Geodesic
Effect algebras and ring-like structures
Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 1, pp. 63-79

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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.
Keywords: generalized Boolean quasiring, effect algebra, ring-like structure, quantum logics, axiomatic quantum mechanics, state-supported probability, symmetric difference 
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Beltrametti, Enrico; Maczyński, Maciej. Effect algebras and ring-like structures. Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 1, pp. 63-79. http://geodesic.mathdoc.fr/item/DMGAA_2003_23_1_a5/