MV* - Algebras

In this paper we make an algebraic study of the variety of M V * –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 M V * –algebras within its own limits using a concept of ideal and of prime ideal that are very naturally related to the corresponding concepts in –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 –group. In the last section we develop a detailed proof a the one–dimensional theorem of McNaughton, that is, the free M V * –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 M V –algebras.