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Generalised states: a multi-sorted algebraic approach to probability

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Abstract

We introduce a generalised notion of state as an additive map from a Boolean algebra of events to an arbitrary MV-algebra. Generalised states become unary operations in two-sorted algebraic structures that we call state algebras. Since these, as we show, form an equationally defined class of algebras, universal-algebraic techniques apply. We discuss free state algebras, their geometric representation, and their connection with the theory of affine representations of lattice groups.

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Notes

  1. Called heterogeneous algebras in Birkhoff and Lipson (1970). We stick to the multi-sorted terminology which seems to have become standard.

  2. Somewhat more substantial changes are needed in the presence of algebras with underlying multi-sorted sets which are not everywhere non-empty. In our case, given that the type of MV-algebras, hence of Boolean algebras, includes constants, neither E nor D can be empty.

  3. Since universal algebraists reserve the name ‘equation’ for fully invariant identities, in the following we use ‘relation’ to mean an equality between two MV-algebraic terms that holds for given elements of a given MV-algebra.

  4. The theory of the relations satisfied by Schauder bases, which leads to the notion of abstract Schauder bases, was developed in the two papers (Manara et al. 2007; Marra 2013). We see here a simple instance of that theory in action.

References

  • Bigard A, Keimel K, Wolfenstein S (1977) Groupes et anneaux réticulés (Lecture notes in mathematics), vol 608. Springer, Berlin

    Book  MATH  Google Scholar 

  • Birkhoff G, Lipson JD (1970) Heterogeneous algebras. J Comb Theory 8:115–133

    Article  MATH  MathSciNet  Google Scholar 

  • Chang CC (1958) Algebraic analysis of many valued logics. Trans AMS 88:467–490

    Article  MATH  MathSciNet  Google Scholar 

  • Cignoli RLO, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning, vol 7 of Trends in Logic—Studia Logica Library. Kluwer Academic Publishers, Dordrecht

  • Ciungu LC, Georgescu G, Mureşan C (2013) Generalized Bosbach states: part II. Arch Math Log 52(7–8):707–732

    Article  MATH  MathSciNet  Google Scholar 

  • Flaminio T, Kroupa T (2015) States of MV-algebras. In: Cintula P, Fermuller C, Noguera C (eds) Handbook of mathematical fuzzy logic. Vol 3, Studies in logic, mathematical logic and foundations. College Publications, London

  • Flaminio T, Montagna F (2009) MV-algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50(1):138–152

    Article  MATH  MathSciNet  Google Scholar 

  • Goodearl KR (1986) Partially ordered abelian groups with interpolation, vol 20 of mathematical surveys and monographs. American Mathematical Society, Providence

  • Hailperin T (1996) Sentential probability logic: origins, development, current status, and technical applications. Lehigh University Press, Bethlehem

    MATH  Google Scholar 

  • Hájek P (1998) Metamathematics of fuzzy logic, vol 4 of Trends in logic—studia logica library. Kluwer Academic Publishers, Dordrecht

  • Manara C, Marra V, Mundici D (2007) Lattice-ordered abelian groups and Schauder bases of unimodular fans. Trans Am Math Soc 359(4):1593–1604

    Article  MATH  MathSciNet  Google Scholar 

  • Marra V (2013) Lattice-ordered abelian groups and Schauder bases of unimodular fans II. Trans Am Math Soc 365(5):2545–2568

    Article  MATH  MathSciNet  Google Scholar 

  • Marra V, Mundici D (2007) The Lebesgue state of a unital abelian lattice-ordered group. J Group Theory 10(5):655–684

    Article  MATH  MathSciNet  Google Scholar 

  • Mundici D (1995) Averaging the truth-value in Łukasiewicz logic. Stud Log 55(1):113–127

    Article  MATH  MathSciNet  Google Scholar 

  • Mundici D (2000) Reasoning on imprecisely defined functions. In: Discovering the world with fuzzy logic, vol 57 of studies in fuzziness and soft computing, pp 331–366. Physica, Heidelberg

  • Mundici D (2011) Advanced Łukasiewicz calculus and MV-algebras, vol 35 of Trends in logic—studia logica library. Springer, Dordrecht

  • Pulmannová S (2000) A note on observables on MV-algebras. Soft Comput 4(1):45–48

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133, Milano, Italy

    Tomáš Kroupa & Vincenzo Marra

Authors
  1. Tomáš Kroupa
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  2. Vincenzo Marra
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Correspondence to Tomáš Kroupa.

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The authors declare that they have no conflict of interest.

Additional information

Communicated by A. Di Nola, D. Mundici, C. Toffalori, A. Ursini.

This paper is dedicated to the memory of Franco Montagna.

The work of T. Kroupa was supported by Marie Curie Intra-European Fellowship OASIG (PIEF-GA-2013-622645).

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Kroupa, T., Marra, V. Generalised states: a multi-sorted algebraic approach to probability. Soft Comput 21, 57–67 (2017). https://doi.org/10.1007/s00500-016-2343-3

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  • DOI: https://doi.org/10.1007/s00500-016-2343-3

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