603 research outputs found
Stone duality above dimension zero: Axiomatising the algebraic theory of C(X)
It has been known since the work of Duskin and Pelletier four decades ago
that KH^op, the category opposite to compact Hausdorff spaces and continuous
maps, is monadic over the category of sets. It follows that KH^op is equivalent
to a possibly infinitary variety of algebras V in the sense of Slominski and
Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of V can
be generated using a finite number of finitary operations, together with a
single operation of countably infinite arity. In 1983, Banaschewski and Rosicky
independently proved a conjecture of Bankston, establishing a strong negative
result on the axiomatisability of KH^op. In particular, V is not a finitary
variety--Isbell's result is best possible. The problem of axiomatising V by
equations has remained open. Using the theory of Chang's MV-algebras as a key
tool, along with Isbell's fundamental insight on the semantic nature of the
infinitary operation, we provide a finite axiomatisation of V.Comment: 26 pages. Presentation improve
Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective
We establish, generalizing Di Nola and Lettieri's categorical equivalence, a
Morita-equivalence between the theory of lattice-ordered abelian groups and
that of perfect MV-algebras. Further, after observing that the two theories are
not bi-interpretable in the classical sense, we identify, by considering
appropriate topos-theoretic invariants on their common classifying topos, three
levels of bi-intepretability holding for particular classes of formulas:
irreducible formulas, geometric sentences and imaginaries. Lastly, by
investigating the classifying topos of the theory of perfect MV-algebras, we
obtain various results on its syntax and semantics also in relation to the
cartesian theory of the variety generated by Chang's MV-algebra, including a
concrete representation for the finitely presentable models of the latter
theory as finite products of finitely presentable perfect MV-algebras. Among
the results established on the way, we mention a Morita-equivalence between the
theory of lattice-ordered abelian groups and that of cancellative
lattice-ordered abelian monoids with bottom element.Comment: 54 page
On EQ-monoids
An EQ-monoid A is a monoid with distinguished subsemilattice L with 1 2 L and such that any a, b 2 A have a largest right equalizer in L. The class of all such monoids equipped with a binary operation that identifies this largest right equalizer is a variety. Examples include Heyting algebras, Cartesian products of monoids with zero, as well as monoids of relations and partial maps on sets. The variety is 0-regular (though not ideal determined and hence congruences do not permute), and we describe the normal subobjects in terms of a global semilattice structure. We give representation theorems for several natural subvarieties in terms of Boolean algebras, Cartesian products and partial maps. The case in which the EQmonoid is assumed to be an inverse semigroup with zero is given particular attention. Finally, we define the derived category associated with a monoid having a distinguished subsemilattice containing the identity (a construction generalising the idea of a monoid category), and show that those monoids for which this derived category has equalizers in the semilattice constitute a variety of EQ-monoids
Semiring and semimodule issues in MV-algebras
In this paper we propose a semiring-theoretic approach to MV-algebras based
on the connection between such algebras and idempotent semirings - such an
approach naturally imposing the introduction and study of a suitable
corresponding class of semimodules, called MV-semimodules.
We present several results addressed toward a semiring theory for
MV-algebras. In particular we show a representation of MV-algebras as a
subsemiring of the endomorphism semiring of a semilattice, the construction of
the Grothendieck group of a semiring and its functorial nature, and the effect
of Mundici categorical equivalence between MV-algebras and lattice-ordered
Abelian groups with a distinguished strong order unit upon the relationship
between MV-semimodules and semimodules over idempotent semifields.Comment: This version contains some corrections to some results at the end of
Section
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 -semigroups, with a
class of topological groupoids, called Hausdorff Boolean groupoids. Much of the
paper is given over to showing that Boolean inverse -semigroups arise
as completions of inverse semigroups we call pre-Boolean. An inverse
-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 . 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 -algebras.Comment: The presentation has been sharpened up and some minor errors
correcte
The semiring-theoretic approach to MV-algebras: a survey
In this paper we review some of the main achievements of the
semiring-theoretic approach to MV-algebras initiated and pursued mainly by the
present authors and their collaborators. The survey focuses mainly on the
connections between MV-algebras and other theories that such a semiringbased
approach enabled, and on an application of such a framework to Digital Image
Processing. We also give some suggestions for further developments by stating
several open problems and possible research lines.Comment: Published versio
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