Brandt Semigroup Research Articles

Sandwich semigroups and Brandt semigroups

Sandwich semigroups and Brandt semigroups

On some module amenability-like properties of Banach algebras with applications

In this paper, we introduce a new notion of module strong pseudo-amenability for Banach algebras. We study the relation between this new concept to other various notions at this issue, module pseudo-contractibility, module pseudo-amenability and module approximate amenability. For an inverse semigroup [Formula: see text] with the set of all idempotents [Formula: see text], we show that [Formula: see text] is module strong pseudo-amenable as an [Formula: see text]-module if and only if [Formula: see text] is amenable. For specific types of semigroups such as Brandt semigroups and bicyclic semigroups, we investigate the module strong pseudo-amenability of [Formula: see text]. We show that for every non-empty set [Formula: see text], [Formula: see text] under this new notion is forced to have a finite index as an [Formula: see text]-module, where [Formula: see text].

Генерический полиномиальный алгоритм для проблемы о сумме подмножеств в полугруппах Брандта

The article proposes a polynomial generic algorithm for the subset-sum problem in the Brandt semigroups over finitely presented groups with a polynomially solvable word problem.

Congruences on the partial automorphism monoid of a free group action

We study congruences on the partial automorphism monoid of a finite rank free group action. We determine a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. The constituent parts are further described in terms of subgroups of direct and semidirect products of groups. We utilize this description to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and the rank of the action.

On strong pseudo-amenability of some Banach algebras

In this paper, we introduce a new notion of strong pseudo-amenability for Banach algebras. We study strong pseudo-amenability of some matrix algebras. Using this tool, we characterize strong pseudo-amenability of [Formula: see text], provided that [Formula: see text] is a uniformly locally finite inverse semigroup. As an application, we show that for a Brandt semigroup [Formula: see text], [Formula: see text] is strong pseudo-amenable if and only if [Formula: see text] is amenable and [Formula: see text] is finite. We give some examples to show the differences between strong pseudo-amenability and the other classical notions of amenability.

From A to B to Z

The variety generated by the Brandt semigroup ${\bf B}_2$ can be defined within the variety generated by the semigroup ${\bf A}_2$ by the single identity $x^2y^2\approx y^2x^2$. Edmond Lee asked whether or not the same is true for the monoids ${\bf B}_2^1$ and ${\bf A}_2^1$. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of ${\bf A}_2^1$ that satisfy $x^2y^2\approx y^2x^2$ and contain ${\bf B}_2^1$. A further consequence is that the variety of ${\bf B}_2^1$ cannot be defined within the variety of ${\bf A}_2^1$ by any finite system of identities. Continuing downward, we then turn to subvarieties of ${\bf B}_2^1$. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2y\approx yx^2$ and containing the monoid $M({\bf z}_\infty)$, where ${\bf z}_\infty$ denotes the infinite limit of the Zimin words ${\bf z}_0=x_0$, ${\bf z}_{n+1}={\bf z}_n x_{n+1}{\bf z}_n$.

Generic complexity of algorithmic problems over Brandt semigroups

In this paper we present a generic polynomial algorithm for the problem about the sum of subset over Brandt semigroup with index set ℕ over any finitely defined group with polynomially decidable word problem. The problem about the sum of subset is a classical combinatorial problem, studied for many decades. This problem is very popular in cryptography, where there are many cryptosystems based on it. Myasnikov, Nikolaev, and Ushakov in 2015 introduced analogs of the problem about the sum of subset for arbitrary groups (semigroups). There are Brandt semigroups for which this problem is NP-complete. Also we propose a polynomial generic algorithm for the problem of identity checking over all finite Brandt monoids . This problem is co-NP-complete in the classical sence. Generic algorithms solve problems for almost all inputs with respect to some natural measures on the set of inputs. Supported by Russian Science Foundation, grant 18-71-10028.

Coherency and Constructions for Monoids

AbstractA monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck–Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of S to be finitely generated.

IDENTITIES IN BRANDT SEMIGROUPS, REVISITED

We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap., 14, no.1 (1985), 38–42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit.

Flows on Classes of Regular Semigroups and Cauchy Categories

We consider the structure of the flow monoid for some classes of regular semigroups (which are special case of flows on categories) and for Cauchy categories. In detail, we characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups and also describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. In fact, we obtain a general structure for the flow monoids on Cauchy categories.

Aleph _0-categoricity of semigroups

In this paper we initiate the study of aleph _0-categorical semigroups, where a countable semigroup S is aleph _0-categorical if, for any natural number n, the action of its group of automorphisms {text {Aut}}(S) on S^n has only finitely many orbits. We show that aleph _0-categoricity transfers to certain important substructures such as maximal subgroups and principal factors. We examine the relationship between aleph _0-categoricity and a number of semigroup and monoid constructions, namely Brandt semigroups, direct sums, 0-direct unions, semidirect products and {mathcal {P}}-semigroups. As a corollary, we determine the aleph _0-categoricity of an E-unitary inverse semigroup with finite semilattice of idempotents in terms of that of the maximal group homomorphic image.

Умови, за яких напівгрупа Брандта неізоморфна варіанту.

In this paper we study the question if the Brandt semigroup can be a variant of another semigroup or not. For a semigroup which does not contain a bicyclic subsemigroup we proved that a variant of such semigroup is not a Brandt semigroup. For an infinite semigroup which contains a bicyclic subsemigroup a variant with a sandwich element from this bicyclic subsemigroup is not a Brandt semigroup.

Pseudo-amenability and pseudo-contractibility of restricted semigroup algebra

Abstract In this article the pseudo-amenability and pseudo-contractibility of restricted semigroup algebra l r 1 ( S ) $l_r^1(S)$ and semigroup algebra, l 1(Sr) on restricted semigroup, Sr are investigated for different classes of inverse semi-groups such as Brandt semigroup, and Clifford semigroup. We particularly show the equivalence between pseudo-amenability and character amenability of restricted semigroup algebra on a Clifford semigroup and semigroup algebra on a restricted semigroup. Moreover, we show that when S = M 0(G, I)is a Brandt semigroup, pseudo-amenability of l 1(Sr) is equivalent to its pseudo-contractibility.

On equations over Brandt semigroups

In this paper, we study equations in one variable over Brandt semigroup Bn. We show that the average number of solutions of these equations is asymptotically equal to n2. We prove that the proportion of unsolvable equations is asymptotically equal to .

THE LATTICE OF VARIETIES OF STRICT LEFT RESTRICTION SEMIGROUPS

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety $\mathbf{B}$ of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety $\mathbf{B}_{R}$ of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices ${\mathcal{L}}(\mathbf{B})$ and ${\mathcal{L}}(\mathbf{B}_{R})$ of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval $[\mathbf{D},\mathbf{D}\vee \mathbf{M}]$, every subvariety of $\mathbf{B}$ is induced from a member of $\mathbf{B}_{R}$ and vice versa. Here $\mathbf{D}$ is generated by the three-element left restriction semigroup $D$ and $\mathbf{M}$ is the variety of monoids. The analogues hold for pseudovarieties.

Isomorphism conditions for Cayley digraphs of Brandt semigroups

A Brandt semigroup is a completely 0-simple inverse semigroup. In this paper, we investigate the structure of Cayley digraphs of a Brandt semigroup relative to a one-element connection set. Moreover, we introduce the conditions for which they are connected and isomorphic to each other.

On the subsemigroup complex of an aperiodic Brandt semigroup

We introduce the subsemigroup complex of a finite semigroup S as a (boolean representable) simplicial complex defined through chains in the lattice of subsemigroups of S. We present a research program for such complexes, illustrated through the particular case of combinatorial Brandt semigroups. The results include alternative characterizations of faces and facets, asymptotical estimates on the number of facets, or establishing when the complex is pure or a matroid.

VARIETIES OF LEFT RESTRICTION SEMIGROUPS

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. They may be defined by a simple set of identities and the author initiated a study of the lattice of varieties of such semigroups, in parallel with the study of the lattice of varieties of two-sided restriction semigroups. In this work we study the subvariety $\mathbf{B}$ generated by Brandt semigroups and the subvarieties generated by the five-element Brandt inverse semigroup $B_{2}$, its four-element restriction subsemigroup $B_{0}$ and its three-element left restriction subsemigroup $D$. These have already been studied in the ‘plain’ semigroup context, in the inverse semigroup context (in the first two instances) and in the two-sided restriction semigroup context (in all but the last instance). The author has previously shown that in the last of these contexts, the behavior is pathological: ‘almost all’ finite restriction semigroups are inherently nonfinitely based. Here we show that this is not the case for left restriction semigroups, by exhibiting identities for the above varieties and for their joins with monoids (the analog of groups in this context). We do so by structural means involving subdirect decompositions into certain primitive semigroups. We also show that each identity has a simple structural interpretation.

Johnson pseudo-contractibility of certain semigroup algebras

In this paper we characterize the Johnson pseudo-contractibility of \(\ell ^{1}(S)\), where S is a uniformly locally finite inverse semigroup. We show that for a Brandt semigroup \(S=M^{0}(G,I)\) over a non-empty set I, \(\ell ^{1}(S)\) is Johnson pseudo-contractible if and only if G is amenable and I is finite. We give some examples to show the difference between Johnson pseudo-contractibility, pseudo-amenability and pseudo-contractibility among the semigroup algebras.

Rank properties of the semigroup of endomorphisms over Brandt semigroup

We investigate small rank, lower rank, intermediate rank, upper rank, and large rank of the semigroup of endomorphisms over Brandt semigroup.