2005/08/24 by Giovanni Panti, Panti, Giovanni
Computer Science · Mathematics · #06D35 #37A05 #Advanced Algebra and Logic #Dynamical Systems (math.DS) #FOS: Mathematics #Logic (math.LO) #Rough Sets and Fuzzy Logic #math.DS #math.LO #msc:06D35 #msc:37A05 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0508445
12 pages, 4 figures. Title changed, motivational section rewritten, mathematics unchanged. To appear in Communications in Algebra
openalex publication_date 2005/08/24 · arxiv created 2007/04/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Freen is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0,1]n. The maximal spectrum of Freen 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.