Morphisms on ${ EMV}$-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.