A Groupoid Approach to Discrete Inverse Semigroup Algebras

Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup $S$ that can be induced from associated groups as precisely those satisfying a certain "finiteness condition". This "finiteness condition" is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent

MacNeille completion and profinite completion can coincide on finitely generated modal algebras

Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if $\mathbb{A}$ is a residually finite, finitely generated modal algebra such that $\operatorname{HSP}(\mathbb{A})$ has equationally definable principal congruences, then the profinite completion of $\mathbb{A}$ is isomorphic to its MacNeille completion, and $\Diamond$ is smooth. Specific examples of such modal algebras are the free $\mathbf{K4}$-algebra and the free $\mathbf{PDL}$-algebra.Comment: 5 page

The possible values of critical points between strongly congruence-proper varieties of algebras

We denote by Conc(A) the semilattice of all finitely generated congruences of an (universal) algebra A, and we define Conc(V) as the class of all isomorphic copies of all Conc(A), for A in V, for any variety V of algebras. Let V and W be locally finite varieties of algebras such that for each finite algebra A in V there are, up to isomorphism, only finitely many B in W such that A and B have isomorphic congruence lattices, and every such B is finite. If Conc(V) is not contained in Conc(W), then there exists a semilattice of cardinality aleph 2 in Conc(V)-Conc(W). Our result extends to quasivarieties of first-order structures, with finitely many relation symbols, and relative congruence lattices. In particular, if W is a finitely generated variety of algebras, then this occurs in case W omits the tame congruence theory types 1 and 5; which, in turn, occurs in case W satisfies a nontrivial congruence identity. The bound aleph 2 is sharp

Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse $\wedge$-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse $\wedge$-semigroups arise as completions of inverse semigroups we call pre-Boolean. An inverse $\wedge$-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where the definition of a tight filter is obtained by combining work of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson-Higman groups $G_{n,r}$. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz-Krieger $C^{\ast}$-algebras.Comment: The presentation has been sharpened up and some minor errors correcte

Principal and syntactic congruences in congruence-distributive and congruence-permutable varieties

We give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even under the additional assumption that the variety is residually very finite