Showing papers on "Bicyclic semigroup published in 2006"
TL;DR: Solomon's approach to the semigroup algebra of a finite semilattice is extended via Mobius functions to arbitrary finite inverse semigroups, which allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroupgebra into a direct product of matrix algebras over group rings.
120 citations
TL;DR: In this article, the ideal structure of the Kauffman monoid is described in terms of the Jones monoid and a purely combinatorial numerical function with linearly ordered ideals.
Abstract: The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kauffman. Borisavljevic, Dosen, and Petric gave a complete proof of its abstract presentation by generators and relations, and suggested the name “Kauffman monoid”. We bring the theory of semigroups to the study of a certain finite homomorphic image of the Kauffman monoid. We show the homomorphic image (the Jones monoid) to be a combinatorial and regular *-semigroup with linearly ordered ideals. The Kauffman monoid is explicitly described in terms of the Jones monoid and a purely combinatorial numerical function. We use this to describe the ideal structure of the Kauffman monoid and two other of its homomorphic images.
62 citations
TL;DR: In this paper, the Brauer monoid and the greatest factorizable inverse submonoid of the dual symmetric inverse monoid are presented as Brauer-type monoids.
Abstract: We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.
50 citations
TL;DR: In this paper, a covering theorem of McAlister type is obtained for banded and free guarded semigroups, and a canonical forgetful functor from guarded semigroup to banded semigroup is given.
Abstract: The variety of guarded semigroups consists of all (S,·, ¯) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup S contains a canonical subsemilattice g⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g⋆ ≅ L is obtained.
31 citations
TL;DR: All subvarieties of the variety B2 generated by the five-element Brandt semigroup are characterized and an algorithm is provided that decides if a finite set of identities defines, within B2, a finitely generated subvariety.
Abstract: All subvarieties of the variety B2 generated by the five-element Brandt semigroup are characterized. Based on this characterization, an algorithm is provided that decides if a finite set of identities defines, within B2, a finitely generated subvariety.
28 citations
TL;DR: The Weierstrass semigroup H (P ) is well known and has been studied in this paper, and a renewed interest in these semigroups because of applications in coding theory.
27 citations
TL;DR: In this paper, it was shown that the semigroup algebra of an inverse semigroup is cellular if the group algebras of its maximal subgroups are cellular under certain compatability assumptions.
27 citations
TL;DR: In this article, a semigroup presentation of the singular part of the symmetric inverse monoid on a finite set was given, along with a monoid presentation of all order-preserving injective partial transformations on the finite chain.
Abstract: We give a semigroup presentation of the singular part of the symmetric inverse monoid on a finite set. Along the way, we derive a monoid presentation of the monoid of all order-preserving injective partial transformations on a finite chain, which differs from the presentation discovered by Fernandes.
26 citations
01 Jan 2006
TL;DR: In this paper, a short survey of the semigroup ring theory is presented, where the ring of coefficient R is a field and, for the main part of the results, a semigroup S is a numerical semigroup, i.e. an additive submonoid of N, with finite complement in N.
Abstract: Given a ring R and a semigroup S the semigroup ring R[S] inherits the properties of S and R. If no restrictions are posed on the semigroup S and on the ring R the class of semigroup rings is very large. There is a wide literature on this subject. Gilmer's book [15] is the classical reference in the commutative case, i.e. when both the ring R and the semigroup S are commutative. The book contains a deep study of the conditions under which the semigroup ring R[S] has given ring theoretic properties. The following short survey considers a very particular class of semigroup rings: the ring of coefficient R is supposed to be a field and, for the main part of the results, the semigroup S is supposed to be a numerical semigroup, i.e. an additive submonoid of N, with finite complement in N. Also within this particular class of semigroup algebras over a field, the paper deals only with some themes and it is far from being complete. Results and proofs are ranged through two different sources. On one side there is the classical ring theory, a rich, historically settled and well known theory. On the other side there are more elementary but simetimes quicker arguments which come from studying a less rich structure as that of semigroups. In doing so I hope to be not too far from R. Gilmer's open research attitude and from his tolerant view. Here, as in many other fields of human life, there is not a "Unique Thought", but several different points of view and techniques may coexist and be reciprocally useful.
19 citations
TL;DR: In this article, it was shown that the automorphisms of the endomorphism semigroup of a free inverse semigroup are inner and determine all isomorphisms between them.
Abstract: We prove that automorphisms of the endomorphism semigroup of a free inverse semigroup are inner and determine all isomorphisms between the endomorphism semigroups of free inverse semigroups.
17 citations
TL;DR: In this article, an algorithm is proposed which determines whether the given identity is satisfied in the bicylic semigroup BC Dh a; b j a 2 b D aba D a; ab 2 D bab D bi.
Abstract: To any given balanced semigroup identityv w a number of polyhedral convex cones are associated. In this setting an algorithm is proposed which determines whether the given identity is satisfied in the bicylic semigroup BC Dh a; b j a 2 b D aba D a; ab 2 D bab D bi
TL;DR: It is proved that a Smale space can be reconstructed from the adjacency semigroup of its Markov partition, using the notion of the limit solenoid of a contracting self-similar semigroup.
Abstract: Self-similar inverse semigroups are defined using automata theory. Adjacency semigroups of s-resolved Markov partitions of Smale spaces are introduced. It is proved that a Smale space can be reconstructed from the adjacency semigroup of its Markov partition, using the notion of the limit solenoid of a contracting self-similar semigroup. The notions of the limit solenoid and a contracting semigroup is described.
TL;DR: The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups and has been studied in this paper, where a new concept of restricted representations and corresponding C *-algebras are introduced.
Abstract: The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. This is the first in a series of papers in which we have investigated a similar relation on inverse semigroups. We use a new concept of “restricted” representations and study the restricted semigroup algebras and corresponding C *-algebras. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: A finite semigroup associated with a conjugacy class of a word in the free monoid over a finite alphabet is introduced and results on combinatorics on words are derived.
Abstract: We introduce a finite semigroup associated with a conjugacy class of a word in the free monoid over a finite alphabet. Using properties of this semigroup we derive results on combinatorics on words.
TL;DR: The topological finiteness condition finite derivation type (FDT) on the class of semigroups is introduced and it is proved that if a Rees matrix semigroup M has F DT then the semigroup S also has FDT.
TL;DR: Using a form of Zel'manov's solution of the restricted Burnside problem, it is shown that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups the authors call WMN.
Abstract: We characterize the structure of linear semigroups satisfying certain global and local nilpotence conditions and deduce various Engel-type results. For example, using a form of Zel'manov's solution of the restricted Burnside problem we are able to show that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups we call WMN. Methods of linear semigroups then allow us to prove that a linear semigroup is Mal'cev nilpotent precisely when it satisfies WMN. As an application, we show that a finitely generated associative algebra is nilpotent when viewed as a Lie algebra if and only if its adjoint semigroup is WMN.
TL;DR: It is proved that the above correspondence, restricted to balanced varieties, leads to a correspondence between balanced varieties and inverse monoids, and it is shown that Dehornoy's geometry monoid of a balanced variety is a special kind of inverse submonoid of the authors'.
Abstract: There is a well-known correspondence between varieties of algebras and fully invariant congruences on the appropriate term algebra A special class of varieties are those which are balanced, meaning they can be described by equations in which the same variables appear on each side In this paper, we prove that the above correspondence, restricted to balanced varieties, leads to a correspondence between balanced varieties and inverse monoids In the case of unary algebras, we recover the theorem of Meakin and Sapir that establishes a bijection between congruences on the free monoid with n generators and wide, positively self-conjugate inverse submonoids of the polycyclic monoid on n generators In the case of varieties generated by linear equations, meaning those equations where each variable occurs exactly once on each side, we can replace the clause monoid above by the linear clause monoid In the case of algebras with a single operation of arity n, we prove that the linear clause monoid is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated essential right ideals of the free monoid on n letters, a monoid previously studied by Birget in the course of work on the Thompson group V and its analogues We show that Dehornoy's geometry monoid of a balanced variety is a special kind of inverse submonoid of ours Finally, we construct groups from the inverse monoids associated with a balanced variety and examine some conditions under which they still reflect the structure of the underlying variety Both free groups and Thompson's groups Vn,1 arise in this way
TL;DR: In this paper, the concept of left wreath products of semigroups was introduced, and it was shown that the ℒ*-inverse semigroup can be described as the left wreaths product of a type A semigroup and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B).
Abstract: The concepts of ℒ*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the ℒ*-inverse semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the ℒ*-inverse semigroups by using the left wreath products.
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TL;DR: The 3 x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2 k + 1 3 k + 2 : k ⩾ 0 } together with { 2 } as mentioned in this paper.
TL;DR: The structure of -class of a finite parabolic monoid and it is shown that such a monoid is generated by its unit group and diagonal idempotents.
Abstract: We determine the closure of a parabolic subgroup of a reductive group in a reductive monoid. This allows us to define parabolic submonoids of a finite monoid of Lie type. These are analogues of the monoid of block upper triangular matrices. We determine the structure of -class of a finite parabolic monoid and show that such a monoid is generated by its unit group and diagonal idempotents.
TL;DR: This paper showed that the syntactic monoid of the loop problem is the inverse hull of the monoid in a semigroup with context-free loop problems, which is the case of semigroups with right cancellative monoids.
Abstract: We propose a way of associating to each finitely generated monoid or semigroup a formal language, called its loop problem. In the case of a group, the loop problem is essentially the same as the word problem in the sense of combinatorial group theory. Like the word problem for groups, the loop problem is regular if and only if the monoid is finite. We study also the case in which the loop problem is context-free, showing that a celebrated group-theoretic result of Muller and Schupp extends to describe completely simple semigroups with context-free loop problems. We consider also right cancellative monoids, establishing connections between the loop problem and the structural theory of these semigroups by showing that the syntactic monoid of the loop problem is the inverse hull of the monoid.
TL;DR: In this article, it was shown that a semigroup variety is D-compatible if and only if it is J-compatible, and the semigroup varieties which are minimal for not being D -compatible are all periodic and countably infinite in number.
TL;DR: In this article, the authors considered divisability and factorization into irreducibel elements in the partial semigroup of 2 × 2-matrices with entries in K[x] and determinant 1, where multiplication is defined as matrix-multiplication if the degrees of the factors add up.
Abstract: Let K be a field. We consider divisability and factorization into irreducibel elements in the partial semigroup of 2 × 2-matrices with entries in K[x] and determinant 1, where multiplication is defined as matrix-multiplication if the degrees of the factors add up, cf. Section 2. Our aim is to establish a unique factorization result, cf. Theorem 3.1. Although our considerations are purely algebraic and in fact quite elementary, they should be seen in connection with some results of complex analysis. Let us explain this motivation: Let W (z) = (wij(z))i,j=1,2 be a 2 × 2-matrix function whose entries are entire functions, i.e. are defined and holomorphic in the whole complex plane. We say that W belongs to the class Mκ where κ is a nonnegative integer, if wij(z) = wij(z), W (0) = I, detW (z) = 1, and if the kernel KW (w, z) := W (z)JW (w) − J z − w has κ negative squares. Thereby
TL;DR: In this paper, an example of a finitely generated semigroup $S$ such that the semigroup embeds in a group and the universal group is not automatic is given.
Abstract: Answering a question of Hoffmann and of Kambites, an example is exhibited of a finitely generated semigroup $S$ such that $S$ embeds in a group and $S$ is not automatic, but the universal group of $S$ is automatic.
TL;DR: In this paper, it was shown that a tightly connected fundamental inverse semigroup with no isolated nontrivial subgroups is lattice determined ''modulo semilattices'' and if the inverse monoid of a semigroup whose partial automorphism monoid is isomorphic to that of the semigroup is a monoid consisting of all isomorphisms between its inverse subsemigroups, then either the two monoids are isomorphic or they are dually isomorphic chains relative to the natural order.
Abstract: The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup $S$ with no isolated nontrivial subgroups is lattice determined `modulo semilattices' and if $T$ is an inverse semigroup whose partial automorphism monoid is isomorphic to that of $S$, then either $S$ and $T$ are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if $T$ is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of $S$ and $T$, respectively, are isomorphic. Moreover, for these results to hold, the conditions that $S$ be tightly connected and have no isolated nontrivial subgroups are essential.
TL;DR: In this article, it was shown that there are exactly three distinct maximal noncryptic semigroup varieties contained in the variety determined by xn ≈ x n+m, n ≥ 2, m ≥ 2.
Abstract: We describe all minimal noncryptic periodic semigroup [monoid] varieties We
prove that there are exactly three distinct maximal cryptic semigroup [monoid] varieties contained in the variety determined by xn ≈ x n+m, n ≥ 2, m ≥ 2 Analogous results are obtained for pseudovarieties: there are exactly three maximal cryptic pseudovarieties of semigroups [monoids] It is shown that if a cryptic variety or pseudovariety of monoids contains a nonabelian group, then it consists of bands of groups only Several characterizations are given of the cryptic overcommutative semigroup [monoid] varieties
TL;DR: In this article, it was shown that the complex semigroup algebra of a free monoid of rank at least two is -primitive, where denotes the involution on the algebra induced by word-reversal on the monoid.
Abstract: It is shown that the complex semigroup algebra of a free monoid of rank at least two is -primitive, where denotes the involution on the algebra induced by word-reversal on the monoid.
TL;DR: In this paper, the structure of a permutable Munn semigroup of finite rank is described, where congruences of which commute as binary relations are defined as permutable semigroups.
Abstract: A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank.