2017/06/02 by Dvurečenskij, Anatolij, Zahiri, Omid
#06C15 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1706.00571
The paper deals with an algebraic extension of MV-algebras based on the definition of generalized Boolean algebras. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-algebra and every EMV-algebra contains an MV-algebra. First, we present basic properties of EMV-algebras, give some examples, introduce and investigate congruence relations, ideals and filters on this algebra. We show that each EMV-algebra can be embedded into an MV-algebra and we characterize EMV-algebras either as MV-algebras or maximal ideals of MV-algebras. We study the lattice of ideals of an EMV-algebra and prove that any EMV-algebra has at least one maximal ideal. We define an EMV-clan of fuzzy sets as a special EMV-algebra. We show any semisimple EMV-algebra is isomorphic to an EMV-clan of fuzzy functions on a set. We consider the variety of EMV-algebra and we present an equational base for each proper subvariety of the variety of EMV-algebras. We establish a categorical equivalencies of 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.