Lukasiewicz logic and Riesz spaces

We initiate a deep study of {\em Riesz MV-algebras} which are MV-algebras endowed with a scalar multiplication with scalars from $[0,1]$. Extending Mundici's equivalence between MV-algebras and $\ell$-groups, we prove that Riesz MV-algebras are categorically equivalent with unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent with the class of commutative unital C$^*$-algebras. The propositional calculus ${\mathbb R}{\cal L}$ that has Riesz MV-algebras as models is a conservative extension of \L ukasiewicz $\infty$-valued propositional calculus and it is complete with respect to evaluations in the standard model $[0,1]$. We prove a normal form theorem for this logic, extending McNaughton theorem for \L ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in ${\mathbb R}{\cal L}$ and we relate them with the analogue of de Finetti's coherence criterion for ${\mathbb R}{\cal L}$.Comment: To appear in Soft Computin

A Unified Algebraic Framework for Fuzzy Image Compression and Mathematical Morphology

In this paper we show how certain techniques of image processing, having different scopes, can be joined together under a common "algebraic roof"

An analysis of the logic of Riesz Spaces with strong unit

We study \L ukasiewicz logic enriched with a scalar multiplication with scalars taken in $[0,1]$. Its algebraic models, called {\em Riesz MV-algebras}, are, up to isomorphism, unit intervals of Riesz spaces with a strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of {\em DMV-algebras} and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MV-algebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective

CoproductMV-Algebras, Nonstandard Reals, and Riesz Spaces

AbstractUp to categorical equivalence,MV-algebras are unit intervals of abelian lattice-ordered groups (for short,l-groups) with strong unit. While the property of being a strong unit is not even definable in first-order logic,MV-algebras are definable by a few simple equations. Accordingly, such notions as ideals and coproducts are definable for anyMV-algebraAas particular cases of the general algebraic notions. The radical RadAis the intersection of all maximal ideals ofA. AnMV-algebraAis said to be local iff it has a unique maximal ideal. Then, by Hoelder's theorem, the quotientA/RadAis isomorphic to a subalgebra of the real unit interval [0,1]. Using nonstandard real numbers we give a concrete representation of those totally orderedMV-algebrasAwhich are isomorphic to the coproduct ofA/RadAand 〈RadA〉, the latter denoting the subalgebra ofAgenerated by its radical. As an application, using several categorical equivalences we describe theMV-algebraic counterparts of Riesz spaces, also known as vector lattices

Compact Representations of BL-Algebras

In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local BL-sheaf spaces

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