We study classes of proper restriction semigroups determined by properties of partial actions underlying them. These properties include strongness, antistrongness, being defined by a homomorphism, being an action etc. Of particular interest is the class determined by homomorphisms, primarily because we observe that its elements, while being close to semidirect products, serve as mediators between general restriction semigroups and semidirect products or W -products in an embedding-covering construction. It is remarkable that this class does not have an adequate analogue if specialized to inverse semigroups. F -restriction monoids of this class, called ultra F -restriction monoids, are determined by homomorphisms from a monoid T to the Munn monoid of a semilattice Y . We show that these are precisely the monoids Y * m T considered by Fountain, Gomes and Gould. We obtain a McAlister-type presentation for the class given by strong dual prehomomorphisms and apply it to construct an embedding of ultra F -restriction monoids, for which the base monoid T is free, into W -products of semilattices by monoids. Our approach yields new and simpler proofs of two recent embedding-covering results by Szendrei.
Abstract. We establish two duality theorems which refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category ofétale spaces over locally compact Boolean spaces whose morphisms areétale space cohomomorphisms over continuous proper maps. In the second theorem we prove that the category of lefthanded skew Boolean ∩-algebras whose morphisms are proper skew Boolean ∩-algebra homomorphisms is equivalent to the category ofétale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injectiveétale space cohomomorphisms over continuous proper maps.
We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.
We compare three approaches to the notion of conjugacy for semigroups, the first one via the transitive closure of the uv ∼ vu relation, the second one via an action of inverse semigroups on themselves by partial transformations, and the third one via characters of finitedimensional representations. points of view by many authors, see for example [Pu,Li,KM1]. However, this approach is not unique. Another approach, which will be discussed in detail in the next section, comes from the equivalence relation generated by the (non-transitive in general) relation on S, which relates the element uv to the element vu for all u, v ∈ S. This notion, which has roots in the study of free monoids, see [La, 11.5], was also studied in [GK, KM1, KM2, Ku] for various special classes of semigroups. In the present paper we show that for some classes of semigroups the latter notion of conjugacy admits, just like for groups, alternative descriptions via the action of a semigroup on itself, and/or via characters of finite-dimensional representations.As usually, we denote Green's relations on a semigroup by J , D, R, L and H. If ϕ : S → End C (V ) is a representation of a semigroup S, then the character of ϕ is the function χ ϕ : S → C for which χ ϕ (s) is defined as the trace of ϕ(s), s ∈ S. A semigroup S is called group-bound provided that for each x ∈ S there exists k ∈ {1, 2, . . . } such that x k lies in a subgroup of S. If S is group-bound, then for x ∈ S we denote by e x the idempotent of the subgroup containing x k as above. For a partial transformation a we denote by dom(a) and im(a) the domain or the image of a respectively.
This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over byétale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We prove several main results. To start with, we establish a duality between the category of complete restriction monoids and the category ofétale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups andétale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. A wider class of quantales, called multiplicative Ehresmann quantal frames, is put into a correspondence with those localic categories where the multiplication structure map is semiopen, and all the other structure maps are open. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction betweenétale localic categories andétale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectralétale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work
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