326:
1294:
1268:
1246:
1205:
1183:
1145:
1119:
1089:
1043:
1008:
976:
948:
917:
890:
858:
782:
739:
658:
620:
584:
558:
514:
476:
340:
314:
282:
250:
164:
138:
112:
106:) is the composition of relations "is a brother of" (
86:
4955:
2164:
1560:{\displaystyle {\mathsf {Rel}}({\mathsf {Mat}}(k))}
5841:
5628:http://www.cs.man.ac.uk/~pt/Practical_Foundations/
5426:
5397:
5375:
5285:
5265:
5237:
5205:
5176:
5135:
5097:
5042:
5010:
4978:
4938:
4906:
4869:
4840:
4808:
4773:
4741:
4627:
4559:
4474:
4454:
4432:
4373:
4297:
4246:
4224:
4137:
4005:
3976:
3954:
3905:
3844:
3808:
3784:
3754:
3715:
3688:
3650:
3627:
3489:
3457:
3427:
3392:
3353:
3319:
3215:
3195:
3175:
3155:
3128:
3108:
3082:
3062:
3039:
3009:
2956:
2908:
2864:
2783:
2763:
2710:
2668:
2636:
2588:
2568:
2542:
2513:
2484:
2464:
2432:
2412:
2380:
2345:
2325:
2305:
2285:
2229:
2197:
2144:
2120:
2100:
2080:
2058:
2035:
2004:
1984:
1962:
1939:
1908:
1888:
1854:
1778:
1745:
1663:
1631:
1585:
1559:
1501:
1473:
1450:
1381:
1361:
1334:
1312:
1280:
1254:
1232:
1191:
1163:
1131:
1105:
1059:
1026:
994:
954:
930:
903:
873:
833:
757:
682:
644:
606:
570:
544:
500:
460:
326:
300:
268:
218:
150:
124:
98:
5896:, De Gruyter Expositions in Mathematics vol. 29,
308:are two binary relations, then their composition
6609:
5626:A free HTML version of the book is available at
965:
3070:{ French, German, Italian } with the relation
5927:
5547:Augustus De Morgan and the Logic of Relations
4948:One can practice the logic of residuals with
3010:{\displaystyle R{\bar {R}}^{\textsf {T}}R=R.}
2909:{\displaystyle R\subseteq RR^{\textsf {T}}R.}
1639:. The category of linear relations over the
1233:{\displaystyle {\mathsf {Rel}}(\mathbb {X} )}
5098:{\displaystyle c\,(<)\,d:H\to A\times B.}
3955:{\displaystyle {\bar {A}}=A^{\complement }.}
3047:{ France, Germany, Italy, Switzerland } and
2856:
2826:
2755:
2725:
1632:{\displaystyle R\subseteq k^{n}\oplus k^{m}}
452:
353:
5607:
2711:{\displaystyle I\subseteq RR^{\textsf {T}}}
2169:Finite binary relations are represented by
706:for composition of relations dates back to
6585:Positive cone of a partially ordered group
5934:
5920:
5614:. Cambridge University Press. p. 24.
4986:has been introduced to fuse two relations
2253:
1083:of sets and functions is a subcategory of
5892:M. Kilp, U. Knauer, A.V. Mikhalev (2000)
5601:
5364:
5340:
5313:
5303:
5226:
5197:
5070:
5060:
4809:{\displaystyle X\subseteq A\backslash B.}
4724:
4679:
4613:
4533:
4403:
4335:
4198:
4111:
4053:
3997:
3743:
3733:
3679:
3673:
3513:
3478:
3416:
3378:
3345:
2989:
2957:{\displaystyle {\bar {R}}^{\textsf {T}}R}
2945:
2894:
2808:
2702:
2660:
2625:
2453:
2404:
2369:
2026:
1930:
1843:
1833:
1827:
1812:
1799:
1769:
1651:
1485:), the category of relations internal to
1328:
1248:
1223:
1185:
600:
596:
592:
588:
53:, the composition of relations is called
6568:Positive cone of an ordered vector space
5787:The Origins of the Calculus of Relations
5638:Michael Barr & Charles Wells (1998)
5105:The construction depends on projections
693:
31:is the forming of a new binary relation
5640:Category Theory for Computer Scientists
1389:. Categories of internal relations are
232:, the traditional form of reasoning by
80:it is said that the relation of Uncle (
6610:
5569:
5567:
5565:
5506:
5504:
5502:
1671:is isomorphic to the phase-free qubit
1540:
1537:
1534:
1524:
1521:
1518:
1443:
1440:
1437:
1424:
1421:
1418:
1408:
1405:
1402:
1214:
1211:
1208:
1098:
1095:
1092:
1052:
1049:
1046:
501:{\displaystyle R;S\subseteq X\times Z}
5915:
5541:
5539:
2286:{\displaystyle R\subseteq A\times B;}
2132:with zero, where the identity map on
2068:The set of binary relations on a set
1199:, its category of internal relations
834:{\displaystyle g(f(x))=(g\circ f)(x)}
5611:Practical Foundations of Mathematics
3755:{\displaystyle R\,;R^{\textsf {T}}.}
2916:The opposite inclusion occurs for a
2637:{\displaystyle x,xRR^{\textsf {T}}x}
1313:{\displaystyle R\subseteq X\times Y}
995:{\displaystyle R\subseteq X\times Y}
301:{\displaystyle S\subseteq Y\times Z}
269:{\displaystyle R\subseteq X\times Y}
5671:
5667:Stanford Encyclopedia of Philosophy
5562:
5499:
3689:{\displaystyle R^{\textsf {T}}\,;R}
552:if and only if there is an element
13:
6095:Properties & Types (
5536:
2258:Consider a heterogeneous relation
14:
6639:
6551:Positive cone of an ordered field
5703:Kilp, Knauer & Mikhalev, p. 7
5177:{\displaystyle b:A\times B\to B,}
4956:Join: another form of composition
3764:
3435:when summation is implemented by
3393:{\displaystyle R^{\textsf {T}};R}
2764:{\displaystyle \{(x,x):x\in A\}.}
2718:where I is the identity relation
508:is defined by the rule that says
6405:Ordered topological vector space
5941:
5682:Fundamentals of Semigroup Theory
5238:{\displaystyle b^{\textsf {T}}.}
5136:{\displaystyle a:A\times B\to A}
4254:in relation inclusions such as
3490:{\displaystyle R^{\textsf {T}}R}
3428:{\displaystyle R^{\textsf {T}}R}
2669:{\displaystyle RR^{\textsf {T}}}
2465:{\displaystyle R^{\textsf {T}}R}
2413:{\displaystyle RR^{\textsf {T}}}
2388:there are homogeneous relations
2381:{\displaystyle R^{\textsf {T}},}
2165:Composition in terms of matrices
1746:{\displaystyle R;(S;T)=(R;S);T.}
1664:{\displaystyle \mathbb {F} _{2}}
41:from two given binary relations
5851:
5795:
5777:
5757:
5732:
5715:
5706:
5697:
5206:{\displaystyle a^{\textsf {T}}}
4907:{\displaystyle Y\subseteq D/C,}
4305:For instance, by Schröder rule
4279:
4273:
4092:
4086:
4046:
4040:
4006:{\displaystyle S^{\textsf {T}}}
3365:, and the relation composition
3354:{\displaystyle R^{\textsf {T}}}
1106:{\displaystyle {\mathsf {Rel}}}
1060:{\displaystyle {\mathsf {Rel}}}
183:
177:
5827:
5824:
5684:, page 16, LMS Monograph #12,
5661:Rick Nouwen and others (2016)
5655:
5632:
5597:Algebra und Logik der Relative
5586:
5576:& Thomas Ströhlein (1993)
5527:
5476:
5310:
5304:
5165:
5127:
5080:
5067:
5061:
5034:
5002:
4973:
4967:
4939:{\displaystyle YC\subseteq D.}
4841:{\displaystyle AX\subseteq B.}
4716:
4691:
4662:
4650:
4601:
4545:
4415:
4362:
4347:
4213:
4186:
4126:
4099:
4080:
4065:
4021:. Then the Schröder rules are
3930:
3836:
3830:
2981:
2937:
2841:
2829:
2740:
2728:
1807:
1793:
1731:
1719:
1713:
1701:
1577:
1554:
1551:
1545:
1529:
1429:
1413:
1272:
1227:
1219:
1155:
1123:
1018:
868:
859:
828:
822:
819:
807:
801:
798:
792:
786:
752:
740:
671:
659:
633:
621:
527:
515:
443:
431:
417:
405:
368:
356:
1:
6362:Series-parallel partial order
5886:
5842:{\displaystyle nM^{-1}))\ l.}
5488:American Mathematical Society
5484:Contributions to Group Theory
4870:{\displaystyle YC\subseteq D}
4774:{\displaystyle AX\subseteq B}
3845:{\displaystyle (\subseteq ).}
1678:
718:(2011). The use of semicolon
607:{\displaystyle x\,R\,y\,S\,z}
239:
180: is equivalent to:
57:, and its result is called a
6041:Cantor's isomorphism theorem
5866:Lecture Notes in Mathematics
5712:ISO/IEC 13568:2002(E), p. 23
4734:
4698:
4620:
4552:
4486:
4422:
4089: is equivalent to
4043: is equivalent to
2230:{\displaystyle 1\times 1=1.}
1684:Composition of relations is
1335:{\displaystyle \mathbb {X} }
1255:{\displaystyle \mathbb {X} }
1192:{\displaystyle \mathbb {X} }
966:Mathematical generalizations
545:{\displaystyle (x,z)\in R;S}
7:
6081:Szpilrajn extension theorem
6056:Hausdorff maximal principle
6031:Boolean prime ideal theorem
5860:and Michael Winter (2018):
5451:
2156:, and the empty set is the
10:
6644:
6427:Topological vector lattice
5516:Cambridge University Press
4381:and complementation gives
3984:is a binary relation, let
3019:
1262:, but now the morphisms
931:{\displaystyle \circ _{r}}
904:{\displaystyle \circ _{l}}
758:{\displaystyle (R\circ S)}
683:{\displaystyle (y,z)\in S}
645:{\displaystyle (x,y)\in R}
201: if and only if
6457:
6385:
6324:
6094:
6023:
5972:
5949:
5768:Journal of Symbolic Logic
5545:Daniel D. Merrill (1990)
4848:Similarly, the inclusion
3458:{\displaystyle 3\times 3}
2088:(that is, relations from
1864:semigroup with involution
1240:has the same objects as
769:in his books considering
6036:Cantor–Bernstein theorem
5469:
5427:{\displaystyle n\geq 2,}
5043:{\displaystyle d:H\to B}
5011:{\displaystyle c:H\to A}
4752:Using Schröder's rules,
3439:. It turns out that the
3223:can be represented by a
1288:are given by subobjects
1164:{\displaystyle f:X\to Y}
1027:{\displaystyle R:X\to Y}
775:composition of functions
388: there exists
132:) and "is a parent of" (
29:composition of relations
6580:Partially ordered group
6400:Specialization preorder
5510:Gunther Schmidt (2011)
2543:{\displaystyle y\in B,}
2254:Heterogeneous relations
1342:. Formally, these are
55:relative multiplication
6628:Mathematical relations
6066:Kruskal's tree theorem
6061:Knaster–Tarski theorem
6051:Dushnik–Miller theorem
5843:
5512:Relational Mathematics
5428:
5399:
5377:
5287:
5267:
5239:
5207:
5178:
5137:
5099:
5044:
5012:
4980:
4979:{\displaystyle (<)}
4940:
4908:
4871:
4842:
4810:
4775:
4743:
4629:
4561:
4476:
4456:
4434:
4375:
4299:
4248:
4226:
4139:
4007:
3978:
3956:
3907:
3846:
3810:
3792:the collection of all
3786:
3756:
3717:
3690:
3652:
3629:
3491:
3459:
3429:
3394:
3355:
3321:
3217:
3197:
3177:
3157:
3130:
3110:
3084:
3064:
3041:
3011:
2958:
2910:
2866:
2785:
2765:
2712:
2670:
2638:
2590:
2570:
2544:
2515:
2514:{\displaystyle x\in A}
2486:
2466:
2434:
2414:
2382:
2347:
2327:
2307:
2287:
2231:
2199:
2146:
2122:
2102:
2082:
2060:
2037:
2006:
1986:
1964:
1941:
1910:
1890:
1856:
1780:
1747:
1665:
1633:
1587:
1586:{\displaystyle n\to m}
1561:
1503:
1483:principal ideal domain
1475:
1452:
1383:
1363:
1336:
1314:
1282:
1281:{\displaystyle X\to Y}
1256:
1234:
1193:
1165:
1133:
1132:{\displaystyle X\to Y}
1107:
1061:
1028:
996:
956:
955:{\displaystyle \circ }
932:
905:
875:
835:
759:
716:Relational Mathematics
684:
646:
608:
572:
571:{\displaystyle y\in Y}
546:
502:
462:
328:
302:
270:
220:
152:
126:
100:
16:Mathematical operation
5844:
5774:, quote from page 203
5429:
5405:-place relations for
5400:
5378:
5288:
5268:
5240:
5208:
5179:
5138:
5100:
5045:
5013:
4981:
4941:
4909:
4872:
4843:
4811:
4776:
4744:
4630:
4562:
4477:
4457:
4435:
4376:
4300:
4249:
4227:
4140:
4008:
3979:
3957:
3915:calculus of relations
3908:
3847:
3811:
3787:
3757:
3718:
3691:
3653:
3630:
3492:
3460:
3430:
3395:
3356:
3322:
3218:
3198:
3178:
3158:
3131:
3111:
3085:
3065:
3042:
3012:
2959:
2911:
2867:
2786:
2766:
2713:
2671:
2639:
2598:(left-)total relation
2591:
2571:
2545:
2516:
2487:
2467:
2435:
2415:
2383:
2348:
2328:
2308:
2288:
2232:
2200:
2198:{\displaystyle 1+1=1}
2147:
2123:
2103:
2083:
2061:
2038:
2036:{\displaystyle R\,;S}
2007:
1987:
1965:
1942:
1940:{\displaystyle R\,;S}
1911:
1891:
1857:
1781:
1779:{\displaystyle R\,;S}
1748:
1666:
1634:
1588:
1562:
1504:
1481:(or more generally a
1476:
1453:
1384:
1364:
1337:
1315:
1283:
1257:
1235:
1194:
1166:
1134:
1108:
1062:
1029:
997:
957:
933:
906:
876:
836:
760:
710:'s textbook of 1895.
694:Notational variations
685:
647:
609:
573:
547:
503:
463:
402: such that
329:
303:
271:
221:
153:
127:
101:
51:calculus of relations
6558:Ordered vector space
5805:
5740:"internal relations"
5608:Paul Taylor (1999).
5578:Relations and Graphs
5486:, K.I. Appel editor
5409:
5389:
5297:
5277:
5257:
5217:
5188:
5147:
5109:
5054:
5022:
4990:
4964:
4918:
4881:
4852:
4820:
4785:
4756:
4641:
4637:Symmetric quotient:
4573:
4507:
4466:
4446:
4440:which is called the
4385:
4309:
4258:
4238:
4160:
4025:
3988:
3968:
3921:
3860:
3856:reverses inclusion:
3827:
3800:
3773:
3727:
3704:
3664:
3639:
3501:
3469:
3443:
3407:
3369:
3336:
3231:
3207:
3187:
3167:
3144:
3120:
3094:
3074:
3051:
3028:
2968:
2927:
2876:
2799:
2775:
2722:
2684:
2648:
2604:
2580:
2554:
2525:
2499:
2476:
2444:
2424:
2392:
2360:
2337:
2317:
2297:
2262:
2209:
2177:
2136:
2112:
2092:
2072:
2047:
2020:
1996:
1976:
1951:
1924:
1900:
1880:
1790:
1763:
1692:
1646:
1597:
1571:
1513:
1493:
1465:
1397:
1373:
1353:
1324:
1292:
1266:
1244:
1203:
1181:
1143:
1117:
1087:
1041:
1006:
974:
946:
915:
888:
874:{\displaystyle (RS)}
856:
847:Graphs and Relations
780:
737:
656:
618:
582:
556:
512:
474:
338:
312:
280:
248:
162:
136:
110:
84:
63:Function composition
6396:Alexandrov topology
6342:Lexicographic order
6301:Well-quasi-ordering
5862:Relational Topology
5791:Stanford University
5458:Demonic composition
4326: implies
4177: implies
3874: implies
3437:logical disjunction
3400:corresponds to the
3361:corresponds to the
3109:{\displaystyle aRb}
2793:surjective relation
2569:{\displaystyle xRy}
1871:(partial) functions
1869:The composition of
327:{\displaystyle R;S}
151:{\displaystyle yPz}
125:{\displaystyle xBy}
99:{\displaystyle xUz}
6377:Transitive closure
6337:Converse/Transpose
6046:Dilworth's theorem
5839:
5729:on FileFormat.info
5645:2016-03-04 at the
5463:Friend of a friend
5442:relational algebra
5424:
5395:
5373:
5283:
5263:
5235:
5203:
5174:
5133:
5095:
5040:
5008:
4976:
4936:
4904:
4867:
4838:
4806:
4771:
4739:
4625:
4557:
4472:
4452:
4430:
4371:
4295:
4244:
4222:
4154:Augustus De Morgan
4135:
4017:, also called the
4003:
3974:
3952:
3903:
3842:
3806:
3785:{\displaystyle V,}
3782:
3752:
3716:{\displaystyle B,}
3713:
3698:universal relation
3686:
3651:{\displaystyle A.}
3648:
3625:
3616:
3487:
3455:
3425:
3390:
3351:
3317:
3308:
3213:
3193:
3173:
3156:{\displaystyle a.}
3153:
3126:
3106:
3080:
3063:{\displaystyle B=}
3060:
3040:{\displaystyle A=}
3037:
3007:
2954:
2906:
2862:
2781:
2761:
2708:
2678:reflexive relation
2666:
2634:
2586:
2566:
2540:
2521:there exists some
2511:
2482:
2462:
2430:
2410:
2378:
2343:
2323:
2303:
2283:
2227:
2195:
2142:
2118:
2098:
2078:
2059:{\displaystyle S.}
2056:
2033:
2002:
1982:
1963:{\displaystyle R.}
1960:
1937:
1906:
1886:
1852:
1776:
1743:
1661:
1629:
1583:
1557:
1499:
1471:
1448:
1379:
1359:
1332:
1310:
1278:
1252:
1230:
1189:
1161:
1129:
1103:
1057:
1024:
992:
952:
928:
901:
871:
831:
755:
680:
642:
604:
568:
542:
498:
458:
324:
298:
266:
230:Augustus De Morgan
216:
148:
122:
96:
6623:Binary operations
6605:
6604:
6563:Partially ordered
6372:Symmetric closure
6357:Reflexive closure
6100:
5898:Walter de Gruyter
5878:978-3-319-74451-3
5832:
5764:Irving Copilowish
5663:Dynamic Semantics
5651:McGill University
5621:978-0-521-63107-5
5523:978-0-521-76268-7
5495:978-0-8218-5035-0
5398:{\displaystyle n}
5366:
5351:
5342:
5327:
5319:
5286:{\displaystyle d}
5266:{\displaystyle c}
5228:
5199:
4877:is equivalent to
4781:is equivalent to
4737:
4726:
4719:
4701:
4694:
4681:
4623:
4615:
4604:
4555:
4548:
4535:
4475:{\displaystyle R}
4455:{\displaystyle S}
4442:left residual of
4425:
4418:
4405:
4365:
4350:
4337:
4327:
4277:
4247:{\displaystyle X}
4216:
4200:
4189:
4178:
4129:
4113:
4102:
4090:
4083:
4068:
4055:
4044:
4015:converse relation
3999:
3977:{\displaystyle S}
3933:
3875:
3809:{\displaystyle V}
3745:
3675:
3660:Correspondingly,
3515:
3480:
3418:
3380:
3363:transposed matrix
3347:
3331:converse relation
3216:{\displaystyle R}
3196:{\displaystyle B}
3176:{\displaystyle A}
3138:national language
3129:{\displaystyle b}
3083:{\displaystyle R}
2991:
2984:
2947:
2940:
2896:
2810:
2784:{\displaystyle R}
2704:
2662:
2627:
2589:{\displaystyle R}
2485:{\displaystyle B}
2455:
2433:{\displaystyle A}
2406:
2371:
2346:{\displaystyle R}
2326:{\displaystyle B}
2306:{\displaystyle A}
2145:{\displaystyle X}
2121:{\displaystyle X}
2101:{\displaystyle X}
2081:{\displaystyle X}
2005:{\displaystyle S}
1985:{\displaystyle R}
1909:{\displaystyle S}
1889:{\displaystyle R}
1845:
1829:
1814:
1757:converse relation
1593:linear subspaces
1502:{\displaystyle k}
1474:{\displaystyle k}
1393:. In particular
1382:{\displaystyle Y}
1362:{\displaystyle X}
1113:where the maps
970:Binary relations
728:dynamic semantics
429:
403:
389:
202:
181:
6635:
6347:Linear extension
6096:
6076:Mirsky's theorem
5936:
5929:
5922:
5913:
5912:
5880:
5855:
5849:
5848:
5846:
5845:
5840:
5830:
5823:
5822:
5799:
5793:
5781:
5775:
5761:
5755:
5754:
5752:
5750:
5736:
5730:
5719:
5713:
5710:
5704:
5701:
5695:
5675:
5669:
5659:
5653:
5636:
5630:
5625:
5605:
5599:
5590:
5584:
5571:
5560:
5543:
5534:
5531:
5525:
5508:
5497:
5480:
5433:
5431:
5430:
5425:
5404:
5402:
5401:
5396:
5382:
5380:
5379:
5374:
5369:
5368:
5367:
5349:
5345:
5344:
5343:
5325:
5324:
5317:
5292:
5290:
5289:
5284:
5272:
5270:
5269:
5264:
5251:
5250:
5244:
5242:
5241:
5236:
5231:
5230:
5229:
5212:
5210:
5209:
5204:
5202:
5201:
5200:
5183:
5181:
5180:
5175:
5142:
5140:
5139:
5134:
5104:
5102:
5101:
5096:
5049:
5047:
5046:
5041:
5017:
5015:
5014:
5009:
4985:
4983:
4982:
4977:
4960:A fork operator
4945:
4943:
4942:
4937:
4913:
4911:
4910:
4905:
4897:
4876:
4874:
4873:
4868:
4847:
4845:
4844:
4839:
4815:
4813:
4812:
4807:
4780:
4778:
4777:
4772:
4748:
4746:
4745:
4740:
4738:
4733:
4729:
4728:
4727:
4721:
4720:
4712:
4707:
4702:
4697:
4696:
4695:
4687:
4684:
4683:
4682:
4671:
4669:
4634:
4632:
4631:
4626:
4624:
4619:
4618:
4617:
4616:
4606:
4605:
4597:
4593:
4591:
4583:
4569:Right residual:
4566:
4564:
4563:
4558:
4556:
4551:
4550:
4549:
4541:
4538:
4537:
4536:
4525:
4523:
4481:
4479:
4478:
4473:
4461:
4459:
4458:
4453:
4439:
4437:
4436:
4431:
4426:
4421:
4420:
4419:
4411:
4408:
4407:
4406:
4395:
4380:
4378:
4377:
4372:
4367:
4366:
4358:
4352:
4351:
4343:
4340:
4339:
4338:
4328:
4325:
4304:
4302:
4301:
4296:
4278:
4275:
4253:
4251:
4250:
4245:
4231:
4229:
4228:
4223:
4218:
4217:
4209:
4203:
4202:
4201:
4191:
4190:
4182:
4179:
4176:
4144:
4142:
4141:
4136:
4131:
4130:
4122:
4116:
4115:
4114:
4104:
4103:
4095:
4091:
4088:
4085:
4084:
4076:
4070:
4069:
4061:
4058:
4057:
4056:
4045:
4042:
4012:
4010:
4009:
4004:
4002:
4001:
4000:
3983:
3981:
3980:
3975:
3961:
3959:
3958:
3953:
3948:
3947:
3935:
3934:
3926:
3912:
3910:
3909:
3904:
3899:
3898:
3886:
3885:
3876:
3873:
3851:
3849:
3848:
3843:
3815:
3813:
3812:
3807:
3794:binary relations
3791:
3789:
3788:
3783:
3769:For a given set
3761:
3759:
3758:
3753:
3748:
3747:
3746:
3722:
3720:
3719:
3714:
3695:
3693:
3692:
3687:
3678:
3677:
3676:
3657:
3655:
3654:
3649:
3634:
3632:
3631:
3626:
3621:
3620:
3518:
3517:
3516:
3496:
3494:
3493:
3488:
3483:
3482:
3481:
3464:
3462:
3461:
3456:
3434:
3432:
3431:
3426:
3421:
3420:
3419:
3399:
3397:
3396:
3391:
3383:
3382:
3381:
3360:
3358:
3357:
3352:
3350:
3349:
3348:
3326:
3324:
3323:
3318:
3313:
3312:
3222:
3220:
3219:
3214:
3202:
3200:
3199:
3194:
3182:
3180:
3179:
3174:
3162:
3160:
3159:
3154:
3135:
3133:
3132:
3127:
3115:
3113:
3112:
3107:
3089:
3087:
3086:
3081:
3069:
3067:
3066:
3061:
3046:
3044:
3043:
3038:
3016:
3014:
3013:
3008:
2994:
2993:
2992:
2986:
2985:
2977:
2963:
2961:
2960:
2955:
2950:
2949:
2948:
2942:
2941:
2933:
2923:The composition
2915:
2913:
2912:
2907:
2899:
2898:
2897:
2871:
2869:
2868:
2863:
2813:
2812:
2811:
2790:
2788:
2787:
2782:
2770:
2768:
2767:
2762:
2717:
2715:
2714:
2709:
2707:
2706:
2705:
2675:
2673:
2672:
2667:
2665:
2664:
2663:
2643:
2641:
2640:
2635:
2630:
2629:
2628:
2600:), then for all
2595:
2593:
2592:
2587:
2575:
2573:
2572:
2567:
2549:
2547:
2546:
2541:
2520:
2518:
2517:
2512:
2491:
2489:
2488:
2483:
2471:
2469:
2468:
2463:
2458:
2457:
2456:
2439:
2437:
2436:
2431:
2419:
2417:
2416:
2411:
2409:
2408:
2407:
2387:
2385:
2384:
2379:
2374:
2373:
2372:
2352:
2350:
2349:
2344:
2332:
2330:
2329:
2324:
2312:
2310:
2309:
2304:
2292:
2290:
2289:
2284:
2237:An entry in the
2236:
2234:
2233:
2228:
2204:
2202:
2201:
2196:
2171:logical matrices
2151:
2149:
2148:
2143:
2127:
2125:
2124:
2119:
2107:
2105:
2104:
2099:
2087:
2085:
2084:
2079:
2065:
2063:
2062:
2057:
2042:
2040:
2039:
2034:
2011:
2009:
2008:
2003:
1991:
1989:
1988:
1983:
1969:
1967:
1966:
1961:
1946:
1944:
1943:
1938:
1915:
1913:
1912:
1907:
1895:
1893:
1892:
1887:
1861:
1859:
1858:
1853:
1848:
1847:
1846:
1832:
1831:
1830:
1817:
1816:
1815:
1785:
1783:
1782:
1777:
1752:
1750:
1749:
1744:
1675:modulo scalars.
1670:
1668:
1667:
1662:
1660:
1659:
1654:
1638:
1636:
1635:
1630:
1628:
1627:
1615:
1614:
1592:
1590:
1589:
1584:
1566:
1564:
1563:
1558:
1544:
1543:
1528:
1527:
1508:
1506:
1505:
1500:
1480:
1478:
1477:
1472:
1457:
1455:
1454:
1449:
1447:
1446:
1428:
1427:
1412:
1411:
1388:
1386:
1385:
1380:
1368:
1366:
1365:
1360:
1341:
1339:
1338:
1333:
1331:
1319:
1317:
1316:
1311:
1287:
1285:
1284:
1279:
1261:
1259:
1258:
1253:
1251:
1239:
1237:
1236:
1231:
1226:
1218:
1217:
1198:
1196:
1195:
1190:
1188:
1176:regular category
1170:
1168:
1167:
1162:
1138:
1136:
1135:
1130:
1112:
1110:
1109:
1104:
1102:
1101:
1073:the objects are
1066:
1064:
1063:
1058:
1056:
1055:
1033:
1031:
1030:
1025:
1001:
999:
998:
993:
961:
959:
958:
953:
937:
935:
934:
929:
927:
926:
910:
908:
907:
902:
900:
899:
880:
878:
877:
872:
840:
838:
837:
832:
764:
762:
761:
756:
689:
687:
686:
681:
651:
649:
648:
643:
613:
611:
610:
605:
577:
575:
574:
569:
551:
549:
548:
543:
507:
505:
504:
499:
470:In other words,
467:
465:
464:
459:
430:
427:
404:
401:
390:
387:
334:is the relation
333:
331:
330:
325:
307:
305:
304:
299:
275:
273:
272:
267:
225:
223:
222:
217:
203:
200:
182:
179:
157:
155:
154:
149:
131:
129:
128:
123:
105:
103:
102:
97:
59:relative product
40:
25:binary relations
6643:
6642:
6638:
6637:
6636:
6634:
6633:
6632:
6618:Algebraic logic
6608:
6607:
6606:
6601:
6597:Young's lattice
6453:
6381:
6320:
6170:Heyting algebra
6118:Boolean algebra
6090:
6071:Laver's theorem
6019:
5985:Boolean algebra
5980:Binary relation
5968:
5945:
5940:
5889:
5884:
5883:
5858:Gunther Schmidt
5856:
5852:
5815:
5811:
5806:
5803:
5802:
5800:
5796:
5782:
5778:
5770:13(4): 193–203
5762:
5758:
5748:
5746:
5738:
5737:
5733:
5720:
5716:
5711:
5707:
5702:
5698:
5686:Clarendon Press
5676:
5672:
5660:
5656:
5649:, page 6, from
5647:Wayback Machine
5637:
5633:
5622:
5606:
5602:
5591:
5587:
5574:Gunther Schmidt
5572:
5563:
5551:Kluwer Academic
5544:
5537:
5532:
5528:
5509:
5500:
5481:
5477:
5472:
5454:
5410:
5407:
5406:
5390:
5387:
5386:
5363:
5362:
5358:
5339:
5338:
5334:
5320:
5298:
5295:
5294:
5278:
5275:
5274:
5258:
5255:
5254:
5248:
5247:
5225:
5224:
5220:
5218:
5215:
5214:
5196:
5195:
5191:
5189:
5186:
5185:
5148:
5145:
5144:
5110:
5107:
5106:
5055:
5052:
5051:
5023:
5020:
5019:
4991:
4988:
4987:
4965:
4962:
4961:
4958:
4919:
4916:
4915:
4893:
4882:
4879:
4878:
4853:
4850:
4849:
4821:
4818:
4817:
4786:
4783:
4782:
4757:
4754:
4753:
4723:
4722:
4711:
4710:
4709:
4708:
4706:
4686:
4685:
4678:
4677:
4673:
4672:
4670:
4665:
4642:
4639:
4638:
4612:
4611:
4607:
4596:
4595:
4594:
4592:
4587:
4579:
4574:
4571:
4570:
4540:
4539:
4532:
4531:
4527:
4526:
4524:
4519:
4508:
4505:
4504:
4503:Left residual:
4489:
4467:
4464:
4463:
4447:
4444:
4443:
4410:
4409:
4402:
4401:
4397:
4396:
4394:
4386:
4383:
4382:
4357:
4356:
4342:
4341:
4334:
4333:
4329:
4324:
4310:
4307:
4306:
4274:
4259:
4256:
4255:
4239:
4236:
4235:
4208:
4207:
4197:
4196:
4192:
4181:
4180:
4175:
4161:
4158:
4157:
4121:
4120:
4110:
4109:
4105:
4094:
4093:
4087:
4075:
4074:
4060:
4059:
4052:
4051:
4047:
4041:
4026:
4023:
4022:
3996:
3995:
3991:
3989:
3986:
3985:
3969:
3966:
3965:
3943:
3939:
3925:
3924:
3922:
3919:
3918:
3894:
3890:
3881:
3877:
3872:
3861:
3858:
3857:
3854:complementation
3828:
3825:
3824:
3818:Boolean lattice
3801:
3798:
3797:
3774:
3771:
3770:
3767:
3742:
3741:
3737:
3728:
3725:
3724:
3705:
3702:
3701:
3672:
3671:
3667:
3665:
3662:
3661:
3640:
3637:
3636:
3615:
3614:
3609:
3604:
3599:
3593:
3592:
3587:
3582:
3577:
3571:
3570:
3565:
3560:
3555:
3549:
3548:
3543:
3538:
3533:
3523:
3522:
3512:
3511:
3507:
3502:
3499:
3498:
3477:
3476:
3472:
3470:
3467:
3466:
3444:
3441:
3440:
3415:
3414:
3410:
3408:
3405:
3404:
3377:
3376:
3372:
3370:
3367:
3366:
3344:
3343:
3339:
3337:
3334:
3333:
3307:
3306:
3301:
3296:
3290:
3289:
3284:
3279:
3273:
3272:
3267:
3262:
3256:
3255:
3250:
3245:
3235:
3234:
3232:
3229:
3228:
3208:
3205:
3204:
3188:
3185:
3184:
3168:
3165:
3164:
3145:
3142:
3141:
3121:
3118:
3117:
3095:
3092:
3091:
3075:
3072:
3071:
3052:
3049:
3048:
3029:
3026:
3025:
3022:
2988:
2987:
2976:
2975:
2974:
2969:
2966:
2965:
2944:
2943:
2932:
2931:
2930:
2928:
2925:
2924:
2893:
2892:
2888:
2877:
2874:
2873:
2807:
2806:
2802:
2800:
2797:
2796:
2776:
2773:
2772:
2723:
2720:
2719:
2701:
2700:
2696:
2685:
2682:
2681:
2659:
2658:
2654:
2649:
2646:
2645:
2624:
2623:
2619:
2605:
2602:
2601:
2581:
2578:
2577:
2555:
2552:
2551:
2526:
2523:
2522:
2500:
2497:
2496:
2477:
2474:
2473:
2452:
2451:
2447:
2445:
2442:
2441:
2425:
2422:
2421:
2403:
2402:
2398:
2393:
2390:
2389:
2368:
2367:
2363:
2361:
2358:
2357:
2338:
2335:
2334:
2318:
2315:
2314:
2298:
2295:
2294:
2293:that is, where
2263:
2260:
2259:
2256:
2210:
2207:
2206:
2178:
2175:
2174:
2167:
2154:neutral element
2137:
2134:
2133:
2113:
2110:
2109:
2093:
2090:
2089:
2073:
2070:
2069:
2048:
2045:
2044:
2021:
2018:
2017:
1997:
1994:
1993:
1977:
1974:
1973:
1952:
1949:
1948:
1925:
1922:
1921:
1901:
1898:
1897:
1881:
1878:
1877:
1842:
1841:
1837:
1826:
1825:
1821:
1811:
1810:
1806:
1791:
1788:
1787:
1764:
1761:
1760:
1693:
1690:
1689:
1681:
1655:
1650:
1649:
1647:
1644:
1643:
1623:
1619:
1610:
1606:
1598:
1595:
1594:
1572:
1569:
1568:
1533:
1532:
1517:
1516:
1514:
1511:
1510:
1494:
1491:
1490:
1466:
1463:
1462:
1436:
1435:
1417:
1416:
1401:
1400:
1398:
1395:
1394:
1374:
1371:
1370:
1354:
1351:
1350:
1327:
1325:
1322:
1321:
1293:
1290:
1289:
1267:
1264:
1263:
1247:
1245:
1242:
1241:
1222:
1207:
1206:
1204:
1201:
1200:
1184:
1182:
1179:
1178:
1144:
1141:
1140:
1118:
1115:
1114:
1091:
1090:
1088:
1085:
1084:
1045:
1044:
1042:
1039:
1038:
1007:
1004:
1003:
975:
972:
971:
968:
947:
944:
943:
922:
918:
916:
913:
912:
895:
891:
889:
886:
885:
857:
854:
853:
781:
778:
777:
738:
735:
734:
733:A small circle
724:category theory
712:Gunther Schmidt
696:
657:
654:
653:
619:
616:
615:
583:
580:
579:
557:
554:
553:
513:
510:
509:
475:
472:
471:
428: and
426:
400:
386:
339:
336:
335:
313:
310:
309:
281:
278:
277:
249:
246:
245:
242:
228:Beginning with
199:
178:
163:
160:
159:
137:
134:
133:
111:
108:
107:
85:
82:
81:
78:algebraic logic
32:
17:
12:
11:
5:
6641:
6631:
6630:
6625:
6620:
6603:
6602:
6600:
6599:
6594:
6589:
6588:
6587:
6577:
6576:
6575:
6570:
6565:
6555:
6554:
6553:
6543:
6538:
6537:
6536:
6531:
6524:Order morphism
6521:
6520:
6519:
6509:
6504:
6499:
6494:
6489:
6488:
6487:
6477:
6472:
6467:
6461:
6459:
6455:
6454:
6452:
6451:
6450:
6449:
6444:
6442:Locally convex
6439:
6434:
6424:
6422:Order topology
6419:
6418:
6417:
6415:Order topology
6412:
6402:
6392:
6390:
6383:
6382:
6380:
6379:
6374:
6369:
6364:
6359:
6354:
6349:
6344:
6339:
6334:
6328:
6326:
6322:
6321:
6319:
6318:
6308:
6298:
6293:
6288:
6283:
6278:
6273:
6268:
6263:
6262:
6261:
6251:
6246:
6245:
6244:
6239:
6234:
6229:
6227:Chain-complete
6219:
6214:
6213:
6212:
6207:
6202:
6197:
6192:
6182:
6177:
6172:
6167:
6162:
6152:
6147:
6142:
6137:
6132:
6127:
6126:
6125:
6115:
6110:
6104:
6102:
6092:
6091:
6089:
6088:
6083:
6078:
6073:
6068:
6063:
6058:
6053:
6048:
6043:
6038:
6033:
6027:
6025:
6021:
6020:
6018:
6017:
6012:
6007:
6002:
5997:
5992:
5987:
5982:
5976:
5974:
5970:
5969:
5967:
5966:
5961:
5956:
5950:
5947:
5946:
5939:
5938:
5931:
5924:
5916:
5910:
5909:
5888:
5885:
5882:
5881:
5870:Springer books
5850:
5838:
5835:
5829:
5826:
5821:
5818:
5814:
5810:
5794:
5776:
5756:
5731:
5714:
5705:
5696:
5678:John M. Howie
5670:
5654:
5631:
5620:
5600:
5593:Ernst Schroder
5585:
5582:Springer books
5561:
5535:
5526:
5498:
5474:
5473:
5471:
5468:
5467:
5466:
5460:
5453:
5450:
5423:
5420:
5417:
5414:
5394:
5372:
5361:
5357:
5354:
5348:
5337:
5333:
5330:
5323:
5316:
5312:
5309:
5306:
5302:
5282:
5262:
5234:
5223:
5194:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5132:
5129:
5126:
5123:
5120:
5117:
5114:
5094:
5091:
5088:
5085:
5082:
5079:
5076:
5073:
5069:
5066:
5063:
5059:
5039:
5036:
5033:
5030:
5027:
5007:
5004:
5001:
4998:
4995:
4975:
4972:
4969:
4957:
4954:
4935:
4932:
4929:
4926:
4923:
4903:
4900:
4896:
4892:
4889:
4886:
4866:
4863:
4860:
4857:
4837:
4834:
4831:
4828:
4825:
4805:
4802:
4799:
4796:
4793:
4790:
4770:
4767:
4764:
4761:
4750:
4749:
4736:
4732:
4718:
4715:
4705:
4700:
4693:
4690:
4676:
4668:
4664:
4661:
4658:
4655:
4652:
4649:
4646:
4635:
4622:
4610:
4603:
4600:
4590:
4586:
4582:
4578:
4567:
4554:
4547:
4544:
4530:
4522:
4518:
4515:
4512:
4488:
4485:
4471:
4451:
4429:
4424:
4417:
4414:
4400:
4393:
4390:
4370:
4364:
4361:
4355:
4349:
4346:
4332:
4323:
4320:
4317:
4314:
4294:
4291:
4288:
4285:
4282:
4272:
4269:
4266:
4263:
4243:
4221:
4215:
4212:
4206:
4195:
4188:
4185:
4174:
4171:
4168:
4165:
4150:Ernst Schröder
4134:
4128:
4125:
4119:
4108:
4101:
4098:
4082:
4079:
4073:
4067:
4064:
4050:
4039:
4036:
4033:
4030:
4013:represent the
3994:
3973:
3951:
3946:
3942:
3938:
3932:
3929:
3902:
3897:
3893:
3889:
3884:
3880:
3871:
3868:
3865:
3841:
3838:
3835:
3832:
3805:
3781:
3778:
3766:
3765:Schröder rules
3763:
3751:
3740:
3736:
3732:
3712:
3709:
3685:
3682:
3670:
3647:
3644:
3624:
3619:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3594:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3572:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3550:
3547:
3544:
3542:
3539:
3537:
3534:
3532:
3529:
3528:
3526:
3521:
3510:
3506:
3486:
3475:
3454:
3451:
3448:
3424:
3413:
3402:matrix product
3389:
3386:
3375:
3342:
3316:
3311:
3305:
3302:
3300:
3297:
3295:
3292:
3291:
3288:
3285:
3283:
3280:
3278:
3275:
3274:
3271:
3268:
3266:
3263:
3261:
3258:
3257:
3254:
3251:
3249:
3246:
3244:
3241:
3240:
3238:
3225:logical matrix
3212:
3192:
3172:
3152:
3149:
3125:
3105:
3102:
3099:
3079:
3059:
3056:
3036:
3033:
3021:
3018:
3006:
3003:
3000:
2997:
2983:
2980:
2973:
2953:
2939:
2936:
2905:
2902:
2891:
2887:
2884:
2881:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2805:
2780:
2771:Similarly, if
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2699:
2695:
2692:
2689:
2657:
2653:
2633:
2622:
2618:
2615:
2612:
2609:
2585:
2565:
2562:
2559:
2539:
2536:
2533:
2530:
2510:
2507:
2504:
2481:
2461:
2450:
2429:
2401:
2397:
2377:
2366:
2342:
2322:
2302:
2282:
2279:
2276:
2273:
2270:
2267:
2255:
2252:
2239:matrix product
2226:
2223:
2220:
2217:
2214:
2194:
2191:
2188:
2185:
2182:
2166:
2163:
2162:
2161:
2141:
2117:
2097:
2077:
2066:
2055:
2052:
2032:
2029:
2025:
2001:
1981:
1970:
1959:
1956:
1936:
1933:
1929:
1905:
1885:
1874:
1867:
1851:
1840:
1836:
1824:
1820:
1809:
1805:
1802:
1798:
1795:
1775:
1772:
1768:
1753:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1680:
1677:
1658:
1653:
1626:
1622:
1618:
1613:
1609:
1605:
1602:
1582:
1579:
1576:
1567:has morphisms
1556:
1553:
1550:
1547:
1542:
1539:
1536:
1531:
1526:
1523:
1520:
1498:
1470:
1445:
1442:
1439:
1434:
1431:
1426:
1423:
1420:
1415:
1410:
1407:
1404:
1378:
1358:
1330:
1309:
1306:
1303:
1300:
1297:
1277:
1274:
1271:
1250:
1229:
1225:
1221:
1216:
1213:
1210:
1187:
1160:
1157:
1154:
1151:
1148:
1139:are functions
1128:
1125:
1122:
1100:
1097:
1094:
1054:
1051:
1048:
1023:
1020:
1017:
1014:
1011:
1002:are morphisms
991:
988:
985:
982:
979:
967:
964:
951:
925:
921:
898:
894:
870:
867:
864:
861:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
754:
751:
748:
745:
742:
708:Ernst Schroder
704:infix notation
695:
692:
679:
676:
673:
670:
667:
664:
661:
641:
638:
635:
632:
629:
626:
623:
603:
599:
595:
591:
587:
567:
564:
561:
541:
538:
535:
532:
529:
526:
523:
520:
517:
497:
494:
491:
488:
485:
482:
479:
457:
454:
451:
448:
445:
442:
439:
436:
433:
425:
422:
419:
416:
413:
410:
407:
399:
396:
393:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
323:
320:
317:
297:
294:
291:
288:
285:
265:
262:
259:
256:
253:
241:
238:
215:
212:
209:
206:
198:
195:
192:
189:
186:
176:
173:
170:
167:
147:
144:
141:
121:
118:
115:
95:
92:
89:
15:
9:
6:
4:
3:
2:
6640:
6629:
6626:
6624:
6621:
6619:
6616:
6615:
6613:
6598:
6595:
6593:
6590:
6586:
6583:
6582:
6581:
6578:
6574:
6571:
6569:
6566:
6564:
6561:
6560:
6559:
6556:
6552:
6549:
6548:
6547:
6546:Ordered field
6544:
6542:
6539:
6535:
6532:
6530:
6527:
6526:
6525:
6522:
6518:
6515:
6514:
6513:
6510:
6508:
6505:
6503:
6502:Hasse diagram
6500:
6498:
6495:
6493:
6490:
6486:
6483:
6482:
6481:
6480:Comparability
6478:
6476:
6473:
6471:
6468:
6466:
6463:
6462:
6460:
6456:
6448:
6445:
6443:
6440:
6438:
6435:
6433:
6430:
6429:
6428:
6425:
6423:
6420:
6416:
6413:
6411:
6408:
6407:
6406:
6403:
6401:
6397:
6394:
6393:
6391:
6388:
6384:
6378:
6375:
6373:
6370:
6368:
6365:
6363:
6360:
6358:
6355:
6353:
6352:Product order
6350:
6348:
6345:
6343:
6340:
6338:
6335:
6333:
6330:
6329:
6327:
6325:Constructions
6323:
6317:
6313:
6309:
6306:
6302:
6299:
6297:
6294:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6272:
6269:
6267:
6264:
6260:
6257:
6256:
6255:
6252:
6250:
6247:
6243:
6240:
6238:
6235:
6233:
6230:
6228:
6225:
6224:
6223:
6222:Partial order
6220:
6218:
6215:
6211:
6210:Join and meet
6208:
6206:
6203:
6201:
6198:
6196:
6193:
6191:
6188:
6187:
6186:
6183:
6181:
6178:
6176:
6173:
6171:
6168:
6166:
6163:
6161:
6157:
6153:
6151:
6148:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6124:
6121:
6120:
6119:
6116:
6114:
6111:
6109:
6108:Antisymmetric
6106:
6105:
6103:
6099:
6093:
6087:
6084:
6082:
6079:
6077:
6074:
6072:
6069:
6067:
6064:
6062:
6059:
6057:
6054:
6052:
6049:
6047:
6044:
6042:
6039:
6037:
6034:
6032:
6029:
6028:
6026:
6022:
6016:
6015:Weak ordering
6013:
6011:
6008:
6006:
6003:
6001:
6000:Partial order
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5981:
5978:
5977:
5975:
5971:
5965:
5962:
5960:
5957:
5955:
5952:
5951:
5948:
5944:
5937:
5932:
5930:
5925:
5923:
5918:
5917:
5914:
5907:
5906:3-11-015248-7
5903:
5899:
5895:
5891:
5890:
5879:
5875:
5871:
5867:
5863:
5859:
5854:
5836:
5833:
5819:
5816:
5812:
5808:
5798:
5792:
5788:
5785:
5780:
5773:
5769:
5765:
5760:
5745:
5741:
5735:
5728:
5724:
5718:
5709:
5700:
5694:
5693:0-19-851194-9
5690:
5687:
5683:
5679:
5674:
5668:
5664:
5658:
5652:
5648:
5644:
5641:
5635:
5629:
5623:
5617:
5613:
5612:
5604:
5598:
5594:
5589:
5583:
5579:
5575:
5570:
5568:
5566:
5559:
5558:9789400920477
5555:
5552:
5548:
5542:
5540:
5530:
5524:
5520:
5517:
5513:
5507:
5505:
5503:
5496:
5492:
5489:
5485:
5479:
5475:
5464:
5461:
5459:
5456:
5455:
5449:
5447:
5443:
5440:operation of
5439:
5438:
5421:
5418:
5415:
5412:
5392:
5383:
5370:
5359:
5355:
5352:
5346:
5335:
5331:
5328:
5321:
5314:
5307:
5300:
5280:
5260:
5252:
5232:
5221:
5192:
5171:
5168:
5162:
5159:
5156:
5153:
5150:
5130:
5124:
5121:
5118:
5115:
5112:
5092:
5089:
5086:
5083:
5077:
5074:
5071:
5064:
5057:
5037:
5031:
5028:
5025:
5005:
4999:
4996:
4993:
4970:
4953:
4951:
4946:
4933:
4930:
4927:
4924:
4921:
4901:
4898:
4894:
4890:
4887:
4884:
4864:
4861:
4858:
4855:
4835:
4832:
4829:
4826:
4823:
4803:
4800:
4794:
4791:
4788:
4768:
4765:
4762:
4759:
4730:
4713:
4703:
4688:
4674:
4666:
4659:
4656:
4653:
4647:
4644:
4636:
4608:
4598:
4588:
4584:
4580:
4576:
4568:
4542:
4528:
4520:
4516:
4510:
4502:
4501:
4500:
4499:Definitions:
4497:
4494:
4484:
4482:
4469:
4449:
4427:
4412:
4398:
4391:
4388:
4368:
4359:
4353:
4344:
4330:
4321:
4318:
4315:
4312:
4292:
4289:
4286:
4283:
4280:
4270:
4267:
4264:
4261:
4241:
4232:
4219:
4210:
4204:
4193:
4183:
4172:
4169:
4166:
4163:
4155:
4151:
4146:
4132:
4123:
4117:
4106:
4096:
4077:
4071:
4062:
4048:
4037:
4034:
4031:
4028:
4020:
4016:
3992:
3971:
3962:
3949:
3944:
3940:
3936:
3927:
3916:
3900:
3895:
3891:
3887:
3882:
3878:
3869:
3866:
3863:
3855:
3839:
3833:
3823:
3819:
3803:
3795:
3779:
3776:
3762:
3749:
3738:
3734:
3730:
3710:
3707:
3699:
3683:
3680:
3668:
3658:
3645:
3642:
3622:
3617:
3611:
3606:
3601:
3596:
3589:
3584:
3579:
3574:
3567:
3562:
3557:
3552:
3545:
3540:
3535:
3530:
3524:
3519:
3508:
3504:
3484:
3473:
3452:
3449:
3446:
3438:
3422:
3411:
3403:
3387:
3384:
3373:
3364:
3340:
3332:
3327:
3314:
3309:
3303:
3298:
3293:
3286:
3281:
3276:
3269:
3264:
3259:
3252:
3247:
3242:
3236:
3226:
3210:
3190:
3170:
3150:
3147:
3139:
3123:
3103:
3100:
3097:
3077:
3057:
3054:
3034:
3031:
3017:
3004:
3001:
2998:
2995:
2978:
2971:
2951:
2934:
2921:
2919:
2903:
2900:
2889:
2885:
2882:
2879:
2872:In this case
2859:
2853:
2850:
2847:
2844:
2838:
2835:
2832:
2823:
2820:
2817:
2814:
2803:
2794:
2778:
2758:
2752:
2749:
2746:
2743:
2737:
2734:
2731:
2697:
2693:
2690:
2687:
2679:
2655:
2651:
2631:
2620:
2616:
2613:
2610:
2607:
2599:
2583:
2563:
2560:
2557:
2537:
2534:
2531:
2528:
2508:
2505:
2502:
2493:
2479:
2459:
2448:
2427:
2399:
2395:
2375:
2364:
2356:
2340:
2320:
2300:
2280:
2277:
2274:
2271:
2268:
2265:
2251:
2249:
2245:
2240:
2224:
2221:
2218:
2215:
2212:
2192:
2189:
2186:
2183:
2180:
2172:
2159:
2155:
2139:
2131:
2115:
2095:
2075:
2067:
2053:
2050:
2030:
2027:
2023:
2015:
1999:
1979:
1971:
1957:
1954:
1934:
1931:
1927:
1919:
1903:
1883:
1875:
1872:
1868:
1865:
1849:
1838:
1834:
1822:
1818:
1803:
1800:
1796:
1773:
1770:
1766:
1758:
1754:
1740:
1737:
1734:
1728:
1725:
1722:
1716:
1710:
1707:
1704:
1698:
1695:
1687:
1683:
1682:
1676:
1674:
1656:
1642:
1624:
1620:
1616:
1611:
1607:
1603:
1600:
1580:
1574:
1548:
1496:
1488:
1484:
1468:
1461:
1432:
1392:
1376:
1356:
1348:
1345:
1344:jointly monic
1307:
1304:
1301:
1298:
1295:
1275:
1269:
1177:
1172:
1158:
1152:
1149:
1146:
1126:
1120:
1082:
1081:
1076:
1072:
1071:
1037:
1021:
1015:
1012:
1009:
989:
986:
983:
980:
977:
963:
949:
941:
923:
919:
896:
892:
882:
865:
862:
852:
851:Juxtaposition
848:
844:
825:
816:
813:
810:
804:
795:
789:
783:
776:
772:
768:
767:John M. Howie
749:
746:
743:
731:
729:
725:
721:
717:
713:
709:
705:
701:
691:
677:
674:
668:
665:
662:
639:
636:
630:
627:
624:
601:
597:
593:
589:
585:
565:
562:
559:
539:
536:
533:
530:
524:
521:
518:
495:
492:
489:
486:
483:
480:
477:
468:
455:
449:
446:
440:
437:
434:
423:
420:
414:
411:
408:
397:
394:
391:
383:
380:
377:
374:
371:
365:
362:
359:
350:
347:
344:
341:
321:
318:
315:
295:
292:
289:
286:
283:
263:
260:
257:
254:
251:
237:
235:
231:
226:
213:
210:
207:
204:
196:
193:
190:
187:
184:
174:
171:
168:
165:
145:
142:
139:
119:
116:
113:
93:
90:
87:
79:
75:
70:
68:
64:
60:
56:
52:
48:
44:
39:
35:
30:
26:
22:
6389:& Orders
6367:Star product
6331:
6296:Well-founded
6249:Prefix order
6205:Distributive
6195:Complemented
6165:Foundational
6130:Completeness
6086:Zorn's lemma
5990:Cyclic order
5973:Key concepts
5943:Order theory
5893:
5861:
5853:
5797:
5784:Vaughn Pratt
5779:
5759:
5749:26 September
5747:. Retrieved
5743:
5734:
5717:
5708:
5699:
5681:
5673:
5657:
5634:
5610:
5603:
5588:
5577:
5549:, page 121,
5546:
5529:
5511:
5483:
5478:
5435:
5384:
5293:is given by
5246:
4959:
4947:
4751:
4498:
4490:
4441:
4233:
4147:
4018:
3963:
3852:Recall that
3768:
3659:
3328:
3023:
2922:
2918:difunctional
2494:
2257:
2243:
2168:
2158:zero element
1641:finite field
1173:
1078:
1068:
969:
883:
846:
842:
732:
715:
697:
469:
243:
227:
71:
58:
54:
46:
42:
37:
33:
28:
18:
6573:Riesz space
6534:Isomorphism
6410:Normal cone
6332:Composition
6266:Semilattice
6175:Homogeneous
6160:Equivalence
6010:Total order
5868:vol. 2208,
5864:, page 26,
5665:§2.2, from
3820:ordered by
3203:is finite,
3163:Since both
2495:If for all
1686:associative
1673:ZX-calculus
614:(that is,
21:mathematics
6612:Categories
6541:Order type
6475:Cofinality
6316:Well-order
6291:Transitive
6180:Idempotent
6113:Asymmetric
5887:References
5772:Jstor link
5446:Join (SQL)
4493:operations
4152:, in fact
2920:relation.
2576:(that is,
2550:such that
2014:surjective
1679:Properties
1458:. Given a
1391:allegories
940:Z notation
771:semigroups
578:such that
240:Definition
6592:Upper set
6529:Embedding
6465:Antichain
6286:Tolerance
6276:Symmetric
6271:Semiorder
6217:Reflexive
6135:Connected
5817:−
5416:≥
5347:∩
5245:Then the
5166:→
5160:×
5128:→
5122:×
5087:×
5081:→
5035:→
5003:→
4928:⊆
4888:⊆
4862:⊆
4830:⊆
4798:∖
4792:⊆
4766:⊆
4735:¯
4717:¯
4704:∩
4699:¯
4692:¯
4648:
4621:¯
4602:¯
4553:¯
4546:¯
4514:∖
4487:Quotients
4423:¯
4416:¯
4392:⊆
4363:¯
4354:⊆
4348:¯
4319:⊆
4287:⊆
4268:⊆
4214:¯
4205:⊆
4187:¯
4170:⊆
4127:¯
4118:⊆
4100:¯
4081:¯
4072:⊆
4066:¯
4035:⊆
4019:transpose
3945:∁
3931:¯
3896:∁
3888:⊆
3883:∁
3867:⊆
3834:⊆
3822:inclusion
3450:×
3090:given by
2982:¯
2938:¯
2883:⊆
2851:∈
2818:⊇
2750:∈
2691:⊆
2532:∈
2506:∈
2353:with its
2275:×
2269:⊆
2244:computing
2216:×
1918:injective
1617:⊕
1604:⊆
1578:→
1433:≅
1305:×
1299:⊆
1273:→
1156:→
1124:→
1019:→
987:×
981:⊆
950:∘
920:∘
893:∘
814:∘
747:∘
720:coincides
700:semicolon
675:∈
637:∈
563:∈
531:∈
493:×
487:⊆
447:∈
421:∈
395:∈
378:×
372:∈
293:×
287:⊆
261:×
255:⊆
234:syllogism
72:The word
67:functions
49:. In the
6387:Topology
6254:Preorder
6237:Eulerian
6200:Complete
6150:Directed
6140:Covering
6005:Preorder
5964:Category
5959:Glossary
5643:Archived
5452:See also
3816:forms a
2644:so that
2355:converse
1487:matrices
1349:between
1174:Given a
1036:category
843:reverses
6492:Duality
6470:Cofinal
6458:Related
6437:Fréchet
6314:)
6190:Bounded
6185:Lattice
6158:)
6156:Partial
6024:Results
5995:Lattice
5789:, from
5680:(1995)
5595:(1895)
5434:is the
3913:In the
3696:is the
3465:matrix
3020:Example
2248:sorites
2152:is the
2016:, then
1920:, then
1034:in the
19:In the
6517:Subnet
6497:Filter
6447:Normed
6432:Banach
6398:&
6305:Better
6242:Strict
6232:Graded
6123:topics
5954:Topics
5904:
5876:
5831:
5727:U+2A1F
5723:U+2A3E
5691:
5618:
5556:
5521:
5493:
5350:
5326:
5318:
4950:Sudoku
2795:then
2440:) and
2130:monoid
841:which
702:as an
27:, the
6507:Ideal
6485:Graph
6281:Total
6259:Total
6145:Dense
5470:Notes
5050:into
3136:is a
3116:when
2791:is a
2676:is a
2596:is a
1489:over
1460:field
1347:spans
1067:. In
74:uncle
6098:list
5902:ISBN
5874:ISBN
5751:2023
5744:nlab
5725:and
5721:See
5689:ISBN
5616:ISBN
5554:ISBN
5519:ISBN
5491:ISBN
5437:join
5308:<
5273:and
5249:fork
5213:and
5143:and
5065:<
5018:and
4971:<
3329:The
3183:and
3024:Let
2472:(on
2420:(on
2313:and
2205:and
2012:are
1992:and
1916:are
1896:and
1755:The
1369:and
1075:sets
911:and
698:The
652:and
276:and
45:and
6512:Net
6312:Pre
5253:of
4645:syq
4462:by
4276:and
3964:If
3796:on
3700:on
3140:of
2680:or
2492:).
2250:."
2108:to
1972:If
1876:If
1786:is
1759:of
1320:in
1080:Set
1070:Rel
690:).
244:If
158:).
23:of
6614::
5872:,
5742:.
5580:,
5564:^
5538:^
5501:^
5448:.
5322::=
4952:.
4667::=
4589::=
4521::=
4483:.
2225:1.
1688::
1509:,
1171:.
942::
730:.
69:.
61:.
36:;
6310:(
6307:)
6303:(
6154:(
6101:)
5935:e
5928:t
5921:v
5908:.
5900:,
5837:.
5834:l
5828:)
5825:)
5820:1
5813:M
5809:n
5753:.
5624:.
5422:,
5419:2
5413:n
5393:n
5371:.
5365:T
5360:b
5356:;
5353:d
5341:T
5336:a
5332:;
5329:c
5315:d
5311:)
5305:(
5301:c
5281:d
5261:c
5233:.
5227:T
5222:b
5198:T
5193:a
5172:,
5169:B
5163:B
5157:A
5154::
5151:b
5131:A
5125:B
5119:A
5116::
5113:a
5093:.
5090:B
5084:A
5078:H
5075::
5072:d
5068:)
5062:(
5058:c
5038:B
5032:H
5029::
5026:d
5006:A
5000:H
4997::
4994:c
4974:)
4968:(
4934:.
4931:D
4925:C
4922:Y
4902:,
4899:C
4895:/
4891:D
4885:Y
4865:D
4859:C
4856:Y
4836:.
4833:B
4827:X
4824:A
4804:.
4801:B
4795:A
4789:X
4769:B
4763:X
4760:A
4731:F
4725:T
4714:E
4689:F
4680:T
4675:E
4663:)
4660:F
4657:,
4654:E
4651:(
4614:T
4609:C
4599:D
4585:C
4581:/
4577:D
4543:B
4534:T
4529:A
4517:B
4511:A
4470:R
4450:S
4428:,
4413:S
4404:T
4399:R
4389:X
4369:,
4360:X
4345:S
4336:T
4331:R
4322:S
4316:X
4313:R
4293:.
4290:S
4284:R
4281:X
4271:S
4265:X
4262:R
4242:X
4220:.
4211:L
4199:T
4194:M
4184:N
4173:N
4167:M
4164:L
4133:.
4124:Q
4112:T
4107:R
4097:S
4078:R
4063:S
4054:T
4049:Q
4038:S
4032:R
4029:Q
3998:T
3993:S
3972:S
3950:.
3941:A
3937:=
3928:A
3901:.
3892:A
3879:B
3870:B
3864:A
3840:.
3837:)
3831:(
3804:V
3780:,
3777:V
3750:.
3744:T
3739:R
3735:;
3731:R
3711:,
3708:B
3684:R
3681:;
3674:T
3669:R
3646:.
3643:A
3623:.
3618:)
3612:1
3607:1
3602:1
3597:1
3590:1
3585:1
3580:0
3575:0
3568:1
3563:0
3558:1
3553:0
3546:1
3541:0
3536:0
3531:1
3525:(
3520:=
3514:T
3509:R
3505:R
3485:R
3479:T
3474:R
3453:3
3447:3
3423:R
3417:T
3412:R
3388:R
3385:;
3379:T
3374:R
3346:T
3341:R
3315:.
3310:)
3304:1
3299:1
3294:1
3287:1
3282:0
3277:0
3270:0
3265:1
3260:0
3253:0
3248:0
3243:1
3237:(
3211:R
3191:B
3171:A
3151:.
3148:a
3124:b
3104:b
3101:R
3098:a
3078:R
3058:=
3055:B
3035:=
3032:A
3005:.
3002:R
2999:=
2996:R
2990:T
2979:R
2972:R
2952:R
2946:T
2935:R
2904:.
2901:R
2895:T
2890:R
2886:R
2880:R
2860:.
2857:}
2854:B
2848:x
2845::
2842:)
2839:x
2836:,
2833:x
2830:(
2827:{
2824:=
2821:I
2815:R
2809:T
2804:R
2779:R
2759:.
2756:}
2753:A
2747:x
2744::
2741:)
2738:x
2735:,
2732:x
2729:(
2726:{
2703:T
2698:R
2694:R
2688:I
2661:T
2656:R
2652:R
2632:x
2626:T
2621:R
2617:R
2614:x
2611:,
2608:x
2584:R
2564:y
2561:R
2558:x
2538:,
2535:B
2529:y
2509:A
2503:x
2480:B
2460:R
2454:T
2449:R
2428:A
2405:T
2400:R
2396:R
2376:,
2370:T
2365:R
2341:R
2321:B
2301:A
2281:;
2278:B
2272:A
2266:R
2222:=
2219:1
2213:1
2193:1
2190:=
2187:1
2184:+
2181:1
2160:.
2140:X
2116:X
2096:X
2076:X
2054:.
2051:S
2031:S
2028:;
2024:R
2000:S
1980:R
1958:.
1955:R
1935:S
1932:;
1928:R
1904:S
1884:R
1866:.
1850:.
1844:T
1839:R
1835:;
1828:T
1823:S
1819:=
1813:T
1808:)
1804:S
1801:;
1797:R
1794:(
1774:S
1771:;
1767:R
1741:.
1738:T
1735:;
1732:)
1729:S
1726:;
1723:R
1720:(
1717:=
1714:)
1711:T
1708:;
1705:S
1702:(
1699:;
1696:R
1657:2
1652:F
1625:m
1621:k
1612:n
1608:k
1601:R
1581:m
1575:n
1555:)
1552:)
1549:k
1546:(
1541:t
1538:a
1535:M
1530:(
1525:l
1522:e
1519:R
1497:k
1469:k
1444:l
1441:e
1438:R
1430:)
1425:t
1422:e
1419:S
1414:(
1409:l
1406:e
1403:R
1377:Y
1357:X
1329:X
1308:Y
1302:X
1296:R
1276:Y
1270:X
1249:X
1228:)
1224:X
1220:(
1215:l
1212:e
1209:R
1186:X
1159:Y
1153:X
1150::
1147:f
1127:Y
1121:X
1099:l
1096:e
1093:R
1053:l
1050:e
1047:R
1022:Y
1016:X
1013::
1010:R
990:Y
984:X
978:R
924:r
897:l
869:)
866:S
863:R
860:(
829:)
826:x
823:(
820:)
817:f
811:g
808:(
805:=
802:)
799:)
796:x
793:(
790:f
787:(
784:g
753:)
750:S
744:R
741:(
678:S
672:)
669:z
666:,
663:y
660:(
640:R
634:)
631:y
628:,
625:x
622:(
602:z
598:S
594:y
590:R
586:x
566:Y
560:y
540:S
537:;
534:R
528:)
525:z
522:,
519:x
516:(
496:Z
490:X
484:S
481:;
478:R
456:.
453:}
450:S
444:)
441:z
438:,
435:y
432:(
424:R
418:)
415:y
412:,
409:x
406:(
398:Y
392:y
384::
381:Z
375:X
369:)
366:z
363:,
360:x
357:(
354:{
351:=
348:S
345:;
342:R
322:S
319:;
316:R
296:Z
290:Y
284:S
264:Y
258:X
252:R
214:.
211:z
208:U
205:x
197:z
194:P
191:y
188:B
185:x
175:P
172:B
169:=
166:U
146:z
143:P
140:y
120:y
117:B
114:x
94:z
91:U
88:x
47:S
43:R
38:S
34:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.