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In this paper, we study partial group actions on Lie algebras. We describe the structure of the inverse semigroup of all partial automorphisms (isomorphisms between ideals) of a finite-dimensional ...

Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the congruence lattice of a graph inverse semigroup is upp...

As part of his study of representations of the polycylic monoids, Lawson described all the closed inverse submonoids of a polycyclic monoid Pn and classified them up to conjugacy. We show that Lawson’s description can be extended to closed inverse subsemigroups of graph inverse semigroups. We then apply Schein’s theory of cosets in inverse semigroups to the closed inverse subsemigroups of graph inverse semigroups: we give necessary and sufficient conditions for a closed inverse subsemigroup of a graph inverse semigroup to have finite index, and determine the value of the index when it is finite.

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R0[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.

Let ℐn be the symmetric inverse semigroup on Xn = {1, 2,…, n}, and let 𝒟𝒫n and 𝒪𝒟𝒫n be its subsemigroups of partial isometries and of order-preserving partial isometries of Xn, respectively. In this article, we investigate the cycle structure of a partial isometry and characterize the Green's relations on 𝒟𝒫n and 𝒪𝒟𝒫n. We show that 𝒪𝒟𝒫n is a 0-E-unitary inverse semigroup. We also investigate the ranks of 𝒟𝒫n and 𝒪𝒟𝒫n.

The aim in this article is to provide a parametrization of the finite dimensional irreducible representations of a compact inverse semigroup in terms of the irreducible representations of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new, and more conceptual, proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Moreover, we also prove that every norm continuous irreducible *-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

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.

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(E S ), or 𝒞o(E S ), respectively, where E S 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.

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 G S , the union of the subgroups of the semigroup (which is necessarily cryptic).

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.

For any locally inverse semigroup S, there exists a maximal dense ideal extension of S within the class LI of all locally inverse semigroups (Pastijn and Oliveira, 2006, Preprint). Here we realize this maximal dense ideal extension in terms of a canonically constructed quotient of a regular Rees matrix semigroup over an inverse semigroup.

In this article we prove that if S is a faithfully projective R-algebra and H is a finite inverse semigroup acting on S as R-linear maps such that the fixed subring S H = R, then any partial isomorphism between ideals of S which are generated by central idempotents can be obtained as restriction of an R-automorphism of S and there exists a finite subgroup of automorphisms G of S with S G = R.

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 = a n or (ab) m = b n , where σ is the minimum group congruence on S.

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.

The study of coset monoids was initiated by Schein in 1966. The coset monoid of a group G, denoted 𝒞(G), consists of all cosets of all subgroups of G. We show how to generalize 𝒞(G) by constructing a monoid 𝒞ℒ(G) of cosets of subgroups from a semilattice (ℒ, ∨ℒ) of subgroups of G which satisfies certain conditions. When (ℒ, ∨ℒ) = (𝒮(G), ∨) is the join semilattice of all subgroups of G, we recover the coset monoid 𝒞(G). The class of semigroups which are isomorphic to some 𝒞ℒ(G) has a very simple description. Elements of the monoids in this class may be represented by cosets of subgroups of their group of units. We show that the factorizable part of the symmetric (resp. dual symmetric) inverse semigroup does not (resp. does) belong to this class.

We introduce a new notion of rank for a semigroup S. The rank is associated with pairs (I,ρ), where ρ is a right congruence and I is a ρ-saturated right ideal. We allow I to be the empty set; in this case the rank of (∅, ρ) is the Cantor-Bendixson rank of ρ in the lattice of right congruences of S, with respect to a topology we title the finite type topology. If all pairs have rank, then we say that S is ranked. Our notion of rank is intimately connected with chain conditions: every right Noetherian semigroup is ranked, and every ranked inverse semigroup is weakly right Noetherian. Our interest in ranked semigroups stems from the study of the class ± bℰ S of existentially closed S-sets over a right coherent monoid S. It is known that for such S the set of sentences in the language of S-sets that are true in every existentially closed S-set, that is, the theory T S of ± bℰ S , has the model theoretic property of being stable. Moreover, T S is superstable if and only if S is weakly right Noetherian...

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.

Maximal inverse subsemigroups of the finite symmetric inverse semigroup In on the finite set are described. Further, we investigate maximal inverse subsemigroups of the inverse semigroup , and completely obtain their classification.

If an Ockham algebra L belongs to a Berman class and its endomorphism semi­group End L is regular then necessarily L ∈ Kp 2 for some p. For a given L∈Kp 2 the question of precisely when End L is regular is solved in the case where L is subdirectly irre­ducible. Using a particular construction, we show that every Berman class Kp 2 contains an algebra L for which End L is an inverse semigroup.

Let G be a group and S an inverse semigroup of endomorphisms of G. The simplicity of the centralizer near- ring MS(G) = {fe M(G)‖foα = αo f, ∀αeS} is characterized. The necessary and sufficient conditions are given for simplicity of Ms(G) in terms of the structure of G and S.