SEMISIMPLE, ARCHIMEDEAN, AND SEMILOCAL PSEUDO MV-ALGEBRAS

Abstract

The concepts of semisimple, Archimedean, and semilocal pseudo MV-algebras are investigated and many interesting facts concerning them are given. Pseudo MV -algebras were introduced by G. Georgescu and A. Iorgulescu in (6) and in- dependently by J. Rachunek in (8) (there they are called generalized MV -algebras or, for short, GMV -algebras) as a non-commutative generalization of MV -algebras. This work was intended as an attempt to order some notions appearing in the theory of these al- gebras. Semisimple pseudo MV -algebras and Archimedean pseudo MV -algebras are ex- amples of such notions. In Section 3 we give some characterizations of semisimple pseudo MV -algebras. Archimedean pseudo MV -algebras are investigated and characterized in Sec- tion 4. It is shown that in the case of pseudo MV -algebras the notion of Archimedean is equivalent with the notion of Archimedean in the Belluce sense, that occurs in the theory of MV -algebras, and both are equivalent with the notion of semisimple. Section 5 is devoted to introduce and characterize semilocal pseudo MV -algebras, the concept generalizing a simi- lar one from the theory of MV -algebras. For the convenience of the reader, in Section 2 we give the relevant material needed in the sequel, thus making our exposition self-contained.