Ring-like structures with unique symmetric difference related to quantum logic. (English) Zbl 1014.81003

Dorninger, Dietmar; Länger, Helmut; Mączyński, Maciej

Discuss. Math., Gen. Algebra Appl. 21, No. 2, 239-253 (2001).

The authors study ring-like quantum structures (so-called generalized Boolean quasirings) generalizing Boolean rings. Particular attention is paid to those structures in which the two normal forms of the symmetric difference in Boolean algebras coincide. This property is related to associativity and weak associativity (which means \((1+x)+y=1+(x+y)\)). Possible physical interpretation is proposed by presenting a model based on generalized Boolean quasirings and following Mackey’s approach to axiomatic quantum mechanics.

Reviewer: Mirko Navara (Praha)

MSC:

81P10	Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12	Quantum logic
06E99	Boolean algebras (Boolean rings)
06C15	Complemented lattices, orthocomplemented lattices and posets

Keywords:

generalized Boolean quasiring; symmetric difference; quantum logic; normal forms of symmetric difference; weak associativity; Boolean quasirings; Mackey’s approach to axiomatic quantum mechanics