<i>E</i>-unitary covers for inverse semigroups

McAlister, D. B.; Reilly, Norman R.

doi:10.2140/pjm.1977.68.161

Cited by 54 publications

(47 citation statements)

References 6 publications

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“…This parallels the case for certain covers of regular semigroups [1] and E-unitary covers of inverse semigroups [12]. In these cases, the monoid T is replaced by a group G and one is interested in idempotent pure surjective relational morphisms τ : S −→ • G and their inverses τ −1 : G −→ • S. In the inverse case, the associated covers can be described in terms of prehomomorphisms from S into the inverse monoid of all cosets of subgroups of G [12]. For each s ∈ S, the subset sτ is a coset of G; in the left ample case, however, the subsets sτ of T do not appear to enjoy any corresponding property, and so we do not have analogues of these particular results.…”

Section: Relational Morphisms Prehomomorphisms and Proper Coversmentioning

confidence: 54%

“…The surjective relational morphisms from G to S and their associated covers are described in [12] in terms of dual prehomomorphisms from G to the inverse monoid of all permissible subsets of S. There are analogous results for the left ample case which we describe in the following sections.…”

Section: Relational Morphisms Prehomomorphisms and Proper Coversmentioning

confidence: 88%

“…An embedding ϕ of an inverse semigroup S into a factorisable inverse monoid F is strict if for each g ∈ G, there is an element s of S such that sϕ g. McAlister and Reilly [12] showed that such an embedding gives rise to an E-unitary cover of S over G, and conversely that every E-unitary cover is isomorphic to one constructed in this way. The situation is more complicated in the case of left ample semigroups.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Relational Morphisms Prehomomorphisms and Proper Coversmentioning

confidence: 54%

Section: Relational Morphisms Prehomomorphisms and Proper Coversmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

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Proper Covers for Left Ample Semigroups

Qallali

Fountain

2005

Semigroup Forum

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“…On the converse side, any covering : P ! S such that P is E-unitary and P= = G may be completed to essentially the same diagram, giving rise to a strict embedding of S into a factorizable inverse monoid GE [28]. There is a thorough exposition of this equivalence of strict factorizable embeddings with E-unitary covers in [18], especially sections 2.2 and 8.2.…”

Section: The Factorizable Part Of the Symmetric Inverse Monoidmentioning

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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“…LEMMA 0. 4 (McAlister (1974) and McAlister and Reilly (1977) (a,g),(b,h)eP (a, g) a(b, h) if and only if g = h.…”

Section: Preliminariesmentioning

confidence: 99%

Regular semigroups, fundamental semigroups and groups

McAlister

1980

J. Aust. Math. Soc.

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In this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomorphism 6: S -> T between regular semigroups. This is a mapping such that (ab) 9 ^a0b9 in the natural partial order on T.

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E-unitary covers for inverse semigroups

Cited by 54 publications

References 6 publications

Proper Covers for Left Ample Semigroups

Proper Covers for Left Ample Semigroups

Factorizable inverse monoids

Regular semigroups, fundamental semigroups and groups

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