Ordered group-valued probability, positive operators, and integral representations
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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 Hájek’s probability logic. In this paper we obtain a correspondence between probability maps and positive operators of 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.
KeywordsMV-algebra Abelian \(\ell \)-group Riesz space Positive operator
Mathematics Subject Classification06D35 97H50 47H07
The author is grateful to Prof. Vincenzo Marra (University of Milan) for many suggestions and inspiring comments.
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