On EMV-algebras

Abstract

The paper deals with an algebraic extension of MV-algebras based on the definition of generalized Boolean algebras. We introduce a new class of structures, not necessarily with a top element, which are called EMV-algebras, in a way that every EMV-algebra contains an MV-algebra. First, we present basic properties of EMV-algebras. We give some examples, introduce and investigate congruence relations, ideals and filters on these algebras. We establish a basic representation result saying that each EMV-algebra can be embedded into an EMV-algebra with top element and we characterize EMV-algebras either as structures which are termwise equivalent to MV-algebras or as maximal ideals of EMV-algebras with top element. We study the lattice of ideals of an EMV-algebra and prove that every EMV-algebra has at least one maximal ideal. We define an EMV-clan of fuzzy sets as a special EMV-algebra where all operations are defined by points. We show that any semisimple EMV-algebra is isomorphic to an EMV-clan of fuzzy functions on a set. The set of EMV-algebras is neither a variety nor a quasivariety, but rather a special class of EMV-algebras which we call a q-variety of EMV-algebras. We present an equational base for each proper q-subvariety of the q-variety of EMV-algebras. We establish categorical equivalencies among the category of proper EMV-algebras, the category of MV-algebras with a fixed special maximal ideal, and a special category of Abelian unital ℓ-groups.