Factorisable right adequate semigroups

El-Qallali, Abdulsalam

doi:10.1017/s0013091500016497

Cited by 6 publications

(9 citation statements)

References 6 publications

(10 reference statements)

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“…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%

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: The Monoids C(s) and Their Representationsmentioning

confidence: 99%

Almost perfect restriction semigroups

Jones

2016

Journal of Algebra

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“…For our later convenience, denote by ν iλ the surjective homomorphism E iλ → Y which assigns α to every element in E α iλ , and put E i = λ∈Λ E iλ , E λ = i∈I E iλ . Here E i = E Ti is a normal band by Lemma 16 (2), and it can be seen by (3) to be a semilattice Y of the rectangular bands E α i = λ∈Λ E α iλ (α ∈ Y ). Dually, E λ = E T λ is a normal band, and a semilattice Y of the rectangular bands…”

Section: Almost Factorizable Locally Inverse Semigroups 1039mentioning

confidence: 99%

“…The notion of almost factorizability and the basic results mentioned for the inverse case have been generalized in several directions: for straight locally inverse semigroups by Dombi [2], for orthodox semigroups by Hartmann [7] and for right adequate and for weakly ample semigroups by El Qallali [3], and by Gomes and the author [5], respectively. In all of these classes, the definition of almost factorizability followed, in some sense, the way having been known in the inverse case: given a member S of the class considered, the property required for S to be almost factorizable is formulated within the semigroup of all permissible sets of S, or within the translational hull of S. Furthermore, in most of these classes, the almost factorizable members turned out to have a role dual to the E-unitary-like members introduced and studied much earlier.…”

Section: Introductionmentioning

confidence: 99%

Almost Factorizable Locally Inverse Semigroups

Szendrei

2011

Int. J. Algebra Comput.

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“…If S and T are inverse semigroups, then we require (5) and (3) together give (2). We say that τ is surjective if the projection to T is surjective.…”

Section: Relational Morphisms Prehomomorphisms and Proper Coversmentioning

confidence: 99%

“…Dualising the notion of factorisable right ample monoid [2], we say that a left ample monoid F is factorisable if F = ET where E = E(F ) and T is the R * -class of the identity. Note that T is a right cancellative submonoid of F .…”

Section: Factorisable Left Ample Monoidsmentioning

confidence: 99%

Proper Covers for Left Ample Semigroups

Qallali

Fountain

2005

Semigroup Forum

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Abstract. A left ample semigroup is a semigroup with a unary operation + which has a (2, 1)-algebra embedding into a symmetric inverse monoid I(X), the operation + on I(X) being defined by α + = αα −1 . We consider some analogues for left ample semigroups of results on E-unitary covers of inverse semigroups due to McAlister and Reilly. The analogue of an E-unitary cover is a proper cover, and we discuss the construction of proper covers in terms of relational homomorphisms, and of dual prehomomorphisms. We observe that our construction gives an E-dense proper cover for an E-dense left ample semigroup. We also consider proper covers constructed from strict embeddings into factorisable left ample monoids. In contrast to the inverse case, not all proper covers arise in this way. However, in the E-dense case, we characterise those E-dense proper covers which can be constructed from such embeddings.

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Factorisable right adequate semigroups

Cited by 6 publications

References 6 publications

Almost perfect restriction semigroups

Almost perfect restriction semigroups

Almost Factorizable Locally Inverse Semigroups

Proper Covers for Left Ample Semigroups

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