MV Algebras in the Treatment of Uncertainty

Abstract

In their paper [29], Rose and Rosser gave a proof of the completeness of the infinite-valued sentential calculus of Lukasiewicz. Their proof is syntactic in nature. In two subsequent papers [7], [8], Chang introduced MV algebras (many-valued algebras), and gave an algebraic proof of the completeness theorem using these structures. Thus, MV algebras were originally introduced as algebraic counterparts of many-valued logic, just as Boolean algebras are the algebraic counterpart of classical, two-valued, logic. Recently, however, MV algebras have found novel and surprising applications, and today they are studied per se. As proved in [19], MV algebras are categorically equivalent to abelian lattice-groups with strong unit. They are also equivalent to bounded commutative BCK algebras [20], and to several other mathematical structures. Composition with the Grothendieck functor K O yields a one-one correspondence between countable MV algebras and approximately finite-dimensional (AF) C*-algebras whose Murray von Neumann ordering of projections is a lattice order. This correspondence has many applications [10], [19], [21], [22], [23]. AF C*-algebras are the mathematical counterpart of quantum spin systems.