arXiv subjects
Publications and source records attributed to Mark V. Lawson.
We show that, in certain circumstances, a Boolean ample monoid may be fully embedded into a Boolean inverse monoid in a way that generalizes how right reversible cancellative monoids may be embedded into groups. We use groupoids of fractions and non-commutative Stone duality to prove the result.
In this paper, we describe étale Boolean right restriction monoids in terms of Boolean inverse monoids.
A countably infinite Boolean inverse monoid that can be written as an increasing union of finite Boolean inverse monoids (suitably embedded) is said to be of finite type. Borrowing terminology from $C^{\ast}$-algebra theory, we say that such a Boolean inverse monoid is AF (approximately finite) if the finite Boolean inverse monoids above are isomorphic to finite direct products of finite symmetric inverse monoids, and we say that it is UHF (uniformly hyperfinite) if the finite Boolean inverse monoids are in fact isomorphic to finite symmetric inverse monoids. We characterize abstractly the Boolean inverse monoids of finite type and those which are AF and, by using MV-algebras, we also characterize the UHF monoids.
This is an account of the theory of inverse semigroups, assuming only that the reader knows the basics of semigroup theory.
We prove that there is an adjunction between what we call étale topological categories and restriction quantal frames that leads to an adjunction with a category of complete restriction monoids. This generalizes the adjunction between étale groupoids and pseudogroups.
We show explicitly that Boolean inverse semigroups are in duality with what we term Boolean groupoids. This generalizes the classical Stone duality, which we refer to as commutative Stone duality, between generalized Boolean algebras and locally compact Hausdorff $0$-dimensional spaces.
We describe the structure of finite Boolean inverse monoids and apply our results to the representation theory of finite inverse semigroups. We then generalize to semisimple Boolean inverse semigroups.
We prove that the forgetful functor from the category of Boolean inverse semigroups to inverse semigroups with zero has a left adjoint. This left adjoint is what we term the `Booleanization'. We establish the exact connection between the Booleanization of an inverse semigroup and Paterson's universal groupoid of the inverse semigroup and we explicitly compute the Booleanization of the polycyclic inverse monoid $P_{n}$ and demonstrate its affiliation with the Cuntz-Toeplitz algebra.
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 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. 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.
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 $\wedge$-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. Such inverse monoids are natural infinite generalizations of finite symmetric inverse monoids. 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 what we term AF inverse monoids, Krieger's ample groups being examples. We characterize such groups as subgroups of particular kinds of the group of homeomorphisms of the Cantor space.
The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods.
The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa-Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych, In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be signficant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.
We define a class of inverse monoids having the property that their lattices of principal ideals naturally form an MV-algebra. We say that an arbitrary MV-algebra can be co-ordinatized if it is isomorphic to one constructed in this way from such a monoid. We prove that every countable MV-algebra can be so co-ordinatized. The particular inverse monoids needed to establish this result are examples of what we term AF inverse monoids. These are constructed from Bratteli diagrams and arise naturally as direct limits of finite direct products of finite symmetric inverse monoids.
Graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of C*-algebras. This paper has two main goals: first, to provide an abstract characterization of graph inverse semigroups; and second, to show how they may be completed, under suitable conditions, to form what we call the Cuntz-Krieger semigroup of the graph. This semigroup is the ample semigroup of a topological groupoid associated with the graph, and the semigroup analogue of the Leavitt path algebra of the graph.
We prove that the structure of right generalized inverse semigroups is determined by free étale actions of inverse semigroups. This leads to a tensor product interpretation of Yamada's classical struture theorem for generalized inverse semigroups.
A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory, C*-algebras and aperiodic tilings. Our starting point is an adjunction between a category of pseudogroups and a category of etale groupoids from which we are able to set up a duality between spatial pseudogroups and sober etale groupoids. As a corollary to this duality, we deduce a non-commutative version of Stone duality involving what we call boolean inverse semigroups and boolean etale groupoids, as well as a generalization of this duality to distributive inverse semigroups. Non-commutative Stone duality has important applications in the theory of C*-algebras: it is the basis for the construction of Cuntz and Cuntz-Krieger algbras and in the case of the Cuntz algebras it can also be used to construct the Thompson groups. We then define coverages on inverse semigroups and the resulting presentations of pseudogroups. As applications, we show that Paterson's universal groupoid is an example of a booleanization, and reconcile Exel's recent work on the theory of tight maps with the work of the second author.
We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister's theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.