Effect algebras and ring-like structures

• Autorzy: Enrico G. Beltrametti , Maciej J. Maczyński
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

generalized Boolean quasiring; effect algebra; ring-like structure; quantum logics; axiomatic quantum mechanics; state-supported probability; symmetric difference

Kategorie tematyczne

• 28E15: Other connections with logic and set theory
• 06C15: Complemented lattices, orthocomplemented lattices and posets
• 06E20: Ring-theoretic properties
• 03G25: Other algebras related to logic
• 03G12: Quantum logic
• 81P10: Logical foundations of quantum mechanics; quantum logic

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2003
• Tom: 23
• Numer: 1
• Strony: 63-79

Daty

wydano

2003

otrzymano

2003-03-14

poprawiono

2003-06-02

Twórcy

autor

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

autor

Maciej J. Maczyński

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, PL 00 661 Warsaw, Poland

Bibliografia

• [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). doi: 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.

Identyfikatory

DOI

10.7151/dmgaa.1064