Zappa–Szép products of bands and groups

Gilbert, N. D.; Wazzan, Suha A.

doi:10.1007/s00233-008-9065-5

Cited by 9 publications

(18 citation statements)

References 10 publications

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“…2 One may speculate that the reason Tolo's paper did not attract much attention is that his de…nition was broader than the one we use today-too general, one may say-even though his paper does consider the case of a chain of groups, which is a special case (group by chain) of the contemporary sense (group by semilattice). Another promising way to impose extra structure is, as in the group case, to have unique factorisation [12], [11] and hence a Zappa-Szép product.…”

Section: Some Historymentioning

confidence: 99%

Factorizable inverse monoids

FitzGerald

2009

Semigroup Forum

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Factorizable inverse monoids constitute the algebraic theory of those partial symmetries which are restrictions of automorphisms; the formal de…nition is that each element is the product of an idempotent and an invertible. This class of monoids has theoretical signi…cance, and includes concrete instances which are important in various contexts. This survey is organised around the idea of group acts on semilattices and contains a large range of examples. Topics also include methods for construction of factorizable inverse monoids, and aspects of their inner structure, morphisms, and presentations.

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Section: Some Historymentioning

confidence: 99%

Factorizable inverse monoids

FitzGerald

2009

Semigroup Forum

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“…Zappa-Szép products were introduced by Zappa [22] and after being widely developed in the context of groups (see for example Szép [20]) were applied to more general structures by Kunze [12] and Brin [4], who used the term Zappa-Szép product. As shown in [12] and explicated in [6], Zappa-Szép products are closely related to the action of Mealy machines (automata with output depending on input and current state). The concept being so natural, a number of other names have been used, in particular that of general product [13].…”

Section: Introductionmentioning

confidence: 94%

Restriction semigroups and $$\lambda $$ λ -Zappa-Szép products

Gould

Zenab

2016

Period Math Hung

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The aim of this paper is to study λ-semidirect and λ-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product S T where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-ScheinNambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are λ-semidirect products and λ-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of λ-semidirect products.

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“…We remark that our construction immediately applied to the regular case if S is an inverse semigroup as shown in [6].…”

Section: (E E)(e S) = (E Es) = (E S)mentioning

confidence: 99%

“…S be the standard Zappa-Szép product of an inverse semigroup S and semilattice of idempotents E(S) in the sense of [6] and let T = {(e, s) : es = s} be a subsemigroup of Z . Then there is a canonical embedding of S into T under s → (ss −1 , s).…”

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

confidence: 99%

“…The following example is inspired by a result that appeared in [6], where Gilbert and Wazzan considered the Zappa-Szép product E G of a group G and a semilattice E. By using the natural involution in a group they were able to define a Zappa-Szép product G E. Left restriction semigroups and monoids do not possess a natural involution in the way that groups and indeed inverse semigroups do. 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%

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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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Zappa–Szép products of bands and groups

Cited by 9 publications

References 10 publications

Factorizable inverse monoids

Factorizable inverse monoids

Restriction semigroups and $$\lambda $$ λ -Zappa-Szép products

Algebraic properties of Zappa–Szép products of semigroups and monoids

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