Generalized MV-algebras
by
Nikolaos Galatos
Vanderbilt University
Coauthors: Constantine Tsinakis (Vanderbilt University)

MV-algebras are models of Lukasiewicz infinite-valued propositional logic. D. Mundici has shown that MV-algebras are intervals [u, e] of abelian l-groups, where u is a strong order unit, and that this correspondence extends to a categorical equivalence.

Generalized MV-algebras (GMV-algebras) are residuated lattices that satisfy the non-commutative, relativised law of double negation x / ( (x \/ y)\x) = x \/ y = (x / (x \/ y)) \x. These algebras have a distributive lattice reduct and are common generalizations of MV-algebras and l-groups. In particular, bounded, commutative GMV-algebras are reducts (they lack the least element constant) of MV-algebras.

We represent a GMV-algebra as a monotone lattice retraction of an l-group, in which it sits nicely as a sublattice. We characterize the aforementioned retractions and generalize D. Mundici’s result to a categorical equivalence between GMV-algebras and pairs of l-groups and such retractions. Various properties of GMV-algebras, including the decidability of the equational theory, are easy corollaries of the representation.

Date received: December 26, 2002