On Automatic Rees Matrix Semigroups

Descalço, Luís; Ruškuc, Nik

doi:10.1081/agb-120004869

Cited by 6 publications

(12 citation statements)

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“…The rest of this section, and the one that follows it, will be dedicated to the proof of Theorem 4. Results analogous to those above for other finiteness properties, including automaticity and finite derivation type, have appeared in the literature; see [14,22,40]. Other important recent work on completely simple semigroups includes [34].…”

Section: Resolutions For Completely Simple Semigroups Imentioning

confidence: 63%

Homological finiteness properties of monoids, their ideals and maximal subgroups

Gray

Pride

2011

Journal of Pure and Applied Algebra

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a b s t r a c tWe consider the general question of how the homological finiteness property left-FP n (resp. right-FP n ) holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. This is done by giving methods for constructing free resolutions of substructures from free resolutions of their containing monoids, and vice versa. In particular, we show that left-FP n is inherited by the maximal subgroups in a completely simple minimal ideal, in the case that the minimal ideal has finitely many left ideals. For completely simple semigroups, we prove the converse, and as a corollary, show that a completely simple semigroup is of type left-and right-FP n if and only if it has finitely many left and right ideals and all of its maximal subgroups are of type FP n . Also, given an ideal of a monoid, we show that if the ideal has a two-sided identity element then the containing monoid is of type left-FP n if and only if the ideal is of type left-FP n . Applying this result, we obtain necessary and sufficient conditions for a Clifford monoid (and more generally a strong semilattice of monoids) to be of type left-FP n . Examples are provided showing that for each of the results all of the hypotheses are necessary.

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Section: Resolutions For Completely Simple Semigroups Imentioning

confidence: 63%

Homological finiteness properties of monoids, their ideals and maximal subgroups

Gray

Pride

2011

Journal of Pure and Applied Algebra

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“…A further application of both directions of Theorem 1.1, along similar lines to the above, can be found in [6], where the automaticity of Rees matrix semigroups is considered.…”

Section: Applicationsmentioning

confidence: 90%

Automatic Semigroups With Subsemigroups of Finite Rees Index

Hoffmann

Thomas

Ruškuc

2002

Int. J. Algebra Comput.

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“…They represent an alternative formulation of certain special cases of the blocked Rees matrix semigroup constructions introduced by Fountain [8] and subsequently employed by Armstrong [1]. In this section, we extend some results of [5] concerning Rees matrix constructions over semigroups to cover similar cosntructions over semigroupoids and small categories. In some cases, our new results are stronger than previously known results, even when specialised to the case of semigroups.…”

Section: Rees Matrices and Automaticitymentioning

confidence: 96%

“…A number of interesting results have been proved concerning the relationship between automaticity properties and Rees matrix constructions over groups [4] and more generally over semigroups [5,21]. At the same time, Lawson [19] has applied to the study of abundant semigroups a form of Rees matrix construction over semigroupoids.…”

Section: Introductionmentioning

confidence: 99%

“…It is observed in [7] that the notion of automaticity extend naturally from groups to groupoids. More recently, Hudson [15] has introduced a definition of automaticity in semigroups and monoids and a theory of automatic semigroups has begun to emerge [2,3,4,5,6,9,10,11,12,13]. In [17] we developed a common framework for these generalisations, in the form of a theory of automaticity for small categories and semigroupoids.…”

Section: Introductionmentioning

confidence: 99%

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