Generators and relations of Rees matrix semigroups

Ayik, H.; Ruškuc, Nik

doi:10.1017/s0013091500020472

Cited by 22 publications

(24 citation statements)

References 18 publications

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“…To be hyperbolic, S must be finitely generated. It is straightforward to show (and is also immediate from a more general result of Ayik and Ruškuc [2]) that this is equivalent to G being finitely generated, and I and J being finite. In other words, the maximal subgroups of S (which are all isomorphic) are finitely generated, and S has finitely many R-and L -classes.…”

Section: Completely Simple Semigroupsmentioning

confidence: 80%

Hyperbolic Groups and Completely Simple Semigroups

Fountain

Kambites

2004

Semigroups and Languages

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We begin with a brief introduction to the theory of word hyperbolic groups. We then consider four possible conditions which might reasonably be used as definitions or partial definitions of hyperbolicity in semigroups: having a hyperbolic Cayley graph; having hyperbolic Schützenberger graphs; having a context-free multiplication table; or having word hyperbolic maximal subgroups. Our main result is that these conditions coincide in the case of finitely generated completely simple semigroups.This paper is based on a lecture given at the workshop on Semigroups and Languages held at the Centro deÁlgebra da Universidade de Lisboa in November 2002. The aim of the paper, as of the talk, is to provide semigroup theorists with a gentle introduction to the concept of a word hyperbolic group (hereafter referred to as simply a 'hyperbolic group'), to discuss recent work of Gilman which makes it possible to introduce a notion of (word) hyperbolic semigroup and then to examine these ideas in the context of completely simple semigroups. The first three sections of the paper are expository while the fourth introduces new results. After briefly discussing Dehn's algorithm, we describe hyperbolic groups in terms of 'slim triangles', and then describe some properties of these groups which we need later in the paper. Hyperbolic groups were introduced by Gromov 1

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Section: Completely Simple Semigroupsmentioning

confidence: 80%

Hyperbolic Groups and Completely Simple Semigroups

Fountain

Kambites

2004

Semigroups and Languages

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“…The problem of relating the properties of the semigroup with the properties of its maximal subgroups was studied in several papers (see for example [2,3,12,15,17,18,19]). They can be defined using the notion of ideal.…”

Section: Applications To Zero Rees Matrix Semigroupsmentioning

confidence: 99%

“…Before stating the main result of the section let us recall that the Rees matrix semigroup M [A; I, J; P ] is finitely presented if and only if A is finitely presented and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P [3]. Theorem 6.1.…”

Section: Applications To Zero Rees Matrix Semigroupsmentioning

confidence: 99%

On Finite Complete Presentations and Exact Decompositions of Semigroups

Araújo

Malheiro

2011

Communications in Algebra

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Abstract. We prove that given a finite (zero) exact right decomposition (M, T ) of a semigroup S, if M is defined by a finite complete presentation then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence we deduce that the Zappa-Szép extension of a monoid defined by a finite complete presentation, by a finite monoid is also defined by such a presentation.It is also shown that when a semigroup A isomorphic to a variant semigroup A(x) that is defined by a finite complete presentation, where x belongs to a sandwich matrix P , together with some other conditions, we deduce that the zero Rees matrix semigroup M 0 [A; I, J; P ] is also defined by a finite complete presentation.

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“…Finite presentability of semigroup constructions has been widely studied in recent years (see, for example [1,2,6,8,9]). One construction is an extension of a semigroup by a congruence.…”

Section: Introductionmentioning

confidence: 99%

Semigroup Presentations for Congruences on Groups

Ayık¹,

Çalişkan²

2013

Bulletin of the Korean Mathematical Society

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Abstract. We consider a congruence ρ on a group G as a subsemigroup of the direct product G × G. It is well known that a relation ρ on G is a congruence if and only if there exists a normal subgroup N of G such that ρ = (s, t) : st −1 ∈ N . In this paper we prove that if G is a finitely presented group, and if N is a normal subgroup of G with finite index, then the congruence ρ = (s, t) : st −1 ∈ N on G is finitely presented.

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Generators and relations of Rees matrix semigroups

Cited by 22 publications

References 18 publications

Hyperbolic Groups and Completely Simple Semigroups

Hyperbolic Groups and Completely Simple Semigroups

On Finite Complete Presentations and Exact Decompositions of Semigroups

Semigroup Presentations for Congruences on Groups

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