Tense Operators on MV-Algebras and Lukasiewicz-Moisil Algebras

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.

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