Brandt Semigroup Research Articles

The interval [B2, NB2] in the lattice of Rees-Sushkevich varieties

We provide new solutions to the word problems for the variety B2 generated by the five element Brandt semigroup B2 with zero divisors and the variety NB2 generated by B2 and the two element left and right zero semigroups. We also provide a finite basis of identities for the variety NB2. This leads to a complete description of the interval [B2, NB2], in the lattice of semigroup varieties. This is a critical interval in the study of the lattice of aperiodic Rees-Sushkevich varieties. In addition, the varieties B2 and NB2 are especially important in the characterization of those Rees-Sushkevich varieties that are exact, that is, generated by completely 0-simple semigroups. Byproducts of this note are techniques that lend themselves to a deeper study of the structure of the free objects in the varieties in the interval [B2, NB2].

A formula for multiple classifiers in data mining based on Brandt semigroups

A general approach to designing multiple classifiers represents them as a combination of several binary classifiers in order to enable correction of classification errors and increase reliability. This method is explained, for example, in Witten and Frank (Data Mining: Practical Machine Learning Tools and Techniques, 2005, Sect. 7.5). The aim of this paper is to investigate representations of this sort based on Brandt semigroups. We give a formula for the maximum number of errors of binary classifiers, which can be corrected by a multiple classifier of this type. Examples show that our formula does not carry over to larger classes of semigroups.

Kernel and Trace Operators for Extensions of Brandt Semigroups

Let S be an (ideal) extension of a Brandt semigroup S0 by a Brandt semigroup S1 and let $\mathcal{C}(S)$ denote the congruence lattice of S. For $\rho \in \mathcal{C}(S),$ denote by $\rho _K$ and $\rho ^K$ the least and the greatest congruences on S with the same kernel as $\rho ,$ respectively, and let $\rho _T$ and $\rho ^T$ have the analogous meaning relative to trace. We establish necessary and sufficient conditions on S in order that one or more of the operators $\rho \longrightarrow \rho _K, \quad \rho \longrightarrow \rho ^K, \quad \rho \longrightarrow \rho _T, \quad \rho \longrightarrow \rho^T$ be $\vee$ - or $\wedge$ -homomorphisms on $\mathcal{C}(S).$ The conditions are expressed directly in terms of a construction of an extension of S0 and S1 and the proofs make use of a construction of congruences on S expressed by means of congruences on S0 and S1.

A rank for right congruences on inverse semigroups

The S-rank (where ‘S’ abbreviates ‘sandwich’) of a right congruence ρ on a semigroup S is the Cantor-Bendixson rank of ρ in the lattice of right congruences ℛ of S with respect to a topology we call the finite type topology. If every ρ ϵ ℛ possesses S-rank, then S is ranked. It is known that every right Noetherian semigroup is ranked and every ranked inverse semigroup is weakly right Noetherian. Moreover, if S is ranked, then so is every maximal subgroup of S. We show that a Brandt semigroup 0(G, I) is ranked if and only if G is ranked and I is finite.We establish a correspondence between the lattice of congruences on a chain E, and the lattice of right congruences contained within the least group congruence on any inverse semigroup S with semilattice of idempotents E(S) ≅ E. Consequently we argue that the (inverse) bicyclic monoid B is not ranked; moreover, a ranked semigroup cannot contain a bicyclic -class. On the other hand, B is weakly right Noetherian, and possesses trivial (hence ranked) subgroups.Our notion of rank arose from considering stability properties of the theory Ts of existentially closed (right) S-sets over a right coherent monoid S. The property of right coherence guarantees that the existentially closed S-sets form an axiomatisable class. We argue that B is right coherent. As a consequence, it follows from known results that TB is a theory of B-sets that is superstable but not totally transcendental.

NORMALIZERS AND FREE GROUPS IN THE UNIT GROUP OF AN INTEGRAL SEMIGROUP RING

In this article, we introduce the normalizer [Formula: see text] of a subset X of a ring R (with identity) in the unit group [Formula: see text] and consider, in particular, the normalizer of the natural basis ±S of the integral semigroup ring ℤ0S of a finite semigroup S. We investigate properties of this normalizer for the class of semigroup rings of inverse semigroups, which contains, for example, matrix rings, in particular, matrix rings over group rings, and partial group rings. We also construct free groups in the unit group of an integral semigroup ring of a Brandt semigroup using a bicyclic unit.

Null Extensions and Generalized Inflations of Brandt Semigroups

Clarke and Monzo defined in [3] a construction called a generalized inflation of a semigroup. It is always the case that any inflation of a semigroup is a generalized inflation, and any generalized inflation of a semigroup is a null extension of the semigroup. Clarke and Monzo proved that any associative null extension of a base semigroup which is a union of groups is in fact a generalized inflation. In this paper we study null extensions and generalized inflations of Brandt semigroups. We first prove that any generalized inflation of a Brandt semigroup is actually an inflation of the semigroup. This answers a question posed by Clarke and Monzo in [3]. Then we characterize associative null extensions of Brandt semigroups, and show that there are associative null extensions of Brandt semigroups which are not generalized inflations.

The Lattice of Full Subsemigroups of an Inverse Semigroup

In this paper, we consider the lattice Subf S of full subsemigroups of an inverse semigroup S. Our first main theorem states that for any inverse semigroup S, Subf S is a subdirect product of the lattices of full subsemigroups of its principal factors, so that Subf S is distributive [meet semidistributive, join semidistributive, modular, semimodular] if and only if the lattice of full subsemigroups of each principal factor is. To examine such inverse semigroups, therefore, we need essentially only consider those which are 0-simple. For a 0-simple inverse semigroup S (not a group with zero), we show that in fact each of modularity, meet semidistributivity and join semidistributivity of Subf S is equivalent to distributivity of S, that is, S is the combinatorial Brandt semigroup with exactly two nonzero idempotents and two nonidempotents. About semimodularity, however, we concentrate only on the completely 0-simple case, that is, Brandt semigroups. For a Brandt semigroup S (not a group with zero), semimodularity of Subf S is equivalent to distributivity of Subf S. Finally, we characterize an inverse semigroup S for which Subf S is a chain.

Free Spectrum of the Variety Generated by the Five Element Combinatorial Brandt Semigroup

We give an asymptotical bound for the logarithm of the free spectrum of the variety generated by the five element Brandt semigroup. Our bound is much better than any previously known estimate.

SUBVARIETIES OF THE VARIETY GENERATED BY THE FIVE-ELEMENT BRANDT SEMIGROUP

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.

Generating Sets of Completely 0-Simple Semigroups

ABSTRACT A formula for the rank of an arbitrary finite completely 0-simple semigroup, represented as a Rees matrix semigroup ℳ0[G; I, Λ; P], is given. The result generalizes that of Ruškuc concerning the rank of connected finite completely 0-simple semigroups. The rank is expressed in terms of |I|, |Λ|, the number of connected components k of P, and a number r min, which we define. We go on to show that the number r min is expressible in terms of a family of subgroups of G, the members of which are in one-to-one correspondence with, and determined by the nonzero entries of, the components of P. A number of applications are given, including a generalization of a result of Gomes and Howie concerning the rank of an arbitrary Brandt semigroup B(G,{1,…,n}).

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra l1(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then l1(S) is approximately amenable if and only if G is amenable. Moreover l1(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every Ma(S)-measurable subset B of S and s ∈ S the set sB is Ma(S)-measurable, it is proved that if the measure algebra Ma(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.

The M�bius Category of Some Combinatorial Inverse Semigroups

\noindent The purpose of this paper is to point out some aspects of the relationship between combinatorial inverse semigroups and their Mobius categories, and to explore combinatorial results arising from combinatorial Brandt semigroups, fundamental simple inverse ω-semigroups and from the free monogenic inverse semigroup.

Identity Bases for Some Non-exact Varieties

Let A2 be the variety generated by the five-element non-orthodox 0-simple semigroup. This paper presents the identity bases for several subvarieties of A2 that are not generated by any completely 0-simple or completely simple semigroups. It will be shown that several subvarieties of A2, including the variety generated by the five-element Brandt semigroup, are hereditarily finitely based.

Semigroups which Have a minimum primitive inverse congruence

We investigate primitive inverse semigroups and Brandt semigroups as analogues of groups for semigroups with zero. We give a description of the minimum primitive inverse congruence on a categorical E *-dense E-semigroup. We show that a categorical semigroup S with a primitive inverse congruence has a minimum such congruence if $$\tilde D\left( S \right)$$ is *-dense in S. Here $$\tilde D\left( S \right)$$ is the least full, weakly self-conjugate, *-unitary and *-reflexive subsemigroup of S. Our results are analogous to those of Fountain, Pin and Weil for general semigroups.

On semidirectly closed pseudovarieties of aperiodic semigroups

The aim of this work is to study the unknown intervals of the lattice of aperiodic pseudovarieties which are semidirectly closed and answer questions proposed by Almeida in his book “Finite Semigroups and Universal Algebra”. The main results state that the intervals [ V ∗(B 2), ER ∩ LR ] and [ V ∗(B 2 1), ER ∩ A ] are not trivial, and that both contain a chain isomorphic to the chain of real numbers. These results are a consequence of the study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B 2.

THE SEMIDIRECTLY CLOSED PSEUDOVARIETY GENERATED BY APERIODIC BRANDT SEMIGROUPS

This paper presents a study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B2, denoted V*(B2). We construct a basis of pseudoidentities for the semidirect powers of the pseudovariety generated by B2 which leads to the main result, which states that V*(B2) is decidable. Independently, using some suggestions given by J. Almeida in his book "Finite Semigroups and Universal Algebra", we constructed an algorithm to solve the membership problem in V* (B2).

The Five-Element Brandt Semigroup as a Forbidden Divisor

AbstractThe purpose of this paper is to describe all semigroups which do not havethe five-element Brandt semigroup B 2 among its divisors. These semigroupswill be described in two ways: as semigroups which have certain property as ahereditary property, and as semigroups which satisfy some variable identities.We also describe all variable identities over the two-element alphabet thatdetermine variable varieties contained in the considered class of semigroups. 1. Introduction and preliminariesThe five-element Brandt semigroup B 2 was studied many times as a forbiddenmember of certain classes of semigroups. For example, pseudovarieties which do notcontain B 2 were described by S. W. Margolis in [11] (see also M. S. Putcha [16] and J.E. Pin [14] and [15]). Varieties which do not contain B 2 were completely characterizedby the authors in [6] and [7], and in the periodic case by M. V. Sapir and E. V.Suhanov in [19]. Epigroups (semigroups equipped with one special unary operation)which do not have B

Rank properties in finite inverse semigroups

AbstractTwo possible concepts of rank in inverse semigroup theory, the intermediate I-rank and the upper I-rank, are investigated for the finite aperiodic Brandt semigroup. The so-called large I-rank is found for an arbitrary finite Brandt semigroup, and the result is used to obtain the large rank of the inverse semigroup of all proper subpermutations of a finite set.

Small semigroup related structures with infinite properties

In mathematics, one frequently encounters constructions of a pathological or critical nature. In this thesis we investigate such structures in semigroup theory with a particular aim of finding small, finite, examples with certain associated infinite characteristics. We begin our investigation with a study of the identities of finite semigroups. A semigroup (or the variety it generates) whose identities admit a finite basis is said to be finitely based. We find examples of pairs of finite (aperiodic) finitely based semigroups whose direct product is not finitely based (answering a question of M. Sapir) and of pairs of finite (aperiodic) semigroups that are not finitely based whose direct product is finitely based. These and other semigroups from a large class (the class of finite Rees quotients of free monoids) are also shown to generate varieties with a chain of finitely generated supervarieties which alternate between being finitely based and not finitely based. Furthermore it is shown that in a natural sense, almost semigroups from this class are not finitely based. Not finitely based semigroups that are locally finite and have the property that every locally finite variety containing them is also not finitely based are said to be inherently not finitely based. We construct all minimal inherently not finitely based divisors in the class of finite semigroups and establish several results concerning a fundamental example with this property; the six element Brandt semigroup with adjoined identity element, B. 1/2. We then find the first examples of finite semigroups admitting a finite basis of identities but generating a variety with uncountably many subvarieties (indeed with a chain of subvarieties with the same ordering as the real numbers). For some well known classes, a complete description of the members with this property are obtained and related examples and results concerning joins of varieties are also found. A connection between these results and the construction of varieties with decidable word problem but undecidable uniform word problem is investigated. Finally we investigate several embedding problems not directly concerned with semigroup varieties and show that they are undecidable. The first and second of these problems concern the fundamental relations of Green; in addition some small examples are found which exhibit unusual related properties and a problem of M. Sapir is solved. The third of the embedding problems concerns the potential embeddability of finite semigroup amalgams. The results are easily extended to the class of rings.

On finitely based pseudovarieties of the forms V [formula omitted] D and V [formula omitted] D[formula omitted

Let D n be the pseudovariety of all finite semigroups in which products of length n are right zeros and let D = ⋃ n≥1 D n . It is shown in this paper that, if V is a pseudovariety of semigroups whose global g V is finitely based, then V ∗ D n ( n≥1) and V ∗ D are also finitely based. Moreover, if V is itself finitely based and contains the aperiodic five-element Brandt semigroup, then g V is also finitely based. As a further application, it is proved that the finite basis properties for g V , V ∗ D and V ∗ D n ( n≥1) are all equivalent for an arbitrary non-group monoidal pseudovariety V .