Sharply dominating effect algebras. (English) Zbl 0939.03073

Gudder, Stanley P.

Tatra Mt. Math. Publ. 15, 23-30 (1998).

Summary: Sharply dominating effect algebras are introduced. It is shown that if an effect algebra \(P\) is sharply dominating, then the set of sharp elements \(P_s\) in \(P\) forms an orthoalgebra in \(P\). It is also shown that if \(P\) is sharply dominating, then there exists a unique Brouwer-complementation \(\sim \) on \(P\) such that the set of BZ-sharp elements \(P_s^\sim \) coincides with \(P_s\). Conversely, if \(P\) is a BZ-effect algebra in which \(P_s^\sim \) = \(P_s\), then \(P\) is sharply dominating. The concept of sharpness on a quotient effect algebra is briefly considered.

MSC:

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

Keywords:

effects; effect algebras; orthoalgebras; sharpness; foundations of quantum mechanics; Brouwer-complementation; BZ-structure