Morphisms on $${ EMV}$$ EMV -algebras and their applications

Abstract

For a new class of algebras, called \({ EMV}\)-algebras, every idempotent element a determines an \({ MV}\)-algebra which is important for the structure of the \({ EMV}\)-algebra. Therefore, instead of standard homomorphisms of \({ EMV}\)-algebras, we introduce \({ EMV}\)-morphisms as a family of \({ MV}\)-homomorphisms from \({ MV}\)-algebras [0, a] into other ones. \({ EMV}\)-morphisms enable us to study categories of \({ EMV}\)-algebras where objects are \({ EMV}\)-algebras and morphisms are special classes of \({ EMV}\)-morphisms. The category is closed under product. In addition, we define free \({ EMV}\)-algebras on a set X with respect to \({ EMV}\)-morphisms. If X is finite, then a free \({ EMV}\)-algebra on X is termwise equivalent to the free \({ MV}\)-algebra on X. For an infinite set X, the same is true introducing a so-called weakly free \({ EMV}\)-algebra.