On Generators and Presentations of Semidirect Products in Inverse Semigroups

Dombi, Erzsebet; Ruškuc, Nik

doi:10.1017/s0004972708000890

Cited by 7 publications

(4 citation statements)

References 6 publications

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“…for all α, β ∈ Y and g, h ∈ G. The significance of this construction is underlined by the following basic result in the theory of inverse semigroups: every (finite) inverse semigroup/monoid S is a homomorphic image of a subalgebra of a (finite) semidirect product Y * ρ G for suitable ρ and (finite) Y and G. In fact, Y can be taken to be the semilattice E(S ). See [8,18] for additional background. The natural order is stable under left and right multiplication.…”

Section: Semidirect Product Of a Semilattice By A Groupmentioning

confidence: 99%

On Free Spectra of a Class of Finite Inverse monoids

Dolinka

2012

J. Aust. Math. Soc.

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For a finite Clifford inverse algebra A, with natural order meet-semilattice Y A and group of units G A , we show that the inverse monoid obtained as the semidirect product Y 1 A * ρ G A has a log-polynomial free spectrum whenever ρ is a term-expressible left action of G A on Y A and all subgroups of A are nilpotent. This yields a number of examples of finite inverse monoids satisfying the Seif conjecture on finite monoids whose free spectra are not doubly exponential.2010 Mathematics subject classification: primary 20M05; secondary 20M18, 08B20.

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Section: Semidirect Product Of a Semilattice By A Groupmentioning

confidence: 99%

On Free Spectra of a Class of Finite Inverse monoids

Dolinka

2012

J. Aust. Math. Soc.

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“…Although several general results exist on presentations of wreath products of monoids (see [20,44,78,79]), these are not applicable to M ≀I n as they concern different kinds of wreath products. As for M ≀I, we are not aware of any previous results on presentations for wreath products involving categories, but we will be aided by tools developed in [30], which allow presentations for certain classes of categories to be constructed from presentations of their endomorphism monoids.…”

Section: Introductionmentioning

confidence: 99%

Presentations for wreath products involving symmetric inverse monoids and categories

Clark¹,

East²

2022

Preprint

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Wreath products involving symmetric inverse monoids/semigroups/categories arise in many areas of algebra and science, and presentations by generators and relations are crucial tools in such studies. The current paper finds such presentations for M ≀ I n , M ≀ Sing(I n ) and M ≀ I. Here M is an arbitrary monoid, I n is the symmetric inverse monoid, Sing(I n ) its singular ideal, and I is the symmetric inverse category.

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“…As discussed above, presentations are extremely important tools when working with any kind of algebraic structure, and several results exist for building presentations for algebras that arise from others via natural constructions. See for example [21,24,41,65,74,[105][106][107]; the introduction to [36] contains a fuller discussion and many more references. Chapter 6 contains many general results on presentations for a semigroup U S arising from an action pair (U, S) in several important cases:…”

Section: Introductionmentioning

confidence: 99%

On a class of semigroup products

Carson¹,

Dolinka²,

East³

et al. 2022

Preprint

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This paper concerns a class of semigroups that arise as products U S, associated to what we call 'action pairs'. Here U and S are subsemigroups of a common monoid and, roughly speaking, S has an action on the monoid completion U 1 that is suitably compatible with the product in the over-monoid.The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure a comprehensive understanding of them. We then apply our abstract results to many special cases of interest.The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup U S is a quotient of a semidirect product U ⋊S, and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (a.k.a. E-unitary) inverse semigroups.The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for U S out of presentations for the constituents U and S in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.

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On Generators and Presentations of Semidirect Products in Inverse Semigroups

Cited by 7 publications

References 6 publications

On Free Spectra of a Class of Finite Inverse monoids

On Free Spectra of a Class of Finite Inverse monoids

Presentations for wreath products involving symmetric inverse monoids and categories

On a class of semigroup products

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