Tense Operators on MV-Algebras and Łukasiewicz-Moisil Algebras

Abstract

We introduce tense LM _n -algebras and tense MV-algebras as algebraic structures for some tense many-valued logics. A representation theorem for tense LM _n -algebras is proved and the polynomial equivalence between tense LM _3 -algebras (resp. tense LM _4 -algebras) and tense MV _3 -algebras (resp. tense MV _4 -algebras) is established. We study the pairs of dually-conjugated operations on MV-algebras and we use their properties in order to investigate how the axioms of tense operators are preserved by the Dedekind-MacNeille completion of an Archimedean MV-algebra. A tense many-valued propositional calculus is developed and a completeness theorem is proved.