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.

Invariant measures in free MV-algebras

PANTI, Giovanni
2008-01-01

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.
File in questo prodotto:
File Dimensione Formato  
LAGB_A_310605.pdf

non disponibili

Tipologia: Altro materiale allegato
Licenza: Non pubblico
Dimensione 197.13 kB
Formato Adobe PDF
197.13 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11390/692294
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 78
  • ???jsp.display-item.citation.isi??? 73
social impact