Effect algebras and ring-like structures

Enrico G. Beltrametti; Maciej J. Maczyński

Discussiones Mathematicae - General Algebra and Applications (2003)

  • Volume: 23, Issue: 1, page 63-79
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Enrico G. Beltrametti, and Maciej J. Maczyński. "Effect algebras and ring-like structures." Discussiones Mathematicae - General Algebra and Applications 23.1 (2003): 63-79. <http://eudml.org/doc/287749>.

@article{EnricoG2003,
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.},
author = {Enrico G. Beltrametti, Maciej J. Maczyński},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {generalized Boolean quasiring; effect algebra; ring-like structure; quantum logics; axiomatic quantum mechanics; state-supported probability; symmetric difference; quantum logic},
language = {eng},
number = {1},
pages = {63-79},
title = {Effect algebras and ring-like structures},
url = {http://eudml.org/doc/287749},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Enrico G. Beltrametti
AU - Maciej J. Maczyński
TI - Effect algebras and ring-like structures
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2003
VL - 23
IS - 1
SP - 63
EP - 79
AB - 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.
LA - eng
KW - generalized Boolean quasiring; effect algebra; ring-like structure; quantum logics; axiomatic quantum mechanics; state-supported probability; symmetric difference; quantum logic
UR - http://eudml.org/doc/287749
ER -

References

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  1. [1] E.G. Beltrametti and M.J. Maczyński, On a characterization of classical and non-classical probabilities, J. Math. Phys. 32 (1991), 1280-1286. Zbl0731.60003
  2. [2] E.G. Beltrametti and M.J. Maczyński, On the range of non-classical probability, Rep. Math. Phys. 36 (1995), 195-213. Zbl0887.60007
  3. [3] E.G. Beltrametti and S. Bugajski, Effect algebras and statistical theories, J. Math. Phys. 38 (1997), 3020-3030. Zbl0874.06009
  4. [4] E.G. Beltrametti, S. Bugajski and V.S. Varadarajan, Extensions of convexity models, J. Math. Phys. 41 (2000). doi: 1415-1429 Zbl1008.46028
  5. [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. Zbl1056.81004
  6. [6] R. Sikorski, Boolean Algebras, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York 1964. 

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