# A preliminary study of MV-algebras with two quantifiers which commute

### Abstract

In this paper we investigate the class of MV-algebras equipp ed with two quantifiers which commute as a natural generalization of diagonal free two dimensional cylindric álgebras (see [18]). In the 40s, Tasrki first introduced cylindric algebras in order to provide an algebraic apparatus for the study of classical predicate calculus. The diagonal free two dimensional cylindric álgebras is a special cylindric algebra. The treatment here of MV-algebras is done in terms of implication and negation. This allows us to simplify some results due to Di Nola and Grigoliain [9] related to the characterization of a quantifier in terms of some special sub-algebra associated to it. On the other hand, we present a topological duality to this class of algebras and we apply it to characterize the congruences of one algebra vi a certain closed sets. Finally, we study the subvariety of this class generated by a chain of length n + 1 (n <)(BMVn+1). We prove that the subvariety is semisimple and we characterize theirsimple algebras. Using a special functional algebra, we determine all the simple finitealgebras of this subvariety,these results show us that de ordered algebraic structure of BMVn+1 is more complex than to monadic MV-algebra generated by an (n+1)-chain.