DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

SJR: 0.203

SNIP: 0.562

MCQ: 0.12

Index Copernicus: 121.02

Discussiones Mathematicae - General Algebra and Applications

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Discussiones Mathematicae General Algebra and Applications 23(1) (2003) 63-79
DOI: https://doi.org/10.7151/dmgaa.1064

EFFECT ALGEBRAS AND RING-LIKE STRUCTURES

Enrico G. Beltrametti

Department of Physics, University of Genova
and Istituto Nazionale di Fisica Nucleare,
Sezione di Genova
via Dodecaneso 33, I-16146 Genova, Italy
e-mail: Enrico.Beltrametti@ge.infn.it

Maciej J. Maczyński

Faculty of Mathematics and Information Science,
Warsaw University of Technology
Plac Politechniki 1, PL 00 661 Warsaw, Poland
e-mail: mamacz@alpha.mini.pw.edu.pl

Abstract

The dichotomic physical quantities, also called propositions, can be naturally associated to maps of the set of states into the real interval [0,1]. We show that the structure of effect algebra associated to such maps can be represented by quasiring structures, which are a generalization of Boolean rings, in such a way that the ring operation of addition can be non-associative and the ring multiplication non-distributive with respect to addition. By some natural assumption on the effect algebra, the associativity of the ring addition implies the distributivity of the lattice structure corresponding to the effect algebra. This can be interpreted as another characterization of the classicality of the logical systems of propositions, independent of the characterizations by Bell-like inequalities.

Keywords: generalized Boolean quasiring, effect algebra, ring-like structure, quantum logics, axiomatic quantum mechanics, state-supported probability, symmetric difference.

2000 Mathematics Subject Classification: 81P10, 06E20, 03G25, 03G12, 06C15, 28E15.

References

[1]E.G. Beltrametti and M.J. Maczyński, On a characterization of classical and non-classical probabilities, J. Math. Phys. 32 (1991), 1280-1286.
[2]E.G. Beltrametti and M.J. Maczyński, On the range of non-classical probability, Rep. Math. Phys. 36 (1995), 195-213.
[3]E.G. Beltrametti and S. Bugajski, Effect algebras and statistical theories, J. Math. Phys. 38 (1997), 3020-3030.
[4]E.G. Beltrametti, S. Bugajski and V.S. Varadarajan, Extensions of convexity models, J. Math. Phys. 41 (2000), 1415-1429.
[5]D. Dorninger, H. Länger and M. Maczyński, On ring-like structures induced by Mackey's probability function, Rep. Math. Phys. 43 (1999), 499-515.
[6]R. Sikorski, Boolean Algebras, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York 1964.

Received 14 March 2003
Revised 2 June 2003


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