We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV 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)

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)

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)

In this paper we study the logic Ł*, introduced by C. C. Chang as a natural extension of Łukasiewicz' logic Ł. This logic has positive and negative truth values in the real number interval [−1,1].We study deductive filters, we prove a deduction theorem and give detailed proofs of the soundness and completeness theorems. In the last section, we prove that the tautology problem for the logic Ł* is co-NP.This paper is to be considered a continuation of the paper MV*-Algebras, by (...) the same authors and appearing in this same volume. In that paper we study a class of algebras introduced by Chang as what is now known as an equivalent algebraic semantics for the logic Ł. For most definitions and other algebraic concepts, the reader is referred to that paper. (shrink)