Ada Lettieri

Perfect MV-algebras are categorically equivalent to abelianl-groups

In this paper we prove that the category of abelianl-groups is equivalent to the category of perfect MV-algebras. Furthermore, we give a finite equational axiomatization of the variety generated by perfect MV-algebras.

Algebraic analysis of fuzzy systems

In this paper, we have developed an algebraic theory, suitable for the analysis of fuzzy systems. We have used the notions of semiring and semimodule, introduced the notion of semilinear space, have given numerous examples of them and defined also the notions of linear dependence and independence. We have shown that the composition operation, which plays an essential role in the analysis of fuzzy systems because of its role in the compositional rule of inference, can be interpreted as a homomorphism between special semimodules. Consequently, this operation is, in a certain sense, a linear operation. This property formally explains why fuzzy systems are attractive for applications.

On varieties of MV-algebras with internal states

In [4,5] the authors introduced the variety SMV of MV-algebras with an internal operator, state MV-algebras. In [2,3] the authors gave a stronger version of state MV-algebras, called state-morphism MV-algebras. In this paper we continue the studies presented in [2,3] just looking at several proper subvarieties of SMV, obtained by imposing suitable conditions on the behavior of the internal operator.

Relation equations in residuated lattices

In questa nota si affronta il problema della risoluzione di equazioni matriciali del tipoAX=B, doveA eB sono matrici a valori su un reticolo distributivo residuato rispetto a una moltiplicazione. In particolare, si individua la più grande soluzione di una tale equazione e si danno condizioni relative alle soluzioni minimali.

Representations of monadic MV -algebras

Representations of monadic MV -algebra, the characterization of locally finite monadic MV -algebras, with axiomatization of them, definability of non-trivial monadic operators on finitely generated free MV -algebras are given. Moreover, it is shown that finitely generated m-relatively complete subalgebra of finitely generated free MV -algebra is projective.

Finite BL-algebras

BL-algebras were introduced by Hajek as algebraic structures of Basic Logic. The aim of this paper is to analyze the structure of finite BL-algebras. Extending the notion of ordinal sum, we characterize a class of finite BL-algebras, actually BL-comets. Then, just using BL-comets, we can represent any finite BL-algebra as a direct product of BL-comets. Furthermore we define a class of labelled trees, which can be one-to-one mapped onto finite BL-algebras.

On normal forms in Łukasiewicz logic

Abstract.Formulas of n variables of Łukasiewicz sentential calculus can be represented, via McNaughton’s theorem, by piecewise linear functions, with integer coefficients, from hypercube [0,1]n to [0,1], called McNaughton functions. As a consequence of the McNaughton representation of a formula it is obtained a canonical form of a formula. Indeed, up to logical equivalence, any formula can be written as an infimum of finite suprema of formulas associated to McNaughton functions which are truncated functions to

Conditional States in Finite-Valued Logics

[0,1]

SUB ALGEBRAS, DIRECT PRODUCTS AND ASSOCIATED LATTICES OF MV-ALGEBRAS

of the restriction to [0,1]n of single hyperplanes, for short, called simple McNaughton functions. In the present paper we will concern with the problem of presenting formulas of Lukasiewicz sentential calculus in normal form. Here we list the main results we obtained: a) we give an axiomatic description of some classes of formulas having the property to be canonically mapped one-to-one onto the class of simple Mc Naughton functions; b) we provide normal forms for Lukasiewicz sentential calculus, making use of formulas defined in a); c) we prove the polynomial complexity of formulas, in normal form, coming from a certain class described as in a); d) we extend the results described in a), b) and c) to Rational Lukasiewicz logic.

L-σ-algebras and L-measures

One of the basic principles of probability theory is that the set of the events of a trial is a Boolean algebra. It is the case when we consider that the trial follows the laws of classical logic. On the other hand, there exist many trials which are based on a many-valued logic. In this case one can accept the hypothesis that the set of the events has a structure of MV-algebra [Di Nola et al., to appear] .