Invariant Measures in Free MV-Algebras

Abstract

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 MValgebra 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. PreliminariesAn MV-algebra is an algebra (A, ⊕, ¬, 0) such that (A, ⊕, 0) is a commutative monoid and the identities ¬¬f = f , f ⊕ ¬0 = ¬0, and ¬(¬f ⊕ g) ⊕ g = ¬(¬g ⊕ f ) ⊕ f are satisfied. MV-algebras can be viewed either as the Lindenbaum algebras of Lukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups (ℓ-groups) in which a strong order unit has been fixed. We recall that a strong unit in an ℓ-group G is a positive element u of G such that for every g ∈ G there exists a positive integer n for which g ≤ nu. The unit interval Γ(G, u) = {g ∈ G : 0 ≤ g ≤ u} is then an MV-algebra under the operations f ⊕ g = (f + g)∧u, ¬f = u − f , 0 = 0 G , and the functor Γ is an equivalence between the category of ℓ-groups with a distinguished strong unit and the category of MValgebras If A is viewed as the Lindenbaum algebra of some theory in Lukasiewicz logic, then m is a function assigning an "average truth-value" to the elements of A, i.e., to the propositional formulas modulo the theory.Key words and phrases. MV-algebras, state, automorphism-invariance, piecewise-linear homeomorphisms.2000 Math. Subj. Class.: 06D35; 37A05. The Γ functor induces a canonical bijection between the states of (G, u) and those of A = Γ(G, u) and a homeomorphism between the maximal spectrum of (G, u) and MaxSpec A. We recall that the maximal spectrum of (G, u) is the set of maximal ℓ-ideals of G, while MaxSpec A is the set of maximal ideals of A (i.e., kernels of homomorphisms from A to Γ(R, 1)); both sets are equipped with the Zariski topology [2, Chapitre 10]. We will formulate our results mainly in terms of MV-algebras, leaving to the reader their straightforward translation in the language of ℓ-groups with strong unit.The set of all states of A is a compact convex subset of [0,1] A , where the latter is given the product topology, and the subsp...

Strict Coherence on Many-Valued Events

“…States and faithful states of MV-algebras. The purpose of this section is to present states of an MV-algebra, a notion which was introduced in [30] and further investigated in [34,22,31,23]. A state s of A is said to be faithful, if s(x) = 0 implies x = 0.…”

How Do $$\ell $$-Groups and Po-Groups Appear in Algebraic and Quantum Structures?

“…which MV-algebras form an important subclass, see Kôpka and Chovanec (1994). An important characterization of states on MV-algebras by regular Borel probability measures was recently done in Kroupa (2006), Panti (2008). States can also be understood as averaging processes for truth-value in Łukasiewicz logic.…”

Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic

“…It has been argued in detail in [18] why states play the same role for MV-algebras as probability measures for Boolean algebras. Independently, Kroupa and Panti [14,20] gave a precise formulation for this claim: Theorem 4.5. Let A be an MV-algebra and G a unital abelian -group such that A = Γ(G).…”

“…As in the classical case of Boolean algebras, the states of an MV-algebra A have the following characterizations (see [14,20,15]). Here X A denotes the space of all valuations, that is, the set of all MV-algebra homomorphisms v from A to I with the topology of pointwise convergence:…”