396 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
Pavelka-style completeness in expansions of \L ukasiewicz logic
An algebraic setting for the validity of Pavelka style completeness for some
natural expansions of \L ukasiewicz logic by new connectives and rational
constants is given. This algebraic approach is based on the fact that the
standard MV-algebra on the real segment is an injective MV-algebra. In
particular the logics associated with MV-algebras with product and with
divisible MV-algebras are considered
Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics
We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer
and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We
give new analytic proof systems for A and use the embeddings to derive
corresponding systems for L. These include: hypersequent calculi for A and L
and terminating versions of these calculi; labelled single sequent calculi for
A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur
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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