Embedding Into Almost Left Factorizable Restriction Semigroups

Szendrei, Mária B.

doi:10.1080/00927872.2011.643839

Cited by 18 publications

(57 citation statements)

References 13 publications

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“…In the standard literature, these relations are usually denoted R E , L E and H E , respectively (more recently, e.g. [22], without the subscripts). The natural partial order on S is defined by a ≤ b if a = eb for some e ∈ P S ; equivalently if a = a + b.…”

Section: Preliminariesmentioning

confidence: 99%

“…We refer the reader to [22] for more details of the basic results cited here. There Szendrei defined C(S) for the case of restriction semigroups in general, closely following [9] (itself based on the one-sided notion of El Qallali [4] and extending the definition in the case of inverse semigroups [20, V. 2…”

Section: The Monoids C(s) and Their Representationsmentioning

confidence: 99%

“…As further evidence of the centrality of almost perfect restriction semigroups, we note there that whenever a variety of restriction semigroups is closed under taking proper covers, the free objects in the variety are necessarily almost perfect. We also provide a unified description of left, right, and two-sided almost factorizable restriction semigroups (cf [22]).…”

Section: Introductionmentioning

confidence: 99%

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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: Preliminariesmentioning

confidence: 99%

Section: The Monoids C(s) and Their Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Almost perfect restriction semigroups

Jones

2016

Journal of Algebra

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“…This may appear a little strange, but is the price to be paid for dealing with all idempotents, not just a special subset. More general contexts have already been considered, as in [2,12,13], but the approach in hand is a natural and minimal extension of the inductive groupoid case, and returns to the spirit of groupoids as dealt with in another landmark paper-Lawson's [7]. Above all, our ultimate intent is to have B as a skew lattice, and we deal with this in the next section.…”

Section: Remarksmentioning

confidence: 99%

Groupoids on a skew lattice of objects

FitzGerald

2019

Art Discrete Appl. Math.

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Motivated by some alternatives to the classical logical model of boolean algebra, this paper deals with algebraic structures which extend skew lattices by locally invertible elements. Following the meme of the Ehresmann-Schein-Nambooripad theorem, we consider a groupoid (small category of isomorphisms) in which the set of objects carries the structure of a skew lattice. The objects act on the morphisms by left and right restriction and extension mappings of the morphisms, imitating those of an inductive groupoid. Conditions are placed on the actions, from which pseudoproducts may be defined. This gives an algebra of signature (2, 2, 1), in which each binary operation has the structure of an orthodox semigroup. In the reverse direction, a groupoid of the kind described may be reconstructed from the algebra.2010 Mathematics Subject Classification. 20L05, 20M19, 06B75.

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“…It has been shown that every restriction semigroup is embeddable into an almost left factorisable restriction semigroup. Also every restriction semigroup has a cover which is embeddable into a W -product (a special kind of semidirect product) [22,23]. Recent development in this theory reveals that restriction semigroups have become a topic of lively research interest and in providing the common frame work for answering different questions in algebra, they will continue to be a key topic.…”

Section: It Is Easy To See Thatmentioning

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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Embedding Into Almost Left Factorizable Restriction Semigroups

Cited by 18 publications

References 13 publications

Almost perfect restriction semigroups

Almost perfect restriction semigroups

Groupoids on a skew lattice of objects

Algebraic properties of Zappa–Szép products of semigroups and monoids

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