1864:
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
830:
3655:
4332:
4364:
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
1962:
2274:
4389:, 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
267:
708:
2631:
1368:
1301:
3923:
3791:
4037:
3874:
3530:
3340:
2023:
1578:
1202:
1141:
2375:
3738:
2425:
4377:; 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
3542:
4345:
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
1821:
1512:
1469:
336:
4113:
4089:
4065:
3480:
3456:
3408:
3372:
3272:
3209:
3164:
3136:
3077:
3033:
2993:
2955:
2927:
2895:
2867:
2841:
2788:
2752:
2725:
2697:
2669:
688:
649:
613:
589:
561:
537:
2084:
4152:
2814:
1985:
2049:
2556:
2528:
2640:
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.
300:
to take account of this. With the addition of this empty transformation, the composition of partial transformations of a set becomes an everywhere-defined
5700:
4425:
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.
1901:
79:
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
5338:
5308:
5279:
5158:
4771:
2194:
2109:, the minimal element (for the partial order defined by inclusion of sets) need not be the smallest element. In the specific case in which
4797:
Since his father was German, Wagner preferred the German transliteration of his name (with a "W", rather than a "V") from
Cyrillic – see
199:
825:{\displaystyle a\,{\mathcal {L}}\,b\Longleftrightarrow a^{-1}a=b^{-1}b,\quad a\,{\mathcal {R}}\,b\Longleftrightarrow aa^{-1}=bb^{-1}}
2572:
288:. It was Wagner, however, who was the first to observe that the composition of partial transformations is a special case of the
1312:
1250:
5564:
5481:
5462:
5021:
4136:
5428:
5710:
5799:
4730:
4677:
3879:
3747:
1742:, however, there is no need to do so, since this property follows from the above definition, via the following theorem:
3996:
3833:
3489:
3299:
1997:
1539:
1163:
1102:
344:
111:
4341:
for free inverse semigroups is much more intricate than that of free groups. A celebrated result in this area due to
2336:
3008:
1752:
of an inverse semigroup is an inverse semigroup; the inverse of an element is always mapped to the inverse of the
5804:
4578:
4349:, which essentially consists of overlapping common portions of the trees. (see Lawson 1998 for further details)
3666:
3650:{\displaystyle P(G,{\mathcal {X}},{\mathcal {Y}})=\{(A,g)\in {\mathcal {Y}}\times G:g^{-1}A\in {\mathcal {Y}}\}}
2386:
4128:
1674:) of inverse semigroups is defined in exactly the same way as for any other semigroup: for inverse semigroups
289:
86:(The convention followed in this article will be that of writing a function on the right of its argument, e.g.
4338:
4607:
1782:
1683:
146:
5015:
4378:
2672:
1869:
is isomorphic to a subsemigroup of the symmetric inverse semigroup closed under inverses if and only if
1795:
1491:
1448:
310:
4720:
2791:
1892:
1643:
1607:
1429:
699:
616:
190:
4984:
Gonçalves, D; Sobottka, M; Starling, C (2017). "Inverse semigroup shifts over countable alphabets".
4390:
4094:
4070:
4046:
3461:
3437:
3389:
3353:
3253:
3190:
3145:
3117:
3058:
3014:
2974:
2936:
2908:
2876:
2848:
2822:
2769:
2733:
2706:
2678:
2650:
1728:. The definition of a morphism of inverse semigroups could be augmented by including the condition
669:
630:
594:
570:
542:
518:
4394:
3004:
2094:
1768:
1764:
1753:
1749:
1631:
1374:
921:
Multiplication table example. It is associative and every element has its own inverse according to
866:
463:
397:
365:
361:
17:
5672:(2002). "Book Review: "Inverse Semigroups: The Theory of Partial Symmetries" by Mark V. Lawson".
5543:
5508:
5405:
5271:
Homological
Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups
4993:
4705:
4690:
4574:
4327:{\displaystyle \{(xx^{-1}x,x),\;(xx^{-1}yy^{-1},yy^{-1}xx^{-1})\;|\;x,y\in (X\cup X^{-1})^{+}\}.}
3969:
2287:
One class of inverse semigroups that has been studied extensively over the years is the class of
1627:
873:
102:), and composing functions from left to right—a convention often observed in semigroup theory.)
80:
4342:
2054:
4761:
4346:
5560:
5477:
5458:
5334:
5304:
5296:
5275:
5154:
5148:
4767:
4740:
4710:
4670:
4530:
4509:
4481:
4433:
4414:
4385:. This close connection between inverse semigroups and inductive groupoids is embodied in the
493:
353:
292:. He recognised also that the domain of composition of two partial transformations may be the
131:
127:
76:
5269:
2799:
1970:
5775:
5683:
5655:
5625:
5604:
5583:
5533:
5500:
5415:
5384:
5361:
5003:
2028:
907:
852:
304:
135:
2541:
2513:
5674:
5646:
4725:
4382:
1619:
369:
357:
1884:
are defined on inverse semigroups in exactly the same way as for any other semigroup: a
914:
123:
119:
352:, with inverse the functional inverse defined from image to domain (equivalently, the
5793:
5695:
5450:
4695:
1530:
1386:
1378:
1241:
1092:
396:. Inverses in an inverse semigroup have many of the same properties as inverses in a
5669:
5637:
4735:
1854:
1667:
115:
5419:
5760:
5432:
5328:
4581:
of the class of locally inverse semigroups and the class of orthodox semigroups.
5717:
Wagner, V. V. (1953). "The theory of generalised heaps and generalised groups".
5396:
Exel, R. (1998). "Partial actions of groups and actions of inverse semigroups".
4715:
4700:
2870:
2755:
2481:
2296:
1881:
900:
509:
372:, every inverse semigroup can be embedded in a symmetric inverse semigroup (see
301:
285:
5629:
5608:
5587:
5375:
Gołab, St. (1939). "Über den
Begriff der "Pseudogruppe von Transformationen"".
5779:
5366:
5349:
5007:
4555:
4541:
4520:
4496:
4418:
4124:
3934:
2303:
1584:
1382:
1208:
1147:
663:
624:
564:
501:
497:
430:
1646:
it is possible to obtain an analogous result under additional hypotheses on
4438:
4140:
1851:
855:
373:
293:
33:
1957:{\displaystyle a\,\rho \,b,\quad c\,\rho \,d\Longrightarrow ac\,\rho \,bd.}
469:
There are a number of equivalent characterisations of an inverse semigroup
5687:
5333:. Mathematical Surveys of the American Mathematical Society. Vol. 7.
5595:
Preston, G. B. (1954b). "Inverse semi-groups with minimal right ideals".
5491:
McAlister, D. B. (1974a). "Groups, semilattices and inverse semigroups".
5410:
4666:
4615:
2169:. The minimum group congruence can be used to give a characterisation of
1096:
relation ≤ (sometimes denoted by ω), which is defined by the following:
277:
5660:
5641:
5547:
5512:
5388:
4373:
are composed if, and only if, the image of α is equal to the domain of
2269:{\displaystyle a\in S,e\in E(S),a\,\rho \,e\Longrightarrow a\in E(S).}
356:). This is the "archetypal" inverse semigroup, in the same way that a
5784:
4584:
Amongst the non-regular generalisations of an inverse semigroup are:
2700:
415:
284:. Partial transformations had already been studied in the context of
126:
in 1954. Both authors arrived at inverse semigroups via the study of
5538:
5521:
5504:
1759:
One of the earliest results proved about inverse semigroups was the
4998:
839:
will denote the semilattice of idempotents of an inverse semigroup
5616:
Preston, G. B. (1954c). "Representations of inverse semi-groups".
4448:
if every element has at least one inverse; equivalently, for each
3968:-unitary monoid. McAlister's covering theorem has been refined by
3956:
maximal element above it in the natural partial order, i.e. every
4428:
Examples of regular generalisations of an inverse semigroup are:
1428:
The natural partial order on an inverse semigroup interacts with
262:{\displaystyle \operatorname {dom} \alpha \beta =\alpha ^{-1}\,}
2647:-unitary inverse semigroups is the following construction. Let
2105:
is a group. In the set of all group congruences on a semigroup
1244:
is compatible with both multiplication and inversion, that is,
374:§ Homomorphisms and representations of inverse semigroups
429:
are not necessarily equal to the identity, but they are both
4100:
4076:
4052:
4021:
4011:
3907:
3897:
3858:
3848:
3775:
3765:
3639:
3604:
3570:
3560:
3514:
3504:
3467:
3443:
3395:
3359:
3324:
3314:
3259:
3196:
3151:
3123:
3064:
3020:
2980:
2942:
2914:
2882:
2854:
2828:
2775:
2739:
2712:
2684:
2656:
1802:
1587:
form a semilattice under the product operation, products on
1498:
1455:
775:
718:
675:
636:
600:
576:
548:
524:
317:
5736:
For a brief introduction to inverse semigroups, see either
1381:
simply reduces to equality, since the identity is the only
4139:, where involution is the taking of the inverse, and then
2626:{\displaystyle a\sim b\Longleftrightarrow ab^{-1},a^{-1}b}
1363:{\displaystyle a\leq b\Longrightarrow a^{-1}\leq b^{-1}.}
3793:
is an inverse semigroup under this multiplication, with
1895:
that is compatible with semigroup multiplication, i.e.,
1296:{\displaystyle a\leq b,c\leq d\Longrightarrow ac\leq bd}
1662:
Homomorphisms and representations of inverse semigroups
4676:
Inverse categories have found various applications in
3536:. A McAlister triple is used to define the following:
4155:
4097:
4073:
4049:
3999:
3882:
3836:
3750:
3669:
3545:
3492:
3464:
3440:
3392:
3356:
3302:
3256:
3193:
3148:
3120:
3061:
3017:
2977:
2939:
2911:
2879:
2851:
2825:
2802:
2772:
2736:
2709:
2681:
2653:
2575:
2544:
2516:
2389:
2339:
2197:
2057:
2031:
2000:
1973:
1904:
1798:
1542:
1494:
1451:
1315:
1253:
1166:
1105:
711:
672:
633:
597:
573:
545:
521:
313:
202:
5474:
Inverse
Semigroups: The Theory of Partial Symmetries
189:
can be composed (from left to right) on the largest
110:
Inverse semigroups were introduced independently by
5642:"Obituary: Viktor Vladimirovich Vagner (1908–1981)"
5150:
4606:This notion of inverse also readily generalizes to
698:. There is therefore a simple characterisation of
5761:"On inverse categories and transfer in cohomology"
4397:and whose poset of objects is a meet-semilattice.
4326:
4107:
4083:
4059:
4031:
3918:{\displaystyle P(G,{\mathcal {X}},{\mathcal {Y}})}
3917:
3868:
3786:{\displaystyle P(G,{\mathcal {X}},{\mathcal {Y}})}
3785:
3732:
3649:
3524:
3474:
3450:
3402:
3366:
3342:is also assumed to have the following properties:
3334:
3266:
3203:
3158:
3130:
3071:
3027:
2987:
2949:
2921:
2889:
2861:
2835:
2808:
2782:
2746:
2719:
2691:
2663:
2625:
2550:
2522:
2419:
2369:
2291:-unitary inverse semigroups: an inverse semigroup
2268:
2078:
2043:
2017:
1979:
1956:
1815:
1572:
1506:
1463:
1362:
1295:
1196:
1135:
824:
682:
643:
607:
583:
555:
531:
330:
261:
5768:Proceedings of the Edinburgh Mathematical Society
5747:More comprehensive introductions can be found in
5526:Transactions of the American Mathematical Society
5493:Transactions of the American Mathematical Society
5354:Proceedings of the Edinburgh Mathematical Society
4597:Weakly (left, right, two-sided) ample semigroups.
4594:(Left, right, two-sided) semiadequate semigroups.
4032:{\displaystyle (G,{\mathcal {X}},{\mathcal {Y}})}
3869:{\displaystyle (G,{\mathcal {X}},{\mathcal {Y}})}
3525:{\displaystyle (G,{\mathcal {X}},{\mathcal {Y}})}
3335:{\displaystyle (G,{\mathcal {X}},{\mathcal {Y}})}
2018:{\displaystyle a\,\sigma \,b\Longleftrightarrow }
5522:"Groups, semilattices and inverse semigroups II"
5398:Proceedings of the American Mathematical Society
3993:-inverse semigroups as well. A McAlister triple
1573:{\displaystyle e\leq f\Longleftrightarrow e=ef,}
1197:{\displaystyle a\leq b\Longleftrightarrow a=bf,}
1136:{\displaystyle a\leq b\Longleftrightarrow a=eb,}
338:of all partial one-one transformations of a set
5737:
5574:Preston, G. B. (1954a). "Inverse semi-groups".
5200:
5198:
5034:
4971:
4814:, then a much more comprehensive exposition in
3929:-unitary inverse semigroup. Conversely, every
2370:{\displaystyle es\in E\Longrightarrow s\in E.}
4588:(Left, right, two-sided) adequate semigroups.
2487:of idempotents, and minimum group congruence
1595:) give least upper bounds with respect to ≤.
941:. It has no identity and is not commutative.
193:upon which it "makes sense" to compose them:
8:
5295:Hines, Peter; Braunstein, Samuel L. (2010).
4318:
4156:
3644:
3581:
862:form an inverse semigroup under composition.
5701:Proceedings of the USSR Academy of Sciences
5303:. Cambridge University Press. p. 369.
5618:Journal of the London Mathematical Society
5597:Journal of the London Mathematical Society
5576:Journal of the London Mathematical Society
5301:Semantic Techniques in Quantum Computation
5189:
4591:(Left, right, two-sided) ample semigroups.
4273:
4267:
4193:
3816:. One of the main results in the study of
3733:{\displaystyle (A,g)(B,h)=(A\wedge gB,gh)}
2420:{\displaystyle se\in E\Rightarrow s\in E.}
1389:reduces to restriction of mappings, i.e.,
5659:
5537:
5409:
5365:
5299:. In Gay and, Simon; Mackie, Ian (eds.).
4997:
4766:(2nd ed.). CRC Press. p. 1528.
4577:of generalised inverse semigroups is the
4312:
4299:
4268:
4255:
4239:
4220:
4204:
4169:
4154:
4099:
4098:
4096:
4075:
4074:
4072:
4051:
4050:
4048:
4020:
4019:
4010:
4009:
3998:
3906:
3905:
3896:
3895:
3881:
3857:
3856:
3847:
3846:
3835:
3774:
3773:
3764:
3763:
3749:
3668:
3638:
3637:
3622:
3603:
3602:
3569:
3568:
3559:
3558:
3544:
3513:
3512:
3503:
3502:
3491:
3466:
3465:
3463:
3442:
3441:
3439:
3394:
3393:
3391:
3358:
3357:
3355:
3323:
3322:
3313:
3312:
3301:
3258:
3257:
3255:
3195:
3194:
3192:
3150:
3149:
3147:
3122:
3121:
3119:
3063:
3062:
3060:
3019:
3018:
3016:
2979:
2978:
2976:
2941:
2940:
2938:
2913:
2912:
2910:
2881:
2880:
2878:
2853:
2852:
2850:
2827:
2826:
2824:
2801:
2774:
2773:
2771:
2738:
2737:
2735:
2711:
2710:
2708:
2683:
2682:
2680:
2655:
2654:
2652:
2611:
2595:
2574:
2543:
2515:
2388:
2338:
2238:
2234:
2196:
2173:-unitary inverse semigroups (see below).
2056:
2030:
2008:
2004:
1999:
1972:
1944:
1940:
1927:
1923:
1912:
1908:
1903:
1807:
1801:
1800:
1797:
1541:
1503:
1497:
1496:
1495:
1493:
1460:
1454:
1453:
1452:
1450:
1348:
1332:
1314:
1252:
1165:
1104:
813:
794:
780:
774:
773:
772:
753:
734:
723:
717:
716:
715:
710:
674:
673:
671:
635:
634:
632:
599:
598:
596:
575:
574:
572:
547:
546:
544:
523:
522:
520:
481:has a unique inverse, in the above sense.
322:
316:
315:
312:
307:. Under this composition, the collection
258:
249:
201:
5327:Clifford, A. H.; Preston, G. B. (1967).
5252:
4827:
1385:. In a symmetric inverse semigroup, the
943:
5748:
4763:CRC Concise Encyclopedia of Mathematics
4752:
4131:of the free inverse semigroup on a set
3989:-theorem has been used to characterize
1967:Of particular interest is the relation
342:forms an inverse semigroup, called the
5752:
5216:
5204:
5134:
5122:
5098:
5086:
5074:
5050:
5013:
4959:
4912:
4852:
4815:
4811:
4798:
4786:
4127:is possible for inverse semigroups. A
2093:is a congruence and, in fact, it is a
5741:
5297:"The Structure of Partial Isometries"
5240:
5228:
5185:
5173:
5110:
5062:
5046:
4947:
4935:
4923:
4900:
4888:
4876:
4864:
4840:
4409:can be defined by the conditions (1)
4405:As noted above, an inverse semigroup
4401:Generalisations of inverse semigroups
2491:. Then the following are equivalent:
7:
4614:is simply a category in which every
4387:Ehresmann–Schein–Nambooripad Theorem
3960:-class has a maximal element. Every
2097:, meaning that the factor semigroup
177:be partial transformations of a set
4135:may be obtained by considering the
3948:An inverse semigroup is said to be
2430:One further characterisation of an
5429:"(Weakly) left E-ample semigroups"
5330:The Algebraic Theory of Semigroups
5053:and, independently, Preston 1954c.
4495:is an inverse semigroup, for each
4043:-inverse semigroup if and only if
2758:, that is, every pair of elements
1987:, defined on an inverse semigroup
1816:{\displaystyle {\mathcal {I}}_{S}}
1781:is an inverse semigroup, then the
1507:{\displaystyle \,{\mathcal {R}}\,}
1464:{\displaystyle \,{\mathcal {L}}\,}
462:inverse monoid), is, of course, a
331:{\displaystyle {\mathcal {I}}_{X}}
14:
5256:
1877:Congruences on inverse semigroups
1207:for some (in general, different)
5455:Fundamentals of Semigroup Theory
5274:. World Scientific. p. 55.
4360:Connections with category theory
3952:-inverse if every element has a
3978:Every inverse semigroup has an
3820:-unitary inverse semigroups is
1919:
1399:if, and only if, the domain of
768:
290:composition of binary relations
5721:. Novaya Seriya (in Russian).
5698:(1952). "Generalised groups".
4810:First a short announcement in
4527:Generalised inverse semigroups
4352:Any free inverse semigroup is
4309:
4286:
4269:
4264:
4194:
4187:
4159:
4137:free semigroup with involution
4108:{\displaystyle {\mathcal {X}}}
4084:{\displaystyle {\mathcal {X}}}
4060:{\displaystyle {\mathcal {Y}}}
4026:
4000:
3933:-unitary inverse semigroup is
3912:
3886:
3863:
3837:
3780:
3754:
3727:
3703:
3697:
3685:
3682:
3670:
3596:
3584:
3575:
3549:
3519:
3493:
3475:{\displaystyle {\mathcal {Y}}}
3451:{\displaystyle {\mathcal {Y}}}
3403:{\displaystyle {\mathcal {Y}}}
3367:{\displaystyle {\mathcal {X}}}
3329:
3303:
3267:{\displaystyle {\mathcal {X}}}
3204:{\displaystyle {\mathcal {X}}}
3159:{\displaystyle {\mathcal {X}}}
3131:{\displaystyle {\mathcal {X}}}
3072:{\displaystyle {\mathcal {X}}}
3028:{\displaystyle {\mathcal {X}}}
2988:{\displaystyle {\mathcal {Y}}}
2950:{\displaystyle {\mathcal {Y}}}
2922:{\displaystyle {\mathcal {X}}}
2890:{\displaystyle {\mathcal {X}}}
2862:{\displaystyle {\mathcal {Y}}}
2836:{\displaystyle {\mathcal {Y}}}
2783:{\displaystyle {\mathcal {Y}}}
2747:{\displaystyle {\mathcal {Y}}}
2720:{\displaystyle {\mathcal {X}}}
2692:{\displaystyle {\mathcal {Y}}}
2664:{\displaystyle {\mathcal {X}}}
2585:
2402:
2352:
2260:
2254:
2242:
2225:
2219:
2012:
1931:
1552:
1403:is contained in the domain of
1325:
1275:
1176:
1115:
847:Examples of inverse semigroups
784:
727:
683:{\displaystyle {\mathcal {R}}}
644:{\displaystyle {\mathcal {L}}}
608:{\displaystyle {\mathcal {R}}}
584:{\displaystyle {\mathcal {L}}}
563:-class contains precisely one
556:{\displaystyle {\mathcal {R}}}
532:{\displaystyle {\mathcal {L}}}
242:
218:
1:
5420:10.1090/S0002-9939-98-04575-4
5231:, Section 2.4 & Chapter 6
5188:, Theorem 5.9.2. Originally,
4731:Special classes of semigroups
4538:generalised inverse semigroup
3660:together with multiplication
2638:McAlister's Covering Theorem.
2480:be an inverse semigroup with
364:. For example, just as every
5049:, Theorem 5.1.7 Originally,
4678:theoretical computer science
4123:A construction similar to a
3876:be a McAlister triple. Then
5738:Clifford & Preston 1967
5457:. Oxford: Clarendon Press.
5035:Clifford & Preston 1967
4972:Clifford & Preston 1967
4760:Weisstein, Eric W. (2002).
4669:. The category of sets and
4393:with respect to its (dual)
3482:have nonempty intersection.
2675:, with ordering ≤, and let
2434:-unitary inverse semigroup
2283:-unitary inverse semigroups
2137:is any other congruence on
345:symmetric inverse semigroup
112:Viktor Vladimirovich Wagner
5821:
5520:McAlister, D. B. (1974b).
5153:. CRC Press. p. 248.
5020:: CS1 maint: postscript (
4631:has a generalized inverse
4544:form a normal band, i.e.,
4478:Locally inverse semigroups
2079:{\displaystyle c\leq a,b.}
1763:, which is an analogue of
488:has at least one inverse (
384:The inverse of an element
5780:10.1017/S0013091512000211
5367:10.1017/S0013091500016230
5008:10.1007/s00233-017-9858-5
4665:. An inverse category is
3964:-inverse semigroup is an
3035:(on the left), such that
2727:with the properties that
1873:is an inverse semigroup.
1081:The natural partial order
835:Unless stated otherwise,
702:in an inverse semigroup:
5759:Linckelmann, M. (2012).
5630:10.1112/jlms/s1-29.4.411
5609:10.1112/jlms/s1-29.4.404
5588:10.1112/jlms/s1-29.4.396
5348:Fountain, J. B. (1979).
4067:is a principal ideal of
2643:Central to the study of
2180:on an inverse semigroup
2163:minimum group congruence
2133:is a group, that is, if
2113:is an inverse semigroup
1606:) is finite and forms a
869:is an inverse semigroup.
388:of an inverse semigroup
24:(occasionally called an
5719:Matematicheskii Sbornik
5268:Grandis, Marco (2012).
5147:Grillet, P. A. (1995).
4119:Free inverse semigroups
2809:{\displaystyle \wedge }
1980:{\displaystyle \sigma }
1775:Wagner–Preston Theorem.
36:in which every element
5472:Lawson, M. V. (1998).
4903:, Proposition 5.1.2(1)
4879:, Proposition 5.1.2(1)
4673:is the prime example.
4328:
4109:
4085:
4061:
4033:
3919:
3870:
3828:McAlister's P-Theorem.
3787:
3734:
3651:
3526:
3476:
3452:
3404:
3368:
3336:
3268:
3205:
3160:
3132:
3073:
3029:
2989:
2951:
2923:
2891:
2863:
2837:
2810:
2784:
2748:
2721:
2693:
2665:
2627:
2560:compatibility relation
2552:
2524:
2421:
2371:
2270:
2080:
2045:
2044:{\displaystyle c\in S}
2019:
1981:
1958:
1817:
1761:Wagner–Preston Theorem
1574:
1508:
1465:
1364:
1297:
1198:
1137:
826:
684:
645:
609:
585:
557:
533:
500:commute (that is, the
332:
296:, so he introduced an
263:
136:partial transformation
5688:10.1007/s002330010132
5377:Mathematische Annalen
5350:"Adequate semigroups"
4523:forms a subsemigroup.
4329:
4110:
4086:
4062:
4034:
3937:to one of this type.
3920:
3871:
3822:McAlister's P-Theorem
3788:
3735:
3652:
3527:
3477:
3453:
3405:
3369:
3337:
3269:
3206:
3161:
3133:
3074:
3030:
2990:
2952:
2924:
2892:
2864:
2838:
2811:
2785:
2749:
2722:
2694:
2673:partially ordered set
2666:
2628:
2553:
2551:{\displaystyle \sim }
2525:
2523:{\displaystyle \sim }
2438:is the following: if
2422:
2372:
2271:
2089:It can be shown that
2081:
2046:
2020:
1982:
1959:
1818:
1575:
1509:
1466:
1365:
1298:
1199:
1138:
1085:An inverse semigroup
827:
685:
646:
610:
586:
558:
534:
368:can be embedded in a
333:
264:
5800:Algebraic structures
5785:Open access preprint
5555:Petrich, M. (1984).
5476:. World Scientific.
4153:
4095:
4071:
4047:
3997:
3880:
3834:
3748:
3667:
3543:
3490:
3462:
3438:
3390:
3354:
3300:
3254:
3191:
3146:
3118:
3059:
3015:
2975:
2937:
2909:
2877:
2849:
2843:(with respect to ≤);
2823:
2800:
2792:greatest lower bound
2770:
2734:
2707:
2679:
2651:
2573:
2542:
2514:
2387:
2337:
2195:
2055:
2029:
1998:
1971:
1902:
1893:equivalence relation
1796:
1540:
1492:
1449:
1313:
1251:
1222:can be taken to be
1164:
1103:
709:
670:
631:
595:
571:
543:
519:
433:. An inverse monoid
311:
298:empty transformation
200:
5711:English translation
5125:, Proposition 2.4.3
4962:, Proposition 3.2.3
4926:, Proposition 5.2.1
4506:Orthodox semigroups
4395:Alexandrov topology
4141:taking the quotient
3944:-inverse semigroups
2508:is idempotent pure;
946:
392:is usually written
26:inversion semigroup
5661:10.1007/BF02676643
5557:Inverse semigroups
5389:10.1007/BF01597390
4839:See, for example,
4706:Partial symmetries
4691:Orthodox semigroup
4671:partial bijections
4434:Regular semigroups
4381:, in the sense of
4379:inductive groupoid
4324:
4115:is a semilattice.
4105:
4081:
4057:
4029:
3915:
3866:
3783:
3730:
3647:
3522:
3472:
3448:
3400:
3364:
3332:
3264:
3201:
3156:
3138:, there exists an
3128:
3069:
3025:
2985:
2947:
2919:
2887:
2859:
2833:
2806:
2780:
2744:
2717:
2689:
2661:
2623:
2548:
2520:
2417:
2367:
2266:
2076:
2041:
2015:
1977:
1954:
1813:
1570:
1504:
1461:
1360:
1293:
1194:
1133:
945:Inverse semigroup
944:
874:bicyclic semigroup
822:
680:
641:
605:
581:
553:
529:
360:is the archetypal
328:
259:
128:partial bijections
81:partial symmetries
55:in the sense that
5340:978-0-8218-0272-4
5310:978-0-521-51374-6
5281:978-981-4407-06-9
5160:978-0-8247-9662-4
4773:978-1-4200-3522-3
4741:Nambooripad order
4721:Green's relations
4711:Regular semigroup
4531:regular semigroup
4519:if its subset of
4510:regular semigroup
4482:regular semigroup
4415:regular semigroup
4145:Vagner congruence
3374:, there exists a
3221:if, and only if,
3089:if, and only if,
2756:lower semilattice
2157:. The congruence
1756:of that element.
1697:is a morphism if
1642:) is an infinite
1430:Green's relations
1421:in the domain of
1078:
1077:
876:is inverse, with
700:Green's relations
617:Green's relations
539:-class and every
494:regular semigroup
484:Every element of
477:Every element of
354:converse relation
77:regular semigroup
22:inverse semigroup
5812:
5805:Semigroup theory
5783:
5765:
5726:
5709:
5691:
5665:
5663:
5633:
5612:
5591:
5570:
5551:
5541:
5516:
5487:
5468:
5446:
5444:
5443:
5437:
5431:. Archived from
5423:
5413:
5411:funct-an/9511003
5392:
5371:
5369:
5344:
5315:
5314:
5292:
5286:
5285:
5265:
5259:
5250:
5244:
5238:
5232:
5226:
5220:
5214:
5208:
5202:
5193:
5183:
5177:
5176:, pp. 193–4
5171:
5165:
5164:
5144:
5138:
5132:
5126:
5120:
5114:
5108:
5102:
5096:
5090:
5084:
5078:
5072:
5066:
5060:
5054:
5044:
5038:
5032:
5026:
5025:
5019:
5011:
5001:
4981:
4975:
4969:
4963:
4957:
4951:
4945:
4939:
4938:, pp. 152–3
4933:
4927:
4921:
4915:
4910:
4904:
4898:
4892:
4886:
4880:
4874:
4868:
4862:
4856:
4850:
4844:
4837:
4831:
4825:
4819:
4808:
4802:
4795:
4789:
4784:
4778:
4777:
4757:
4664:
4654:
4644:
4630:
4612:inverse category
4602:Inverse category
4553:
4473:
4333:
4331:
4330:
4325:
4317:
4316:
4307:
4306:
4272:
4263:
4262:
4247:
4246:
4228:
4227:
4212:
4211:
4177:
4176:
4114:
4112:
4111:
4106:
4104:
4103:
4090:
4088:
4087:
4082:
4080:
4079:
4066:
4064:
4063:
4058:
4056:
4055:
4038:
4036:
4035:
4030:
4025:
4024:
4015:
4014:
3982:-inverse cover.
3924:
3922:
3921:
3916:
3911:
3910:
3901:
3900:
3875:
3873:
3872:
3867:
3862:
3861:
3852:
3851:
3815:
3792:
3790:
3789:
3784:
3779:
3778:
3769:
3768:
3739:
3737:
3736:
3731:
3656:
3654:
3653:
3648:
3643:
3642:
3630:
3629:
3608:
3607:
3574:
3573:
3564:
3563:
3534:McAlister triple
3531:
3529:
3528:
3523:
3518:
3517:
3508:
3507:
3481:
3479:
3478:
3473:
3471:
3470:
3457:
3455:
3454:
3449:
3447:
3446:
3419:
3409:
3407:
3406:
3401:
3399:
3398:
3373:
3371:
3370:
3365:
3363:
3362:
3341:
3339:
3338:
3333:
3328:
3327:
3318:
3317:
3291:
3273:
3271:
3270:
3265:
3263:
3262:
3230:
3220:
3210:
3208:
3207:
3202:
3200:
3199:
3175:
3165:
3163:
3162:
3157:
3155:
3154:
3137:
3135:
3134:
3129:
3127:
3126:
3098:
3088:
3078:
3076:
3075:
3070:
3068:
3067:
3034:
3032:
3031:
3026:
3024:
3023:
2994:
2992:
2991:
2986:
2984:
2983:
2966:
2956:
2954:
2953:
2948:
2946:
2945:
2928:
2926:
2925:
2920:
2918:
2917:
2896:
2894:
2893:
2888:
2886:
2885:
2868:
2866:
2865:
2860:
2858:
2857:
2842:
2840:
2839:
2834:
2832:
2831:
2815:
2813:
2812:
2807:
2789:
2787:
2786:
2781:
2779:
2778:
2753:
2751:
2750:
2745:
2743:
2742:
2726:
2724:
2723:
2718:
2716:
2715:
2698:
2696:
2695:
2690:
2688:
2687:
2670:
2668:
2667:
2662:
2660:
2659:
2632:
2630:
2629:
2624:
2619:
2618:
2603:
2602:
2557:
2555:
2554:
2549:
2529:
2527:
2526:
2521:
2426:
2424:
2423:
2418:
2376:
2374:
2373:
2368:
2275:
2273:
2272:
2267:
2153:is contained in
2095:group congruence
2085:
2083:
2082:
2077:
2050:
2048:
2047:
2042:
2024:
2022:
2021:
2016:
1986:
1984:
1983:
1978:
1963:
1961:
1960:
1955:
1822:
1820:
1819:
1814:
1812:
1811:
1806:
1805:
1765:Cayley's theorem
1748:The homomorphic
1741:
1715:
1579:
1577:
1576:
1571:
1517:
1513:
1511:
1510:
1505:
1502:
1501:
1484:. Similarly, if
1483:
1470:
1468:
1467:
1462:
1459:
1458:
1441:
1416:
1398:
1369:
1367:
1366:
1361:
1356:
1355:
1340:
1339:
1302:
1300:
1299:
1294:
1203:
1201:
1200:
1195:
1157:. Equivalently,
1142:
1140:
1139:
1134:
947:
940:
930:
908:Brandt semigroup
895:
831:
829:
828:
823:
821:
820:
802:
801:
779:
778:
761:
760:
742:
741:
722:
721:
689:
687:
686:
681:
679:
678:
650:
648:
647:
642:
640:
639:
614:
612:
611:
606:
604:
603:
590:
588:
587:
582:
580:
579:
562:
560:
559:
554:
552:
551:
538:
536:
535:
530:
528:
527:
449:
414:. In an inverse
413:
337:
335:
334:
329:
327:
326:
321:
320:
305:binary operation
268:
266:
265:
260:
257:
256:
118:in 1952, and by
74:
64:
5820:
5819:
5815:
5814:
5813:
5811:
5810:
5809:
5790:
5789:
5763:
5758:
5740:, Chapter 7 or
5733:
5731:Further reading
5716:
5694:
5675:Semigroup Forum
5668:
5647:Semigroup Forum
5636:
5615:
5594:
5573:
5567:
5554:
5539:10.2307/1997032
5519:
5505:10.2307/1996831
5490:
5484:
5471:
5465:
5449:
5441:
5439:
5435:
5426:
5395:
5374:
5347:
5341:
5326:
5323:
5318:
5311:
5294:
5293:
5289:
5282:
5267:
5266:
5262:
5251:
5247:
5239:
5235:
5227:
5223:
5215:
5211:
5203:
5196:
5190:McAlister 1974a
5184:
5180:
5172:
5168:
5161:
5146:
5145:
5141:
5137:, Theorem 2.4.6
5133:
5129:
5121:
5117:
5109:
5105:
5097:
5093:
5089:, Theorem 2.4.1
5085:
5081:
5073:
5069:
5061:
5057:
5045:
5041:
5033:
5029:
5012:
4986:Semigroup Forum
4983:
4982:
4978:
4970:
4966:
4958:
4954:
4946:
4942:
4934:
4930:
4922:
4918:
4911:
4907:
4899:
4895:
4891:, Theorem 5.1.1
4887:
4883:
4875:
4871:
4863:
4859:
4851:
4847:
4838:
4834:
4826:
4822:
4809:
4805:
4796:
4792:
4785:
4781:
4774:
4759:
4758:
4754:
4750:
4745:
4726:Category theory
4686:
4656:
4646:
4632:
4618:
4604:
4545:
4489:locally inverse
4465:
4403:
4383:category theory
4362:
4308:
4295:
4251:
4235:
4216:
4200:
4165:
4151:
4150:
4121:
4093:
4092:
4069:
4068:
4045:
4044:
3995:
3994:
3946:
3878:
3877:
3832:
3831:
3794:
3746:
3745:
3665:
3664:
3618:
3541:
3540:
3488:
3487:
3460:
3459:
3436:
3435:
3411:
3388:
3387:
3352:
3351:
3298:
3297:
3275:
3252:
3251:
3222:
3212:
3189:
3188:
3167:
3144:
3143:
3116:
3115:
3090:
3080:
3057:
3056:
3013:
3012:
2973:
2972:
2958:
2935:
2934:
2907:
2906:
2897:, that is, for
2875:
2874:
2847:
2846:
2821:
2820:
2798:
2797:
2768:
2767:
2732:
2731:
2705:
2704:
2677:
2676:
2649:
2648:
2633:are idempotent.
2607:
2591:
2571:
2570:
2540:
2539:
2512:
2511:
2385:
2384:
2335:
2334:
2285:
2193:
2192:
2186:idempotent pure
2053:
2052:
2027:
2026:
2025:there exists a
1996:
1995:
1969:
1968:
1900:
1899:
1879:
1799:
1794:
1793:
1729:
1698:
1664:
1620:totally ordered
1538:
1537:
1529:), the natural
1490:
1489:
1485:
1475:
1447:
1446:
1433:
1432:as follows: if
1408:
1390:
1344:
1328:
1311:
1310:
1249:
1248:
1162:
1161:
1101:
1100:
1083:
932:
922:
877:
849:
809:
790:
749:
730:
707:
706:
668:
667:
629:
628:
593:
592:
569:
568:
541:
540:
517:
516:
438:
401:
400:, for example,
382:
370:symmetric group
358:symmetric group
348:(or monoid) on
314:
309:
308:
245:
198:
197:
165:are subsets of
108:
66:
56:
12:
11:
5:
5818:
5816:
5808:
5807:
5802:
5792:
5791:
5788:
5787:
5756:
5745:
5732:
5729:
5728:
5727:
5725:(74): 545–632.
5714:
5704:(in Russian).
5692:
5666:
5634:
5613:
5603:(4): 404–411.
5592:
5582:(4): 396–403.
5571:
5565:
5552:
5517:
5488:
5482:
5469:
5463:
5447:
5424:
5404:(12): 3481–4.
5393:
5372:
5360:(2): 113–125.
5345:
5339:
5322:
5319:
5317:
5316:
5309:
5287:
5280:
5260:
5245:
5233:
5221:
5209:
5194:
5178:
5166:
5159:
5139:
5127:
5115:
5103:
5091:
5079:
5067:
5055:
5039:
5037:, Theorem 7.36
5027:
4992:(2): 203–240.
4976:
4964:
4952:
4940:
4928:
4916:
4905:
4893:
4881:
4869:
4857:
4845:
4832:
4820:
4803:
4790:
4779:
4772:
4751:
4749:
4746:
4744:
4743:
4738:
4733:
4728:
4723:
4718:
4713:
4708:
4703:
4698:
4693:
4687:
4685:
4682:
4603:
4600:
4599:
4598:
4595:
4592:
4589:
4571:
4570:
4524:
4503:
4475:
4456:, there is an
4417:, and (2) the
4402:
4399:
4391:étale groupoid
4361:
4358:
4335:
4334:
4323:
4320:
4315:
4311:
4305:
4302:
4298:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4271:
4266:
4261:
4258:
4254:
4250:
4245:
4242:
4238:
4234:
4231:
4226:
4223:
4219:
4215:
4210:
4207:
4203:
4199:
4196:
4192:
4189:
4186:
4183:
4180:
4175:
4172:
4168:
4164:
4161:
4158:
4120:
4117:
4102:
4078:
4054:
4028:
4023:
4018:
4013:
4008:
4005:
4002:
3945:
3939:
3914:
3909:
3904:
3899:
3894:
3891:
3888:
3885:
3865:
3860:
3855:
3850:
3845:
3842:
3839:
3782:
3777:
3772:
3767:
3762:
3759:
3756:
3753:
3742:
3741:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3658:
3657:
3646:
3641:
3636:
3633:
3628:
3625:
3621:
3617:
3614:
3611:
3606:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3572:
3567:
3562:
3557:
3554:
3551:
3548:
3521:
3516:
3511:
3506:
3501:
3498:
3495:
3486:Such a triple
3484:
3483:
3469:
3445:
3421:
3397:
3361:
3331:
3326:
3321:
3316:
3311:
3308:
3305:
3294:
3293:
3261:
3232:
3198:
3177:
3153:
3125:
3100:
3066:
3022:
2997:
2996:
2982:
2944:
2916:
2884:
2856:
2844:
2830:
2805:
2777:
2741:
2714:
2686:
2658:
2635:
2634:
2622:
2617:
2614:
2610:
2606:
2601:
2598:
2594:
2590:
2587:
2584:
2581:
2578:
2547:
2536:
2535:
2519:
2509:
2503:
2428:
2427:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2380:Equivalently,
2378:
2377:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2284:
2278:
2277:
2276:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2237:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2161:is called the
2149:a group, then
2121:congruence on
2087:
2086:
2075:
2072:
2069:
2066:
2063:
2060:
2040:
2037:
2034:
2014:
2011:
2007:
2003:
1976:
1965:
1964:
1953:
1950:
1947:
1943:
1939:
1936:
1933:
1930:
1926:
1922:
1918:
1915:
1911:
1907:
1878:
1875:
1855:representation
1848:
1847:
1810:
1804:
1663:
1660:
1583:so, since the
1581:
1580:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1500:
1457:
1371:
1370:
1359:
1354:
1351:
1347:
1343:
1338:
1335:
1331:
1327:
1324:
1321:
1318:
1304:
1303:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1205:
1204:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1144:
1143:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1082:
1079:
1076:
1075:
1072:
1069:
1066:
1063:
1060:
1054:
1053:
1050:
1047:
1044:
1041:
1038:
1032:
1031:
1028:
1025:
1022:
1019:
1016:
1010:
1009:
1006:
1003:
1000:
997:
994:
988:
987:
984:
981:
978:
975:
972:
966:
965:
962:
959:
956:
953:
950:
919:
918:
915:Munn semigroup
911:
904:
897:
870:
863:
848:
845:
833:
832:
819:
816:
812:
808:
805:
800:
797:
793:
789:
786:
783:
777:
771:
767:
764:
759:
756:
752:
748:
745:
740:
737:
733:
729:
726:
720:
714:
677:
662:, whilst the
638:
621:
620:
602:
578:
550:
526:
513:
482:
381:
378:
325:
319:
270:
269:
255:
252:
248:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
124:United Kingdom
120:Gordon Preston
107:
104:
13:
10:
9:
6:
4:
3:
2:
5817:
5806:
5803:
5801:
5798:
5797:
5795:
5786:
5781:
5777:
5773:
5769:
5762:
5757:
5754:
5750:
5746:
5743:
5739:
5735:
5734:
5730:
5724:
5720:
5715:
5712:
5707:
5703:
5702:
5697:
5696:Wagner, V. V.
5693:
5689:
5685:
5681:
5677:
5676:
5671:
5670:Schein, B. M.
5667:
5662:
5657:
5653:
5649:
5648:
5643:
5639:
5638:Schein, B. M.
5635:
5631:
5627:
5623:
5619:
5614:
5610:
5606:
5602:
5598:
5593:
5589:
5585:
5581:
5577:
5572:
5568:
5562:
5558:
5553:
5549:
5545:
5540:
5535:
5531:
5527:
5523:
5518:
5514:
5510:
5506:
5502:
5498:
5494:
5489:
5485:
5479:
5475:
5470:
5466:
5460:
5456:
5452:
5448:
5438:on 2005-08-26
5434:
5430:
5425:
5421:
5417:
5412:
5407:
5403:
5399:
5394:
5390:
5386:
5382:
5379:(in German).
5378:
5373:
5368:
5363:
5359:
5355:
5351:
5346:
5342:
5336:
5332:
5331:
5325:
5324:
5320:
5312:
5306:
5302:
5298:
5291:
5288:
5283:
5277:
5273:
5272:
5264:
5261:
5258:
5254:
5253:Fountain 1979
5249:
5246:
5243:, p. 222
5242:
5237:
5234:
5230:
5225:
5222:
5218:
5213:
5210:
5207:, p. 230
5206:
5201:
5199:
5195:
5191:
5187:
5182:
5179:
5175:
5170:
5167:
5162:
5156:
5152:
5151:
5143:
5140:
5136:
5131:
5128:
5124:
5119:
5116:
5113:, p. 192
5112:
5107:
5104:
5100:
5095:
5092:
5088:
5083:
5080:
5076:
5071:
5068:
5064:
5059:
5056:
5052:
5048:
5043:
5040:
5036:
5031:
5028:
5023:
5017:
5010:Corollary 4.9
5009:
5005:
5000:
4995:
4991:
4987:
4980:
4977:
4974:, Theorem 7.5
4973:
4968:
4965:
4961:
4956:
4953:
4950:, p. 153
4949:
4944:
4941:
4937:
4932:
4929:
4925:
4920:
4917:
4914:
4909:
4906:
4902:
4897:
4894:
4890:
4885:
4882:
4878:
4873:
4870:
4867:, p. 149
4866:
4861:
4858:
4855:, p. 152
4854:
4849:
4846:
4842:
4836:
4833:
4829:
4828:Preston 1954a
4824:
4821:
4817:
4813:
4807:
4804:
4800:
4794:
4791:
4788:
4783:
4780:
4775:
4769:
4765:
4764:
4756:
4753:
4747:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4702:
4699:
4697:
4696:Biordered set
4694:
4692:
4689:
4688:
4683:
4681:
4679:
4674:
4672:
4668:
4663:
4659:
4653:
4649:
4643:
4639:
4635:
4629:
4625:
4621:
4617:
4613:
4609:
4601:
4596:
4593:
4590:
4587:
4586:
4585:
4582:
4580:
4576:
4568:
4564:
4560:
4557:
4552:
4548:
4543:
4539:
4535:
4532:
4528:
4525:
4522:
4518:
4514:
4511:
4507:
4504:
4501:
4498:
4494:
4490:
4486:
4483:
4479:
4476:
4472:
4468:
4463:
4459:
4455:
4451:
4447:
4443:
4440:
4436:
4435:
4431:
4430:
4429:
4426:
4424:
4420:
4416:
4412:
4408:
4400:
4398:
4396:
4392:
4388:
4384:
4380:
4376:
4372:
4368:
4359:
4357:
4355:
4350:
4348:
4344:
4340:
4321:
4313:
4303:
4300:
4296:
4292:
4289:
4283:
4280:
4277:
4274:
4259:
4256:
4252:
4248:
4243:
4240:
4236:
4232:
4229:
4224:
4221:
4217:
4213:
4208:
4205:
4201:
4197:
4190:
4184:
4181:
4178:
4173:
4170:
4166:
4162:
4149:
4148:
4147:
4146:
4142:
4138:
4134:
4130:
4126:
4118:
4116:
4042:
4016:
4006:
4003:
3992:
3988:
3983:
3981:
3977:
3973:
3971:
3967:
3963:
3959:
3955:
3951:
3943:
3940:
3938:
3936:
3932:
3928:
3902:
3892:
3889:
3883:
3853:
3843:
3840:
3829:
3825:
3823:
3819:
3813:
3809:
3806:
3802:
3798:
3770:
3760:
3757:
3751:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3700:
3694:
3691:
3688:
3679:
3676:
3673:
3663:
3662:
3661:
3634:
3631:
3626:
3623:
3619:
3615:
3612:
3609:
3599:
3593:
3590:
3587:
3578:
3565:
3555:
3552:
3546:
3539:
3538:
3537:
3535:
3509:
3499:
3496:
3434:
3430:
3426:
3422:
3418:
3414:
3385:
3381:
3377:
3349:
3345:
3344:
3343:
3319:
3309:
3306:
3290:
3286:
3282:
3278:
3249:
3245:
3241:
3237:
3233:
3229:
3225:
3219:
3215:
3186:
3182:
3178:
3174:
3170:
3141:
3113:
3109:
3105:
3101:
3097:
3093:
3087:
3083:
3054:
3050:
3046:
3042:
3038:
3037:
3036:
3010:
3006:
3002:
2970:
2965:
2961:
2932:
2904:
2900:
2872:
2845:
2818:
2803:
2796:
2793:
2765:
2761:
2757:
2730:
2729:
2728:
2702:
2674:
2646:
2641:
2639:
2620:
2615:
2612:
2608:
2604:
2599:
2596:
2592:
2588:
2582:
2579:
2576:
2569:
2568:
2567:
2566:, defined by
2565:
2561:
2545:
2533:
2517:
2510:
2507:
2504:
2501:
2497:
2494:
2493:
2492:
2490:
2486:
2483:
2479:
2475:
2471:
2469:
2465:
2461:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2414:
2411:
2408:
2405:
2399:
2396:
2393:
2390:
2383:
2382:
2381:
2364:
2361:
2358:
2355:
2349:
2346:
2343:
2340:
2333:
2332:
2331:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2301:
2298:
2294:
2290:
2282:
2279:
2263:
2257:
2251:
2248:
2245:
2239:
2235:
2231:
2228:
2222:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2191:
2190:
2189:
2187:
2183:
2179:
2176:A congruence
2174:
2172:
2168:
2164:
2160:
2156:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2073:
2070:
2067:
2064:
2061:
2058:
2038:
2035:
2032:
2009:
2005:
2001:
1994:
1993:
1992:
1990:
1974:
1951:
1948:
1945:
1941:
1937:
1934:
1928:
1924:
1920:
1916:
1913:
1909:
1905:
1898:
1897:
1896:
1894:
1890:
1887:
1883:
1876:
1874:
1872:
1868:
1862:
1860:
1856:
1853:
1846:
1842:
1838:
1834:
1830:
1826:
1825:
1824:
1808:
1791:
1787:
1784:
1780:
1776:
1772:
1770:
1766:
1762:
1757:
1755:
1751:
1747:
1743:
1740:
1737:
1733:
1727:
1723:
1719:
1714:
1710:
1706:
1702:
1696:
1692:
1688:
1685:
1681:
1677:
1673:
1669:
1661:
1659:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1596:
1594:
1590:
1586:
1567:
1564:
1561:
1558:
1555:
1549:
1546:
1543:
1536:
1535:
1534:
1532:
1531:partial order
1528:
1524:
1519:
1516:
1488:
1482:
1478:
1473:
1445:
1440:
1436:
1431:
1426:
1424:
1420:
1415:
1411:
1406:
1402:
1397:
1393:
1388:
1387:partial order
1384:
1380:
1379:partial order
1376:
1357:
1352:
1349:
1345:
1341:
1336:
1333:
1329:
1322:
1319:
1316:
1309:
1308:
1307:
1290:
1287:
1284:
1281:
1278:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1247:
1246:
1245:
1243:
1242:partial order
1238:
1236:
1233:
1229:
1225:
1221:
1217:
1213:
1210:
1191:
1188:
1185:
1182:
1179:
1173:
1170:
1167:
1160:
1159:
1158:
1156:
1152:
1149:
1130:
1127:
1124:
1121:
1118:
1112:
1109:
1106:
1099:
1098:
1097:
1095:
1094:
1093:partial order
1088:
1080:
1073:
1070:
1067:
1064:
1061:
1059:
1056:
1055:
1051:
1048:
1045:
1042:
1039:
1037:
1034:
1033:
1029:
1026:
1023:
1020:
1017:
1015:
1012:
1011:
1007:
1004:
1001:
998:
995:
993:
990:
989:
985:
982:
979:
976:
973:
971:
968:
967:
963:
960:
957:
954:
951:
949:
948:
942:
939:
935:
929:
925:
916:
912:
909:
905:
902:
898:
893:
889:
885:
881:
875:
871:
868:
864:
861:
857:
854:
851:
850:
846:
844:
842:
838:
817:
814:
810:
806:
803:
798:
795:
791:
787:
781:
769:
765:
762:
757:
754:
750:
746:
743:
738:
735:
731:
724:
712:
705:
704:
703:
701:
697:
693:
665:
661:
658:
654:
626:
618:
566:
514:
511:
507:
503:
499:
495:
491:
487:
483:
480:
476:
475:
474:
472:
467:
465:
461:
457:
453:
448:
445:
441:
436:
432:
428:
425:
421:
417:
412:
409:
405:
399:
395:
391:
387:
379:
377:
375:
371:
367:
363:
359:
355:
351:
347:
346:
341:
323:
306:
303:
299:
295:
291:
287:
283:
279:
275:
253:
250:
246:
239:
236:
233:
230:
227:
224:
221:
215:
212:
209:
206:
203:
196:
195:
194:
192:
188:
184:
180:
176:
172:
168:
164:
160:
156:
152:
148:
144:
140:
137:
133:
129:
125:
121:
117:
113:
105:
103:
101:
97:
93:
89:
84:
82:
78:
73:
69:
63:
59:
54:
50:
47:
44:has a unique
43:
39:
35:
31:
27:
23:
19:
5771:
5767:
5749:Petrich 1984
5744:, Chapter 5.
5722:
5718:
5708:: 1119–1122.
5705:
5699:
5679:
5673:
5651:
5645:
5624:(4): 411–9.
5621:
5617:
5600:
5596:
5579:
5575:
5556:
5529:
5525:
5496:
5492:
5473:
5454:
5451:Howie, J. M.
5440:. Retrieved
5436:(Postscript)
5433:the original
5401:
5397:
5380:
5376:
5357:
5353:
5329:
5300:
5290:
5270:
5263:
5248:
5236:
5224:
5212:
5181:
5169:
5149:
5142:
5130:
5118:
5106:
5101:, p. 65
5094:
5082:
5077:, p. 62
5070:
5065:, p. 22
5058:
5042:
5030:
5016:cite journal
4989:
4985:
4979:
4967:
4955:
4943:
4931:
4919:
4908:
4896:
4884:
4872:
4860:
4848:
4835:
4823:
4806:
4793:
4782:
4762:
4755:
4736:Weak inverse
4675:
4661:
4657:
4651:
4647:
4641:
4637:
4633:
4627:
4623:
4619:
4611:
4605:
4583:
4579:intersection
4572:
4566:
4562:
4558:
4550:
4546:
4537:
4536:is called a
4533:
4526:
4516:
4512:
4505:
4499:
4492:
4488:
4484:
4477:
4470:
4466:
4461:
4457:
4453:
4449:
4445:
4441:
4432:
4427:
4422:
4410:
4406:
4404:
4386:
4374:
4370:
4366:
4363:
4353:
4351:
4339:word problem
4336:
4144:
4132:
4129:presentation
4122:
4040:
3990:
3986:
3985:McAlister's
3984:
3979:
3975:
3974:
3965:
3961:
3957:
3953:
3949:
3947:
3941:
3930:
3926:
3827:
3826:
3821:
3817:
3811:
3807:
3804:
3800:
3796:
3743:
3659:
3533:
3532:is called a
3485:
3432:
3428:
3424:
3416:
3412:
3383:
3379:
3375:
3347:
3295:
3288:
3284:
3280:
3276:
3247:
3243:
3239:
3235:
3227:
3223:
3217:
3213:
3184:
3180:
3172:
3168:
3139:
3111:
3107:
3103:
3095:
3091:
3085:
3081:
3052:
3048:
3044:
3040:
3000:
2998:
2968:
2963:
2959:
2930:
2902:
2898:
2816:
2794:
2763:
2759:
2644:
2642:
2637:
2636:
2563:
2559:
2537:
2531:
2505:
2499:
2495:
2488:
2484:
2477:
2473:
2472:
2467:
2463:
2459:
2455:
2451:
2447:
2443:
2439:
2435:
2431:
2429:
2379:
2327:
2323:
2319:
2315:
2314:if, for all
2311:
2307:
2299:
2292:
2288:
2286:
2280:
2185:
2181:
2177:
2175:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2106:
2102:
2098:
2090:
2088:
1988:
1966:
1888:
1885:
1880:
1870:
1866:
1863:
1858:
1849:
1844:
1840:
1836:
1832:
1828:
1789:
1785:
1778:
1774:
1773:
1760:
1758:
1745:
1744:
1738:
1735:
1731:
1725:
1721:
1717:
1712:
1708:
1704:
1700:
1694:
1690:
1686:
1679:
1675:
1671:
1668:homomorphism
1665:
1655:
1651:
1647:
1639:
1635:
1623:
1622:by ≤), then
1615:
1611:
1603:
1599:
1597:
1592:
1588:
1582:
1526:
1522:
1520:
1514:
1486:
1480:
1476:
1471:
1443:
1438:
1434:
1427:
1422:
1418:
1413:
1409:
1404:
1400:
1395:
1391:
1372:
1305:
1240:The natural
1239:
1234:
1231:
1227:
1223:
1219:
1215:
1211:
1206:
1154:
1150:
1145:
1090:
1089:possesses a
1086:
1084:
1057:
1035:
1013:
991:
969:
937:
933:
927:
923:
920:
891:
887:
883:
879:
859:
840:
836:
834:
695:
691:
659:
656:
652:
622:
505:
489:
485:
478:
470:
468:
459:
455:
451:
446:
443:
439:
434:
426:
423:
419:
410:
407:
403:
393:
389:
385:
383:
349:
343:
339:
297:
286:pseudogroups
281:
276:denotes the
273:
271:
186:
182:
178:
174:
170:
166:
162:
158:
154:
150:
142:
138:
116:Soviet Union
109:
99:
95:
94:rather than
91:
87:
85:
71:
67:
61:
57:
52:
48:
45:
41:
37:
29:
25:
21:
15:
5753:Lawson 1998
5682:: 149–158.
5654:: 189–200.
5532:: 351–370.
5499:: 227–244.
5383:: 768–780.
5217:Lawson 1998
5205:Lawson 1998
5135:Lawson 1998
5123:Lawson 1998
5099:Lawson 1998
5087:Lawson 1998
5075:Lawson 1998
5051:Wagner 1952
4960:Lawson 1998
4913:Wagner 1952
4853:Schein 2002
4816:Wagner 1953
4812:Wagner 1952
4799:Schein 1981
4787:Lawson 1998
4716:Semilattice
4701:Pseudogroup
4556:idempotents
4542:idempotents
4521:idempotents
4419:idempotents
3970:M.V. Lawson
3296:The triple
2871:order ideal
2482:semilattice
2454:, for some
2304:idempotents
2297:semilattice
1882:Congruences
1823:, given by
1585:idempotents
1218:. In fact,
917:is inverse.
910:is inverse.
903:is inverse.
901:semilattice
615:are two of
510:semilattice
502:idempotents
498:idempotents
302:associative
280:under
20:theory, an
5794:Categories
5742:Howie 1995
5566:0471875457
5483:9810233167
5464:0198511949
5442:2006-08-28
5427:Gould, V.
5321:References
5241:Howie 1995
5229:Howie 1995
5186:Howie 1995
5174:Howie 1995
5111:Howie 1995
5063:Howie 1995
5047:Howie 1995
4999:1510.04117
4948:Howie 1995
4936:Howie 1995
4924:Howie 1995
4901:Howie 1995
4889:Howie 1995
4877:Howie 1995
4865:Howie 1995
4841:Gołab 1939
4645:such that
4608:categories
4554:, for all
4497:idempotent
4464:such that
4356:-inverse.
4347:Munn trees
4343:W. D. Munn
4125:free group
3935:isomorphic
3410:such that
3346:for every
3166:such that
2184:is called
2125:such that
1886:congruence
1716:, for all
1417:, for all
1383:idempotent
1209:idempotent
1148:idempotent
856:bijections
690:-class of
664:idempotent
651:-class of
625:idempotent
565:idempotent
450:, for all
431:idempotent
380:The basics
5559:. Wiley.
4439:semigroup
4301:−
4293:∪
4284:∈
4257:−
4241:−
4222:−
4206:−
4171:−
3710:∧
3635:∈
3624:−
3610:×
3600:∈
3110:and each
3102:for each
2804:∧
2613:−
2597:−
2586:⟺
2580:∼
2546:∼
2518:∼
2502:-unitary;
2409:∈
2403:⇒
2397:∈
2359:∈
2353:⟹
2347:∈
2249:∈
2243:⟹
2236:ρ
2214:∈
2202:∈
2062:≤
2036:∈
2013:⟺
2006:σ
1975:σ
1942:ρ
1932:⟹
1925:ρ
1910:ρ
1553:⟺
1547:≤
1533:becomes:
1350:−
1342:≤
1334:−
1326:⟹
1320:≤
1285:≤
1276:⟹
1270:≤
1258:≤
1177:⟺
1171:≤
1146:for some
1116:⟺
1110:≤
858:on a set
815:−
796:−
785:⟺
755:−
736:−
728:⟺
460:unipotent
437:in which
294:empty set
251:−
247:α
240:β
237:
231:∩
228:α
225:
213:β
210:α
207:
141:of a set
75:, i.e. a
34:semigroup
5640:(1981).
5453:(1995).
4684:See also
4667:selfdual
4636: :
4622: :
4616:morphism
4517:orthodox
3976:Theorem.
3423:for all
3246:and all
3234:for all
3179:for all
3047:and all
3039:for all
2999:Now let
2474:Theorem.
2322:and all
2119:smallest
1852:faithful
1783:function
1746:Theorem.
1684:function
1672:morphism
1091:natural
567:, where
376:below).
278:preimage
157:, where
147:function
5774:: 187.
5548:1997032
5513:1996831
5219:, 4.1.8
4540:if its
4446:regular
4143:by the
3382:and an
2967:, then
2558:is the
2462:, then
2312:unitary
2117:is the
1610:(i.e.,
1474:, then
1377:, this
853:Partial
666:in the
627:in the
508:form a
122:in the
114:in the
106:Origins
46:inverse
5563:
5546:
5511:
5480:
5461:
5337:
5307:
5278:
5157:
4770:
4039:is an
3954:unique
3925:is an
2971:is in
2933:is in
2869:is an
2790:has a
2701:subset
2538:where
2466:is in
2442:is in
2295:(with
1891:is an
1769:groups
1632:groups
1230:to be
899:Every
865:Every
515:Every
496:) and
442:= 1 =
416:monoid
272:where
191:domain
169:. Let
90:
5764:(PDF)
5713:(PDF)
5544:JSTOR
5509:JSTOR
5406:arXiv
5257:Gould
4994:arXiv
4830:,b,c.
4748:Notes
4610:. An
4575:class
4413:is a
3803:) = (
3744:Then
3283:) = (
3007:that
3005:group
3003:be a
2929:, if
2754:is a
2699:be a
2671:be a
2446:and
2306:) is
2141:with
2051:with
1850:is a
1827:dom (
1788:from
1754:image
1750:image
1707:) = (
1689:from
1644:chain
1634:. If
1628:union
1626:is a
1618:) is
1608:chain
1375:group
1373:In a
886:) = (
867:group
492:is a
464:group
398:group
366:group
362:group
149:from
145:is a
130:of a
32:is a
18:group
5751:and
5561:ISBN
5478:ISBN
5459:ISBN
5335:ISBN
5305:ISBN
5276:ISBN
5155:ISBN
5022:link
4768:ISBN
4655:and
4573:The
4551:xzyx
4547:xyzx
4529:: a
4508:: a
4480:: a
4437:: a
4369:and
4337:The
4091:and
3972:to:
3830:Let
3458:and
3009:acts
2957:and
2476:Let
2188:if
1843:) =
1835:and
1831:) =
1767:for
1734:) =
1682:, a
1678:and
1670:(or
1650:and
1442:and
1407:and
1306:and
1226:and
913:The
906:The
872:The
837:E(S)
623:The
591:and
422:and
406:) =
185:and
173:and
161:and
134:: a
65:and
5776:doi
5684:doi
5656:doi
5626:doi
5605:doi
5584:doi
5534:doi
5530:196
5501:doi
5497:192
5416:doi
5402:126
5385:doi
5381:116
5362:doi
5192:,b.
5004:doi
4658:gfg
4648:fgf
4515:is
4493:eSe
4491:if
4487:is
4467:axa
4460:in
4452:in
4444:is
4421:in
3427:in
3386:in
3378:in
3350:in
3250:in
3242:in
3187:in
3142:in
3114:in
3106:in
3055:in
3043:in
3011:on
2905:in
2873:of
2819:in
2766:in
2703:of
2562:on
2498:is
2458:in
2326:in
2318:in
2302:of
2165:on
1991:by
1857:of
1792:to
1777:If
1724:in
1693:to
1658:).
1630:of
1598:If
1521:On
1214:in
1153:in
934:bab
924:aba
694:is
655:is
504:of
458:(a
454:in
234:dom
204:dom
153:to
132:set
72:yxy
62:xyx
51:in
40:in
16:In
5796::
5772:56
5770:.
5766:.
5723:32
5706:84
5680:65
5678:.
5652:28
5650:.
5644:.
5622:29
5620:.
5601:29
5599:.
5580:29
5578:.
5542:.
5528:.
5524:.
5507:.
5495:.
5414:.
5400:.
5358:22
5356:.
5352:.
5255:,
5197:^
5018:}}
5014:{{
5002:.
4990:96
4988:.
4680:.
4660:=
4650:=
4640:→
4626:→
4565:,
4561:,
4549:=
4469:=
3824::
3810:,
3799:,
3431:,
3415:=
3413:gA
3285:gh
3281:hA
3274:,
3238:,
3228:gB
3226:≤
3224:gA
3216:≤
3211:,
3183:,
3171:=
3169:gA
3094:=
3086:gB
3084:=
3082:gA
3079:,
3051:,
2962:≤
2901:,
2762:,
2530:=
2470:.
2450:≤
2330:,
1861:.
1845:xa
1841:aφ
1833:Sa
1829:aφ
1771::
1732:sθ
1709:st
1705:tθ
1703:)(
1701:sθ
1666:A
1518:.
1479:=
1437:≤
1425:.
1414:xβ
1412:=
1410:xα
1394:≤
1237:.
1224:aa
1074:e
1052:a
1030:c
1008:a
986:a
964:e
936:=
931:,
926:=
890:,
882:,
843:.
696:ss
512:).
473::
466:.
440:xx
420:xx
418:,
404:ab
222:im
181:;
83:.
70:=
60:=
28:)
5782:.
5778::
5755:.
5690:.
5686::
5664:.
5658::
5632:.
5628::
5611:.
5607::
5590:.
5586::
5569:.
5550:.
5536::
5515:.
5503::
5486:.
5467:.
5445:.
5422:.
5418::
5408::
5391:.
5387::
5370:.
5364::
5343:.
5313:.
5284:.
5163:.
5024:)
5006::
4996::
4843:.
4818:.
4801:.
4776:.
4662:g
4652:f
4642:X
4638:Y
4634:g
4628:Y
4624:X
4620:f
4569:.
4567:z
4563:y
4559:x
4534:S
4513:S
4502:.
4500:e
4485:S
4474:.
4471:a
4462:S
4458:x
4454:S
4450:a
4442:S
4423:S
4411:S
4407:S
4375:β
4371:β
4367:α
4354:F
4322:.
4319:}
4314:+
4310:)
4304:1
4297:X
4290:X
4287:(
4281:y
4278:,
4275:x
4270:|
4265:)
4260:1
4253:x
4249:x
4244:1
4237:y
4233:y
4230:,
4225:1
4218:y
4214:y
4209:1
4202:x
4198:x
4195:(
4191:,
4188:)
4185:x
4182:,
4179:x
4174:1
4167:x
4163:x
4160:(
4157:{
4133:X
4101:X
4077:X
4053:Y
4041:F
4027:)
4022:Y
4017:,
4012:X
4007:,
4004:G
4001:(
3991:F
3987:P
3980:F
3966:E
3962:F
3958:σ
3950:F
3942:F
3931:E
3927:E
3913:)
3908:Y
3903:,
3898:X
3893:,
3890:G
3887:(
3884:P
3864:)
3859:Y
3854:,
3849:X
3844:,
3841:G
3838:(
3818:E
3814:)
3812:g
3808:A
3805:g
3801:g
3797:A
3795:(
3781:)
3776:Y
3771:,
3766:X
3761:,
3758:G
3755:(
3752:P
3740:.
3728:)
3725:h
3722:g
3719:,
3716:B
3713:g
3707:A
3704:(
3701:=
3698:)
3695:h
3692:,
3689:B
3686:(
3683:)
3680:g
3677:,
3674:A
3671:(
3645:}
3640:Y
3632:A
3627:1
3620:g
3616::
3613:G
3605:Y
3597:)
3594:g
3591:,
3588:A
3585:(
3582:{
3579:=
3576:)
3571:Y
3566:,
3561:X
3556:,
3553:G
3550:(
3547:P
3520:)
3515:Y
3510:,
3505:X
3500:,
3497:G
3494:(
3468:Y
3444:Y
3433:g
3429:G
3425:g
3420:;
3417:X
3396:Y
3384:A
3380:G
3376:g
3360:X
3348:X
3330:)
3325:Y
3320:,
3315:X
3310:,
3307:G
3304:(
3292:.
3289:A
3287:)
3279:(
3277:g
3260:X
3248:A
3244:G
3240:h
3236:g
3231:;
3218:B
3214:A
3197:X
3185:B
3181:A
3176:;
3173:B
3152:X
3140:A
3124:X
3112:B
3108:G
3104:g
3099:;
3096:B
3092:A
3065:X
3053:B
3049:A
3045:G
3041:g
3021:X
3001:G
2995:.
2981:Y
2969:B
2964:A
2960:B
2943:Y
2931:A
2915:X
2903:B
2899:A
2883:X
2855:Y
2829:Y
2817:B
2795:A
2776:Y
2764:B
2760:A
2740:Y
2713:X
2685:Y
2657:X
2645:E
2621:b
2616:1
2609:a
2605:,
2600:1
2593:b
2589:a
2583:b
2577:a
2564:S
2534:,
2532:σ
2506:σ
2500:E
2496:S
2489:σ
2485:E
2478:S
2468:E
2464:s
2460:S
2456:s
2452:s
2448:e
2444:E
2440:e
2436:S
2432:E
2415:.
2412:E
2406:s
2400:E
2394:e
2391:s
2365:.
2362:E
2356:s
2350:E
2344:s
2341:e
2328:S
2324:s
2320:E
2316:e
2310:-
2308:E
2300:E
2293:S
2289:E
2281:E
2264:.
2261:)
2258:S
2255:(
2252:E
2246:a
2240:e
2232:a
2229:,
2226:)
2223:S
2220:(
2217:E
2211:e
2208:,
2205:S
2199:a
2182:S
2178:ρ
2171:E
2167:S
2159:σ
2155:τ
2151:σ
2147:τ
2145:/
2143:S
2139:S
2135:τ
2131:σ
2129:/
2127:S
2123:S
2115:σ
2111:S
2107:S
2103:σ
2101:/
2099:S
2091:σ
2074:.
2071:b
2068:,
2065:a
2059:c
2039:S
2033:c
2010:b
2002:a
1989:S
1952:.
1949:d
1946:b
1938:c
1935:a
1929:d
1921:c
1917:,
1914:b
1906:a
1889:ρ
1871:S
1867:S
1859:S
1839:(
1837:x
1809:S
1803:I
1790:S
1786:φ
1779:S
1739:θ
1736:s
1730:(
1726:S
1722:t
1720:,
1718:s
1713:θ
1711:)
1699:(
1695:T
1691:S
1687:θ
1680:T
1676:S
1656:S
1654:(
1652:E
1648:S
1640:S
1638:(
1636:E
1624:S
1616:S
1614:(
1612:E
1604:S
1602:(
1600:E
1593:S
1591:(
1589:E
1568:,
1565:f
1562:e
1559:=
1556:e
1550:f
1544:e
1527:S
1525:(
1523:E
1515:t
1499:R
1487:s
1481:t
1477:s
1472:t
1456:L
1444:s
1439:t
1435:s
1423:α
1419:x
1405:β
1401:α
1396:β
1392:α
1358:.
1353:1
1346:b
1337:1
1330:a
1323:b
1317:a
1291:d
1288:b
1282:c
1279:a
1273:d
1267:c
1264:,
1261:b
1255:a
1235:a
1232:a
1228:f
1220:e
1216:S
1212:f
1192:,
1189:f
1186:b
1183:=
1180:a
1174:b
1168:a
1155:S
1151:e
1131:,
1128:b
1125:e
1122:=
1119:a
1113:b
1107:a
1087:S
1071:d
1068:a
1065:a
1062:a
1058:e
1049:a
1046:e
1043:d
1040:a
1036:d
1027:b
1024:a
1021:a
1018:a
1014:c
1005:a
1002:c
999:b
996:a
992:b
983:a
980:a
977:a
974:a
970:a
961:d
958:c
955:b
952:a
938:b
928:a
896:.
894:)
892:a
888:b
884:b
880:a
878:(
860:X
841:S
818:1
811:b
807:b
804:=
799:1
792:a
788:a
782:b
776:R
770:a
766:,
763:b
758:1
751:b
747:=
744:a
739:1
732:a
725:b
719:L
713:a
692:s
676:R
660:s
657:s
653:s
637:L
619:.
601:R
577:L
549:R
525:L
506:S
490:S
486:S
479:S
471:S
456:S
452:x
447:x
444:x
435:S
427:x
424:x
411:a
408:b
402:(
394:x
390:S
386:x
350:X
340:X
324:X
318:I
282:α
274:α
254:1
243:]
219:[
216:=
187:β
183:α
179:X
175:β
171:α
167:X
163:B
159:A
155:B
151:A
143:X
139:α
100:x
98:(
96:f
92:f
88:x
68:y
58:x
53:S
49:y
42:S
38:x
30:S
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