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

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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 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.

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Acknowledgements

The author is grateful to Prof. Vincenzo Marra (University of Milan) for many suggestions and inspiring comments.

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Authors and Affiliations

  1. The Czech Academy of Sciences, Institute of Information Theory and Automation, Prague, Czech Republic

    Tomáš Kroupa

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  1. Tomáš Kroupa
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Correspondence to Tomáš Kroupa.

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Presented by C. Tsinakis.

The work on this paper has been supported from the GAČR grant project GA17-04630S.

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Kroupa, T. Ordered group-valued probability, positive operators, and integral representations. Algebra Univers. 79, 86 (2018). https://doi.org/10.1007/s00012-018-0569-x

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