Bernoulli automorphisms of finitely generated free MV-algebras

Abstract

MV-algebras can be viewed either as the Lindenbaum algebras of Łukasiewicz infinite-valued logic, or as unit intervals [ 0 , u ] of lattice-ordered abelian groups in which a strong order unit u > 0 has been fixed. They form an equational class, and the free n -generated free MV-algebra is representable as an algebra of piecewise-linear continuous functions with integer coefficients over the unit n -dimensional cube. In this paper we show that the automorphism group of such a free algebra contains elements having strongly chaotic behaviour, in the sense that their duals are measure-theoretically isomorphic to a Bernoulli shift. This fact is noteworthy from the viewpoint of algebraic logic, since it gives a distinguished status to Lebesgue measure as an averaging measure on the space of valuations. As an ergodic theory fact, it provides explicit examples of volume-preserving homeomorphisms of the unit cube which are piecewise-linear with integer coefficients, preserve the denominators of rational points, and enjoy the Bernoulli property.