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Mark V Lawson

Publications and source records attributed to Mark V Lawson.

17 recordsLinked to original sources
arXiv2026

Finitary Cartesian closed varieties and semigroup actions

We build on some ideas of Richard Garner. Let $M$ be a monoid and $B$ a Boolean algebra. A `matched pair' $[B|M]$ consists of $B$ and $M$ and some mutual interactions. Garner showed that every such matched pair determines (what we shall call) a Boolean left restriction monoid $S = S[B|M]$. In this paper, we show that the data of a $[B|M]$-set (defined later) may be encoded by means of a certain kind of action by $S$. This means that the category $[B|M]$-{\bf sets} is equivalent to a category of {\bf $S$-actions}. We deduce, as a result of Garner's work, that every non-degenerate finitary Cartesian closed variety is equivalent to a special category of $S$-actions where $S$ is a Boolean left restriction monoid.

math.CT↗
arXiv2021

On Ehresmann semigroups

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

math.CT↗
arXiv2020

The polycyclic inverse monoids and the Thompson groups revisited

We revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group $G_{n,1}$ is the group of units of a Boolean inverse monoid $C_{n}$ called the Cuntz inverse monoid. This inverse monoid is proved to be the tight completion of the polycyclic inverse monoid $P_{n}$. The étale topological groupoid associated with $C_{n}$ under non-commutative Stone duality is the usual groupoid associated with the corresponding Cuntz $C^{\ast}$-algebra. We then show that the group $G_{n,1}$ is also the group of automorphisms of a specific $n$-ary Cantor algebra: this $n$-ary Cantor algebra is constructed first as the monoid of total maps of a restriction semigroup à la Statman and then in terms of labelled trees à la Higman.

math.GR↗
arXiv2020

Primer on inverse semigroups II

This report assumes the basics of inverse semigroup theory as described in the first primer but goes on to show how they may be analysed using ideas from category theory.

math.CT↗
arXiv2020

Primer on inverse semigroups I

This is an overview of the basics of inverse semigroup theory written for the Workshop on Semigroups and Categories held at the University of Ottawa in 2010.

math.GR↗
arXiv2020

Higher dimensional generalizations of the Thompson groups

We show how to construct a family of groups with simple commutator subgroups from aperiodic 1-vertex, finitely aligned higher rank graphs (which are, in fact, a class of cancellative monoids). Inverse semigroups form the intermediary between these cancellative monoids and the family of groups we are interested in. These groups can naturally be viewed as higher-dimensional generalizations of the classical Thompson groups since the finite direct products of free monoids are examples of the appropriate 1-vertex higher rank graphs.

math.RA↗
arXiv2019

The universal Boolean inverse semigroup presented by the abstract Cuntz-Krieger relations

This paper is a contribution to the theory of what might be termed $0$-dimensional non-commutative spaces. We prove that associated with each inverse semigroup $S$ is a Boolean inverse semigroup presented by the abstract versions of the Cuntz-Krieger relations. We call this Boolean inverse semigroup the Exel completion of $S$ and show that it arises from Exel's tight groupoid under non-commutative Stone duality.

math.OA↗
arXiv2017

Tarski monoids: Matui's spatial realization theorem

We introduce a class of inverse monoids, called Tarski monoids, that can be regarded as non-commutative generalizations of the unique countable, atomless Boolean algebra. These inverse monoids are related to a class of etale topological groupoids under a non-commutative generalization of classical Stone duality and, significantly, they arise naturally in the theory of dynamical systems as developed by Matui. We are thereby able to reinterpret a theorem of Matui on a class of étale groupoids as an equivalent theorem about a class of Tarski monoids: two simple Tarski monoids are isomorphic if and only if their groups of units are isomorphic. The inverse monoids in question may also be viewed as countably infinite generalizations of finite symmetric inverse monoids. Their groups of units therefore generalize the finite symmetric groups and include amongst their number the classical Thompson groups.

math.CT↗
arXiv2014

On a class of countable Boolean inverse monoids and Matui's spatial realization theorem

We introduce a class of inverse monoids that can be regarded as non-commutative generalizations of Boolean algebras. These inverse monoids are related to a class of étale topological groupoids, under a non-commutative generalization of classical Stone duality. Furthermore, and significantly for this paper, they arise naturally in the theory of dynamical systems as developed by Matui. We are thereby able to reinterpret a theorem of Matui on a class of étale groupoids, in the spirit of Rubin's theorem, as an equivalent theorem about a class of inverse monoids. The inverse monoids in question may be viewed as the countably infinite generalizations of finite symmetric inverse monoids. Their groups of units therefore generalize the finite symmetric groups and include amongst their number the Thompson groups $G_{n,1}$.

math.CT↗
arXiv2013

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.

math.CT↗
arXiv2012

The classifying space of an inverse semigroup

We refine Funk's description of the classifying space of an inverse semigroup by replacing his *-semigroups by right generalized inverse *-semigroups. Our proof uses the idea that presheaves of sets over meet semilattices may be characterized algebraically as right normal bands.

math.CT↗
arXiv2012

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.

math.CT↗
arXiv2010

Compactable semilattices

We characterize those semilattices that give rise to Boolean spaces on their associated spaces of ultrafilters. The class of 0-disjunctive semilattices, important in the theory of congruence-free inverse semigroups, plays a distinguished role in this theory.

math.GM↗
arXiv2009

A non-commutative generalization of Stone duality

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid associated with the Cuntz groupoid is the strong orthogonal completion of the polycyclic (or Cuntz) monoid and so its group of units is a Thompson group.

math.CT↗
arXiv2002

Universal groups for point-sets and tilings

We study the universal groups of inverse semigroups associated with point sets and with tilings. We focus our attention on two classes of examples. The first class consists of point sets which are obtained by a cut and projection scheme (so-called model sets). Here we introduce another inverse semigroup which is given in terms of the defining data of the projection scheme and related to the model set by the empire congruence. The second class is given by one-dimensional tilings.

math.GR↗