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Erschienen in: Soft Computing 1/2017

15.09.2016 | Focus

Generalised states: a multi-sorted algebraic approach to probability

verfasst von: Tomáš Kroupa, Vincenzo Marra

Erschienen in: Soft Computing | Ausgabe 1/2017

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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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Fußnoten
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.
 
Literatur
Zurück zum Zitat Bigard A, Keimel K, Wolfenstein S (1977) Groupes et anneaux réticulés (Lecture notes in mathematics), vol 608. Springer, BerlinCrossRefMATH Bigard A, Keimel K, Wolfenstein S (1977) Groupes et anneaux réticulés (Lecture notes in mathematics), vol 608. Springer, BerlinCrossRefMATH
Zurück zum Zitat 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 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
Zurück zum Zitat 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, 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
Zurück zum Zitat Flaminio T, Montagna F (2009) MV-algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50(1):138–152CrossRefMATHMathSciNet Flaminio T, Montagna F (2009) MV-algebras with internal states and probabilistic fuzzy logics. Int J Approx Reason 50(1):138–152CrossRefMATHMathSciNet
Zurück zum Zitat Goodearl KR (1986) Partially ordered abelian groups with interpolation, vol 20 of mathematical surveys and monographs. American Mathematical Society, Providence Goodearl KR (1986) Partially ordered abelian groups with interpolation, vol 20 of mathematical surveys and monographs. American Mathematical Society, Providence
Zurück zum Zitat Hailperin T (1996) Sentential probability logic: origins, development, current status, and technical applications. Lehigh University Press, BethlehemMATH Hailperin T (1996) Sentential probability logic: origins, development, current status, and technical applications. Lehigh University Press, BethlehemMATH
Zurück zum Zitat Hájek P (1998) Metamathematics of fuzzy logic, vol 4 of Trends in logic—studia logica library. Kluwer Academic Publishers, Dordrecht Hájek P (1998) Metamathematics of fuzzy logic, vol 4 of Trends in logic—studia logica library. Kluwer Academic Publishers, Dordrecht
Zurück zum Zitat Manara C, Marra V, Mundici D (2007) Lattice-ordered abelian groups and Schauder bases of unimodular fans. Trans Am Math Soc 359(4):1593–1604CrossRefMATHMathSciNet Manara C, Marra V, Mundici D (2007) Lattice-ordered abelian groups and Schauder bases of unimodular fans. Trans Am Math Soc 359(4):1593–1604CrossRefMATHMathSciNet
Zurück zum Zitat 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 (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
Zurück zum Zitat Mundici D (2011) Advanced Łukasiewicz calculus and MV-algebras, vol 35 of Trends in logic—studia logica library. Springer, Dordrecht Mundici D (2011) Advanced Łukasiewicz calculus and MV-algebras, vol 35 of Trends in logic—studia logica library. Springer, Dordrecht
Metadaten
Titel
Generalised states: a multi-sorted algebraic approach to probability
verfasst von
Tomáš Kroupa
Vincenzo Marra
Publikationsdatum
15.09.2016
Verlag
Springer Berlin Heidelberg
Erschienen in
Soft Computing / Ausgabe 1/2017
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-016-2343-3

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