MV-pairs and states – p. 1/2

Similarities are an extension of equivalence relations to a fuzzy context. In this paper we introduce the class of similarity MV-algebras obtained as a generalization of the variety of MValgebras by adding a binary operator playing the role of similarity. We further introduce the similarity Łukasiewicz logic and we prove a completeness theorem.

An MV-pair is a pair (B, G) where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain conditions. Let ∼G be the equivalence relation on B naturally associated with G. We prove that for every MV-pair (B, G), the effect algebra B/ ∼G is an MV-effect algebra. Moreover, for every MV-effect algebra M there is an MV-pair (B, G) such that M is isomorphic to B/ ∼G.

In this paper, we propose extensions of affine classical linear algebra(ACL-algebra) and many-valued algebra(MV-algebra), the emphasis is being on the latter. MV-algebras are algebraic models of à Lukasiewicz א0-valued logic(à Lא0) and that was established long back in 1958 by C.C. Chang. MV-algebra is recently studied from the angle of fuzzy set theory. Also it is investigated in connection wi...