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In this paper, we investigate certain submodules in C*-algebras associated to effective étale groupoids. First, we show that a submodule generated by normalizers is a closure of the set of compactly supported continuous functions on some open set. As a corollary, we show that discrete group coactions on groupoid C*-algebras are induced by cocycles of étale groupoids if the fixed point algebras contain C*-subalgebras of continuous functions vanishing at infinity on the unit spaces. In the latter part, we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras. [ABSTRACT FROM AUTHOR]

AbstractLet P={Xi:i∈I} be a partition of a set X. Denote by T(X) the full transformation semigroup on X, and T(X,P) the subsemigroup of T(X) consisting of all transformations that preserve P . For every subsemigroup S(I) of T(I), let TS(I)(X,P) be the semigroup of all transformations f∈T(X,P) such that χ(f)∈S(I) , where χ(f)∈T(I) defined by iχ(f)=j whenever Xif⊆Xj . We describe regular and idempotent elements in TS(I)(X,P) , and determine when TS(I)(X,P) is a regular semigroup [inverse semigroup]. We characterize Green’s relations on TS(I)(X,P) , describe unit-regular elements in TS(I)(X,P) , and determine when TS(I)(X,P) is unit-regular. [ABSTRACT FROM AUTHOR]

We consider HNN extensions S * = [ S ; U 1 , U 2 ; ϕ ] where U1 and U2 are inverse monoids of an inverse semigroup S such that, for any u ∈ U i and e ∈ E (S) with u ≥ e in S, there exists f ∈ E (U i) with u ≥ f ≥ e in S, for i ∈ { 1 , 2 } ; we say that U1 and U2 are lower bounded in S. We construct and describe the Schützenberger automata of S * and give conditions for S * to have decidable word problem. Homomorphisms of the Schützenberger graphs of S * are studied and conditions are given for S * to be completely semisimple. When S has decidable word problem and U1 and U2 are finite, we show that S * has decidable word problem. The class of HNN extensions considered here is surprisingly useful and generalizes the class introduced by Jajcayová. A future paper intends to show that any HNN extension of an inverse semigroup can be embedded into an HNN extension where the subsemigroups are lower bounded. [ABSTRACT FROM AUTHOR]

For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et al., we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSpace -complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSpace -complete. Additionally, we provide simpler constructions for the uniform word problems of these classes. For the uniform word problem for automaton groups (without rational constraints), we show NL -hardness. Finally, we investigate a question asked by Cain about a better upper bound for the length of a word on which two distinct elements of an automaton semigroup must act differently. A detailed listing of the contributions of this paper can be found at the end of this paper. [ABSTRACT FROM AUTHOR]

This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by étale 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 of étale 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 and étale 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 between étale localic categories and étale 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 spectral étale 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 by Lawson and Lenz and by Resende. [ABSTRACT FROM AUTHOR]

Under non-commutative Stone duality, there is a correspondence between second countable Hausdorff étale groupoids which have a Cantor space of identities and what we call Tarski inverse monoids: that is, countable Boolean inverse ∧-monoids with semilattices of idempotents which are countable and atomless. Tarski inverse monoids are therefore the algebraic counterparts of the étale groupoids studied by Matui and provide a natural setting for many of his calculations. Under this duality, we prove that natural properties of the étale groupoid correspond to natural algebraic properties of the Tarski inverse monoid: effective groupoids correspond to fundamental Tarski inverse monoids and minimal groupoids correspond to 0-simplifying Tarski inverse monoids. Particularly interesting are the principal groupoids which correspond to Tarski inverse monoids where every element is a finite join of infinitesimals and idempotents. Here an infinitesimal is simply a non-zero element with square zero. The groups of units of fundamental Tarski inverse monoids generalize the finite symmetric groups and include amongst their number the Thompson groups G n , 1 as well as the groups of units of AF inverse monoids, Krieger's ample groups being examples. [ABSTRACT FROM AUTHOR]

An inverse semigroupSis a Howson inverse semigroup if the intersection of finitely generated inverse subsemigroups ofSis finitely generated. Given a locally finite action θ of a groupGon a semilatticeE, it is proved thatE*θGis a Howson inverse semigroup if and only ifGis a Howson group. It is also shown that this equivalence fails for arbitrary actions. [ABSTRACT FROM PUBLISHER]

This paper studies simplicity, primitivity and semiprimitivity of algebras associated to étale groupoids. Applications to inverse semigroup algebras are presented. The results also recover the semiprimitivity of Leavitt path algebras and can be used to recover the known primitivity criterion for Leavitt path algebras. [ABSTRACT FROM AUTHOR]

For a given integer ( k ≥ 2), we give here a class of finitely presented finite monoids of order n. Indeed the monoids Mon(π), where As a result of this study we are able to classify a wide family of the k-generated p-monoids (finite monoids of order a power of a prime p). An interesting di erence between the center of finite p-groups and the center of finite p-monoids has been achieved as well. All of these monoids are regular and non-commutative. [ABSTRACT FROM AUTHOR]

In this work, we generalize to hypercubic tilings of dimension n the description of tiling semigroups as inverse semigroups associated to factorial languages and the representation of this semigroup as a P*-semigroup. In addition, we show that, in contrast with the one-dimensional case, the tiling semigroup of any n-dimensional hypercubic tiling is always infinitely presented (even as a strongly E*-unitary inverse semigroup) and give a necessary and sufficient condition for two hypercubic tiling semigroups to be isomorphic. [ABSTRACT FROM AUTHOR]

The description by Johnston-Thom and the second author of the inverse semigroups S for which the lattice LF(S) of full inverse subsemigroups of S is join semidistributive is used to describe those for which (a) the lattice L(S) of all inverse subsemigroups or (b) the lattice Co(S) of convex inverse subsemigroups have that property. In contrast with the methods used by the authors to investigate lower semimodularity, the methods are based on decompositions via GS, the union of the subgroups of the semigroup (which is necessarily cryptic). [ABSTRACT FROM AUTHOR]

The authors' description of the inverse semigroups S for which the lattice LF(S) of full inverse subsemigroups is lower semimodular is used to describe those for which (a) the lattice L(S) of all inverse subsemigroups or (b) the lattice Co(S) of convex inverse subsemigroups has that property. In each case, we show that this occurs if and only if the entire lattice is a subdirect product of LF(S) with L(ES), or Co(ES), respectively, where ES is the semilattice of idempotents of S; a simple necessary and sufficient condition is found for each decomposition. For a semilattice E, L(E) is in fact always lower semimodular, and Co(E) is lower semimodular if and only if E is a tree. The conjunction of these results leads to quite a divergence between the ultimate descriptions in the two cases, L(S) and Co(S), with the latter being substantially richer. [ABSTRACT FROM AUTHOR]

In this article, we investigate some further properties of one-dimensional tiling semigroups as a particular case of the inverse semigroup associated with a factorial language. Namely, a presentation for the semigroup and its description as a P*-semigroup are obtained. Since both cons-truc-tions rely on the language, these properties highlight the deep connection between the semigroup and the language associated with a one-dimensional tiling semigroup. [ABSTRACT FROM AUTHOR]

An inverse semigroup S is said to be 0-semidistributive if its lattice LF(S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a,b ∈ S/σ with ab ≠ 1, there exist nonzero integers n and m such that (ab)m = an or (ab)m = bn, where σ is the minimum group congruence on S. [ABSTRACT FROM AUTHOR]

We introduce the notion of an orthogonal completion of an inverse monoid with zero. We show that the orthogonal completion of the polycyclic monoid on n generators is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated right ideals of the free monoid on n generators, and so we can make a direct connection with the Thompson groups Vn,1. [ABSTRACT FROM AUTHOR]

We prove that automorphisms of the endomorphism semigroup of a free inverse semigroup are inner and determine all isomorphisms between the endomorphism semigroups of free inverse semigroups. [ABSTRACT FROM AUTHOR]

It is shown that every finitely generated inverse subsemigroup (submonoid) of the monogenic free inverse semigroup (monoid) is finitely presented. As a consequence, the homomorphism and the isomorphism problems for the monogenic free inverse semigroup (monoid) are proven to be decidable. [ABSTRACT FROM AUTHOR]

We describe maximal Clifford subsemigroups of the finite symmetric inverse semigroup In of the set Xn - {1,2, ..., n} and obtain their complete classification, determine the number and the order of such subsemigroups, give a criterion for two maximal Clifford subsemigroups of In to be isomorphic, and determine (up to isomorphism) the number of such subsemigroups. Further, both maximal ideal extensions and maximal nil extensions of any Clifford subsemigroup S of In are described. In particular, we show that S has a unique maximal ideal extension in In and, up to isomorphism, S has a unique maximal nil extension in In. [ABSTRACT FROM AUTHOR]