Proper two-sided restriction semigroups and partial actions

Cornock, Claire; Gould, Victoria

doi:10.1016/j.jpaa.2011.10.015

Cited by 36 publications

(60 citation statements)

References 17 publications

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“…Although we believe that this paper incorporates a new approach, there is a already a body of work on the structure of proper restriction semigroups. Cornock and Gould [3] obtained a structure theorem for proper restriction semigroups in general, based more on the work of Petrich and Reilly [21], cited above, than on McAlister's theorem per se, using as parameters a monoid that acts on both sides of a semilattice. Of course their results and those in Section 9 of our paper must be deducible from each other, but we leave that exercise to the reader, as we leave the exercise of specializing the results therein to the perfect and almost perfect cases.…”

Section: Introductionmentioning

confidence: 99%

Almost perfect restriction semigroups

Jones

2016

Journal of Algebra

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Abstract:We call a restriction semigroup almost perfect if it is proper and the least congruence that identifies all its projections is perfect. We show that any such semigroup is isomorphic to a 'W-product' W(T,Y), where T is a monoid, Y is a semilattice and there is a homomorphism from T into the inverse semigroup TIY of isomorphisms between ideals of Y. Conversely, all such W-products are almost perfect. Since we also show that every restriction semigroup has an easily computed cover of this type, the combination yields a 'McAlister-type' theorem for all restriction semigroups. It is one of the theses of this work that almost perfection and perfection, the analogue of this definition for restriction monoids, are the appropriate settings for such a theorem. That these theorems do not reduce to a general theorem for inverse semigroups illustrates a second thesis of this work: that restriction (and, by extension, Ehresmann) semigroups have a rich theory that does not consist merely of generalizations of inverse semigroup theory. It is then with some ambivalence that we show that all the main results of this work easily generalize to encompass all proper restriction semigroups.The notation W(T,Y) recognizes that it is a far-reaching generalization of a long-known similarly titled construction. As a result, our work generalizes Szendrei's description of almost factorizable semigroups while at the same time including certain classes of free restriction semigroups in its realm.

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Section: Introductionmentioning

confidence: 99%

Almost perfect restriction semigroups

Jones

2016

Journal of Algebra

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“…and ϕ(t)(e) = (ae) + for any e ∈ dom(ϕ(t)) and a ∈ t such that a * ≥ e. The premorphism ϕ satisfies axioms (A), (B), (C). The following theorem is a specialization to monoids of the result due to Cornock and Gould [7].…”

Section: Definition 22mentioning

confidence: 86%

“…Structure of proper restriction monoids. In this subsection we recall the Cornock-Gould structure result on proper restriction monoids [7]. We remark that in [7] a pair of partial actions, called a double action, satisfying certain compatibility conditions, was considered, and in [23] we reformulated this using one partial action by partial bijections.…”

Section: Definition 22mentioning

confidence: 99%

Two-sided expansions of monoids

Kudryavtseva

2019

Int. J. Algebra Comput.

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We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget-Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of respective relatively free inverse monoids.For a monoid M , we define F R R (M ) to be the freest two-sided restriction monoid generated by a bijective copy, M ′ , of the underlying set of M , such that the inclusion map ι : M → F R R (M ) is determined by a set of relations, R, so that ι is a premorphism which is weaker than a homomorphism. Our main result states that F R R (M ) can be constructed, by means of a partial action product construction, from M and the idempotent semilattice of F I R (M ), the free M ′ -generated inverse monoid subject to relations R. In particular, the semilattice of projections of F R R (M ) is isomorphic to the idempotent semilattice of F I R (M ). The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result.We show that important properties of F R R (M ) are well agreed with suitable properties of M , such as being cancellative or embeddable into a group. We observe that if M is an inverse monoid, then F I s (M ), the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson-Margolis-Steinberg generalized prefix expansion M pr . This gives a presentation of M pr and leads to a model for F R s (M ) in terms of the known model for M pr .

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“…For (left) restriction semigroups, many authors have produced work using constructions similar (at least on the surface) to those of McAlister, replacing groups by monoids of various kinds. Covering results for restriction semigroups are known due to Cornock, Fountain, Gomes and Gould [3,5,7,8]. The structure of proper restriction semigroups is determined by analogues of P-semigroups.…”

Section: It Is Easy To See Thatmentioning

confidence: 99%

“…To overcome this, we use the notion of a double action (introduced in [5]) and compatibility conditions (introduced in [3]). …”

Section: Corollary 35 Let Z = E(s)mentioning

confidence: 99%

Algebraic properties of Zappa–Szép products of semigroups and monoids

Zenab

2017

Semigroup Forum

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Direct, semidirect and Zappa-Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa-Szép products of monoids and semigroups and investigate generalised Greens relations R * , L * , R E and L E for these Zappa-Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa-Szép product E S. We further investigate properties of E S and show that S is a retract of E S. We also find a subset T of E S which is left restriction.

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Proper two-sided restriction semigroups and partial actions

Cited by 36 publications

References 17 publications

Almost perfect restriction semigroups

Almost perfect restriction semigroups

Two-sided expansions of monoids

Algebraic properties of Zappa–Szép products of semigroups and monoids

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