172 research outputs found
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 . Extending
Mundici's equivalence between MV-algebras and -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 that has Riesz MV-algebras as
models is a conservative extension of \L ukasiewicz -valued
propositional calculus and it is complete with respect to evaluations in the
standard model . 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 and we relate them with the analogue of de Finetti's coherence
criterion for .Comment: To appear in Soft Computin
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 . 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
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
Entropy on effect algebras with the Riesz decomposition property I: Basic properties
summary:We define the entropy, lower and upper entropy, and the conditional entropy of a dynamical system consisting of an effect algebra with the Riesz decomposition property, a state, and a transformation. Such effect algebras allow many refinements of two partitions. We present the basic properties of these entropies and these notions are illustrated by many examples. Entropy on MV-algebras is postponed to Part II
Dynamic Ćukasiewicz logic and its application to immune system
AbstractIt is introduced an immune dynamicn-valued Ćukasiewicz logicID{\L }_nIDĆnon the base ofn-valued Ćukasiewicz logic{\L }_nĆnand corresponding to it immune dynamicMVn-algebra (IDLn-algebra),1<n<Ï, which are algebraic counterparts of the logic, that in turn represent two-sorted algebras(M,R,â)that combine the varieties ofMVn-algebrasM=(M,â,â,âŒ,0,1)and regular algebrasR=(R,âȘ,;,â)into a single finitely axiomatized variety resemblingR-module with "scalar" multiplicationâ. Kripke semantics is developed for immune dynamic Ćukasiewicz logicID{\L }_nIDĆnwith application in immune system
Entropy on effect algebras with Riesz decomposition property II: MV-algebras
summary:We study the entropy mainly on special effect algebras with (RDP), namely on tribes of fuzzy sets and sigma-complete MV-algebras. We generalize results from [RiMu] and [RiNe] which were known only for special tribes
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
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