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

Distributive inverse semigroups and non-commutative Stone dualities

We develop the theory of distributive inverse semigroups as the analogue of distributive lattices without top element and prove that they are in a duality with those etale groupoids having a spectral space of identities, where our spectral spaces are not necessarily compact. We prove that Boolean inverse semigroups can be characterized as those distributive inverse semigroups in which every prime filter is an ultrafilter; we also provide a topological characterization in terms of Hausdorffness. We extend the notion of the patch topology to distributive inverse semigroups and prove that every distributive inverse semigroup has a Booleanization. As applications of this result, we give a new interpretation of Paterson's universal groupoid of an inverse semigroup and by developing the theory of what we call tight coverages, we also provide a conceptual foundation for Exel's tight groupoid.Comment: arXiv admin note: substantial text overlap with arXiv:1107.551

Filtering germs: Groupoids associated to inverse semigroups

We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action of the inverse semigroup on the space of idempotent filters. We also investigate the restriction of this isomorphism to the groupoid of tight filters and to the groupoid of ultrafilters.Comment: 9 pages. This version matches the version in Expositiones Mathematica