1875:
Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup, and with image closed under the inverse operation on partial bijections. Conversely, any subsemigroup of the symmetric inverse semigroup closed under the inverse operation is an inverse semigroup. Hence a semigroup
841:
3666:
4343:
4375:
The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. There is another way of composing partial transformations, which is more restrictive than that used above: two partial transformations
1973:
2285:
4400:, which states that an inductive groupoid can always be constructed from an inverse semigroup, and conversely. More precisely, an inverse semigroup is precisely a groupoid in the category of posets that is an
278:
719:
2642:
1379:
1312:
3934:
3802:
4048:
3885:
3541:
3351:
2034:
1589:
1213:
1152:
2386:
3749:
2436:
4388:; otherwise, the composition αβ is undefined. Under this alternative composition, the collection of all partial one-one transformations of a set forms not an inverse semigroup but an
3553:
4356:
who showed that elements of the free inverse semigroup can be naturally regarded as trees, known as Munn trees. Multiplication in the free inverse semigroup has a correspondent on
1832:
1523:
1480:
347:
4124:
4100:
4076:
3491:
3467:
3419:
3383:
3283:
3220:
3175:
3147:
3088:
3044:
3004:
2966:
2938:
2906:
2878:
2852:
2799:
2763:
2736:
2708:
2680:
699:
660:
624:
600:
572:
548:
2095:
4163:
2825:
1996:
2060:
2567:
2539:
2651:
Every inverse semigroup S has a E-unitary cover; that is there exists an idempotent separating surjective homomorphism from some E-unitary semigroup T onto S.
311:
to take account of this. With the addition of this empty transformation, the composition of partial transformations of a set becomes an everywhere-defined
5711:
4436:
commute; this has led to two distinct classes of generalisations of an inverse semigroup: semigroups in which (1) holds, but (2) does not, and vice versa.
1912:
90:
in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of
5349:
5319:
5290:
5169:
4782:
2205:
2120:, the minimal element (for the partial order defined by inclusion of sets) need not be the smallest element. In the specific case in which
4808:
Since his father was German, Wagner preferred the German transliteration of his name (with a "W", rather than a "V") from
Cyrillic – see
210:
836:{\displaystyle a\,{\mathcal {L}}\,b\Longleftrightarrow a^{-1}a=b^{-1}b,\quad a\,{\mathcal {R}}\,b\Longleftrightarrow aa^{-1}=bb^{-1}}
2583:
299:. It was Wagner, however, who was the first to observe that the composition of partial transformations is a special case of the
1323:
1261:
5575:
5492:
5473:
5032:
4147:
5439:
5721:
5810:
4741:
4688:
3890:
3758:
1753:, however, there is no need to do so, since this property follows from the above definition, via the following theorem:
4007:
3844:
3500:
3310:
2008:
1550:
1174:
1113:
355:
122:
4352:
for free inverse semigroups is much more intricate than that of free groups. A celebrated result in this area due to
2347:
3019:
1763:
of an inverse semigroup is an inverse semigroup; the inverse of an element is always mapped to the inverse of the
5815:
4589:
4360:, which essentially consists of overlapping common portions of the trees. (see Lawson 1998 for further details)
3677:
3661:{\displaystyle P(G,{\mathcal {X}},{\mathcal {Y}})=\{(A,g)\in {\mathcal {Y}}\times G:g^{-1}A\in {\mathcal {Y}}\}}
2397:
4139:
1685:) of inverse semigroups is defined in exactly the same way as for any other semigroup: for inverse semigroups
300:
97:(The convention followed in this article will be that of writing a function on the right of its argument, e.g.
4349:
4618:
1793:
1694:
157:
5026:
4389:
2683:
1880:
is isomorphic to a subsemigroup of the symmetric inverse semigroup closed under inverses if and only if
1806:
1502:
1459:
321:
4731:
2802:
1903:
1654:
1618:
1440:
710:
627:
201:
4995:
Gonçalves, D; Sobottka, M; Starling, C (2017). "Inverse semigroup shifts over countable alphabets".
4401:
4105:
4081:
4057:
3472:
3448:
3400:
3364:
3264:
3201:
3156:
3128:
3069:
3025:
2985:
2947:
2919:
2887:
2859:
2833:
2780:
2744:
2717:
2689:
2661:
1739:. The definition of a morphism of inverse semigroups could be augmented by including the condition
680:
641:
605:
581:
553:
529:
4405:
3015:
2105:
1779:
1775:
1764:
1760:
1642:
1385:
932:
Multiplication table example. It is associative and every element has its own inverse according to
877:
474:
408:
376:
372:
28:
5683:(2002). "Book Review: "Inverse Semigroups: The Theory of Partial Symmetries" by Mark V. Lawson".
5554:
5519:
5416:
5282:
Homological
Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups
5004:
4716:
4701:
4585:
4338:{\displaystyle \{(xx^{-1}x,x),\;(xx^{-1}yy^{-1},yy^{-1}xx^{-1})\;|\;x,y\in (X\cup X^{-1})^{+}\}.}
3980:
2298:
One class of inverse semigroups that has been studied extensively over the years is the class of
1638:
884:
113:), and composing functions from left to right—a convention often observed in semigroup theory.)
91:
4353:
2065:
4772:
4357:
5571:
5488:
5469:
5345:
5315:
5307:
5286:
5165:
5159:
4778:
4751:
4721:
4681:
4541:
4520:
4492:
4444:
4425:
4396:. This close connection between inverse semigroups and inductive groupoids is embodied in the
504:
364:
303:. He recognised also that the domain of composition of two partial transformations may be the
142:
138:
87:
5280:
2810:
1981:
5786:
5694:
5666:
5636:
5615:
5594:
5544:
5511:
5426:
5395:
5372:
5014:
2039:
918:
863:
315:
146:
2552:
2524:
5685:
5657:
4736:
4393:
1630:
380:
368:
1895:
are defined on inverse semigroups in exactly the same way as for any other semigroup: a
925:
134:
130:
363:, with inverse the functional inverse defined from image to domain (equivalently, the
5804:
5706:
5461:
4706:
1541:
1397:
1389:
1252:
1103:
407:. Inverses in an inverse semigroup have many of the same properties as inverses in a
5680:
5648:
4746:
1865:
1678:
126:
5430:
5771:
5443:
5339:
4592:
of the class of locally inverse semigroups and the class of orthodox semigroups.
5728:
Wagner, V. V. (1953). "The theory of generalised heaps and generalised groups".
5407:
Exel, R. (1998). "Partial actions of groups and actions of inverse semigroups".
4726:
4711:
2881:
2766:
2492:
2307:
1892:
911:
520:
383:, every inverse semigroup can be embedded in a symmetric inverse semigroup (see
312:
296:
17:
5640:
5619:
5598:
5386:
Gołab, St. (1939). "Über den
Begriff der "Pseudogruppe von Transformationen"".
5790:
5377:
5360:
5018:
4566:
4552:
4531:
4507:
4429:
4135:
3945:
2314:
1595:
1393:
1219:
1158:
674:
635:
575:
512:
508:
441:
1657:
it is possible to obtain an analogous result under additional hypotheses on
4449:
4151:
1862:
866:
384:
304:
44:
1968:{\displaystyle a\,\rho \,b,\quad c\,\rho \,d\Longrightarrow ac\,\rho \,bd.}
480:
There are a number of equivalent characterisations of an inverse semigroup
5698:
5344:. Mathematical Surveys of the American Mathematical Society. Vol. 7.
5606:
Preston, G. B. (1954b). "Inverse semi-groups with minimal right ideals".
5502:
McAlister, D. B. (1974a). "Groups, semilattices and inverse semigroups".
5421:
4677:
4626:
2180:. The minimum group congruence can be used to give a characterisation of
1107:
relation ≤ (sometimes denoted by ω), which is defined by the following:
288:
5671:
5652:
5558:
5523:
5399:
4384:
are composed if, and only if, the image of α is equal to the domain of
2280:{\displaystyle a\in S,e\in E(S),a\,\rho \,e\Longrightarrow a\in E(S).}
367:). This is the "archetypal" inverse semigroup, in the same way that a
5795:
4595:
Amongst the non-regular generalisations of an inverse semigroup are:
2711:
426:
295:. Partial transformations had already been studied in the context of
137:
in 1954. Both authors arrived at inverse semigroups via the study of
5549:
5532:
5515:
1770:
One of the earliest results proved about inverse semigroups was the
5009:
850:
will denote the semilattice of idempotents of an inverse semigroup
5627:
Preston, G. B. (1954c). "Representations of inverse semi-groups".
4459:
if every element has at least one inverse; equivalently, for each
3979:-unitary monoid. McAlister's covering theorem has been refined by
3967:
maximal element above it in the natural partial order, i.e. every
4439:
Examples of regular generalisations of an inverse semigroup are:
1439:
The natural partial order on an inverse semigroup interacts with
273:{\displaystyle \operatorname {dom} \alpha \beta =\alpha ^{-1}\,}
2658:-unitary inverse semigroups is the following construction. Let
2116:
is a group. In the set of all group congruences on a semigroup
1255:
is compatible with both multiplication and inversion, that is,
385:§ Homomorphisms and representations of inverse semigroups
440:
are not necessarily equal to the identity, but they are both
4111:
4087:
4063:
4032:
4022:
3918:
3908:
3869:
3859:
3786:
3776:
3650:
3615:
3581:
3571:
3525:
3515:
3478:
3454:
3406:
3370:
3335:
3325:
3270:
3207:
3162:
3134:
3075:
3031:
2991:
2953:
2925:
2893:
2865:
2839:
2786:
2750:
2723:
2695:
2667:
1813:
1598:
form a semilattice under the product operation, products on
1509:
1466:
786:
729:
686:
647:
611:
587:
559:
535:
328:
5747:
For a brief introduction to inverse semigroups, see either
1392:
simply reduces to equality, since the identity is the only
4150:, where involution is the taking of the inverse, and then
2637:{\displaystyle a\sim b\Longleftrightarrow ab^{-1},a^{-1}b}
1374:{\displaystyle a\leq b\Longrightarrow a^{-1}\leq b^{-1}.}
3804:
is an inverse semigroup under this multiplication, with
1906:
that is compatible with semigroup multiplication, i.e.,
1307:{\displaystyle a\leq b,c\leq d\Longrightarrow ac\leq bd}
1673:
Homomorphisms and representations of inverse semigroups
4687:
Inverse categories have found various applications in
3547:. A McAlister triple is used to define the following:
4166:
4108:
4084:
4060:
4010:
3893:
3847:
3761:
3680:
3556:
3503:
3475:
3451:
3403:
3367:
3313:
3267:
3204:
3159:
3131:
3072:
3028:
2988:
2950:
2922:
2890:
2862:
2836:
2813:
2783:
2747:
2720:
2692:
2664:
2586:
2555:
2527:
2400:
2350:
2208:
2068:
2042:
2011:
1984:
1915:
1809:
1553:
1505:
1462:
1326:
1264:
1177:
1116:
722:
683:
644:
608:
584:
556:
532:
324:
213:
5485:
Inverse
Semigroups: The Theory of Partial Symmetries
200:
can be composed (from left to right) on the largest
121:
Inverse semigroups were introduced independently by
5653:"Obituary: Viktor Vladimirovich Vagner (1908–1981)"
5161:
4617:This notion of inverse also readily generalizes to
709:. There is therefore a simple characterisation of
5772:"On inverse categories and transfer in cohomology"
4408:and whose poset of objects is a meet-semilattice.
4337:
4118:
4094:
4070:
4042:
3929:{\displaystyle P(G,{\mathcal {X}},{\mathcal {Y}})}
3928:
3879:
3797:{\displaystyle P(G,{\mathcal {X}},{\mathcal {Y}})}
3796:
3743:
3660:
3535:
3485:
3461:
3413:
3377:
3353:is also assumed to have the following properties:
3345:
3277:
3214:
3169:
3141:
3082:
3038:
2998:
2960:
2932:
2900:
2872:
2846:
2819:
2793:
2757:
2730:
2702:
2674:
2636:
2561:
2533:
2430:
2380:
2302:-unitary inverse semigroups: an inverse semigroup
2279:
2089:
2054:
2028:
1990:
1967:
1826:
1583:
1517:
1474:
1373:
1306:
1207:
1146:
835:
693:
654:
618:
594:
566:
542:
341:
272:
5779:Proceedings of the Edinburgh Mathematical Society
5758:More comprehensive introductions can be found in
5537:Transactions of the American Mathematical Society
5504:Transactions of the American Mathematical Society
5365:Proceedings of the Edinburgh Mathematical Society
4608:Weakly (left, right, two-sided) ample semigroups.
4605:(Left, right, two-sided) semiadequate semigroups.
4043:{\displaystyle (G,{\mathcal {X}},{\mathcal {Y}})}
3880:{\displaystyle (G,{\mathcal {X}},{\mathcal {Y}})}
3536:{\displaystyle (G,{\mathcal {X}},{\mathcal {Y}})}
3346:{\displaystyle (G,{\mathcal {X}},{\mathcal {Y}})}
2029:{\displaystyle a\,\sigma \,b\Longleftrightarrow }
5533:"Groups, semilattices and inverse semigroups II"
5409:Proceedings of the American Mathematical Society
4004:-inverse semigroups as well. A McAlister triple
1584:{\displaystyle e\leq f\Longleftrightarrow e=ef,}
1208:{\displaystyle a\leq b\Longleftrightarrow a=bf,}
1147:{\displaystyle a\leq b\Longleftrightarrow a=eb,}
349:of all partial one-one transformations of a set
5748:
5585:Preston, G. B. (1954a). "Inverse semi-groups".
5211:
5209:
5045:
4982:
4825:, then a much more comprehensive exposition in
3940:-unitary inverse semigroup. Conversely, every
2381:{\displaystyle es\in E\Longrightarrow s\in E.}
4599:(Left, right, two-sided) adequate semigroups.
2498:of idempotents, and minimum group congruence
1606:) give least upper bounds with respect to ≤.
952:. It has no identity and is not commutative.
204:upon which it "makes sense" to compose them:
8:
5306:Hines, Peter; Braunstein, Samuel L. (2010).
4329:
4167:
3655:
3592:
873:form an inverse semigroup under composition.
5712:Proceedings of the USSR Academy of Sciences
5314:. Cambridge University Press. p. 369.
5629:Journal of the London Mathematical Society
5608:Journal of the London Mathematical Society
5587:Journal of the London Mathematical Society
5312:Semantic Techniques in Quantum Computation
5200:
4602:(Left, right, two-sided) ample semigroups.
4284:
4278:
4204:
3827:. One of the main results in the study of
3744:{\displaystyle (A,g)(B,h)=(A\wedge gB,gh)}
2431:{\displaystyle se\in E\Rightarrow s\in E.}
1400:reduces to restriction of mappings, i.e.,
5670:
5548:
5420:
5376:
5310:. In Gay and, Simon; Mackie, Ian (eds.).
5008:
4777:(2nd ed.). CRC Press. p. 1528.
4588:of generalised inverse semigroups is the
4323:
4310:
4279:
4266:
4250:
4231:
4215:
4180:
4165:
4110:
4109:
4107:
4086:
4085:
4083:
4062:
4061:
4059:
4031:
4030:
4021:
4020:
4009:
3917:
3916:
3907:
3906:
3892:
3868:
3867:
3858:
3857:
3846:
3785:
3784:
3775:
3774:
3760:
3679:
3649:
3648:
3633:
3614:
3613:
3580:
3579:
3570:
3569:
3555:
3524:
3523:
3514:
3513:
3502:
3477:
3476:
3474:
3453:
3452:
3450:
3405:
3404:
3402:
3369:
3368:
3366:
3334:
3333:
3324:
3323:
3312:
3269:
3268:
3266:
3206:
3205:
3203:
3161:
3160:
3158:
3133:
3132:
3130:
3074:
3073:
3071:
3030:
3029:
3027:
2990:
2989:
2987:
2952:
2951:
2949:
2924:
2923:
2921:
2892:
2891:
2889:
2864:
2863:
2861:
2838:
2837:
2835:
2812:
2785:
2784:
2782:
2749:
2748:
2746:
2722:
2721:
2719:
2694:
2693:
2691:
2666:
2665:
2663:
2622:
2606:
2585:
2554:
2526:
2399:
2349:
2249:
2245:
2207:
2184:-unitary inverse semigroups (see below).
2067:
2041:
2019:
2015:
2010:
1983:
1955:
1951:
1938:
1934:
1923:
1919:
1914:
1818:
1812:
1811:
1808:
1552:
1514:
1508:
1507:
1506:
1504:
1471:
1465:
1464:
1463:
1461:
1359:
1343:
1325:
1263:
1176:
1115:
824:
805:
791:
785:
784:
783:
764:
745:
734:
728:
727:
726:
721:
685:
684:
682:
646:
645:
643:
610:
609:
607:
586:
585:
583:
558:
557:
555:
534:
533:
531:
492:has a unique inverse, in the above sense.
333:
327:
326:
323:
318:. Under this composition, the collection
269:
260:
212:
5338:Clifford, A. H.; Preston, G. B. (1967).
5263:
4838:
1396:. In a symmetric inverse semigroup, the
954:
5759:
4774:CRC Concise Encyclopedia of Mathematics
4763:
4142:of the free inverse semigroup on a set
4000:-theorem has been used to characterize
1978:Of particular interest is the relation
353:forms an inverse semigroup, called the
5763:
5227:
5215:
5145:
5133:
5109:
5097:
5085:
5061:
5024:
4970:
4923:
4863:
4826:
4822:
4809:
4797:
4138:is possible for inverse semigroups. A
2104:is a congruence and, in fact, it is a
5752:
5308:"The Structure of Partial Isometries"
5251:
5239:
5196:
5184:
5121:
5073:
5057:
4958:
4946:
4934:
4911:
4899:
4887:
4875:
4851:
4420:can be defined by the conditions (1)
4416:As noted above, an inverse semigroup
4412:Generalisations of inverse semigroups
2502:. Then the following are equivalent:
7:
4625:is simply a category in which every
4398:Ehresmann–Schein–Nambooripad Theorem
3971:-class has a maximal element. Every
2108:, meaning that the factor semigroup
188:be partial transformations of a set
4146:may be obtained by considering the
3959:An inverse semigroup is said to be
2441:One further characterisation of an
5440:"(Weakly) left E-ample semigroups"
5341:The Algebraic Theory of Semigroups
5064:and, independently, Preston 1954c.
4506:is an inverse semigroup, for each
4054:-inverse semigroup if and only if
2769:, that is, every pair of elements
1998:, defined on an inverse semigroup
1827:{\displaystyle {\mathcal {I}}_{S}}
1792:is an inverse semigroup, then the
1518:{\displaystyle \,{\mathcal {R}}\,}
1475:{\displaystyle \,{\mathcal {L}}\,}
473:inverse monoid), is, of course, a
342:{\displaystyle {\mathcal {I}}_{X}}
25:
5267:
1888:Congruences on inverse semigroups
1218:for some (in general, different)
5466:Fundamentals of Semigroup Theory
5285:. World Scientific. p. 55.
4371:Connections with category theory
3963:-inverse if every element has a
3989:Every inverse semigroup has an
3831:-unitary inverse semigroups is
1930:
1410:if, and only if, the domain of
779:
301:composition of binary relations
5732:. Novaya Seriya (in Russian).
5709:(1952). "Generalised groups".
4821:First a short announcement in
4538:Generalised inverse semigroups
4363:Any free inverse semigroup is
4320:
4297:
4280:
4275:
4205:
4198:
4170:
4148:free semigroup with involution
4119:{\displaystyle {\mathcal {X}}}
4095:{\displaystyle {\mathcal {X}}}
4071:{\displaystyle {\mathcal {Y}}}
4037:
4011:
3944:-unitary inverse semigroup is
3923:
3897:
3874:
3848:
3791:
3765:
3738:
3714:
3708:
3696:
3693:
3681:
3607:
3595:
3586:
3560:
3530:
3504:
3486:{\displaystyle {\mathcal {Y}}}
3462:{\displaystyle {\mathcal {Y}}}
3414:{\displaystyle {\mathcal {Y}}}
3378:{\displaystyle {\mathcal {X}}}
3340:
3314:
3278:{\displaystyle {\mathcal {X}}}
3215:{\displaystyle {\mathcal {X}}}
3170:{\displaystyle {\mathcal {X}}}
3142:{\displaystyle {\mathcal {X}}}
3083:{\displaystyle {\mathcal {X}}}
3039:{\displaystyle {\mathcal {X}}}
2999:{\displaystyle {\mathcal {Y}}}
2961:{\displaystyle {\mathcal {Y}}}
2933:{\displaystyle {\mathcal {X}}}
2901:{\displaystyle {\mathcal {X}}}
2873:{\displaystyle {\mathcal {Y}}}
2847:{\displaystyle {\mathcal {Y}}}
2794:{\displaystyle {\mathcal {Y}}}
2758:{\displaystyle {\mathcal {Y}}}
2731:{\displaystyle {\mathcal {X}}}
2703:{\displaystyle {\mathcal {Y}}}
2675:{\displaystyle {\mathcal {X}}}
2596:
2413:
2363:
2271:
2265:
2253:
2236:
2230:
2023:
1942:
1563:
1414:is contained in the domain of
1336:
1286:
1187:
1126:
858:Examples of inverse semigroups
795:
738:
694:{\displaystyle {\mathcal {R}}}
655:{\displaystyle {\mathcal {L}}}
619:{\displaystyle {\mathcal {R}}}
595:{\displaystyle {\mathcal {L}}}
574:-class contains precisely one
567:{\displaystyle {\mathcal {R}}}
543:{\displaystyle {\mathcal {L}}}
253:
229:
1:
5431:10.1090/S0002-9939-98-04575-4
5242:, Section 2.4 & Chapter 6
5199:, Theorem 5.9.2. Originally,
4742:Special classes of semigroups
4549:generalised inverse semigroup
3671:together with multiplication
2649:McAlister's Covering Theorem.
2491:be an inverse semigroup with
375:. For example, just as every
5060:, Theorem 5.1.7 Originally,
4689:theoretical computer science
4134:A construction similar to a
3887:be a McAlister triple. Then
5749:Clifford & Preston 1967
5468:. Oxford: Clarendon Press.
5046:Clifford & Preston 1967
4983:Clifford & Preston 1967
4771:Weisstein, Eric W. (2002).
4680:. The category of sets and
4404:with respect to its (dual)
3493:have nonempty intersection.
2686:, with ordering ≤, and let
2445:-unitary inverse semigroup
2294:-unitary inverse semigroups
2148:is any other congruence on
356:symmetric inverse semigroup
123:Viktor Vladimirovich Wagner
5832:
5531:McAlister, D. B. (1974b).
5164:. CRC Press. p. 248.
5031:: CS1 maint: postscript (
4642:has a generalized inverse
4555:form a normal band, i.e.,
4489:Locally inverse semigroups
2090:{\displaystyle c\leq a,b.}
1774:, which is an analogue of
499:has at least one inverse (
395:The inverse of an element
5791:10.1017/S0013091512000211
5378:10.1017/S0013091500016230
5019:10.1007/s00233-017-9858-5
4676:. An inverse category is
3975:-inverse semigroup is an
3046:(on the left), such that
2738:with the properties that
1884:is an inverse semigroup.
1092:The natural partial order
846:Unless stated otherwise,
713:in an inverse semigroup:
5770:Linckelmann, M. (2012).
5641:10.1112/jlms/s1-29.4.411
5620:10.1112/jlms/s1-29.4.404
5599:10.1112/jlms/s1-29.4.396
5359:Fountain, J. B. (1979).
4078:is a principal ideal of
2654:Central to the study of
2191:on an inverse semigroup
2174:minimum group congruence
2144:is a group, that is, if
2124:is an inverse semigroup
1617:) is finite and forms a
880:is an inverse semigroup.
399:of an inverse semigroup
35:(occasionally called an
5730:Matematicheskii Sbornik
5279:Grandis, Marco (2012).
5158:Grillet, P. A. (1995).
4130:Free inverse semigroups
2820:{\displaystyle \wedge }
1991:{\displaystyle \sigma }
1786:Wagner–Preston Theorem.
47:in which every element
5483:Lawson, M. V. (1998).
4914:, Proposition 5.1.2(1)
4890:, Proposition 5.1.2(1)
4684:is the prime example.
4339:
4120:
4096:
4072:
4044:
3930:
3881:
3839:McAlister's P-Theorem.
3798:
3745:
3662:
3537:
3487:
3463:
3415:
3379:
3347:
3279:
3216:
3171:
3143:
3084:
3040:
3000:
2962:
2934:
2902:
2874:
2848:
2821:
2795:
2759:
2732:
2704:
2676:
2638:
2571:compatibility relation
2563:
2535:
2432:
2382:
2281:
2091:
2056:
2055:{\displaystyle c\in S}
2030:
1992:
1969:
1828:
1772:Wagner–Preston Theorem
1585:
1519:
1476:
1375:
1308:
1209:
1148:
837:
695:
656:
620:
596:
568:
544:
511:commute (that is, the
343:
307:, so he introduced an
274:
147:partial transformation
5699:10.1007/s002330010132
5388:Mathematische Annalen
5361:"Adequate semigroups"
4534:forms a subsemigroup.
4340:
4121:
4097:
4073:
4045:
3948:to one of this type.
3931:
3882:
3833:McAlister's P-Theorem
3799:
3746:
3663:
3538:
3488:
3464:
3416:
3380:
3348:
3280:
3217:
3172:
3144:
3085:
3041:
3001:
2963:
2935:
2903:
2875:
2849:
2822:
2796:
2760:
2733:
2705:
2684:partially ordered set
2677:
2639:
2564:
2562:{\displaystyle \sim }
2536:
2534:{\displaystyle \sim }
2449:is the following: if
2433:
2383:
2282:
2100:It can be shown that
2092:
2057:
2031:
1993:
1970:
1829:
1586:
1520:
1477:
1376:
1309:
1210:
1149:
1096:An inverse semigroup
838:
696:
657:
621:
597:
569:
545:
379:can be embedded in a
344:
275:
5811:Algebraic structures
5796:Open access preprint
5566:Petrich, M. (1984).
5487:. World Scientific.
4164:
4106:
4082:
4058:
4008:
3891:
3845:
3759:
3678:
3554:
3501:
3473:
3449:
3401:
3365:
3311:
3265:
3202:
3157:
3129:
3070:
3026:
2986:
2948:
2920:
2888:
2860:
2854:(with respect to ≤);
2834:
2811:
2803:greatest lower bound
2781:
2745:
2718:
2690:
2662:
2584:
2553:
2525:
2398:
2348:
2206:
2066:
2040:
2009:
1982:
1913:
1904:equivalence relation
1807:
1551:
1503:
1460:
1324:
1262:
1233:can be taken to be
1175:
1114:
720:
681:
642:
606:
582:
554:
530:
444:. An inverse monoid
322:
309:empty transformation
211:
5722:English translation
5136:, Proposition 2.4.3
4973:, Proposition 3.2.3
4937:, Proposition 5.2.1
4517:Orthodox semigroups
4406:Alexandrov topology
4152:taking the quotient
3955:-inverse semigroups
2519:is idempotent pure;
957:
403:is usually written
37:inversion semigroup
5672:10.1007/BF02676643
5568:Inverse semigroups
5400:10.1007/BF01597390
4850:See, for example,
4717:Partial symmetries
4702:Orthodox semigroup
4682:partial bijections
4445:Regular semigroups
4392:, in the sense of
4390:inductive groupoid
4335:
4126:is a semilattice.
4116:
4092:
4068:
4040:
3926:
3877:
3794:
3741:
3658:
3533:
3483:
3459:
3411:
3375:
3343:
3275:
3212:
3167:
3149:, there exists an
3139:
3080:
3036:
2996:
2958:
2930:
2898:
2870:
2844:
2817:
2791:
2755:
2728:
2700:
2672:
2634:
2559:
2531:
2428:
2378:
2277:
2087:
2052:
2026:
1988:
1965:
1824:
1581:
1515:
1472:
1371:
1304:
1205:
1144:
956:Inverse semigroup
955:
885:bicyclic semigroup
833:
691:
652:
616:
592:
564:
540:
371:is the archetypal
339:
270:
139:partial bijections
92:partial symmetries
66:in the sense that
5351:978-0-8218-0272-4
5321:978-0-521-51374-6
5292:978-981-4407-06-9
5171:978-0-8247-9662-4
4784:978-1-4200-3522-3
4752:Nambooripad order
4732:Green's relations
4722:Regular semigroup
4542:regular semigroup
4530:if its subset of
4521:regular semigroup
4493:regular semigroup
4426:regular semigroup
4156:Vagner congruence
3385:, there exists a
3232:if, and only if,
3100:if, and only if,
2767:lower semilattice
2168:. The congruence
1767:of that element.
1708:is a morphism if
1653:) is an infinite
1441:Green's relations
1432:in the domain of
1089:
1088:
887:is inverse, with
711:Green's relations
628:Green's relations
550:-class and every
505:regular semigroup
495:Every element of
488:Every element of
365:converse relation
88:regular semigroup
33:inverse semigroup
16:(Redirected from
5823:
5816:Semigroup theory
5794:
5776:
5737:
5720:
5702:
5676:
5674:
5644:
5623:
5602:
5581:
5562:
5552:
5527:
5498:
5479:
5457:
5455:
5454:
5448:
5442:. Archived from
5434:
5424:
5422:funct-an/9511003
5403:
5382:
5380:
5355:
5326:
5325:
5303:
5297:
5296:
5276:
5270:
5261:
5255:
5249:
5243:
5237:
5231:
5225:
5219:
5213:
5204:
5194:
5188:
5187:, pp. 193–4
5182:
5176:
5175:
5155:
5149:
5143:
5137:
5131:
5125:
5119:
5113:
5107:
5101:
5095:
5089:
5083:
5077:
5071:
5065:
5055:
5049:
5043:
5037:
5036:
5030:
5022:
5012:
4992:
4986:
4980:
4974:
4968:
4962:
4956:
4950:
4949:, pp. 152–3
4944:
4938:
4932:
4926:
4921:
4915:
4909:
4903:
4897:
4891:
4885:
4879:
4873:
4867:
4861:
4855:
4848:
4842:
4836:
4830:
4819:
4813:
4806:
4800:
4795:
4789:
4788:
4768:
4675:
4665:
4655:
4641:
4623:inverse category
4613:Inverse category
4564:
4484:
4344:
4342:
4341:
4336:
4328:
4327:
4318:
4317:
4283:
4274:
4273:
4258:
4257:
4239:
4238:
4223:
4222:
4188:
4187:
4125:
4123:
4122:
4117:
4115:
4114:
4101:
4099:
4098:
4093:
4091:
4090:
4077:
4075:
4074:
4069:
4067:
4066:
4049:
4047:
4046:
4041:
4036:
4035:
4026:
4025:
3993:-inverse cover.
3935:
3933:
3932:
3927:
3922:
3921:
3912:
3911:
3886:
3884:
3883:
3878:
3873:
3872:
3863:
3862:
3826:
3803:
3801:
3800:
3795:
3790:
3789:
3780:
3779:
3750:
3748:
3747:
3742:
3667:
3665:
3664:
3659:
3654:
3653:
3641:
3640:
3619:
3618:
3585:
3584:
3575:
3574:
3545:McAlister triple
3542:
3540:
3539:
3534:
3529:
3528:
3519:
3518:
3492:
3490:
3489:
3484:
3482:
3481:
3468:
3466:
3465:
3460:
3458:
3457:
3430:
3420:
3418:
3417:
3412:
3410:
3409:
3384:
3382:
3381:
3376:
3374:
3373:
3352:
3350:
3349:
3344:
3339:
3338:
3329:
3328:
3302:
3284:
3282:
3281:
3276:
3274:
3273:
3241:
3231:
3221:
3219:
3218:
3213:
3211:
3210:
3186:
3176:
3174:
3173:
3168:
3166:
3165:
3148:
3146:
3145:
3140:
3138:
3137:
3109:
3099:
3089:
3087:
3086:
3081:
3079:
3078:
3045:
3043:
3042:
3037:
3035:
3034:
3005:
3003:
3002:
2997:
2995:
2994:
2977:
2967:
2965:
2964:
2959:
2957:
2956:
2939:
2937:
2936:
2931:
2929:
2928:
2907:
2905:
2904:
2899:
2897:
2896:
2879:
2877:
2876:
2871:
2869:
2868:
2853:
2851:
2850:
2845:
2843:
2842:
2826:
2824:
2823:
2818:
2800:
2798:
2797:
2792:
2790:
2789:
2764:
2762:
2761:
2756:
2754:
2753:
2737:
2735:
2734:
2729:
2727:
2726:
2709:
2707:
2706:
2701:
2699:
2698:
2681:
2679:
2678:
2673:
2671:
2670:
2643:
2641:
2640:
2635:
2630:
2629:
2614:
2613:
2568:
2566:
2565:
2560:
2540:
2538:
2537:
2532:
2437:
2435:
2434:
2429:
2387:
2385:
2384:
2379:
2286:
2284:
2283:
2278:
2164:is contained in
2106:group congruence
2096:
2094:
2093:
2088:
2061:
2059:
2058:
2053:
2035:
2033:
2032:
2027:
1997:
1995:
1994:
1989:
1974:
1972:
1971:
1966:
1833:
1831:
1830:
1825:
1823:
1822:
1817:
1816:
1776:Cayley's theorem
1759:The homomorphic
1752:
1726:
1590:
1588:
1587:
1582:
1528:
1524:
1522:
1521:
1516:
1513:
1512:
1495:. Similarly, if
1494:
1481:
1479:
1478:
1473:
1470:
1469:
1452:
1427:
1409:
1380:
1378:
1377:
1372:
1367:
1366:
1351:
1350:
1313:
1311:
1310:
1305:
1214:
1212:
1211:
1206:
1168:. Equivalently,
1153:
1151:
1150:
1145:
958:
951:
941:
919:Brandt semigroup
906:
842:
840:
839:
834:
832:
831:
813:
812:
790:
789:
772:
771:
753:
752:
733:
732:
700:
698:
697:
692:
690:
689:
661:
659:
658:
653:
651:
650:
625:
623:
622:
617:
615:
614:
601:
599:
598:
593:
591:
590:
573:
571:
570:
565:
563:
562:
549:
547:
546:
541:
539:
538:
460:
425:. In an inverse
424:
348:
346:
345:
340:
338:
337:
332:
331:
316:binary operation
279:
277:
276:
271:
268:
267:
129:in 1952, and by
85:
75:
21:
18:Inverse category
5831:
5830:
5826:
5825:
5824:
5822:
5821:
5820:
5801:
5800:
5774:
5769:
5751:, Chapter 7 or
5744:
5742:Further reading
5727:
5705:
5686:Semigroup Forum
5679:
5658:Semigroup Forum
5647:
5626:
5605:
5584:
5578:
5565:
5550:10.2307/1997032
5530:
5516:10.2307/1996831
5501:
5495:
5482:
5476:
5460:
5452:
5450:
5446:
5437:
5406:
5385:
5358:
5352:
5337:
5334:
5329:
5322:
5305:
5304:
5300:
5293:
5278:
5277:
5273:
5262:
5258:
5250:
5246:
5238:
5234:
5226:
5222:
5214:
5207:
5201:McAlister 1974a
5195:
5191:
5183:
5179:
5172:
5157:
5156:
5152:
5148:, Theorem 2.4.6
5144:
5140:
5132:
5128:
5120:
5116:
5108:
5104:
5100:, Theorem 2.4.1
5096:
5092:
5084:
5080:
5072:
5068:
5056:
5052:
5044:
5040:
5023:
4997:Semigroup Forum
4994:
4993:
4989:
4981:
4977:
4969:
4965:
4957:
4953:
4945:
4941:
4933:
4929:
4922:
4918:
4910:
4906:
4902:, Theorem 5.1.1
4898:
4894:
4886:
4882:
4874:
4870:
4862:
4858:
4849:
4845:
4837:
4833:
4820:
4816:
4807:
4803:
4796:
4792:
4785:
4770:
4769:
4765:
4761:
4756:
4737:Category theory
4697:
4667:
4657:
4643:
4629:
4615:
4556:
4500:locally inverse
4476:
4414:
4394:category theory
4373:
4319:
4306:
4262:
4246:
4227:
4211:
4176:
4162:
4161:
4132:
4104:
4103:
4080:
4079:
4056:
4055:
4006:
4005:
3957:
3889:
3888:
3843:
3842:
3805:
3757:
3756:
3676:
3675:
3629:
3552:
3551:
3499:
3498:
3471:
3470:
3447:
3446:
3422:
3399:
3398:
3363:
3362:
3309:
3308:
3286:
3263:
3262:
3233:
3223:
3200:
3199:
3178:
3155:
3154:
3127:
3126:
3101:
3091:
3068:
3067:
3024:
3023:
2984:
2983:
2969:
2946:
2945:
2918:
2917:
2908:, that is, for
2886:
2885:
2858:
2857:
2832:
2831:
2809:
2808:
2779:
2778:
2743:
2742:
2716:
2715:
2688:
2687:
2660:
2659:
2644:are idempotent.
2618:
2602:
2582:
2581:
2551:
2550:
2523:
2522:
2396:
2395:
2346:
2345:
2296:
2204:
2203:
2197:idempotent pure
2064:
2063:
2038:
2037:
2036:there exists a
2007:
2006:
1980:
1979:
1911:
1910:
1890:
1810:
1805:
1804:
1740:
1709:
1675:
1631:totally ordered
1549:
1548:
1540:), the natural
1501:
1500:
1496:
1486:
1458:
1457:
1444:
1443:as follows: if
1419:
1401:
1355:
1339:
1322:
1321:
1260:
1259:
1173:
1172:
1112:
1111:
1094:
943:
933:
888:
860:
820:
801:
760:
741:
718:
717:
679:
678:
640:
639:
604:
603:
580:
579:
552:
551:
528:
527:
449:
412:
411:, for example,
393:
381:symmetric group
369:symmetric group
359:(or monoid) on
325:
320:
319:
256:
209:
208:
176:are subsets of
119:
77:
67:
23:
22:
15:
12:
11:
5:
5829:
5827:
5819:
5818:
5813:
5803:
5802:
5799:
5798:
5767:
5756:
5743:
5740:
5739:
5738:
5736:(74): 545–632.
5725:
5715:(in Russian).
5703:
5677:
5645:
5624:
5614:(4): 404–411.
5603:
5593:(4): 396–403.
5582:
5576:
5563:
5528:
5499:
5493:
5480:
5474:
5458:
5435:
5415:(12): 3481–4.
5404:
5383:
5371:(2): 113–125.
5356:
5350:
5333:
5330:
5328:
5327:
5320:
5298:
5291:
5271:
5256:
5244:
5232:
5220:
5205:
5189:
5177:
5170:
5150:
5138:
5126:
5114:
5102:
5090:
5078:
5066:
5050:
5048:, Theorem 7.36
5038:
5003:(2): 203–240.
4987:
4975:
4963:
4951:
4939:
4927:
4916:
4904:
4892:
4880:
4868:
4856:
4843:
4831:
4814:
4801:
4790:
4783:
4762:
4760:
4757:
4755:
4754:
4749:
4744:
4739:
4734:
4729:
4724:
4719:
4714:
4709:
4704:
4698:
4696:
4693:
4614:
4611:
4610:
4609:
4606:
4603:
4600:
4582:
4581:
4535:
4514:
4486:
4467:, there is an
4428:, and (2) the
4413:
4410:
4402:étale groupoid
4372:
4369:
4346:
4345:
4334:
4331:
4326:
4322:
4316:
4313:
4309:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4282:
4277:
4272:
4269:
4265:
4261:
4256:
4253:
4249:
4245:
4242:
4237:
4234:
4230:
4226:
4221:
4218:
4214:
4210:
4207:
4203:
4200:
4197:
4194:
4191:
4186:
4183:
4179:
4175:
4172:
4169:
4131:
4128:
4113:
4089:
4065:
4039:
4034:
4029:
4024:
4019:
4016:
4013:
3956:
3950:
3925:
3920:
3915:
3910:
3905:
3902:
3899:
3896:
3876:
3871:
3866:
3861:
3856:
3853:
3850:
3793:
3788:
3783:
3778:
3773:
3770:
3767:
3764:
3753:
3752:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3669:
3668:
3657:
3652:
3647:
3644:
3639:
3636:
3632:
3628:
3625:
3622:
3617:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3583:
3578:
3573:
3568:
3565:
3562:
3559:
3532:
3527:
3522:
3517:
3512:
3509:
3506:
3497:Such a triple
3495:
3494:
3480:
3456:
3432:
3408:
3372:
3342:
3337:
3332:
3327:
3322:
3319:
3316:
3305:
3304:
3272:
3243:
3209:
3188:
3164:
3136:
3111:
3077:
3033:
3008:
3007:
2993:
2955:
2927:
2895:
2867:
2855:
2841:
2816:
2788:
2752:
2725:
2697:
2669:
2646:
2645:
2633:
2628:
2625:
2621:
2617:
2612:
2609:
2605:
2601:
2598:
2595:
2592:
2589:
2558:
2547:
2546:
2530:
2520:
2514:
2439:
2438:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2391:Equivalently,
2389:
2388:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2295:
2289:
2288:
2287:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2248:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2172:is called the
2160:a group, then
2132:congruence on
2098:
2097:
2086:
2083:
2080:
2077:
2074:
2071:
2051:
2048:
2045:
2025:
2022:
2018:
2014:
1987:
1976:
1975:
1964:
1961:
1958:
1954:
1950:
1947:
1944:
1941:
1937:
1933:
1929:
1926:
1922:
1918:
1889:
1886:
1866:representation
1859:
1858:
1821:
1815:
1674:
1671:
1594:so, since the
1592:
1591:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1511:
1468:
1382:
1381:
1370:
1365:
1362:
1358:
1354:
1349:
1346:
1342:
1338:
1335:
1332:
1329:
1315:
1314:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1216:
1215:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1155:
1154:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1093:
1090:
1087:
1086:
1083:
1080:
1077:
1074:
1071:
1065:
1064:
1061:
1058:
1055:
1052:
1049:
1043:
1042:
1039:
1036:
1033:
1030:
1027:
1021:
1020:
1017:
1014:
1011:
1008:
1005:
999:
998:
995:
992:
989:
986:
983:
977:
976:
973:
970:
967:
964:
961:
930:
929:
926:Munn semigroup
922:
915:
908:
881:
874:
859:
856:
844:
843:
830:
827:
823:
819:
816:
811:
808:
804:
800:
797:
794:
788:
782:
778:
775:
770:
767:
763:
759:
756:
751:
748:
744:
740:
737:
731:
725:
688:
673:, whilst the
649:
632:
631:
613:
589:
561:
537:
524:
493:
392:
389:
336:
330:
281:
280:
266:
263:
259:
255:
252:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
135:United Kingdom
131:Gordon Preston
118:
115:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5828:
5817:
5814:
5812:
5809:
5808:
5806:
5797:
5792:
5788:
5784:
5780:
5773:
5768:
5765:
5761:
5757:
5754:
5750:
5746:
5745:
5741:
5735:
5731:
5726:
5723:
5718:
5714:
5713:
5708:
5707:Wagner, V. V.
5704:
5700:
5696:
5692:
5688:
5687:
5682:
5681:Schein, B. M.
5678:
5673:
5668:
5664:
5660:
5659:
5654:
5650:
5649:Schein, B. M.
5646:
5642:
5638:
5634:
5630:
5625:
5621:
5617:
5613:
5609:
5604:
5600:
5596:
5592:
5588:
5583:
5579:
5573:
5569:
5564:
5560:
5556:
5551:
5546:
5542:
5538:
5534:
5529:
5525:
5521:
5517:
5513:
5509:
5505:
5500:
5496:
5490:
5486:
5481:
5477:
5471:
5467:
5463:
5459:
5449:on 2005-08-26
5445:
5441:
5436:
5432:
5428:
5423:
5418:
5414:
5410:
5405:
5401:
5397:
5393:
5390:(in German).
5389:
5384:
5379:
5374:
5370:
5366:
5362:
5357:
5353:
5347:
5343:
5342:
5336:
5335:
5331:
5323:
5317:
5313:
5309:
5302:
5299:
5294:
5288:
5284:
5283:
5275:
5272:
5269:
5265:
5264:Fountain 1979
5260:
5257:
5254:, p. 222
5253:
5248:
5245:
5241:
5236:
5233:
5229:
5224:
5221:
5218:, p. 230
5217:
5212:
5210:
5206:
5202:
5198:
5193:
5190:
5186:
5181:
5178:
5173:
5167:
5163:
5162:
5154:
5151:
5147:
5142:
5139:
5135:
5130:
5127:
5124:, p. 192
5123:
5118:
5115:
5111:
5106:
5103:
5099:
5094:
5091:
5087:
5082:
5079:
5075:
5070:
5067:
5063:
5059:
5054:
5051:
5047:
5042:
5039:
5034:
5028:
5021:Corollary 4.9
5020:
5016:
5011:
5006:
5002:
4998:
4991:
4988:
4985:, Theorem 7.5
4984:
4979:
4976:
4972:
4967:
4964:
4961:, p. 153
4960:
4955:
4952:
4948:
4943:
4940:
4936:
4931:
4928:
4925:
4920:
4917:
4913:
4908:
4905:
4901:
4896:
4893:
4889:
4884:
4881:
4878:, p. 149
4877:
4872:
4869:
4866:, p. 152
4865:
4860:
4857:
4853:
4847:
4844:
4840:
4839:Preston 1954a
4835:
4832:
4828:
4824:
4818:
4815:
4811:
4805:
4802:
4799:
4794:
4791:
4786:
4780:
4776:
4775:
4767:
4764:
4758:
4753:
4750:
4748:
4745:
4743:
4740:
4738:
4735:
4733:
4730:
4728:
4725:
4723:
4720:
4718:
4715:
4713:
4710:
4708:
4707:Biordered set
4705:
4703:
4700:
4699:
4694:
4692:
4690:
4685:
4683:
4679:
4674:
4670:
4664:
4660:
4654:
4650:
4646:
4640:
4636:
4632:
4628:
4624:
4620:
4612:
4607:
4604:
4601:
4598:
4597:
4596:
4593:
4591:
4587:
4579:
4575:
4571:
4568:
4563:
4559:
4554:
4550:
4546:
4543:
4539:
4536:
4533:
4529:
4525:
4522:
4518:
4515:
4512:
4509:
4505:
4501:
4497:
4494:
4490:
4487:
4483:
4479:
4474:
4470:
4466:
4462:
4458:
4454:
4451:
4447:
4446:
4442:
4441:
4440:
4437:
4435:
4431:
4427:
4423:
4419:
4411:
4409:
4407:
4403:
4399:
4395:
4391:
4387:
4383:
4379:
4370:
4368:
4366:
4361:
4359:
4355:
4351:
4332:
4324:
4314:
4311:
4307:
4303:
4300:
4294:
4291:
4288:
4285:
4270:
4267:
4263:
4259:
4254:
4251:
4247:
4243:
4240:
4235:
4232:
4228:
4224:
4219:
4216:
4212:
4208:
4201:
4195:
4192:
4189:
4184:
4181:
4177:
4173:
4160:
4159:
4158:
4157:
4153:
4149:
4145:
4141:
4137:
4129:
4127:
4053:
4027:
4017:
4014:
4003:
3999:
3994:
3992:
3988:
3984:
3982:
3978:
3974:
3970:
3966:
3962:
3954:
3951:
3949:
3947:
3943:
3939:
3913:
3903:
3900:
3894:
3864:
3854:
3851:
3840:
3836:
3834:
3830:
3824:
3820:
3817:
3813:
3809:
3781:
3771:
3768:
3762:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3711:
3705:
3702:
3699:
3690:
3687:
3684:
3674:
3673:
3672:
3645:
3642:
3637:
3634:
3630:
3626:
3623:
3620:
3610:
3604:
3601:
3598:
3589:
3576:
3566:
3563:
3557:
3550:
3549:
3548:
3546:
3520:
3510:
3507:
3445:
3441:
3437:
3433:
3429:
3425:
3396:
3392:
3388:
3360:
3356:
3355:
3354:
3330:
3320:
3317:
3301:
3297:
3293:
3289:
3260:
3256:
3252:
3248:
3244:
3240:
3236:
3230:
3226:
3197:
3193:
3189:
3185:
3181:
3152:
3124:
3120:
3116:
3112:
3108:
3104:
3098:
3094:
3065:
3061:
3057:
3053:
3049:
3048:
3047:
3021:
3017:
3013:
2981:
2976:
2972:
2943:
2915:
2911:
2883:
2856:
2829:
2814:
2807:
2804:
2776:
2772:
2768:
2741:
2740:
2739:
2713:
2685:
2657:
2652:
2650:
2631:
2626:
2623:
2619:
2615:
2610:
2607:
2603:
2599:
2593:
2590:
2587:
2580:
2579:
2578:
2577:, defined by
2576:
2572:
2556:
2544:
2528:
2521:
2518:
2515:
2512:
2508:
2505:
2504:
2503:
2501:
2497:
2494:
2490:
2486:
2482:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2425:
2422:
2419:
2416:
2410:
2407:
2404:
2401:
2394:
2393:
2392:
2375:
2372:
2369:
2366:
2360:
2357:
2354:
2351:
2344:
2343:
2342:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2309:
2305:
2301:
2293:
2290:
2274:
2268:
2262:
2259:
2256:
2250:
2246:
2242:
2239:
2233:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2202:
2201:
2200:
2198:
2194:
2190:
2187:A congruence
2185:
2183:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2151:
2147:
2143:
2139:
2135:
2131:
2127:
2123:
2119:
2115:
2111:
2107:
2103:
2084:
2081:
2078:
2075:
2072:
2069:
2049:
2046:
2043:
2020:
2016:
2012:
2005:
2004:
2003:
2001:
1985:
1962:
1959:
1956:
1952:
1948:
1945:
1939:
1935:
1931:
1927:
1924:
1920:
1916:
1909:
1908:
1907:
1905:
1901:
1898:
1894:
1887:
1885:
1883:
1879:
1873:
1871:
1867:
1864:
1857:
1853:
1849:
1845:
1841:
1837:
1836:
1835:
1819:
1802:
1798:
1795:
1791:
1787:
1783:
1781:
1777:
1773:
1768:
1766:
1762:
1758:
1754:
1751:
1748:
1744:
1738:
1734:
1730:
1725:
1721:
1717:
1713:
1707:
1703:
1699:
1696:
1692:
1688:
1684:
1680:
1672:
1670:
1668:
1664:
1660:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1612:
1607:
1605:
1601:
1597:
1578:
1575:
1572:
1569:
1566:
1560:
1557:
1554:
1547:
1546:
1545:
1543:
1542:partial order
1539:
1535:
1530:
1527:
1499:
1493:
1489:
1484:
1456:
1451:
1447:
1442:
1437:
1435:
1431:
1426:
1422:
1417:
1413:
1408:
1404:
1399:
1398:partial order
1395:
1391:
1390:partial order
1387:
1368:
1363:
1360:
1356:
1352:
1347:
1344:
1340:
1333:
1330:
1327:
1320:
1319:
1318:
1301:
1298:
1295:
1292:
1289:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1258:
1257:
1256:
1254:
1253:partial order
1249:
1247:
1244:
1240:
1236:
1232:
1228:
1224:
1221:
1202:
1199:
1196:
1193:
1190:
1184:
1181:
1178:
1171:
1170:
1169:
1167:
1163:
1160:
1141:
1138:
1135:
1132:
1129:
1123:
1120:
1117:
1110:
1109:
1108:
1106:
1105:
1104:partial order
1099:
1091:
1084:
1081:
1078:
1075:
1072:
1070:
1067:
1066:
1062:
1059:
1056:
1053:
1050:
1048:
1045:
1044:
1040:
1037:
1034:
1031:
1028:
1026:
1023:
1022:
1018:
1015:
1012:
1009:
1006:
1004:
1001:
1000:
996:
993:
990:
987:
984:
982:
979:
978:
974:
971:
968:
965:
962:
960:
959:
953:
950:
946:
940:
936:
927:
923:
920:
916:
913:
909:
904:
900:
896:
892:
886:
882:
879:
875:
872:
868:
865:
862:
861:
857:
855:
853:
849:
828:
825:
821:
817:
814:
809:
806:
802:
798:
792:
780:
776:
773:
768:
765:
761:
757:
754:
749:
746:
742:
735:
723:
716:
715:
714:
712:
708:
704:
676:
672:
669:
665:
637:
629:
577:
525:
522:
518:
514:
510:
506:
502:
498:
494:
491:
487:
486:
485:
483:
478:
476:
472:
468:
464:
459:
456:
452:
447:
443:
439:
436:
432:
428:
423:
420:
416:
410:
406:
402:
398:
390:
388:
386:
382:
378:
374:
370:
366:
362:
358:
357:
352:
334:
317:
314:
310:
306:
302:
298:
294:
290:
286:
264:
261:
257:
250:
247:
244:
241:
238:
235:
232:
226:
223:
220:
217:
214:
207:
206:
205:
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
151:
148:
144:
140:
136:
132:
128:
124:
116:
114:
112:
108:
104:
100:
95:
93:
89:
84:
80:
74:
70:
65:
61:
58:
55:has a unique
54:
50:
46:
42:
38:
34:
30:
19:
5782:
5778:
5760:Petrich 1984
5755:, Chapter 5.
5733:
5729:
5719:: 1119–1122.
5716:
5710:
5690:
5684:
5662:
5656:
5635:(4): 411–9.
5632:
5628:
5611:
5607:
5590:
5586:
5567:
5540:
5536:
5507:
5503:
5484:
5465:
5462:Howie, J. M.
5451:. Retrieved
5447:(Postscript)
5444:the original
5412:
5408:
5391:
5387:
5368:
5364:
5340:
5311:
5301:
5281:
5274:
5259:
5247:
5235:
5223:
5192:
5180:
5160:
5153:
5141:
5129:
5117:
5112:, p. 65
5105:
5093:
5088:, p. 62
5081:
5076:, p. 22
5069:
5053:
5041:
5027:cite journal
5000:
4996:
4990:
4978:
4966:
4954:
4942:
4930:
4919:
4907:
4895:
4883:
4871:
4859:
4846:
4834:
4817:
4804:
4793:
4773:
4766:
4747:Weak inverse
4686:
4672:
4668:
4662:
4658:
4652:
4648:
4644:
4638:
4634:
4630:
4622:
4616:
4594:
4590:intersection
4583:
4577:
4573:
4569:
4561:
4557:
4548:
4547:is called a
4544:
4537:
4527:
4523:
4516:
4510:
4503:
4499:
4495:
4488:
4481:
4477:
4472:
4468:
4464:
4460:
4456:
4452:
4443:
4438:
4433:
4421:
4417:
4415:
4397:
4385:
4381:
4377:
4374:
4364:
4362:
4350:word problem
4347:
4155:
4143:
4140:presentation
4133:
4051:
4001:
3997:
3996:McAlister's
3995:
3990:
3986:
3985:
3976:
3972:
3968:
3964:
3960:
3958:
3952:
3941:
3937:
3838:
3837:
3832:
3828:
3822:
3818:
3815:
3811:
3807:
3754:
3670:
3544:
3543:is called a
3496:
3443:
3439:
3435:
3427:
3423:
3394:
3390:
3386:
3358:
3306:
3299:
3295:
3291:
3287:
3258:
3254:
3250:
3246:
3238:
3234:
3228:
3224:
3195:
3191:
3183:
3179:
3150:
3122:
3118:
3114:
3106:
3102:
3096:
3092:
3063:
3059:
3055:
3051:
3011:
3009:
2979:
2974:
2970:
2941:
2913:
2909:
2827:
2805:
2774:
2770:
2655:
2653:
2648:
2647:
2574:
2570:
2548:
2542:
2516:
2510:
2506:
2499:
2495:
2488:
2484:
2483:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2440:
2390:
2338:
2334:
2330:
2326:
2325:if, for all
2322:
2318:
2310:
2303:
2299:
2297:
2291:
2196:
2192:
2188:
2186:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2125:
2121:
2117:
2113:
2109:
2101:
2099:
1999:
1977:
1899:
1896:
1891:
1881:
1877:
1874:
1869:
1860:
1855:
1851:
1847:
1843:
1839:
1800:
1796:
1789:
1785:
1784:
1771:
1769:
1756:
1755:
1749:
1746:
1742:
1736:
1732:
1728:
1723:
1719:
1715:
1711:
1705:
1701:
1697:
1690:
1686:
1682:
1679:homomorphism
1676:
1666:
1662:
1658:
1650:
1646:
1634:
1633:by ≤), then
1626:
1622:
1614:
1610:
1608:
1603:
1599:
1593:
1537:
1533:
1531:
1525:
1497:
1491:
1487:
1482:
1454:
1449:
1445:
1438:
1433:
1429:
1424:
1420:
1415:
1411:
1406:
1402:
1383:
1316:
1251:The natural
1250:
1245:
1242:
1238:
1234:
1230:
1226:
1222:
1217:
1165:
1161:
1156:
1101:
1100:possesses a
1097:
1095:
1068:
1046:
1024:
1002:
980:
948:
944:
938:
934:
931:
902:
898:
894:
890:
870:
851:
847:
845:
706:
702:
670:
667:
663:
633:
516:
500:
496:
489:
481:
479:
470:
466:
462:
457:
454:
450:
445:
437:
434:
430:
421:
418:
414:
404:
400:
396:
394:
360:
354:
350:
308:
297:pseudogroups
292:
287:denotes the
284:
282:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
153:
149:
127:Soviet Union
120:
110:
106:
105:rather than
102:
98:
96:
82:
78:
72:
68:
63:
59:
56:
52:
48:
40:
36:
32:
26:
5764:Lawson 1998
5693:: 149–158.
5665:: 189–200.
5543:: 351–370.
5510:: 227–244.
5394:: 768–780.
5228:Lawson 1998
5216:Lawson 1998
5146:Lawson 1998
5134:Lawson 1998
5110:Lawson 1998
5098:Lawson 1998
5086:Lawson 1998
5062:Wagner 1952
4971:Lawson 1998
4924:Wagner 1952
4864:Schein 2002
4827:Wagner 1953
4823:Wagner 1952
4810:Schein 1981
4798:Lawson 1998
4727:Semilattice
4712:Pseudogroup
4567:idempotents
4553:idempotents
4532:idempotents
4430:idempotents
3981:M.V. Lawson
3307:The triple
2882:order ideal
2493:semilattice
2465:, for some
2315:idempotents
2308:semilattice
1893:Congruences
1834:, given by
1596:idempotents
1229:. In fact,
928:is inverse.
921:is inverse.
914:is inverse.
912:semilattice
626:are two of
521:semilattice
513:idempotents
509:idempotents
313:associative
291:under
31:theory, an
5805:Categories
5753:Howie 1995
5577:0471875457
5494:9810233167
5475:0198511949
5453:2006-08-28
5438:Gould, V.
5332:References
5252:Howie 1995
5240:Howie 1995
5197:Howie 1995
5185:Howie 1995
5122:Howie 1995
5074:Howie 1995
5058:Howie 1995
5010:1510.04117
4959:Howie 1995
4947:Howie 1995
4935:Howie 1995
4912:Howie 1995
4900:Howie 1995
4888:Howie 1995
4876:Howie 1995
4852:Gołab 1939
4656:such that
4619:categories
4565:, for all
4508:idempotent
4475:such that
4367:-inverse.
4358:Munn trees
4354:W. D. Munn
4136:free group
3946:isomorphic
3421:such that
3357:for every
3177:such that
2195:is called
2136:such that
1897:congruence
1727:, for all
1428:, for all
1394:idempotent
1220:idempotent
1159:idempotent
867:bijections
701:-class of
675:idempotent
662:-class of
636:idempotent
576:idempotent
461:, for all
442:idempotent
391:The basics
5570:. Wiley.
4450:semigroup
4312:−
4304:∪
4295:∈
4268:−
4252:−
4233:−
4217:−
4182:−
3721:∧
3646:∈
3635:−
3621:×
3611:∈
3121:and each
3113:for each
2815:∧
2624:−
2608:−
2597:⟺
2591:∼
2557:∼
2529:∼
2513:-unitary;
2420:∈
2414:⇒
2408:∈
2370:∈
2364:⟹
2358:∈
2260:∈
2254:⟹
2247:ρ
2225:∈
2213:∈
2073:≤
2047:∈
2024:⟺
2017:σ
1986:σ
1953:ρ
1943:⟹
1936:ρ
1921:ρ
1564:⟺
1558:≤
1544:becomes:
1361:−
1353:≤
1345:−
1337:⟹
1331:≤
1296:≤
1287:⟹
1281:≤
1269:≤
1188:⟺
1182:≤
1157:for some
1127:⟺
1121:≤
869:on a set
826:−
807:−
796:⟺
766:−
747:−
739:⟺
471:unipotent
448:in which
305:empty set
262:−
258:α
251:β
248:
242:∩
239:α
236:
224:β
221:α
218:
152:of a set
86:, i.e. a
45:semigroup
5651:(1981).
5464:(1995).
4695:See also
4678:selfdual
4647: :
4633: :
4627:morphism
4528:orthodox
3987:Theorem.
3434:for all
3257:and all
3245:for all
3190:for all
3058:and all
3050:for all
3010:Now let
2485:Theorem.
2333:and all
2130:smallest
1863:faithful
1794:function
1757:Theorem.
1695:function
1683:morphism
1102:natural
578:, where
387:below).
289:preimage
168:, where
158:function
5785:: 187.
5559:1997032
5524:1996831
5230:, 4.1.8
4551:if its
4457:regular
4154:by the
3393:and an
2978:, then
2569:is the
2473:, then
2323:unitary
2128:is the
1621:(i.e.,
1485:, then
1388:, this
864:Partial
677:in the
638:in the
519:form a
133:in the
125:in the
117:Origins
57:inverse
5574:
5557:
5522:
5491:
5472:
5348:
5318:
5289:
5168:
4781:
4050:is an
3965:unique
3936:is an
2982:is in
2944:is in
2880:is an
2801:has a
2712:subset
2549:where
2477:is in
2453:is in
2306:(with
1902:is an
1780:groups
1643:groups
1241:to be
910:Every
876:Every
526:Every
507:) and
453:= 1 =
427:monoid
283:where
202:domain
180:. Let
101:
5775:(PDF)
5724:(PDF)
5555:JSTOR
5520:JSTOR
5417:arXiv
5268:Gould
5005:arXiv
4841:,b,c.
4759:Notes
4621:. An
4586:class
4424:is a
3814:) = (
3755:Then
3294:) = (
3018:that
3016:group
3014:be a
2940:, if
2765:is a
2710:be a
2682:be a
2457:and
2317:) is
2152:with
2062:with
1861:is a
1838:dom (
1799:from
1765:image
1761:image
1718:) = (
1700:from
1655:chain
1645:. If
1639:union
1637:is a
1629:) is
1619:chain
1386:group
1384:In a
897:) = (
878:group
503:is a
475:group
409:group
377:group
373:group
160:from
156:is a
141:of a
43:is a
29:group
5762:and
5572:ISBN
5489:ISBN
5470:ISBN
5346:ISBN
5316:ISBN
5287:ISBN
5166:ISBN
5033:link
4779:ISBN
4666:and
4584:The
4562:xzyx
4558:xyzx
4540:: a
4519:: a
4491:: a
4448:: a
4380:and
4348:The
4102:and
3983:to:
3841:Let
3469:and
3020:acts
2968:and
2487:Let
2199:if
1854:) =
1846:and
1842:) =
1778:for
1745:) =
1693:, a
1689:and
1681:(or
1661:and
1453:and
1418:and
1317:and
1237:and
924:The
917:The
883:The
848:E(S)
634:The
602:and
433:and
417:) =
196:and
184:and
172:and
145:: a
76:and
5787:doi
5695:doi
5667:doi
5637:doi
5616:doi
5595:doi
5545:doi
5541:196
5512:doi
5508:192
5427:doi
5413:126
5396:doi
5392:116
5373:doi
5203:,b.
5015:doi
4669:gfg
4659:fgf
4526:is
4504:eSe
4502:if
4498:is
4478:axa
4471:in
4463:in
4455:is
4432:in
3438:in
3397:in
3389:in
3361:in
3261:in
3253:in
3198:in
3153:in
3125:in
3117:in
3066:in
3054:in
3022:on
2916:in
2884:of
2830:in
2777:in
2714:of
2573:on
2509:is
2469:in
2337:in
2329:in
2313:of
2176:on
2002:by
1868:of
1803:to
1788:If
1735:in
1704:to
1669:).
1641:of
1609:If
1532:On
1225:in
1164:in
945:bab
935:aba
705:is
666:is
515:of
469:(a
465:in
245:dom
215:dom
164:to
143:set
83:yxy
73:xyx
62:in
51:in
27:In
5807::
5783:56
5781:.
5777:.
5734:32
5717:84
5691:65
5689:.
5663:28
5661:.
5655:.
5633:29
5631:.
5612:29
5610:.
5591:29
5589:.
5553:.
5539:.
5535:.
5518:.
5506:.
5425:.
5411:.
5369:22
5367:.
5363:.
5266:,
5208:^
5029:}}
5025:{{
5013:.
5001:96
4999:.
4691:.
4671:=
4661:=
4651:→
4637:→
4576:,
4572:,
4560:=
4480:=
3835::
3821:,
3810:,
3442:,
3426:=
3424:gA
3296:gh
3292:hA
3285:,
3249:,
3239:gB
3237:≤
3235:gA
3227:≤
3222:,
3194:,
3182:=
3180:gA
3105:=
3097:gB
3095:=
3093:gA
3090:,
3062:,
2973:≤
2912:,
2773:,
2541:=
2481:.
2461:≤
2341:,
1872:.
1856:xa
1852:aφ
1844:Sa
1840:aφ
1782::
1743:sθ
1720:st
1716:tθ
1714:)(
1712:sθ
1677:A
1529:.
1490:=
1448:≤
1436:.
1425:xβ
1423:=
1421:xα
1405:≤
1248:.
1235:aa
1085:e
1063:a
1041:c
1019:a
997:a
975:e
947:=
942:,
937:=
901:,
893:,
854:.
707:ss
523:).
484::
477:.
451:xx
431:xx
429:,
415:ab
233:im
192:;
94:.
81:=
71:=
39:)
5793:.
5789::
5766:.
5701:.
5697::
5675:.
5669::
5643:.
5639::
5622:.
5618::
5601:.
5597::
5580:.
5561:.
5547::
5526:.
5514::
5497:.
5478:.
5456:.
5433:.
5429::
5419::
5402:.
5398::
5381:.
5375::
5354:.
5324:.
5295:.
5174:.
5035:)
5017::
5007::
4854:.
4829:.
4812:.
4787:.
4673:g
4663:f
4653:X
4649:Y
4645:g
4639:Y
4635:X
4631:f
4580:.
4578:z
4574:y
4570:x
4545:S
4524:S
4513:.
4511:e
4496:S
4485:.
4482:a
4473:S
4469:x
4465:S
4461:a
4453:S
4434:S
4422:S
4418:S
4386:β
4382:β
4378:α
4365:F
4333:.
4330:}
4325:+
4321:)
4315:1
4308:X
4301:X
4298:(
4292:y
4289:,
4286:x
4281:|
4276:)
4271:1
4264:x
4260:x
4255:1
4248:y
4244:y
4241:,
4236:1
4229:y
4225:y
4220:1
4213:x
4209:x
4206:(
4202:,
4199:)
4196:x
4193:,
4190:x
4185:1
4178:x
4174:x
4171:(
4168:{
4144:X
4112:X
4088:X
4064:Y
4052:F
4038:)
4033:Y
4028:,
4023:X
4018:,
4015:G
4012:(
4002:F
3998:P
3991:F
3977:E
3973:F
3969:σ
3961:F
3953:F
3942:E
3938:E
3924:)
3919:Y
3914:,
3909:X
3904:,
3901:G
3898:(
3895:P
3875:)
3870:Y
3865:,
3860:X
3855:,
3852:G
3849:(
3829:E
3825:)
3823:g
3819:A
3816:g
3812:g
3808:A
3806:(
3792:)
3787:Y
3782:,
3777:X
3772:,
3769:G
3766:(
3763:P
3751:.
3739:)
3736:h
3733:g
3730:,
3727:B
3724:g
3718:A
3715:(
3712:=
3709:)
3706:h
3703:,
3700:B
3697:(
3694:)
3691:g
3688:,
3685:A
3682:(
3656:}
3651:Y
3643:A
3638:1
3631:g
3627::
3624:G
3616:Y
3608:)
3605:g
3602:,
3599:A
3596:(
3593:{
3590:=
3587:)
3582:Y
3577:,
3572:X
3567:,
3564:G
3561:(
3558:P
3531:)
3526:Y
3521:,
3516:X
3511:,
3508:G
3505:(
3479:Y
3455:Y
3444:g
3440:G
3436:g
3431:;
3428:X
3407:Y
3395:A
3391:G
3387:g
3371:X
3359:X
3341:)
3336:Y
3331:,
3326:X
3321:,
3318:G
3315:(
3303:.
3300:A
3298:)
3290:(
3288:g
3271:X
3259:A
3255:G
3251:h
3247:g
3242:;
3229:B
3225:A
3208:X
3196:B
3192:A
3187:;
3184:B
3163:X
3151:A
3135:X
3123:B
3119:G
3115:g
3110:;
3107:B
3103:A
3076:X
3064:B
3060:A
3056:G
3052:g
3032:X
3012:G
3006:.
2992:Y
2980:B
2975:A
2971:B
2954:Y
2942:A
2926:X
2914:B
2910:A
2894:X
2866:Y
2840:Y
2828:B
2806:A
2787:Y
2775:B
2771:A
2751:Y
2724:X
2696:Y
2668:X
2656:E
2632:b
2627:1
2620:a
2616:,
2611:1
2604:b
2600:a
2594:b
2588:a
2575:S
2545:,
2543:σ
2517:σ
2511:E
2507:S
2500:σ
2496:E
2489:S
2479:E
2475:s
2471:S
2467:s
2463:s
2459:e
2455:E
2451:e
2447:S
2443:E
2426:.
2423:E
2417:s
2411:E
2405:e
2402:s
2376:.
2373:E
2367:s
2361:E
2355:s
2352:e
2339:S
2335:s
2331:E
2327:e
2321:-
2319:E
2311:E
2304:S
2300:E
2292:E
2275:.
2272:)
2269:S
2266:(
2263:E
2257:a
2251:e
2243:a
2240:,
2237:)
2234:S
2231:(
2228:E
2222:e
2219:,
2216:S
2210:a
2193:S
2189:ρ
2182:E
2178:S
2170:σ
2166:τ
2162:σ
2158:τ
2156:/
2154:S
2150:S
2146:τ
2142:σ
2140:/
2138:S
2134:S
2126:σ
2122:S
2118:S
2114:σ
2112:/
2110:S
2102:σ
2085:.
2082:b
2079:,
2076:a
2070:c
2050:S
2044:c
2021:b
2013:a
2000:S
1963:.
1960:d
1957:b
1949:c
1946:a
1940:d
1932:c
1928:,
1925:b
1917:a
1900:ρ
1882:S
1878:S
1870:S
1850:(
1848:x
1820:S
1814:I
1801:S
1797:φ
1790:S
1750:θ
1747:s
1741:(
1737:S
1733:t
1731:,
1729:s
1724:θ
1722:)
1710:(
1706:T
1702:S
1698:θ
1691:T
1687:S
1667:S
1665:(
1663:E
1659:S
1651:S
1649:(
1647:E
1635:S
1627:S
1625:(
1623:E
1615:S
1613:(
1611:E
1604:S
1602:(
1600:E
1579:,
1576:f
1573:e
1570:=
1567:e
1561:f
1555:e
1538:S
1536:(
1534:E
1526:t
1510:R
1498:s
1492:t
1488:s
1483:t
1467:L
1455:s
1450:t
1446:s
1434:α
1430:x
1416:β
1412:α
1407:β
1403:α
1369:.
1364:1
1357:b
1348:1
1341:a
1334:b
1328:a
1302:d
1299:b
1293:c
1290:a
1284:d
1278:c
1275:,
1272:b
1266:a
1246:a
1243:a
1239:f
1231:e
1227:S
1223:f
1203:,
1200:f
1197:b
1194:=
1191:a
1185:b
1179:a
1166:S
1162:e
1142:,
1139:b
1136:e
1133:=
1130:a
1124:b
1118:a
1098:S
1082:d
1079:a
1076:a
1073:a
1069:e
1060:a
1057:e
1054:d
1051:a
1047:d
1038:b
1035:a
1032:a
1029:a
1025:c
1016:a
1013:c
1010:b
1007:a
1003:b
994:a
991:a
988:a
985:a
981:a
972:d
969:c
966:b
963:a
949:b
939:a
907:.
905:)
903:a
899:b
895:b
891:a
889:(
871:X
852:S
829:1
822:b
818:b
815:=
810:1
803:a
799:a
793:b
787:R
781:a
777:,
774:b
769:1
762:b
758:=
755:a
750:1
743:a
736:b
730:L
724:a
703:s
687:R
671:s
668:s
664:s
648:L
630:.
612:R
588:L
560:R
536:L
517:S
501:S
497:S
490:S
482:S
467:S
463:x
458:x
455:x
446:S
438:x
435:x
422:a
419:b
413:(
405:x
401:S
397:x
361:X
351:X
335:X
329:I
293:α
285:α
265:1
254:]
230:[
227:=
198:β
194:α
190:X
186:β
182:α
178:X
174:B
170:A
166:B
162:A
154:X
150:α
111:x
109:(
107:f
103:f
99:x
79:y
69:x
64:S
60:y
53:S
49:x
41:S
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.