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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

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Received 14 March 2003
Revised 2 June 2003