Groups, semilattices and inverse semigroups. II

McAlister, D. B.

doi:10.1090/s0002-9947-74-99950-4

Cited by 103 publications

(61 citation statements)

References 16 publications

(6 reference statements)

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“…This map is, in fact, an immersion [20] on the 1-skeleton; also, any based 2-cell has at most one based lift to any vertex. McAlister's P -theorem [10] can easily be shown to be equivalent to stating that I is E-unitary (that is, the natural projection to G is idempotent pure) if and only there is an ordered 2-complex Z containing X and an extension of ψ to Z which is a covering such that Π 1 (Z) is an enlargement of Π 1 (X) in the sense of [7,21]. In fact, all the results of [21] can be restated and proved more generally in the context of ordered 2-complexes where the role of the derived ordered groupoid is replaced by what is called in homotopy theory, the mapping fiber.…”

Section: The Schützenberger Complexmentioning

confidence: 99%

A topological approach to inverse and regular semigroups

Steinberg¹

2003

Pacific J. Math.

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Work of Ehresmann and Schein shows that an inverse semigroup can be viewed as a groupoid with an order structure; this approach was generalized by Nambooripad to apply to arbitrary regular semigroups. This paper introduces the notion of an ordered 2-complex and shows how to represent any ordered groupoid as the fundamental groupoid of an ordered 2-complex. This approach then allows us to construct a standard 2-complex for an inverse semigroup presentation.Our primary applications are to calculating the maximal subgroups of an inverse semigroup which, under our topological approach, turn out to be the fundamental groups of the various connected components of the standard 2-complex. Our main results generalize results of Haatja, Margolis, and Meakin giving a graph of groups decomposition for the maximal subgroups of certain regular semigroup amalgams. We also generalize a theorem of Hall by showing the strong embeddability of certain regular semigroup amalgams as well as structural results of Nambooripad and Pastijn on such amalgams.

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Section: The Schützenberger Complexmentioning

confidence: 99%

A topological approach to inverse and regular semigroups

Steinberg¹

2003

Pacific J. Math.

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“…of an order ideal and subsemigroup of a semigroup which, in a sense, generalizes the construction of a P-semigroup, which has proved to be useful in describing inverse semigroups (McAlister [7], [8]), by replacing the group which acts on a semilattice by a completely simple semigroup. The homomorphism which arises in Pastijn's theorem is special in that the inverse image of each idempotent is a completely simple semigroup.…”

Section: Introductionmentioning

confidence: 99%

Some covering theorems for locally inverse semigroups

McAlister

1985

J Aust Math Soc A

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A regular semigroup 5 is said to be locally inverse if each local submonoid eSe, with e an idempotent, is an inverse semigroup. In this paper we apply known covering theorems for inverse semigroups and a covering theorem for locally inverse semigroups due to the author to obtain some covering theorems for locally inverse semigroups. The techniques developed here permit us to give an alternative proof for, and slight strengthening of, an important covering theorem for locally inverse semigroups due to F. Pastijn.

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“…McAlister [17,18] demonstrated the utility of semidirect products in understanding inverse semigroups, results subsequently extended by a number of authors to the broader context of (left) restriction semigroups. 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.…”

Section: Introductionmentioning

confidence: 99%

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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Groups, semilattices and inverse semigroups. II

Cited by 103 publications

References 16 publications

A topological approach to inverse and regular semigroups

A topological approach to inverse and regular semigroups

Some covering theorems for locally inverse semigroups

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

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