Invariant measures in free MV-algebras

Abstract

MV-algebras can be viewed either as the Lindenbaum algebras of Lukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free_n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0,1]^n. The maximal spectrum of Free_n is canonically homeomorphic to [0,1]^n, and the automorphisms of the algebra are in 1-1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this paper we prove that the only probability measure on [0,1]^n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure.