MV*—Algebras

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.