Showing papers on "Cancellative semigroup published in 2014"
01 Jun 2014
TL;DR: In this paper, the full and reduced C *-algebras of I l (S) are related to Li's C * -algesia, and conditions for their isomorphicity are given.
Abstract: To each discrete left cancellative semigroup S one may associate an inverse semigroup I l (S) , often called the left inverse hull of S . We show how the full and reduced C *-algebras of I l (S) are related to the full and reduced semigroup C *-algebras for S , recently introduced by Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.
47 citations
TL;DR: In this paper, the rank of a finite semigroup is defined as the smallest number of elements required to generate the semigroup, and the problem of determining the maximum rank of the subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.
Abstract: The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (not necessarily regular) Rees matrix semigroup over a group. The formula is expressed in terms of the dimensions of the structure matrix, and the relative rank of a certain subset of the structure group obtained from subgroups generated by entries in the structure matrix, which is assumed to be in Graham normal form. This formula is then applied to answer questions about minimal generating sets of certain natural families of transformation semigroups. In particular, the problem of determining the maximum rank of a subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.
40 citations
TL;DR: In this article, it was shown that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup.
40 citations
TL;DR: In this article, the question of when a semigroup is the semigroup of a valuation dominating a two-dimensional noetherian domain was considered, and a necessary and sufficient condition was given for the pair of semigroup S and a field extension L/k to be semigroup and residue field of a regular local ring R of dimension two with residue field k.
Abstract: We consider the question of when a semigroup is the semigroup of a valuation dominating a two dimensional noetherian domain, giving some surprising examples. We give a necessary and sufficient condition for the pair of a semigroup S and a field extension L/k to be the semigroup and residue field of a valuation dominating a regular local ring R of dimension two with residue field k, generalizing the theorem of Spivakovsky for the case when there is no residue field extension.
37 citations
TL;DR: In this paper, the authors introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra, which implies that near a dynamical equilibrium, the local normal form of a semi-giant network is the same as the original network itself.
Abstract: We introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra. This implies that near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself. Networks without the semigroup property will support normal forms with a more general network architecture, but these normal forms nevertheless possess the same symmetries and synchronous solutions as the original network. We explain how to compute Lie brackets and normal forms of coupled cell networks and we characterize the SN-decomposition that determines the normal form symmetry. This paper concludes with a generalization to nonhomogeneous networks with the structure of a semigroupoid.
37 citations
TL;DR: In this paper, the uniqueness or non-uniqueness of these factorizations is measured in terms of the uniqueness of factorizations in the integral domain of the cancellative semigroup of upper triangular matrices.
31 citations
TL;DR: In this paper, the submonoid of all n×n triangular tropical matrices satisfies a nontrivial semigroup identity and a generic construction for classes of such identities is provided.
Abstract: We show that the submonoid of all n×n triangular tropical matrices satisfies a nontrivial semigroup identity and provide a generic construction for classes of such identities The utilization of the Fibonacci number formula gives us an upper bound on the length of these 2-variable semigroup identities
30 citations
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TL;DR: In this paper, the authors studied C*-algebras associated to right LCM semigroups and showed that the C *-algebra admits a natural boundary quotient.
Abstract: We study C*-algebras associated to right LCM semigroups, that is, semigroups which are left cancellative and for which any two principal right ideals are either disjoint or intersect in another principal right ideal. If $P$ is such a semigroup, its C*-algebra admits a natural boundary quotient $\mathcal{Q}(P)$. We show that $\mathcal{Q}(P)$ is isomorphic to the tight C*-algebra of a certain inverse semigroup associated to $P$, and thus is isomorphic to the C*-algebra of an \'etale groupoid. We use this to give conditions on $P$ which guarantee that $\mathcal{Q}(P)$ is simple and purely infinite, and give applications to self-similar groups and Zappa-Sz\'ep products of semigroups.
29 citations
TL;DR: In this article, it was shown that the Cuntz semigroup of C(T,A) is determined by its Murray-von Neumann semigroup and a certain semigroup for lower semicontinuous functions.
Abstract: Let A be a simple, separable C � -algebra of stable rank one. We prove that the Cuntz semigroup of C(T,A) is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of A). This result has two consequences. First, specializing to the case that A is simple, finite, separable and Z-stable, this yields a description of the Cuntz semigroup of C(T,A) in terms of the Elliott invariant of A. Second, suitably interpreted, it shows that the Elliott functor and the functor de- fined by the Cuntz semigroup of the tensor product with the algebra of continuous functions on the circle are naturally equivalent.
27 citations
TL;DR: In this paper, the authors consider the problem of embedding semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra, and present A. I. Maltsev's example of a cancellative semigroup which may not be embedded in a group.
Abstract: We consider the investigation of the embedding of semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra. Although this is a simple problem to state, it has proved rather harder to solve, and its apparent simplicity caused some of its would-be solvers to go awry. We begin with the analogous problem for rings, as dealt with by Ernst Steinitz, B. L. van der Waerden and Oystein Ore. After disposing of A. K. Sushkevich’s erroneous contribution in this area, we present A. I. Maltsev’s example of a cancellative semigroup which may not be embedded in a group, which showed for the first time that such an embedding is not possible in general. We then look at the various conditions that were derived for such an embedding to take place: the sufficient conditions of Paul Dubreil and others, and the necessary and sufficient conditions obtained by A. I. Maltsev, Vlastimil Ptak and Joachim Lambek. We conclude with some comments on the place of this problem within the theory of semigroups, and also within abstract algebra more generally.
13 citations
TL;DR: In this paper, the Frobenius number can be removed from an irreducible numerical semigroup by removing some minimal generators from a semigroup with the same Frobenii number.
Abstract: Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number
TL;DR: In this paper, the affine near-semiring over a Brandt semigroup has been studied in terms of the Green's classes of its semigroup reducts and the size of A +(B n ).
Abstract: In order to study the structure of A +(B n )—the affine near-semiring over a Brandt semigroup—this work completely characterizes the Green's classes of its semigroup reducts. In this connection, this work classifies the elements of A +(B n ) and reports the size of A +(B n ). Further, idempotents and regular elements of the semigroup reducts of A +(B n ) have also been characterized and studied some relevant semigroups in A +(B n ).
TL;DR: In this paper, the authors study the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroup or semigroups that satisfy Clifford's condition.
Abstract: We initiate the study of the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Clifford's condition. Our main findings are results about uniqueness of the full semigroup C*-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on S under which C*(S) is purely infinite and simple.
TL;DR: In this paper, the authors characterize Cohen-Macaulay and Gorenstein rings obtained from certain types of convex body semigroups and provide algorithms to check if a polygonal or circle semigroup is CoMHA/Gorenstein.
Abstract: We characterize Cohen-Macaulay and Gorenstein rings obtained from certain types of convex body semigroups. Algorithmic methods to check if a polygonal or circle semigroup is Cohen-Macaulay/Gorenstein are given. We also provide some families of Cohen-Macaulay and Gorenstein rings.
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TL;DR: In this paper, the authors introduced a (conformable) fractional semigroup of operators whose generator will be the fractional derivative of the semigroup at a constant factor of 0.
Abstract: Let $X$ be a Banach space, and $T:[0,\infty)\rightarrow {\mathcal{L}}(X,X),$ the bounded linear operators on $X.$ A family $\{T(t)\}_{t\ge 0}\subseteq {% \mathcal{L}}(X,X)$ is called a one-parameter semigroup if $T(s+t)=T(s)T(t),$ and $T(0)=I,$ the identity operator on $X.$ The infinitesimal generator of the semigroup is the derivative of the semigroup at $t=0.$ The object of this paper is to introduce a (conformable) fractional semigroup of operators whose generator will be the fractional derivative of the semigroup at $t=0.$ The basic properties of such semigroups will be studied.
TL;DR: In this article, it was shown that any almost unitary subsemigroup of a semigroup is closed in the containing semigroup and that the class of all left [right] regular semigroups is closed.
Abstract: We show that any almost unitary subsemigroup of a semigroup is closed in the containing semigroup and that the class of all left [right] regular semigroups is closed. Finally, we show that every globally idempotent ideal satisfying a seminormal permutation identity of a supersaturated semigroup is supersaturated.
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TL;DR: The main purpose of this paper is to consider some methods for constructing Q-algebras and to give the concrete form of a free (unital) Q- algebra over a set (a semigroup, an ordered semigroup).
TL;DR: In this paper, the authors give characterizations of the closed subsemigroups of a Clifford semigroup and show that the class of all Clifford semigroups satisfies the strong isomorphism property and so is globally determined.
Abstract: In this paper we shall give characterizations of the closed subsemigroups of a Clifford semigroup. Also, we shall show that the class of all Clifford semigroups satisfies the strong isomorphism property and so is globally determined. Thus the results obtained by Kobayashi [‘Semilattices are globally determined’, Semigroup Forum29 (1984), 217–222] and by Gould and Iskra [‘Globally determined classes of semigroups’ Semigroup Forum28 (1984), 1–11] are generalized.
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TL;DR: In this paper, a semi-direct product groupoid with a Haar system was constructed for compact topological spaces, and it was shown that it is equivalent to a transformation groupoid.
Abstract: In this paper, we consider topological semigroup actions on compact topological spaces. Under mild assumptions on the semigroup and the action, we construct a semi-direct product groupoid with a Haar system. We also show that it is equivalent to a transformation groupoid. We apply this construction to the Wiener-Hopf $C^{*}$-algebras.
Journal Article•
TL;DR: In this paper, it was shown that if the set of idempotents of an inverse semigroup S is �nite, then S is Arens regular if and only if S is notite.
Abstract: We present a characterization of Arens regular semi- group algebras l 1 ( S ), for a large class of semigroups. Mainly, we show that if the set of idempotents of an inverse semigroup S is �nite, then l 1 ( S ) is Arens regular if and only if S isnite.
TL;DR: In this paper, a necessary and sufficient condition for a restriction semigroup to be embeddable in a W-product of a semilattice by a monoid is provided.
Abstract: A necessary and sufficient condition is provided for a (two-sided) restriction semigroup to be embeddable in a W-product of a semilattice by a monoid.
TL;DR: In this paper, the authors investigated the question of when the members of a finite regular semigroup may be permuted in such a way that each member is mapped to one of its inverses.
Abstract: We investigate the question as to when the members of a finite regular semigroup may be permuted in such a way that each member is mapped to one of its inverses. In general this is not possible. However we reformulate the problem in terms of a related graph and, using an application of Hall’s Marriage Lemma, we show in particular that the finite full transformation semigroup does enjoy this property.
TL;DR: In this article, it was proved that any indecomposable good matrix representation of an ample semigroup can be constructed by using those of weak Brandt semigroups, which is known as ample matrix representations.
Abstract: An adequate semigroup S is said to be ample if for any e2 = e, a ∈ S, ae = (ae)†a and ea = a(ea)*. It is well known that inverse semigroups are ample semigroups. The purpose of this paper is to study matrix representations of an ample semigroup. Some properties of ample semigroups are obtained. It is proved that any indecomposable good matrix representations of an ample semigroup can be constructed by using those of weak Brandt semigroups.
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TL;DR: In this paper, it was shown that the semigroup algebra is always a 2n-weakly module amenable as an inverse semigroup with the set of idempotents $E$.
Abstract: Let $S$ be an inverse semigroup with the set of idempotents $E$. We prove that the semigroup algebra $\ell^{1}(S)$ is always $2n$-weakly module amenable as an $\ell^{1}(E)$-module, for any $n\in \mathbb{N}$, where $E$ acts on $S$ trivially from the left and by multiplication from the right.
01 Jan 2014
TL;DR: Gorbatkov as discussed by the authors described the interassociates of the free semigroup over the two-element alphabet, and showed that a semigroup is called an interassociate if the following identities hold x · (y ◦ z) = (x · y) · z.
Abstract: A. B. Gorbatkov. Interassociates of a free semigroup on two generators, Mat. Stud. 41 (2014), 139–145. For any semigroup (S; ·) let (S; ◦) be a semigroup defined on the same set. Semigroup (S; ◦) is called an interassociate of (S; ·) if the following identities hold x · (y ◦ z) = (x · y) ◦ z and x ◦ (y · z) = (x ◦ y) · z. All interassociates of the free semigroup over the two-element alphabet are described.
TL;DR: An algorithm to compute the set of primitive elements for an embedding dimension three numerical semigroups is given and it is shown how this procedure is used in the study of the construction of L-shapes and the tame degree of the semigroup.
TL;DR: In this article, it is shown that the word problem is solvable for semigroup presentations with one defining relation if the thickness of the derivation diagram is bounded and if the number of regions in a presentation over a presentation with one relation is bounded.
Abstract: Briefly, a feather is a semigroup derivation diagram with the labels on the edges removed. In this paper, we are concerned with possible definitions for the thickness of a feather. A major open problem is whether the word problem is solvable for semigroup presentations with one defining relation. It is known that word problems for semigroup presentations are solvable if the number of regions in minimal derivation diagrams is bounded. For some definitions for thickness, the number of regions in a derivation diagram over a presentation with one relation will be bounded if the thickness of the diagram is bounded.
TL;DR: In this article, it was shown that the full transformation semigroup T ∃(X) is not abundant if X/E is infinite, and the semigroup is a subsemigroup of.
Abstract: Let be the full transformation semigroup on a nonempty set X and E be an equivalence relation on X. We write Then T∃(X) is a subsemigroup of . In this paper, we proved that the semigroup T∃(X) is not abundant if X/E is infinite.
TL;DR: In this paper, it was shown that the semigroup algebra of a finite locally ample semigroup is isomorphic to the semi-abstraction algebra of an associate primitive abundant semigroup.
Abstract: An idempotent-connected abundant semigroup S is a locally ample semigroup if for any idempotent e of S, the local submonoid eSe of S is an ample subsemigroup of S. Clearly, an ample semigroup is a locally ample semigroup. In this paper, it is proved that the semigroup algebra of a finite locally ample semigroup is isomorphic to the semigroup algebra of an associate primitive abundant semigroup. As an application of this result, we characterize Jacobson radicals of finite locally ample semigroup algebras.
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18 Feb 2014
TL;DR: In this article, the authors studied the non-uniqueness of non-zero-divisors into atoms (irreducibles) in non-commutative rings and gave necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization.
Abstract: We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. Several notions of factorizations as well as distances between them are intro- duced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the !-invariant, and the tame degree, are extended from commutative to noncom- mutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of n×n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.