Ordered group-valued probability, positive operators, and integral representations

TL;DR

This paper establishes a correspondence between probability maps valued in lattice-ordered groups and positive operators on Riesz spaces, extending classical representation theorems and characterizing extreme points as homomorphisms.

Contribution

It extends the representation theorem of MV-algebraic states to probability maps into lattice-ordered groups and characterizes extreme points as homomorphisms.

Findings

Probability maps correspond to positive operators between Riesz spaces.

Extreme points of the convex set of probability maps are homomorphisms.

Probability maps can be viewed as collections of states indexed by maximal ideals.

Abstract

Probability maps are additive and normalised maps taking values in the unit interval of a lattice ordered Abelian group. They appear in theory of affine representations and they are also a semantic counterpart of Hajek's probability logic. In this paper we obtain a correspondence between probability maps and positive operators between certain Riesz spaces, which extends the well-known representation theorem of real-valued MV-algebraic states by positive linear functionals. When the codomain algebra contains all continuous functions, the set of all probability maps is convex, and we prove that its extreme points coincide with homomorphisms. We also show that probability maps can be viewed as a collection of states indexed by maximal ideals of a codomain algebra and we characterise this collection in special cases.