MV-Algebras and Abelian l-Groups: a Fruitful Interaction

Abstract

Introduced by Chang in the late fifties, MV-algebras stand to Łukasiewicz’s infinite-valued propositional logic as boolean algebras stand to the classical propositional calculus. As stated by Chang in his original paper, “MV is supposed to suggest many-valued logics … for want of a better name”. The name has stuck. After some decades of relative quiescency, MV-algebras are today intensely investigated. On the one hand, these algebras find applications in such diverse fields as error-correcting feedback codes and logic-based control theory. On the other hand, MV-algebras are interesting mathematical objects in their own right. The main aim of this paper is to show that the interaction between MV-algebras and lattice-ordered abelian groups — including the timehonored theory of magnitudes — has much to offer, not only to specialists in these two fields, but also to people interested in the fan-theoretic description of toric varieties, and in the K0-theory of AF C*-algebras.