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Mathematica Slovaca

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Volume 68, Issue 3


Structural properties of algebras of S-probabilities

Dietmar Dorninger / Helmut Länger
  • Institute of Discrete Mathematics and Geometry TU Wien Wiedner Hauptstraβe 8–10 A–1040 Vienna Austria
  • Department of Algebra and Geometry Palacký University Olomouc 17. listopadu 12 CZ–771 46 Olomouc Czech Republic
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Published Online: 2018-05-18 | DOI: https://doi.org/10.1515/ms-2017-0118


Let S be a set of states of a physical system. The probabilities p(s) of the occurrence of an event when the system is in different states sS define a function from S to [0, 1] called a numerical event or, more precisely, an S-probability. A set of S-probabilities comprising the constant functions 0 and 1 which is structured by means of the addition and order of real functions in such a way that an orthomodular partially ordered set arises is called an algebra of S-probabilities, a structure significant as a quantum-logic with a full set of states. The main goal of this paper is to describe algebraic properties of algebras of S-probabilities through operations with real functions. In particular, we describe lattice characteristics and characterize Boolean features. Moreover, representations by sets are considered and pertinent examples provided.

MSC 2010: Primary 06C15; Secondary 03G12, 81P16

Keywords: algebra of S-probabilities; lattice properties; concrete quantum logic; infimum faithful logic; Boolean algebra

Support of the research of the second author by the Austrian Science Fund (FWF), project I 1923-N25, by ÖAD, project CZ 04/2017, as well as by IGA, project PřF 2018 012, is gratefully acknowledged.


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About the article

Received: 2016-06-14

Accepted: 2017-03-01

Published Online: 2018-05-18

Published in Print: 2018-06-26

Citation Information: Mathematica Slovaca, Volume 68, Issue 3, Pages 485–490, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0118.

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