179 research outputs found
The Booleanization of an inverse semigroup
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 and demonstrate its affiliation with
the Cuntz-Toeplitz algebra.Comment: This is an updated version of the previous paper. Typos where found
have been corrected and a new section added that shows how to construct the
Booleanization directly from an arbitrary inverse semigroup with zero
(without having to use its distributive completion
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 \'etale 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.Comment: arXiv admin note: text overlap with arXiv:1407.147
Entailment systems for stably locally compact locales
The category SCFrU of stably continuous frames and preframe ho-momorphisms (preserving ¯nite meets and directed joins) is dual to the Karoubi envelope of a category Ent whose objects are sets and whose
morphisms X ! Y are upper closed relations between the ¯nite powersets FX and FY . Composition of these morphisms is the \cut composition" of Jung et al. that interfaces disjunction in the codomains with conjunctions in the domains, and thereby relates to their multi-lingual sequent
calculus. Thus stably locally compact locales are represented by \entailment systems" (X; `) in which `, a generalization of entailment relations,is idempotent for cut composition.
Some constructions on stably locally compact locales are represented
in terms of entailment systems: products, duality and powerlocales.
Relational converse provides Ent with an involution, and this gives a simple treatment of the duality of stably locally compact locales. If A and B are stably continuous frames, then the internal preframe hom A t B is isomorphic to e A Â B where e A is the Hofmann-Lawson dual.
For a stably locally compact locale X, the lower powerlocale of X is shown to be the dual of the upper powerlocale of the dual of X
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
Restriction categories III: colimits, partial limits, and extensivity
A restriction category is an abstract formulation for a category of partial
maps, defined in terms of certain specified idempotents called the restriction
idempotents. All categories of partial maps are restriction categories;
conversely, a restriction category is a category of partial maps if and only if
the restriction idempotents split. Restriction categories facilitate reasoning
about partial maps as they have a purely algebraic formulation.
In this paper we consider colimits and limits in restriction categories. As
the notion of restriction category is not self-dual, we should not expect
colimits and limits in restriction categories to behave in the same manner. The
notion of colimit in the restriction context is quite straightforward, but
limits are more delicate. The suitable notion of limit turns out to be a kind
of lax limit, satisfying certain extra properties.
Of particular interest is the behaviour of the coproduct both by itself and
with respect to partial products. We explore various conditions under which the
coproducts are ``extensive'' in the sense that the total category (of the
related partial map category) becomes an extensive category. When partial
limits are present, they become ordinary limits in the total category. Thus,
when the coproducts are extensive we obtain as the total category a lextensive
category. This provides, in particular, a description of the extensive
completion of a distributive category.Comment: 39 page
Sheaf representation of monoidal categories
Every small monoidal category with universal (finite) joins of central
idempotents is monoidally equivalent to the category of global sections of a
sheaf of (sub)local monoidal categories on a topological space. Every small
stiff monoidal category monoidally embeds into such a category of global
sections. These representation results are functorial and subsume the
Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone
representation of Boolean algebras, and the Takahashi representation of Hilbert
modules as continuous fields of Hilbert spaces. Many properties of a monoidal
category carry over to the stalks of its sheaf, including having a trace,
having exponential objects, having dual objects, having limits of some shape,
and the central idempotents forming a Boolean algebra.Comment: 39 page
Introduction to inverse semigroups
This is an account of the theory of inverse semigroups, assuming only that
the reader knows the basics of semigroup theory.Comment: arXiv admin note: text overlap with arXiv:2006.0162
A categorical foundation for structured reversible flowchart languages: Soundness and adequacy
Structured reversible flowchart languages is a class of imperative reversible
programming languages allowing for a simple diagrammatic representation of
control flow built from a limited set of control flow structures. This class
includes the reversible programming language Janus (without recursion), as well
as more recently developed reversible programming languages such as R-CORE and
R-WHILE.
In the present paper, we develop a categorical foundation for this class of
languages based on inverse categories with joins. We generalize the notion of
extensivity of restriction categories to one that may be accommodated by
inverse categories, and use the resulting decisions to give a reversible
representation of predicates and assertions. This leads to a categorical
semantics for structured reversible flowcharts, which we show to be
computationally sound and adequate, as well as equationally fully abstract with
respect to the operational semantics under certain conditions
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