In this paper we make an algebraic study of the variety of MV*-algebras introduced by C. C. Chang as an algebraic counterpart for a logic with positive and negative truth values.We build the algebraic theory of MV*-algebras within its own limits using a concept of ideal and of prime ideal that are very naturally related to the corresponding concepts in l-groups. The main results are a subdirect representation theorem, a completeness theorem, a study of simple and semisimple algebras, and a (...) characterization of the ideal of infinitesimals as an l-group. In the last section we develop a detailed proof a the one-dimensional theorem of McNaughton, that is, the free MV*-algebra in one generator is the algebra of McNaughton functions over [−1,1]. In contrast with the rest of the paper, this last result is based on work done for MV-algebras. (shrink)

This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such as (...) a simple proof of the completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic. Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean partitions can be defined and computed in classical logic. Detailed bibliographic remarks at the end of each chapter and an extensive bibliography lead the reader on to further specialised topics. (shrink)

We investigate an expansion of quasi-MV algebras ([10]) by a genuine quantum unary operator. The variety of such quasi-MV algebras has a subquasivariety whose members—called cartesian—can be obtained in an appropriate way out of MV algebras. After showing that cartesian . quasi-MV algebras generate ,we prove a standard completeness theorem for w.r.t. an algebra over the complex numbers.

In this paper we make an algebraic study of the variety of MV*-algebras introduced by C. C. Chang as an algebraic counterpart for a logic with positive and negative truth values.We build the algebraic theory of MV*-algebras within its own limits using a concept of ideal and of prime ideal that are very naturally related to the corresponding concepts in l-groups. The main results are a subdirect representation theorem, a completeness theorem, a study of simple and semisimple algebras, and a (...) characterization of the ideal of infinitesimals as an l-group. In the last section we develop a detailed proof a the one-dimensional theorem of McNaughton, that is, the free MV*-algebra in one generator is the algebra of McNaughton functions over [−1,1]. In contrast with the rest of the paper, this last result is based on work done for MV-algebras. (shrink)

We investigate some properties of two varieties of algebras arising from quantum computation - quasi-MV algebras and $\sqrt{^{\prime }}$ quasi-MV algebras - first introduced in \cite{Ledda et al. 2006}, \cite{Giuntini et al. 200+} and tightly connected with fuzzy logic. We establish the finite model property and the congruence extension property for both varieties; we characterize the quasi-MV reducts and subreducts of $\sqrt{^{\prime }}$ quasi-MV algebras; we give a representation of semisimple $\sqrt{^{\prime }}$ quasi-MV algebras in terms of algebras of functions; (...) finally, we describe the structure of free algebras with one generator in both varieties. (shrink)