31:
3285:
1084:. A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup. If a semigroup has both a left identity and a right identity, then it has a two-sided identity (which is therefore the unique one-sided identity).
4560:, the semigroups based on multiplication modulo 2 (choosing a or b as the identity element 1), the groups equivalent to addition modulo 2 (choosing a or b to be the identity element 0), and the semigroups in which the elements are either both left identities or both right identities.
248:
do form a monoid. A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with
3002:
2095:
and at least one idempotent. The number of finite semigroups of a given size (greater than 1) is (obviously) larger than the number of groups of the same size. For example, of the sixteen possible "multiplication tables" for a set of two elements
3705:
obtained the first non-trivial results about semigroups. His 1928 paper "Über die endlichen
Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" ("On finite groups without the rule of unique invertibility") determined the structure of finite
3458:
3582:
326:, semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention:
2074:. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotent
4529:
The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup.
4448:, in which the requirement that the binary relation be total is lifted. As categories generalize monoids in the same way, a semigroupoid behaves much like a category but lacks identities.
3280:{\displaystyle {\begin{cases}\partial _{t}u(t,x)=\partial _{x}^{2}u(t,x),&x\in (0,1),t>0;\\u(t,x)=0,&x\in \{0,1\},t>0;\\u(t,x)=u_{0}(x),&x\in (0,1),t=0.\end{cases}}}
2624:
Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup. The structure theorem says that for any commutative semigroup
2949:
of the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups.
4997:
693:
3329:
2408:
are regular semigroups where every element has exactly one inverse. Alternatively, a regular semigroup is inverse if and only if any two idempotents commute.
3791:
In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, like
2891:
An important question is to characterize those semigroups for which this map is an embedding. This need not always be the case: for example, take
3471:. Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space
3481:
4853:
2100:}, eight form semigroups whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, see
177:, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be
5515:
5485:
5466:
5436:
5280:
5253:
5215:
5195:
5166:
5145:
5105:
5069:
4810:
4785:
4693:
4658:
2082:. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the term
359:
272:, in which arbitrary functions replace the role of bijections in group theory. A deep result in the classification of finite semigroups is
5017:
277:
5347:
3741:. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodical called
686:
253:, which are generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups
93:
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic
2411:
Affine semigroup: a semigroup that is isomorphic to a finitely-generated subsemigroup of Z. These semigroups have applications to
4935:
4635:
5329:
4650:
Monoids, Acts, and
Categories: With Applications to Wreath Products and Graphs : a Handbook for Students and Researchers
2972:
2968:
2091:
More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal
316:
228:, thus obeying all but one of the axioms of a group: existence of inverses is not required of a monoid. A natural example is
5548:
5339:
5137:
4490:
2665:
2115:
861:
679:
323:
1783:
of any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of the
312:
288:
4577:, in M. Hazewinkel, Handbook of Algebra, Vol. 1, Elsevier, 1996. Notice that in this context the authors use the term
3714:
J-class) of a finite semigroup is simple. From that point on, the foundations of semigroup theory were further laid by
1458:. But there are semigroup homomorphisms that are not monoid homomorphisms, e.g. the canonical embedding of a semigroup
1345:
An example of a semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a
4500:
854:
548:
4996:
B. M. Schein & R. McKenzie (1997) "Every semigroup is isomorphic to a transitive semigroup of binary relations",
3715:
2900:
974:
823:
5543:
1826:
904:
335:
237:
229:
3796:
3771:
3723:
2235:
1366:
The subset with the property that every element commutes with any other element of the semigroup is called the
1250:
940:
923:
269:
265:
2971:. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an
2102:
273:
1363:
they generate, are important tools for analysing the ideals of a semigroup and related notions of structure.
4959:
3695:(Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton's
2801:
1788:
639:
3683:
The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as
3890:
2980:
2151:
967:
723:
343:
258:
1078:) is an element that is both a left and right identity. Semigroups with a two-sided identity are called
5370:
5057:
4451:
Infinitary generalizations of commutative semigroups have sometimes been considered by various authors.
3834:
3819:
3755:
3719:
2938:
2437:
2155:
1841:
944:
202:
4379:
instead of a binary operation. The associative law is generalized as follows: ternary associativity is
1731:
Because ~ is a congruence, the set of all congruence classes of ~ forms a semigroup with ∘, called the
1447:, i.e. the result is the same when performing the semigroup operation after or before applying the map
3839:
3824:
3711:
3321:
2455:
2321:
2313:
1784:
1619:
1356:
1352:
741:
626:
618:
590:
585:
576:
533:
475:
281:
51:
4859:
3490:
3011:
2049:
if all of its elements are of finite order. A semigroup generated by a single element is said to be
4480:
4318:
4272:
3800:
3684:
2765:
2412:
2133:
2092:
2050:
1780:
1455:
951:
644:
634:
485:
385:
377:
368:
308:
254:
224:
is an algebraic structure intermediate between semigroups and groups, and is a semigroup having an
174:
75:
39:
3774:
for the semigroup product. At an algebraic conference in 1972 Schein surveyed the literature on B
5413:
5232:
5123:
5087:
4900:
4250:
3734:
2975:
on a function space. For example, consider the following initial/boundary value problem for the
2946:
2854:
2828:
2328:
886:
450:
441:
399:
351:
331:
300:
2061:
with the operation of addition. If it is finite and nonempty, then it must contain at least one
236:
as the binary operation, and the empty string as the identity element. Restricting to non-empty
5511:
5510:. Encyclopedia of Mathematics and Its Applications. Vol. 90. Cambridge University Press.
5481:
5462:
5432:
5397:
5343:
5276:
5249:
5191:
5162:
5141:
5101:
5065:
4806:
4781:
4689:
4654:
4470:
4460:
4434:
adjacent elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to an
3792:
2405:
2363:
2204:. Every group is a cancellative semigroup, and every finite cancellative semigroup is a group.
1684:
1172:
959:
955:
930:
712:
339:
327:
79:
5213:
Hollings, Christopher (2009). "The Early
Development of the Algebraic Theory of Semigroups".
4683:
4648:
3691:. A number of sources attribute the first use of the term (in French) to J.-A. de Séguier in
1199:
The semigroup operation induces an operation on the collection of its subsets: given subsets
346:. There are also interesting classes of semigroups that do not contain any groups except the
5521:
5491:
5442:
5426:
5387:
5379:
5365:
5286:
5259:
5224:
5201:
5172:
5111:
5075:
5033:
4958:
B. M. Schein (1963) "Representations of semigroups by means of binary relations" (Russian),
4892:
4664:
4495:
4485:
4326:
3846:
3829:
3707:
3702:
2126:
2084:
2054:
1368:
1339:
719:
470:
319:, a semigroup is associated to any equation whose spatial evolution is independent of time.
296:
225:
87:
63:
5409:
5357:
5317:
5004:
4985:
4966:
2057:). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive
1067:. A semigroup may have one or more left identities but no right identity, and vice versa.
882:
with minimum or maximum. (With positive/negative infinity included, this becomes a monoid.)
5525:
5495:
5454:
5446:
5405:
5353:
5313:
5290:
5263:
5205:
5176:
5115:
5079:
5001:
4982:
4963:
4880:
4668:
4475:
3980:
3957:
3763:
3748:
3743:
2950:
2137:
1360:
1002:
809:
797:
562:
556:
543:
523:
514:
480:
417:
292:
244:
with addition form a commutative semigroup that is not a monoid, whereas the non-negative
170:
47:
35:
3788:
proved that every semigroup is isomorphic to a transitive semigroup of binary relations.
3453:{\displaystyle D(A)={\big \{}u\in H^{2}((0,1);\mathbf {R} ){\big |}u(0)=u(1)=0{\big \}},}
1822:
4942:
5305:
5127:
5091:
4849:
4505:
4115:
4092:
3785:
3738:
3728:
3688:
3609:
2332:
2208:
1884:
The following notions introduce the idea that a semigroup is contained in another one.
1033:
912:
897:
817:
604:
304:
94:
264:
The formal study of semigroups began in the early 20th century. Early results include
5537:
5417:
5236:
4904:
4734:
4465:
4294:
3468:
2976:
2930:
2223:
993:
908:
490:
455:
412:
347:
233:
210:
178:
5368:(1928). "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit".
5246:
Mathematics across the Iron
Curtain: A History of the Algebraic Theory of Semigroups
3751:) became one of the few mathematical journals devoted entirely to semigroup theory.
1713:
and the semigroup operation induces a binary operation ∘ on the congruence classes:
4510:
4445:
4070:
4047:
3868:
3759:
3672:
2962:
2353:
1383:
963:
664:
595:
429:
112:
2700:
The
Archimedean property follows immediately from the ordering in the semilattice
201:; a well-known example of an operation that is associative but non-commutative is
5333:
5185:
5156:
5131:
5095:
2065:. It follows that every nonempty periodic semigroup has at least one idempotent.
5503:
5325:
4435:
2473:
2425:
2219:
1927:, using the morphism consisting of taking the remainder modulo 2 of an integer.
1589:
1346:
916:
830:
654:
649:
538:
528:
502:
495:
355:
83:
17:
5392:
5228:
4206:
4183:
4025:
4002:
2212:
2062:
1763:
1573:
893:
404:
250:
30:
5401:
2632:
by this equivalence relation is a semilattice. Denoting this semilattice by
2317:
1594:
1466:. Conditions characterizing monoid homomorphisms are discussed further. Let
1326:
is a semigroup, then the intersection of any collection of subsemigroups of
1092:
937:, with convolution as the operation. This is called a convolution semigroup.
813:
659:
465:
422:
205:. If the semigroup operation is commutative, then the semigroup is called a
27:
Algebraic structure consisting of a set with an associative binary operation
3577:{\displaystyle {\begin{cases}{\dot {u}}(t)=Au(t);\\u(0)=u_{0}.\end{cases}}}
864:
representing the three operations on a switch – set, reset, and do nothing.
2621:. This equivalence relation is a semigroup congruence, as defined above.
291:
since the 1950s because of the natural link between finite semigroups and
3935:
3912:
3306:
2585:
2229:
2070:
460:
1372:
of the semigroup. The center of a semigroup is actually a subsemigroup.
5383:
4896:
2058:
1829:
holds: there is no infinite strictly ascending chain of congruences on
1454:
A semigroup homomorphism between monoids preserves identity if it is a
879:
868:
280:
for finite groups. Some other techniques for studying semigroups, like
245:
241:
4317:
If the associativity axiom of a semigroup is dropped, the result is a
2903:
as the binary operation (this is an example of a semilattice). Since
4228:
3608:. However, for a rigorous treatment, a meaning must be given to the
2312:. Sequencing clearly is an associative operation, here equivalent to
2144:
2122:
1787:. Congruence classes and factor monoids are the objects of study in
1080:
872:
394:
287:
The theory of finite semigroups has been of particular importance in
261:. Division in semigroups (or in monoids) is not possible in general.
221:
55:
3312:
of square-integrable real-valued functions with domain the interval
2967:
Semigroup theory can be used to study some problems in the field of
2953:
gave necessary and sufficient conditions for embeddability in 1937.
2668:. An Archimedean semigroup is one where given any pair of elements
2424:
There is a structure theorem for commutative semigroups in terms of
1964:, while it is not necessarily the case that there are a quotient of
915:
over Σ". With the empty string included, this semigroup becomes the
4918:
1622:
that is compatible with the semigroup operation. That is, a subset
4373:
3587:
On an heuristic level, the solution to this problem "ought" to be
1683:. Like any equivalence relation, a semigroup congruence ~ induces
827:
29:
2884:
as the "most general" group that contains a homomorphic image of
2507:
from an arbitrary semigroup to a semilattice, each inverse image
958:
to itself with composition of functions forms a monoid with the
173:, where the operation is associative, or as a generalization of
111:, denotes the result of applying the semigroup operation to the
2078:
of the semigroup there is a unique maximal subgroup containing
240:
gives an example of a semigroup that is not a monoid. Positive
4883:(1937). "On the immersion of an algebraic ring into a field".
1775:
is a monoid then quotient semigroup is a monoid with identity
1387:
is a function that preserves semigroup structure. A function
4936:"An account of Suschkewitsch's paper by Christopher Hollings"
2147:
of a semigroup that is closed under the semigroup operation.
1311:
is both a left ideal and a right ideal then it is called an
3570:
3273:
2628:, there is a finest congruence ~ such that the quotient of
1179:. Analogous to the above construction, for every semigroup
5310:
An introduction to probability theory and its applications
4569:
See references in Udo
Hebisch and Hanns Joachim Weinert,
4919:"Earliest Known Uses of Some of the Words of Mathematics"
4855:
Personal reminiscences of the early history of semigroups
2945:
is commutative this condition is also sufficient and the
1401:
between two semigroups is a homomorphism if the equation
2242:
can be represented by transformations of a (state-) set
826:: there is essentially only one (specifically, only one
4647:
Kilp, Mati; Knauer, U.; Mikhalev, Aleksandr V. (2000).
2428:. A semilattice (or more precisely a meet-semilattice)
911:
of strings as the semigroup operation – the so-called "
3484:
3332:
3005:
5478:
Inverse semigroups: the theory of partial symmetries
4627:
4625:
3795:, as well as monographs focusing on applications in
5158:
4778:
Problems on mapping class groups and related topics
2957:
Semigroup methods in partial differential equations
2331:is in fact a monoid, which can be described as the
1821:, ordered by inclusion, has a maximal element. By
3576:
3452:
3279:
2933:. It is clearly necessary for embeddability that
4998:Transactions of the American Mathematical Society
2222:is a semigroup whose operation is idempotent and
1895:if there is a surjective semigroup morphism from
1611:. Isomorphic semigroups have the same structure.
4538:Namely: the trivial semigroup in which (for all
3799:, particularly for finite automata, and also in
2068:A subsemigroup that is also a group is called a
1802:is one that is the kernel of an endomorphism of
871:with addition. (With 0 included, this becomes a
857:: there are five that are essentially different.
4825:
4752:
2472:into a semigroup that satisfies the additional
2088:differs from its standard use in group theory.
1840:of a semigroup induces a factor semigroup, the
1359:that characterise the elements in terms of the
796:More succinctly, a semigroup is an associative
169:Semigroups may be considered a special case of
5100:. Vol. 1. American Mathematical Society.
4368:) is a generalization of a semigroup to a set
2804:. There is an obvious semigroup homomorphism
1785:first isomorphism theorem in universal algebra
126:. Associativity is formally expressed as that
4837:
3442:
3402:
3350:
687:
8:
4415:with any three adjacent elements bracketed.
3693:Élements de la Théorie des Groupes Abstraits
3170:
3158:
2529:is a (possibly empty) semigroup. Moreover,
2420:Structure theorem for commutative semigroups
1063:. Left and right identities are both called
962:acting as the identity. More generally, the
284:, do not resemble anything in group theory.
4930:
4928:
4736:Mathematical Foundations of Automata Theory
2895:to be the semigroup of subsets of some set
2827:to the corresponding generator. This has a
1568:is a monoid homomorphism. Particularly, if
1095:in a monoid formed by adjoining an element
889:of a given size with matrix multiplication.
4346:Generalizing in a different direction, an
3810:
2921:, this must be true for all generators of
2136:is a monoid in which every element has an
1526:is also a monoid with an identity element
1515:) is the identity element in the image of
1486:be a semigroup homomorphism. The image of
694:
680:
364:
5391:
4688:. American Mathematical Soc. p. 96.
4419:-ary associativity is a string of length
3558:
3494:
3493:
3485:
3483:
3441:
3440:
3401:
3400:
3392:
3365:
3349:
3348:
3331:
3217:
3054:
3049:
3018:
3006:
3004:
4801:Auslander, M.; Buchsbaum, D. A. (1974).
4720:
4708:
4616:
2592:by the equivalence relation ~ such that
2402:are sometimes called "mutually inverse".
1971:Both of those relations are transitive.
1825:, this is equivalent to saying that the
1756:is a semigroup homomorphism, called the
1253:.) In terms of this operation, a subset
1175:, which in semigroup theory is called a
311:, semigroups are fundamental models for
4764:
4598:
4522:
3758:of semigroups was developed in 1963 by
3320:be the second-derivative operator with
2316:. This representation is basic for any
1349:semigroup, when it exists, is a group.
1249:(This notion is defined identically as
1171:Similarly, every magma has at most one
367:
5312:. Vol. II (2nd ed.). Wiley.
4631:
4605:
3710:and showed that the minimal ideal (or
5461:. Vol. 1 (2nd ed.). Dover.
5216:Archive for History of Exact Sciences
1497:is a monoid with an identity element
7:
4733:Pin, Jean-Éric (November 30, 2016).
3628:to itself, taking the initial state
2652:becomes graded by this semilattice.
1633:that is an equivalence relation and
1576:, then it is a monoid homomorphism.
900:with the multiplication of the ring.
5335:Functional analysis and semi-groups
4321:, which is nothing more than a set
3624:) is a semigroup of operators from
2704:, since with this ordering we have
5133:The algebraic theory of semigroups
5097:The Algebraic Theory of Semigroups
4979:Miniconference on semigroup Theory
3046:
3015:
2929:) as well, which is therefore the
2211:is a semigroup whose operation is
1956:. In particular, subsemigroups of
1844:, via the congruence ρ defined by
350:; examples of the latter kind are
25:
5248:. American Mathematical Society.
4653:. Walter de Gruyter. p. 25.
1987:there is a smallest subsemigroup
1187:, a semigroup with 0 that embeds
354:and their commutative subclass –
303:, semigroups are associated with
5508:Algebraic combinatorics on words
5184:Grillet, Pierre Antoine (2001).
5155:Grillet, Pierre Antoine (1995).
5062:Fundamentals of Semigroup Theory
4805:. Harper & Row. p. 50.
4780:. Amer. Math. Soc. p. 357.
3780:, the semigroup of relations on
3697:Theory of Groups of Finite Order
3393:
2786:as generators and all equations
1952:is a quotient of a subsemigroup
1817:if any family of congruences on
1815:maximal condition on congruences
970:form a monoid under composition.
5026:Quasigroups and Related Systems
2839:and any semigroup homomorphism
1150:denotes a monoid obtained from
266:a Cayley theorem for semigroups
5244:Hollings, Christopher (2014).
4575:Semirings with infinite sums
4444:A third generalization is the
3548:
3542:
3529:
3523:
3511:
3505:
3431:
3425:
3416:
3410:
3397:
3386:
3374:
3371:
3342:
3336:
3255:
3243:
3229:
3223:
3207:
3195:
3138:
3126:
3101:
3089:
3075:
3063:
3039:
3027:
2973:ordinary differential equation
2969:partial differential equations
317:partial differential equations
1:
5428:Topics in Operator Semigroups
5340:American Mathematical Society
5138:American Mathematical Society
4573:, in particular, Section 10,
4553:and its counterpart in which
2835:to a group: given any group
2782:generated by the elements of
2440:where every pair of elements
2116:Special classes of semigroups
2110:Special classes of semigroups
1891:is a quotient of a semigroup
1376:Homomorphisms and congruences
907:over a fixed alphabet Σ with
862:semigroup with three elements
324:special classes of semigroups
313:linear time-invariant systems
268:realizing any semigroup as a
34:Algebraic structures between
5425:Kantorovitz, Shmuel (2009).
5124:Clifford, Alfred Hoblitzelle
5088:Clifford, Alfred Hoblitzelle
4205:Commutative-and-associative
2655:Furthermore, the components
2045:. A semigroup is said to be
360:ordered algebraic structures
289:theoretical computer science
191:is not necessarily equal to
4826:Clifford & Preston 1961
4753:Clifford & Preston 2010
4501:Quantum dynamical semigroup
2823:that sends each element of
2017:generates the subsemigroup
973:The product of faces of an
855:Semigroup with two elements
816:forms a semigroup with the
278:Jordan–Hölder decomposition
209:or (less often than in the
5565:
5273:Introduction to Semigroups
4571:Semirings and Semifields
4491:Light's associativity test
2960:
2917:holds for all elements of
2901:set-theoretic intersection
2676:, there exists an element
2113:
2033:. If this is finite, then
1334:. So the subsemigroups of
1330:is also a subsemigroup of
975:arrangement of hyperplanes
860:The "flip-flop" monoid: a
824:Semigroup with one element
336:semigroups with involution
5229:10.1007/s00407-009-0044-3
5018:"On some old problems in
3797:algebraic automata theory
3724:Evgenii Sergeevich Lyapin
2580:is onto, the semilattice
2468:. The operation ∧ makes
2370:has at least one inverse
2236:Transformation semigroups
1827:ascending chain condition
1154:by adjoining an identity
941:Transformation semigroups
5476:Lawson, Mark V. (1998).
3772:composition of relations
2853:, there exists a unique
2636:, we get a homomorphism
1789:string rewriting systems
1540:belongs to the image of
1490:is also a semigroup. If
1195:Subsemigroups and ideals
1091:without identity may be
924:probability distribution
820:as the binary operation.
270:transformation semigroup
211:analogous case of groups
5271:Petrich, Mario (1973).
5128:Preston, Gordon Bamford
5092:Preston, Gordon Bamford
5032:: 15–36. Archived from
4960:Matematicheskii Sbornik
4682:Li͡apin, E. S. (1968).
3673:infinitesimal generator
2238:: any finite semigroup
2125:is a semigroup with an
1975:Structure of semigroups
1880:Quotients and divisions
1597:semigroup homomorphism
1435:holds for all elements
344:cancellative semigroups
5275:. Charles E. Merrill.
5187:Commutative Semigroups
4803:Groups, rings, modules
3812:Group-like structures
3578:
3454:
3281:
2666:Archimedean semigroups
2152:cancellative semigroup
1462:without identity into
1221:, written commonly as
903:The set of all finite
804:Examples of semigroups
213:) it may be called an
67:
5371:Mathematische Annalen
3784:. In 1997 Schein and
3756:representation theory
3747:(currently edited by
3720:James Alexander Green
3579:
3455:
3282:
2961:Further information:
2939:cancellation property
2584:is isomorphic to the
2493:Given a homomorphism
2438:partially ordered set
2343:, under the relation
2256:states. Each element
2156:cancellation property
2041:, otherwise it is of
1842:Rees factor semigroup
1357:equivalence relations
740:) that satisfies the
207:commutative semigroup
203:matrix multiplication
33:
5549:Algebraic structures
5480:. World Scientific.
5366:Suschkewitsch, Anton
5016:Dudek, W.A. (2001).
4977:B. M. Schein (1972)
3616:. As a function of
3482:
3330:
3003:
2541:, in the sense that
2456:greatest lower bound
2322:finite-state machine
2314:function composition
2143:A subsemigroup is a
1934:divides a semigroup
1620:equivalence relation
1616:semigroup congruence
1065:one-sided identities
1001:(or more generally,
952:continuous functions
887:nonnegative matrices
867:The set of positive
742:associative property
591:Group with operators
534:Complemented lattice
369:Algebraic structures
307:. In other areas of
255:preserve from groups
5299:Specific references
5190:. Springer Verlag.
5064:. Clarendon Press.
4619:, p. 30, ex. 5
4481:Generalized inverse
3813:
3801:functional analysis
3059:
2831:for morphisms from
2413:commutative algebra
2103:Krohn–Rhodes theory
2009:. A single element
1456:monoid homomorphism
966:of any object of a
645:Composition algebra
405:Quasigroup and loop
332:orthodox semigroups
322:There are numerous
309:applied mathematics
276:, analogous to the
274:Krohn–Rhodes theory
76:algebraic structure
5393:10338.dmlcz/100078
5384:10.1007/BF01459084
5330:Phillips, Ralph S.
5051:General references
4897:10.1007/BF01571659
4838:Suschkewitsch 1928
4411:, i.e. the string
4366:multiary semigroup
4362:polyadic semigroup
4251:Commutative monoid
3811:
3793:inverse semigroups
3735:Alfred H. Clifford
3675:of the semigroup.
3671:is said to be the
3574:
3569:
3450:
3277:
3272:
3045:
2947:Grothendieck group
2880:. We may think of
2855:group homomorphism
2829:universal property
2796:that hold true in
2754:group of fractions
2748:Group of fractions
2406:Inverse semigroups
2364:Regular semigroups
2335:on two generators
2329:bicyclic semigroup
2186:and similarly for
2154:is one having the
1999:, and we say that
1796:nuclear congruence
1779:. Conversely, the
1733:quotient semigroup
1685:congruence classes
1593:if there exists a
1072:two-sided identity
1041:such that for all
1009:such that for all
931:convolution powers
929:together with all
833:), the singleton {
340:inverse semigroups
328:regular semigroups
301:probability theory
166:in the semigroup.
70:In mathematics, a
68:
5517:978-0-521-18071-9
5487:978-981-02-3316-7
5468:978-0-486-47189-1
5438:978-0-8176-4932-6
5282:978-0-675-09062-9
5255:978-1-4704-1493-1
5197:978-0-7923-7067-3
5168:978-0-8247-9662-4
5161:. Marcel Dekker.
5147:978-0-8218-0272-4
5107:978-0-8218-0271-7
5071:978-0-19-851194-6
4812:978-0-06-040387-4
4787:978-0-8218-3838-9
4776:Farb, B. (2006).
4695:978-0-8218-8641-0
4660:978-3-11-015248-7
4471:Compact semigroup
4461:Absorbing element
4315:
4314:
3712:Green's relations
3708:simple semigroups
3502:
2648:. As mentioned,
2037:is said to be of
1915:is a quotient of
1353:Green's relations
1183:, one can define
1173:absorbing element
987:Identity and zero
960:identity function
956:topological space
837:} with operation
711:A semigroup is a
704:
703:
358:, which are also
282:Green's relations
215:abelian semigroup
82:together with an
16:(Redirected from
5556:
5544:Semigroup theory
5529:
5499:
5472:
5455:Jacobson, Nathan
5450:
5421:
5395:
5361:
5321:
5294:
5267:
5240:
5209:
5180:
5151:
5119:
5083:
5038:
5037:
5013:
5007:
4994:
4988:
4975:
4969:
4956:
4950:
4949:
4947:
4941:. Archived from
4940:
4932:
4923:
4922:
4915:
4909:
4908:
4877:
4871:
4870:
4868:
4867:
4858:. Archived from
4846:
4840:
4835:
4829:
4823:
4817:
4816:
4798:
4792:
4791:
4773:
4767:
4762:
4756:
4750:
4744:
4743:
4741:
4730:
4724:
4718:
4712:
4706:
4700:
4699:
4679:
4673:
4672:
4644:
4638:
4629:
4620:
4614:
4608:
4603:
4586:
4567:
4561:
4559:
4552:
4536:
4530:
4527:
4496:Principal factor
4486:Identity element
4429:
4410:
4342:
4327:binary operation
4325:equipped with a
3814:
3764:binary relations
3732:
3703:Anton Sushkevich
3667:. The operator
3662:
3641:
3607:
3583:
3581:
3580:
3575:
3573:
3572:
3563:
3562:
3504:
3503:
3495:
3459:
3457:
3456:
3451:
3446:
3445:
3406:
3405:
3396:
3370:
3369:
3354:
3353:
3315:
3304:
3286:
3284:
3283:
3278:
3276:
3275:
3222:
3221:
3058:
3053:
3023:
3022:
2995:
2988:
2916:
2879:
2869:
2852:
2822:
2795:
2781:
2758:group completion
2743:
2732:
2722:
2696:
2686:
2620:
2601:
2572:
2528:
2522:
2506:
2489:
2467:
2453:
2435:
2393:
2383:
2366:. Every element
2349:
2303:
2281:
2255:
2253:
2203:
2185:
2175:
2127:identity element
2099:
2085:maximal subgroup
2032:
1947:
1926:
1914:
1863:
1853:
1755:
1745:
1737:factor semigroup
1662:
1652:
1642:
1632:
1610:
1563:
1485:
1430:
1400:
1361:principal ideals
1355:, a set of five
1340:complete lattice
1251:it is for groups
1248:
1220:
1211:, their product
1167:
1145:
1130:
1104:
1062:
1031:. Similarly, a
1030:
1005:) is an element
850:
791:
739:
720:binary operation
718:together with a
696:
689:
682:
471:Commutative ring
400:Rack and quandle
365:
305:Markov processes
297:syntactic monoid
226:identity element
200:
190:
153:
125:
106:
88:binary operation
78:consisting of a
64:identity element
21:
18:Semigroup theory
5564:
5563:
5559:
5558:
5557:
5555:
5554:
5553:
5534:
5533:
5532:
5518:
5502:
5488:
5475:
5469:
5453:
5439:
5424:
5364:
5350:
5324:
5306:Feller, William
5304:
5301:
5283:
5270:
5256:
5243:
5212:
5198:
5183:
5169:
5154:
5148:
5136:. Vol. 2.
5122:
5108:
5086:
5072:
5056:
5053:
5047:
5042:
5041:
5015:
5014:
5010:
5000:349(1): 271–85
4995:
4991:
4976:
4972:
4957:
4953:
4945:
4938:
4934:
4933:
4926:
4917:
4916:
4912:
4879:
4878:
4874:
4865:
4863:
4848:
4847:
4843:
4836:
4832:
4824:
4820:
4813:
4800:
4799:
4795:
4788:
4775:
4774:
4770:
4763:
4759:
4751:
4747:
4739:
4732:
4731:
4727:
4719:
4715:
4707:
4703:
4696:
4681:
4680:
4676:
4661:
4646:
4645:
4641:
4630:
4623:
4615:
4611:
4604:
4600:
4595:
4590:
4589:
4568:
4564:
4554:
4547:
4537:
4533:
4528:
4524:
4519:
4476:Empty semigroup
4457:
4420:
4380:
4330:
4329:that is closed
3809:
3807:Generalizations
3779:
3749:Springer Verlag
3744:Semigroup Forum
3726:
3681:
3661:
3643:
3636:
3634:
3606:
3588:
3568:
3567:
3554:
3536:
3535:
3486:
3480:
3479:
3361:
3328:
3327:
3313:
3291:
3271:
3270:
3235:
3213:
3189:
3188:
3150:
3120:
3119:
3081:
3014:
3007:
3001:
3000:
2990:
2983:
2979:on the spatial
2965:
2959:
2951:Anatoly Maltsev
2904:
2871:
2857:
2840:
2805:
2787:
2768:
2760:of a semigroup
2750:
2738:
2724:
2723:if and only if
2705:
2688:
2681:
2663:
2603:
2602:if and only if
2593:
2571:
2558:
2550:
2542:
2518:
2516:
2508:
2494:
2477:
2459:
2441:
2429:
2422:
2394:; the elements
2385:
2375:
2374:that satisfies
2357:
2344:
2287:
2269:
2249:
2247:
2187:
2177:
2159:
2138:inverse element
2118:
2112:
2097:
2018:
1979:For any subset
1977:
1939:
1916:
1904:
1903:. For example,
1882:
1855:
1845:
1778:
1754:
1747:
1740:
1727:
1723:
1719:
1692:
1654:
1644:
1634:
1623:
1598:
1587:are said to be
1579:Two semigroups
1562:
1555:
1545:
1539:
1532:
1525:
1514:
1503:
1496:
1484:
1477:
1467:
1405:
1388:
1378:
1317:two-sided ideal
1299:is a subset of
1284:is a subset of
1269:is a subset of
1226:
1212:
1207:of a semigroup
1197:
1168:for a monoid).
1159:
1146:. The notation
1132:
1110:
1096:
1050:
1018:
997:of a semigroup
989:
984:
838:
810:Empty semigroup
806:
765:
764:, the equation
726:
709:
700:
671:
670:
669:
640:Non-associative
622:
611:
610:
600:
580:
569:
568:
557:Map of lattices
553:
549:Boolean algebra
544:Heyting algebra
518:
507:
506:
500:
481:Integral domain
445:
434:
433:
427:
381:
293:finite automata
192:
182:
127:
115:
98:
28:
23:
22:
15:
12:
11:
5:
5562:
5560:
5552:
5551:
5546:
5536:
5535:
5531:
5530:
5516:
5500:
5486:
5473:
5467:
5451:
5437:
5422:
5362:
5349:978-0821874646
5348:
5322:
5300:
5297:
5296:
5295:
5281:
5268:
5254:
5241:
5223:(5): 497–536.
5210:
5196:
5181:
5167:
5152:
5146:
5120:
5106:
5084:
5070:
5058:Howie, John M.
5052:
5049:
5048:
5046:
5043:
5040:
5039:
5036:on 2009-07-14.
5008:
4989:
4970:
4951:
4948:on 2009-10-25.
4924:
4910:
4872:
4850:Preston, G. B.
4841:
4830:
4818:
4811:
4793:
4786:
4768:
4757:
4745:
4725:
4713:
4701:
4694:
4674:
4659:
4639:
4621:
4609:
4597:
4596:
4594:
4591:
4588:
4587:
4562:
4531:
4521:
4520:
4518:
4515:
4514:
4513:
4508:
4506:Semigroup ring
4503:
4498:
4493:
4488:
4483:
4478:
4473:
4468:
4463:
4456:
4453:
4377:-ary operation
4351:-ary semigroup
4313:
4312:
4309:
4306:
4303:
4300:
4297:
4291:
4290:
4287:
4284:
4281:
4278:
4275:
4269:
4268:
4265:
4262:
4259:
4256:
4253:
4247:
4246:
4243:
4240:
4237:
4234:
4231:
4225:
4224:
4221:
4218:
4215:
4212:
4209:
4202:
4201:
4198:
4195:
4192:
4189:
4186:
4179:
4178:
4175:
4172:
4169:
4166:
4163:
4156:
4155:
4152:
4149:
4146:
4143:
4140:
4134:
4133:
4130:
4127:
4124:
4121:
4118:
4111:
4110:
4107:
4104:
4101:
4098:
4095:
4089:
4088:
4085:
4082:
4079:
4076:
4073:
4066:
4065:
4062:
4059:
4056:
4053:
4050:
4044:
4043:
4040:
4037:
4034:
4031:
4028:
4021:
4020:
4017:
4014:
4011:
4008:
4005:
3999:
3998:
3995:
3992:
3989:
3986:
3983:
3976:
3975:
3972:
3969:
3966:
3963:
3960:
3954:
3953:
3950:
3947:
3944:
3941:
3938:
3931:
3930:
3927:
3924:
3921:
3918:
3915:
3909:
3908:
3905:
3902:
3899:
3896:
3893:
3891:Small category
3887:
3886:
3883:
3880:
3877:
3874:
3871:
3865:
3864:
3861:
3858:
3855:
3852:
3849:
3843:
3842:
3837:
3832:
3827:
3822:
3817:
3808:
3805:
3786:Ralph McKenzie
3775:
3739:Gordon Preston
3680:
3677:
3659:
3632:
3604:
3585:
3584:
3571:
3566:
3561:
3557:
3553:
3550:
3547:
3544:
3541:
3538:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3501:
3498:
3492:
3491:
3489:
3461:
3460:
3449:
3444:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3404:
3399:
3395:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3368:
3364:
3360:
3357:
3352:
3347:
3344:
3341:
3338:
3335:
3288:
3287:
3274:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3234:
3231:
3228:
3225:
3220:
3216:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3057:
3052:
3048:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3021:
3017:
3013:
3012:
3010:
2958:
2955:
2749:
2746:
2659:
2563:
2554:
2546:
2512:
2421:
2418:
2417:
2416:
2409:
2403:
2361:
2355:
2351:
2333:free semigroup
2325:
2286:is defined by
2233:
2227:
2216:
2205:
2148:
2141:
2130:
2114:Main article:
2111:
2108:
2043:infinite order
1995:that contains
1976:
1973:
1881:
1878:
1813:satisfies the
1776:
1752:
1746:. The mapping
1739:, and denoted
1729:
1728:
1725:
1721:
1717:
1711:
1710:
1690:
1560:
1553:
1537:
1530:
1523:
1512:
1501:
1494:
1482:
1475:
1433:
1432:
1377:
1374:
1305:
1304:
1289:
1274:
1196:
1193:
1037:is an element
1034:right identity
988:
985:
983:
982:Basic concepts
980:
979:
978:
971:
948:
938:
920:
913:free semigroup
901:
890:
883:
876:
865:
858:
852:
821:
818:empty function
805:
802:
794:
793:
722:⋅ (that is, a
708:
705:
702:
701:
699:
698:
691:
684:
676:
673:
672:
668:
667:
662:
657:
652:
647:
642:
637:
631:
630:
629:
623:
617:
616:
613:
612:
609:
608:
605:Linear algebra
599:
598:
593:
588:
582:
581:
575:
574:
571:
570:
567:
566:
563:Lattice theory
559:
552:
551:
546:
541:
536:
531:
526:
520:
519:
513:
512:
509:
508:
499:
498:
493:
488:
483:
478:
473:
468:
463:
458:
453:
447:
446:
440:
439:
436:
435:
426:
425:
420:
415:
409:
408:
407:
402:
397:
388:
382:
376:
375:
372:
371:
257:the notion of
95:multiplication
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5561:
5550:
5547:
5545:
5542:
5541:
5539:
5527:
5523:
5519:
5513:
5509:
5505:
5501:
5497:
5493:
5489:
5483:
5479:
5474:
5470:
5464:
5460:
5459:Basic algebra
5456:
5452:
5448:
5444:
5440:
5434:
5430:
5429:
5423:
5419:
5415:
5411:
5407:
5403:
5399:
5394:
5389:
5385:
5381:
5377:
5373:
5372:
5367:
5363:
5359:
5355:
5351:
5345:
5341:
5337:
5336:
5331:
5327:
5323:
5319:
5315:
5311:
5307:
5303:
5302:
5298:
5292:
5288:
5284:
5278:
5274:
5269:
5265:
5261:
5257:
5251:
5247:
5242:
5238:
5234:
5230:
5226:
5222:
5218:
5217:
5211:
5207:
5203:
5199:
5193:
5189:
5188:
5182:
5178:
5174:
5170:
5164:
5160:
5159:
5153:
5149:
5143:
5139:
5135:
5134:
5129:
5125:
5121:
5117:
5113:
5109:
5103:
5099:
5098:
5093:
5089:
5085:
5081:
5077:
5073:
5067:
5063:
5059:
5055:
5054:
5050:
5044:
5035:
5031:
5027:
5023:
5021:
5012:
5009:
5006:
5003:
4999:
4993:
4990:
4987:
4984:
4980:
4974:
4971:
4968:
4965:
4961:
4955:
4952:
4944:
4937:
4931:
4929:
4925:
4920:
4914:
4911:
4906:
4902:
4898:
4894:
4890:
4886:
4885:Math. Annalen
4882:
4876:
4873:
4862:on 2009-01-09
4861:
4857:
4856:
4851:
4845:
4842:
4839:
4834:
4831:
4827:
4822:
4819:
4814:
4808:
4804:
4797:
4794:
4789:
4783:
4779:
4772:
4769:
4766:
4761:
4758:
4754:
4749:
4746:
4742:. p. 19.
4738:
4737:
4729:
4726:
4723:, p. 465
4722:
4721:Lothaire 2011
4717:
4714:
4711:, p. 463
4710:
4709:Lothaire 2011
4705:
4702:
4697:
4691:
4687:
4686:
4678:
4675:
4670:
4666:
4662:
4656:
4652:
4651:
4643:
4640:
4637:
4633:
4628:
4626:
4622:
4618:
4617:Jacobson 2009
4613:
4610:
4607:
4602:
4599:
4592:
4584:
4580:
4576:
4572:
4566:
4563:
4557:
4550:
4545:
4541:
4535:
4532:
4526:
4523:
4516:
4512:
4509:
4507:
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4467:
4466:Biordered set
4464:
4462:
4459:
4458:
4454:
4452:
4449:
4447:
4442:
4440:
4438:
4433:
4427:
4423:
4418:
4414:
4408:
4404:
4400:
4396:
4392:
4388:
4384:
4378:
4376:
4371:
4367:
4363:
4359:
4357:
4352:
4350:
4344:
4341:
4337:
4333:
4328:
4324:
4320:
4310:
4307:
4304:
4301:
4298:
4296:
4295:Abelian group
4293:
4292:
4288:
4285:
4282:
4279:
4276:
4274:
4271:
4270:
4266:
4263:
4260:
4257:
4254:
4252:
4249:
4248:
4244:
4241:
4238:
4235:
4232:
4230:
4227:
4226:
4222:
4219:
4216:
4213:
4210:
4208:
4204:
4203:
4199:
4196:
4193:
4190:
4187:
4185:
4181:
4180:
4176:
4173:
4170:
4167:
4164:
4162:
4158:
4157:
4153:
4150:
4147:
4144:
4141:
4139:
4136:
4135:
4131:
4128:
4125:
4122:
4119:
4117:
4113:
4112:
4108:
4105:
4102:
4099:
4096:
4094:
4091:
4090:
4086:
4083:
4080:
4077:
4074:
4072:
4068:
4067:
4063:
4060:
4057:
4054:
4051:
4049:
4046:
4045:
4041:
4038:
4035:
4032:
4029:
4027:
4023:
4022:
4018:
4015:
4012:
4009:
4006:
4004:
4001:
4000:
3996:
3993:
3990:
3987:
3984:
3982:
3978:
3977:
3973:
3970:
3967:
3964:
3961:
3959:
3956:
3955:
3951:
3948:
3945:
3942:
3939:
3937:
3933:
3932:
3928:
3925:
3922:
3919:
3916:
3914:
3911:
3910:
3906:
3903:
3900:
3897:
3894:
3892:
3889:
3888:
3884:
3881:
3878:
3875:
3872:
3870:
3867:
3866:
3862:
3859:
3856:
3853:
3850:
3848:
3847:Partial magma
3845:
3844:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3815:
3806:
3804:
3802:
3798:
3794:
3789:
3787:
3783:
3778:
3773:
3769:
3765:
3761:
3757:
3752:
3750:
3746:
3745:
3740:
3736:
3730:
3725:
3721:
3717:
3713:
3709:
3704:
3700:
3698:
3694:
3690:
3686:
3678:
3676:
3674:
3670:
3666:
3658:
3654:
3650:
3646:
3642:to the state
3639:
3631:
3627:
3623:
3619:
3615:
3611:
3603:
3599:
3595:
3591:
3564:
3559:
3555:
3551:
3545:
3539:
3532:
3526:
3520:
3517:
3514:
3508:
3499:
3496:
3487:
3478:
3477:
3476:
3474:
3470:
3469:Sobolev space
3466:
3447:
3437:
3434:
3428:
3422:
3419:
3413:
3407:
3389:
3383:
3380:
3377:
3366:
3362:
3358:
3355:
3345:
3339:
3333:
3326:
3325:
3324:
3323:
3319:
3311:
3309:
3302:
3298:
3294:
3267:
3264:
3261:
3258:
3252:
3249:
3246:
3240:
3237:
3232:
3226:
3218:
3214:
3210:
3204:
3201:
3198:
3192:
3185:
3182:
3179:
3176:
3173:
3167:
3164:
3161:
3155:
3152:
3147:
3144:
3141:
3135:
3132:
3129:
3123:
3116:
3113:
3110:
3107:
3104:
3098:
3095:
3092:
3086:
3083:
3078:
3072:
3069:
3066:
3060:
3055:
3050:
3042:
3036:
3033:
3030:
3024:
3019:
3008:
2999:
2998:
2997:
2993:
2987:
2982:
2978:
2977:heat equation
2974:
2970:
2964:
2956:
2954:
2952:
2948:
2944:
2940:
2936:
2932:
2931:trivial group
2928:
2924:
2920:
2915:
2911:
2907:
2902:
2898:
2894:
2889:
2887:
2883:
2878:
2874:
2868:
2864:
2860:
2856:
2851:
2847:
2843:
2838:
2834:
2830:
2826:
2820:
2816:
2812:
2808:
2803:
2799:
2794:
2790:
2785:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2747:
2745:
2741:
2736:
2731:
2727:
2720:
2716:
2712:
2708:
2703:
2698:
2695:
2691:
2684:
2679:
2675:
2671:
2667:
2662:
2658:
2653:
2651:
2647:
2643:
2639:
2635:
2631:
2627:
2622:
2618:
2614:
2610:
2606:
2600:
2596:
2591:
2587:
2583:
2579:
2574:
2570:
2566:
2562:
2557:
2553:
2549:
2545:
2540:
2536:
2532:
2526:
2521:
2515:
2511:
2505:
2501:
2497:
2491:
2488:
2484:
2480:
2475:
2471:
2466:
2462:
2457:
2452:
2448:
2444:
2439:
2433:
2427:
2419:
2414:
2410:
2407:
2404:
2401:
2397:
2392:
2388:
2382:
2378:
2373:
2369:
2365:
2362:
2359:
2352:
2347:
2342:
2338:
2334:
2330:
2326:
2323:
2319:
2315:
2311:
2307:
2302:
2298:
2294:
2290:
2285:
2282:and sequence
2280:
2276:
2272:
2267:
2263:
2259:
2252:
2245:
2241:
2237:
2234:
2231:
2228:
2225:
2221:
2217:
2214:
2210:
2206:
2202:
2198:
2194:
2190:
2184:
2180:
2174:
2170:
2166:
2162:
2157:
2153:
2149:
2146:
2142:
2139:
2135:
2131:
2128:
2124:
2120:
2119:
2117:
2109:
2107:
2105:
2104:
2094:
2089:
2087:
2086:
2081:
2077:
2073:
2072:
2066:
2064:
2060:
2056:
2052:
2048:
2044:
2040:
2036:
2030:
2026:
2022:
2016:
2012:
2008:
2005:
2002:
1998:
1994:
1990:
1986:
1982:
1974:
1972:
1969:
1967:
1963:
1959:
1955:
1951:
1946:
1942:
1937:
1933:
1928:
1924:
1920:
1912:
1908:
1902:
1898:
1894:
1890:
1885:
1879:
1877:
1875:
1871:
1867:
1862:
1858:
1852:
1848:
1843:
1839:
1834:
1832:
1828:
1824:
1820:
1816:
1812:
1807:
1805:
1801:
1797:
1792:
1790:
1786:
1782:
1774:
1770:
1766:
1765:
1759:
1750:
1743:
1738:
1734:
1716:
1715:
1714:
1708:
1704:
1700:
1696:
1689:
1688:
1687:
1686:
1682:
1678:
1674:
1670:
1666:
1661:
1657:
1651:
1647:
1641:
1637:
1631:
1627:
1621:
1617:
1612:
1609:
1605:
1601:
1596:
1592:
1591:
1586:
1582:
1577:
1575:
1571:
1567:
1559:
1552:
1548:
1543:
1536:
1529:
1522:
1518:
1511:
1507:
1500:
1493:
1489:
1481:
1474:
1470:
1465:
1461:
1457:
1452:
1450:
1446:
1442:
1438:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1403:
1402:
1399:
1395:
1391:
1386:
1385:
1375:
1373:
1371:
1370:
1364:
1362:
1358:
1354:
1350:
1348:
1343:
1341:
1337:
1333:
1329:
1325:
1320:
1318:
1314:
1310:
1302:
1298:
1294:
1290:
1287:
1283:
1279:
1275:
1272:
1268:
1264:
1260:
1259:
1258:
1256:
1252:
1246:
1242:
1238:
1234:
1230:
1225:, is the set
1224:
1219:
1215:
1210:
1206:
1202:
1194:
1192:
1190:
1186:
1182:
1178:
1174:
1169:
1166:
1162:
1157:
1153:
1149:
1143:
1139:
1135:
1129:
1125:
1121:
1117:
1113:
1109:and defining
1108:
1103:
1099:
1094:
1090:
1085:
1083:
1082:
1077:
1073:
1068:
1066:
1061:
1057:
1053:
1048:
1044:
1040:
1036:
1035:
1029:
1025:
1021:
1016:
1012:
1008:
1004:
1000:
996:
995:
994:left identity
986:
981:
976:
972:
969:
965:
964:endomorphisms
961:
957:
953:
949:
946:
942:
939:
936:
932:
928:
925:
921:
918:
914:
910:
909:concatenation
906:
902:
899:
895:
891:
888:
884:
881:
877:
874:
870:
866:
863:
859:
856:
853:
849:
845:
841:
836:
832:
829:
825:
822:
819:
815:
811:
808:
807:
803:
801:
799:
789:
785:
781:
777:
773:
769:
763:
759:
755:
751:
747:
746:
745:
743:
738:
734:
730:
725:
721:
717:
714:
706:
697:
692:
690:
685:
683:
678:
677:
675:
674:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
632:
628:
625:
624:
620:
615:
614:
607:
606:
602:
601:
597:
594:
592:
589:
587:
584:
583:
578:
573:
572:
565:
564:
560:
558:
555:
554:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
525:
522:
521:
516:
511:
510:
505:
504:
497:
494:
492:
491:Division ring
489:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
457:
454:
452:
449:
448:
443:
438:
437:
432:
431:
424:
421:
419:
416:
414:
413:Abelian group
411:
410:
406:
403:
401:
398:
396:
392:
389:
387:
384:
383:
379:
374:
373:
370:
366:
363:
361:
357:
353:
349:
348:trivial group
345:
341:
337:
333:
329:
325:
320:
318:
314:
310:
306:
302:
298:
294:
290:
285:
283:
279:
275:
271:
267:
262:
260:
256:
252:
247:
243:
239:
235:
234:concatenation
231:
227:
223:
218:
216:
212:
208:
204:
199:
195:
189:
185:
180:
176:
172:
167:
165:
161:
157:
151:
147:
143:
139:
135:
131:
123:
119:
114:
110:
105:
101:
96:
91:
89:
85:
81:
77:
73:
65:
61:
57:
53:
52:associativity
49:
45:
41:
37:
32:
19:
5507:
5504:Lothaire, M.
5477:
5458:
5431:. Springer.
5427:
5378:(1): 30–50.
5375:
5369:
5334:
5326:Hille, Einar
5309:
5272:
5245:
5220:
5214:
5186:
5157:
5132:
5096:
5061:
5034:the original
5029:
5025:
5022:-ary groups"
5019:
5011:
4992:
4978:
4973:
4962:60: 292–303
4954:
4943:the original
4913:
4888:
4884:
4875:
4864:. Retrieved
4860:the original
4854:
4844:
4833:
4828:, p. 34
4821:
4802:
4796:
4777:
4771:
4765:Grillet 2001
4760:
4748:
4735:
4728:
4716:
4704:
4684:
4677:
4649:
4642:
4612:
4601:
4582:
4581:in place of
4578:
4574:
4570:
4565:
4555:
4548:
4543:
4539:
4534:
4525:
4511:Weak inverse
4450:
4446:semigroupoid
4443:
4436:
4431:
4425:
4421:
4416:
4412:
4406:
4402:
4398:
4394:
4390:
4386:
4382:
4374:
4369:
4365:
4361:
4355:
4354:
4348:
4347:
4345:
4339:
4335:
4331:
4322:
4316:
4182:Associative
4160:
4159:Commutative
4137:
4114:Commutative
4071:unital magma
4069:Commutative
4048:Unital magma
4024:Commutative
3979:Commutative
3934:Commutative
3869:Semigroupoid
3835:Cancellation
3790:
3781:
3776:
3767:
3760:Boris Schein
3753:
3742:
3701:
3696:
3692:
3682:
3668:
3664:
3656:
3652:
3648:
3644:
3637:
3629:
3625:
3621:
3617:
3613:
3601:
3597:
3593:
3589:
3586:
3472:
3464:
3462:
3317:
3307:
3300:
3296:
3292:
3289:
2991:
2985:
2966:
2963:C0-semigroup
2942:
2934:
2926:
2922:
2918:
2913:
2909:
2905:
2896:
2892:
2890:
2885:
2881:
2876:
2872:
2866:
2862:
2858:
2849:
2845:
2841:
2836:
2832:
2824:
2818:
2814:
2810:
2806:
2797:
2792:
2788:
2783:
2777:
2773:
2769:
2761:
2757:
2753:
2751:
2739:
2734:
2729:
2725:
2718:
2714:
2710:
2706:
2701:
2699:
2693:
2689:
2682:
2677:
2673:
2669:
2660:
2656:
2654:
2649:
2645:
2641:
2637:
2633:
2629:
2625:
2623:
2616:
2612:
2608:
2604:
2598:
2594:
2589:
2581:
2577:
2575:
2568:
2564:
2560:
2555:
2551:
2547:
2543:
2538:
2534:
2530:
2524:
2519:
2513:
2509:
2503:
2499:
2495:
2492:
2486:
2482:
2478:
2469:
2464:
2460:
2450:
2446:
2442:
2431:
2426:semilattices
2423:
2399:
2395:
2390:
2386:
2380:
2376:
2371:
2367:
2345:
2340:
2336:
2309:
2305:
2300:
2296:
2292:
2288:
2283:
2278:
2274:
2270:
2268:into itself
2265:
2261:
2257:
2250:
2243:
2239:
2200:
2196:
2192:
2188:
2182:
2178:
2172:
2168:
2164:
2160:
2101:
2090:
2083:
2079:
2075:
2069:
2067:
2046:
2042:
2039:finite order
2038:
2034:
2028:
2024:
2020:
2014:
2010:
2006:
2003:
2000:
1996:
1992:
1988:
1984:
1980:
1978:
1970:
1965:
1961:
1957:
1953:
1949:
1944:
1940:
1935:
1931:
1930:A semigroup
1929:
1922:
1918:
1910:
1906:
1900:
1896:
1892:
1888:
1887:A semigroup
1886:
1883:
1873:
1869:
1865:
1860:
1856:
1850:
1846:
1837:
1836:Every ideal
1835:
1830:
1823:Zorn's lemma
1818:
1814:
1810:
1809:A semigroup
1808:
1803:
1799:
1795:
1793:
1772:
1768:
1761:
1758:quotient map
1757:
1748:
1741:
1736:
1732:
1730:
1712:
1706:
1702:
1698:
1694:
1680:
1676:
1672:
1668:
1664:
1659:
1655:
1649:
1645:
1639:
1635:
1629:
1625:
1615:
1613:
1607:
1603:
1599:
1588:
1584:
1580:
1578:
1569:
1565:
1557:
1550:
1546:
1541:
1534:
1527:
1520:
1516:
1509:
1505:
1498:
1491:
1487:
1479:
1472:
1468:
1463:
1459:
1453:
1448:
1444:
1440:
1436:
1434:
1426:
1422:
1418:
1414:
1410:
1406:
1397:
1393:
1389:
1384:homomorphism
1381:
1379:
1367:
1365:
1351:
1344:
1335:
1331:
1327:
1323:
1321:
1316:
1312:
1308:
1306:
1300:
1296:
1292:
1285:
1281:
1277:
1270:
1266:
1263:subsemigroup
1262:
1254:
1244:
1240:
1236:
1232:
1228:
1222:
1217:
1213:
1208:
1204:
1200:
1198:
1188:
1184:
1180:
1176:
1170:
1164:
1160:
1156:if necessary
1155:
1151:
1147:
1141:
1137:
1133:
1127:
1123:
1119:
1115:
1111:
1106:
1101:
1097:
1088:
1087:A semigroup
1086:
1079:
1075:
1071:
1069:
1064:
1059:
1055:
1051:
1046:
1042:
1038:
1032:
1027:
1023:
1019:
1014:
1010:
1006:
998:
992:
990:
934:
926:
847:
843:
839:
834:
795:
787:
783:
779:
775:
771:
767:
761:
757:
753:
749:
736:
732:
728:
715:
710:
665:Hopf algebra
603:
596:Vector space
561:
501:
430:Group theory
428:
393: /
390:
356:semilattices
321:
286:
263:
219:
214:
206:
197:
193:
187:
183:
168:
163:
159:
155:
149:
145:
141:
137:
133:
129:
121:
117:
113:ordered pair
108:
107:, or simply
103:
99:
92:
71:
69:
59:
43:
4891:: 686–691.
4881:Maltsev, A.
4755:, p. 3
4632:Lawson 1998
4606:Feller 1971
3840:Commutative
3825:Associative
3727: [
3610:exponential
2474:idempotence
2358:-semigroups
2246:of at most
2232:semigroups.
2224:commutative
2220:semilattice
1347:commutative
1278:right ideal
950:The set of
917:free monoid
878:The set of
831:isomorphism
650:Lie algebra
635:Associative
539:Total order
529:Semilattice
503:Ring theory
251:quasigroups
179:commutative
84:associative
5538:Categories
5526:1221.68183
5496:1079.20505
5447:1187.47003
5291:0321.20037
5264:1317.20001
5206:1040.20048
5177:0830.20079
5116:0111.03403
5080:0835.20077
5045:References
4866:2009-05-12
4685:Semigroups
4669:0945.20036
4634:, p.
4579:semimodule
4439:-ary group
4358:-semigroup
4207:quasigroup
4184:quasigroup
4026:quasigroup
4003:Quasigroup
3716:David Rees
2989:and times
2687:such that
2458:, denoted
2264:then maps
2254:| + 1
2213:idempotent
2063:idempotent
1938:, denoted
1864:, or both
1854:if either
1769:projection
1764:surjection
1762:canonical
1663:for every
1590:isomorphic
1574:surjective
1382:semigroup
1293:left ideal
1257:is called
707:Definition
5506:(2011) .
5418:121081075
5402:0025-5831
5237:123422715
5130:(2010) .
4905:122295935
4593:Citations
4583:semigroup
4430:with any
4311:Required
4289:Unneeded
4267:Required
4245:Unneeded
4223:Required
4200:Unneeded
4177:Required
4161:semigroup
4154:Unneeded
4138:Semigroup
4132:Required
4109:Unneeded
4087:Required
4064:Unneeded
4042:Required
4019:Unneeded
3997:Required
3974:Unneeded
3952:Required
3929:Unneeded
3907:Unneeded
3885:Unneeded
3863:Unneeded
3766:on a set
3500:˙
3359:∈
3241:∈
3156:∈
3087:∈
3047:∂
3016:∂
2984:(0, 1) ⊂
2937:have the
2802:relations
2733:for some
2318:automaton
2304:for each
2051:monogenic
2004:generates
1595:bijective
1074:(or just
814:empty set
727:⋅ :
660:Bialgebra
466:Near-ring
423:Lie group
391:Semigroup
86:internal
72:semigroup
60:semigroup
44:semigroup
5457:(2009).
5332:(1974).
5308:(1971).
5094:(1961).
5060:(1995).
4852:(1990).
4455:See also
4308:Required
4305:Required
4302:Required
4299:Required
4286:Required
4283:Required
4280:Required
4277:Required
4264:Unneeded
4261:Required
4258:Required
4255:Required
4242:Unneeded
4239:Required
4236:Required
4233:Required
4220:Required
4217:Unneeded
4214:Required
4211:Required
4197:Required
4194:Unneeded
4191:Required
4188:Required
4174:Unneeded
4171:Unneeded
4168:Required
4165:Required
4151:Unneeded
4148:Unneeded
4145:Required
4142:Required
4129:Required
4126:Required
4123:Unneeded
4120:Required
4106:Required
4103:Required
4100:Unneeded
4097:Required
4084:Unneeded
4081:Required
4078:Unneeded
4075:Required
4061:Unneeded
4058:Required
4055:Unneeded
4052:Required
4039:Required
4036:Unneeded
4033:Unneeded
4030:Required
4016:Required
4013:Unneeded
4010:Unneeded
4007:Required
3994:Unneeded
3991:Unneeded
3988:Unneeded
3985:Required
3971:Unneeded
3968:Unneeded
3965:Unneeded
3962:Required
3949:Required
3946:Required
3943:Required
3940:Unneeded
3936:Groupoid
3926:Required
3923:Required
3920:Required
3917:Unneeded
3913:Groupoid
3904:Unneeded
3901:Required
3898:Required
3895:Unneeded
3882:Unneeded
3879:Unneeded
3876:Required
3873:Unneeded
3860:Unneeded
3857:Unneeded
3854:Unneeded
3851:Unneeded
3830:Identity
3663:at time
3651:) = exp(
3635:at time
3596:) = exp(
3316:and let
3299:((0, 1)
2981:interval
2941:. When
2861: :
2844: :
2809: :
2664:are all
2586:quotient
2533:becomes
2498: :
2273: :
2230:0-simple
2176:implies
2071:subgroup
2059:integers
2047:periodic
1960:divides
1653:implies
1618:~ is an
1602: :
1471: :
1392: :
1131:for all
1093:embedded
1076:identity
968:category
880:integers
869:integers
748:For all
724:function
496:Lie ring
461:Semiring
295:via the
259:division
246:integers
242:integers
154:for all
62:with an
5410:1512437
5358:0423094
5318:0270403
5005:1370647
4986:0401970
4967:0153760
4372:with a
3820:Closure
3679:History
3305:be the
2764:is the
1872:are in
1701:|
1564:, i.e.
1544:, then
1504:, then
1338:form a
1081:monoids
954:from a
945:monoids
919:over Σ.
905:strings
885:Square
627:Algebra
619:Algebra
524:Lattice
515:Lattice
238:strings
230:strings
90:on it.
5524:
5514:
5494:
5484:
5465:
5445:
5435:
5416:
5408:
5400:
5356:
5346:
5316:
5289:
5279:
5262:
5252:
5235:
5204:
5194:
5175:
5165:
5144:
5114:
5104:
5078:
5068:
4903:
4809:
4784:
4692:
4667:
4657:
4353:(also
4229:Monoid
3762:using
3685:groups
3620:, exp(
3463:where
3322:domain
3314:(0, 1)
2742:> 0
2685:> 0
2535:graded
2454:has a
2324:(FSM).
2248:|
2145:subset
2123:monoid
2055:cyclic
1781:kernel
1369:center
1315:(or a
873:monoid
812:: the
792:holds.
655:Graded
586:Module
577:Module
476:Domain
395:Monoid
315:. In
299:. In
222:monoid
175:groups
171:magmas
74:is an
56:monoid
40:groups
36:magmas
5414:S2CID
5233:S2CID
4946:(PDF)
4939:(PDF)
4901:S2CID
4740:(PDF)
4517:Notes
4413:abcde
4319:magma
4273:Group
3981:magma
3958:Magma
3731:]
3689:rings
3467:is a
3310:space
2899:with
2870:with
2766:group
2644:onto
2640:from
2436:is a
2295:) = (
2134:group
2098:{a, b
2093:ideal
1771:; if
1519:. If
1313:ideal
1288:, and
1003:magma
896:of a
894:ideal
828:up to
798:magma
621:-like
579:-like
517:-like
486:Field
444:-like
418:Magma
386:Group
380:-like
378:Group
352:bands
232:with
181:, so
58:is a
50:with
48:magma
46:is a
5512:ISBN
5482:ISBN
5463:ISBN
5433:ISBN
5398:ISSN
5344:ISBN
5277:ISBN
5250:ISBN
5192:ISBN
5163:ISBN
5142:ISBN
5102:ISBN
5066:ISBN
4807:ISBN
4782:ISBN
4690:ISBN
4655:ISBN
4542:and
4428:− 1)
4116:loop
4093:Loop
3770:and
3754:The
3737:and
3290:Let
3180:>
3111:>
2752:The
2737:and
2713:) ≤
2680:and
2611:) =
2476:law
2434:, ≤)
2398:and
2384:and
2339:and
2327:The
2209:band
2053:(or
1925:, +)
1913:, +)
1868:and
1643:and
1624:~ ⊆
1583:and
1556:) =
1533:and
1413:) =
1239:and
1203:and
1177:zero
943:and
898:ring
892:Any
774:) ⋅
451:Ring
442:Ring
342:and
162:and
136:) ⋅
54:. A
42:: A
38:and
5522:Zbl
5492:Zbl
5443:Zbl
5388:hdl
5380:doi
5287:Zbl
5260:Zbl
5225:doi
5202:Zbl
5173:Zbl
5112:Zbl
5076:Zbl
4893:doi
4889:113
4665:Zbl
4558:= b
4551:= a
4424:+ (
4407:cde
4395:bcd
4383:abc
4364:or
3687:or
3640:= 0
3612:of
2994:≥ 0
2800:as
2756:or
2588:of
2576:If
2537:by
2387:yxy
2377:xyx
2348:= 1
2320:or
2308:in
2260:of
2013:of
1991:of
1983:of
1948:if
1899:to
1798:on
1767:or
1744:/ ~
1735:or
1693:= {
1679:in
1572:is
1443:in
1322:If
1319:).
1307:If
1295:if
1280:if
1265:if
1243:in
1235:in
1140:∪ {
1105:to
1045:in
1013:in
933:of
782:⋅ (
713:set
456:Rng
144:⋅ (
97:):
80:set
5540::
5520:.
5490:.
5441:.
5412:.
5406:MR
5404:.
5396:.
5386:.
5376:99
5374:.
5354:MR
5352:.
5342:.
5338:.
5328:;
5314:MR
5285:.
5258:.
5231:.
5221:63
5219:.
5200:.
5171:.
5140:.
5126:;
5110:.
5090:;
5074:.
5028:.
5024:.
5002:MR
4983:MR
4981:,
4964:MR
4927:^
4899:.
4887:.
4663:.
4636:20
4624:^
4556:xy
4549:xy
4546:)
4441:.
4403:ab
4401:=
4389:=
4387:de
4360:,
4343:.
4338:→
4334:×
3803:.
3733:,
3729:fr
3722:,
3718:,
3699:.
3653:tA
3622:tA
3614:tA
3598:tA
3475::
3295:=
3268:0.
2996::
2912:=
2888:.
2877:fj
2875:=
2865:→
2848:→
2813:→
2791:=
2789:xy
2772:=
2744:.
2730:yz
2728:=
2697:.
2694:yz
2692:=
2674:y
2672:,
2597:~
2573:.
2559:⊆
2517:=
2502:→
2490:.
2485:=
2481:∧
2463:∧
2449:∈
2445:,
2389:=
2379:=
2346:pq
2297:qx
2293:xy
2284:xy
2277:→
2218:A
2207:A
2199:·
2195:=
2191:·
2181:=
2171:·
2167:=
2163:·
2158::
2150:A
2132:A
2121:A
2106:.
2027:∈
2023:|
2019:{
1968:.
1943:≼
1921:/4
1909:/2
1876:.
1859:=
1849:ρ
1833:.
1806:.
1794:A
1791:.
1760:,
1751:↦
1724:=
1720:∘
1705:~
1697:∈
1675:,
1671:,
1667:,
1660:yv
1658:~
1656:xu
1648:~
1638:~
1628:×
1614:A
1606:→
1478:→
1451:.
1439:,
1411:ab
1396:→
1380:A
1342:.
1297:SA
1291:a
1282:AS
1276:a
1267:AA
1261:a
1247:}.
1231:|
1229:ab
1227:{
1223:AB
1216:·
1191:.
1163:=
1136:∈
1126:=
1122:⋅
1118:=
1114:⋅
1100:∉
1070:A
1058:=
1054:⋅
1049:,
1026:=
1022:⋅
1017:,
991:A
922:A
875:.)
846:=
842:·
800:.
786:⋅
778:=
770:⋅
760:∈
756:,
752:,
744::
735:→
731:×
362:.
338:,
334:,
330:,
220:A
217:.
196:⋅
186:⋅
158:,
148:⋅
140:=
132:⋅
120:,
109:xy
102:⋅
5528:.
5498:.
5471:.
5449:.
5420:.
5390::
5382::
5360:.
5320:.
5293:.
5266:.
5239:.
5227::
5208:.
5179:.
5150:.
5118:.
5082:.
5030:8
5020:n
4921:.
4907:.
4895::
4869:.
4815:.
4790:.
4698:.
4671:.
4585:.
4544:y
4540:x
4437:n
4432:n
4426:n
4422:n
4417:n
4409:)
4405:(
4399:e
4397:)
4393:(
4391:a
4385:)
4381:(
4375:n
4370:G
4356:n
4349:n
4340:M
4336:M
4332:M
4323:M
3782:A
3777:A
3768:A
3669:A
3665:t
3660:0
3657:u
3655:)
3649:t
3647:(
3645:u
3638:t
3633:0
3630:u
3626:X
3618:t
3605:0
3602:u
3600:)
3594:t
3592:(
3590:u
3565:.
3560:0
3556:u
3552:=
3549:)
3546:0
3543:(
3540:u
3533:;
3530:)
3527:t
3524:(
3521:u
3518:A
3515:=
3512:)
3509:t
3506:(
3497:u
3488:{
3473:X
3465:H
3448:,
3443:}
3438:0
3435:=
3432:)
3429:1
3426:(
3423:u
3420:=
3417:)
3414:0
3411:(
3408:u
3403:|
3398:)
3394:R
3390:;
3387:)
3384:1
3381:,
3378:0
3375:(
3372:(
3367:2
3363:H
3356:u
3351:{
3346:=
3343:)
3340:A
3337:(
3334:D
3318:A
3308:L
3303:)
3301:R
3297:L
3293:X
3265:=
3262:t
3259:,
3256:)
3253:1
3250:,
3247:0
3244:(
3238:x
3233:,
3230:)
3227:x
3224:(
3219:0
3215:u
3211:=
3208:)
3205:x
3202:,
3199:t
3196:(
3193:u
3186:;
3183:0
3177:t
3174:,
3171:}
3168:1
3165:,
3162:0
3159:{
3153:x
3148:,
3145:0
3142:=
3139:)
3136:x
3133:,
3130:t
3127:(
3124:u
3117:;
3114:0
3108:t
3105:,
3102:)
3099:1
3096:,
3093:0
3090:(
3084:x
3079:,
3076:)
3073:x
3070:,
3067:t
3064:(
3061:u
3056:2
3051:x
3043:=
3040:)
3037:x
3034:,
3031:t
3028:(
3025:u
3020:t
3009:{
2992:t
2986:R
2943:S
2935:S
2927:S
2925:(
2923:G
2919:S
2914:A
2910:A
2908:.
2906:A
2897:X
2893:S
2886:S
2882:G
2873:k
2867:H
2863:G
2859:f
2850:H
2846:S
2842:k
2837:H
2833:S
2825:S
2821:)
2819:S
2817:(
2815:G
2811:S
2807:j
2798:S
2793:z
2784:S
2780:)
2778:S
2776:(
2774:G
2770:G
2762:S
2740:n
2735:z
2726:x
2721:)
2719:y
2717:(
2715:f
2711:x
2709:(
2707:f
2702:L
2690:x
2683:n
2678:z
2670:x
2661:a
2657:S
2650:S
2646:L
2642:S
2638:f
2634:L
2630:S
2626:S
2619:)
2617:y
2615:(
2613:f
2609:x
2607:(
2605:f
2599:y
2595:x
2590:S
2582:L
2578:f
2569:b
2567:∧
2565:a
2561:S
2556:b
2552:S
2548:a
2544:S
2539:L
2531:S
2527:}
2525:a
2523:{
2520:f
2514:a
2510:S
2504:L
2500:S
2496:f
2487:a
2483:a
2479:a
2470:L
2465:b
2461:a
2451:L
2447:b
2443:a
2432:L
2430:(
2415:.
2400:y
2396:x
2391:y
2381:x
2372:y
2368:x
2360:.
2356:0
2354:C
2350:.
2341:q
2337:p
2310:Q
2306:q
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