Gaetano Vitale

Riesz–McNaughton functions and Riesz MV-algebras of nonlinear functions

Abstract We focus on Riesz MV-algebras, which are MV-algebras equipped with a multiplication by numbers in the real interval [ 0 , 1 ] . We consider for every integer n the Riesz MV-algebra of all continuous functions from the n-th power of [ 0 , 1 ] to [ 0 , 1 ] and the Riesz MV-subalgebras thereof. In particular we study the Riesz MV-subalgebras isomorphic to free Riesz MV-algebras with finitely many generators, possibly different from the usual linear models given by what we call Riesz–McNaughton functions (and which generalize McNaughton functions used in the case of MV-algebras). In doing this we characterize zerosets of Riesz–McNaughton functions by means of polyhedra, and we extend to Riesz MV-algebras a duality for MV-algebras exposed in a paper by Marra and Spada.

Łukasiewicz Equivalent Neural Networks

In this paper we propose a particular class of multilayer perceptrons, which describes possibly non-linear phenomena, linked with Łukasiewicz logic; we show how we can name a neural network with a formula and, viceversa, how we can associate a class of neural networks to each formula. Moreover, we introduce the definition of Łukasiewicz Equivalent Neural Networks to stress the strong connection between different neural networks via Łukasiewicz logical objects.

A General Framework for Individual and Social Choices

Suitable algebraic structures for individual and social choices are proposed. Some relevant properties are illustrated.

Social Preferences Through Riesz Spaces: A First Approach

In this paper we propose Riesz spaces as general framework in the context of pairwise comparison matrices, to deal with definable properties, real situations and aggregation of preferences. Some significant examples are presented to describe how properties of Riesz spaces can be used to express preferences. Riesz spaces allow us to combine the advantages of many approaches. We also provide a characterization of collective choice rules which satisfy some classical criteria in social choice theory and an abstract approach to social welfare functions.

On free MV algebras and a problem of Tarski

In this paper we prove that the first order theories of the free MV algebras with finitely many generators are different from each other. This answers to the MV algebra analogue of a well-known (now solved) problem of Tarski for groups.

Riesz-McNaughton functions and Riesz MV-algebras of nonlinear functions

We focus on Riesz MV-algebras, which are MV-algebras equipped with a multiplication by numbers in the real interval [0,1]. In analogy with a work in preparation for MV-algebras by the same authors, we consider for every integer n the Riesz MV-algebra of all continuous functions from the n-th power of [0,1] to [0,1] and the Riesz MV-subalgebras thereof. In particular we study the Riesz MV-subalgebras isomorphic to free Riesz MV-algebras with finitely many generators, possibly different from the usual linear models given by what we call Riesz-McNaughton functions (and which generalize McNaughton functions used in the case of MV-algebras). In doing this we characterise zerosets of Riesz-McNaughton functions by means of polyhedra, and we extend to Riesz MV-algebras a duality for MV-algebras exposed in a paper by Marra and Spada.