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On EMV-algebras.
- Source :
-
Fuzzy Sets & Systems . Oct2019, Vol. 373, p116-148. 33p. - Publication Year :
- 2019
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01650114
- Volume :
- 373
- Database :
- Academic Search Index
- Journal :
- Fuzzy Sets & Systems
- Publication Type :
- Academic Journal
- Accession number :
- 138252929
- Full Text :
- https://doi.org/10.1016/j.fss.2019.02.013