\(S\)-dominating effect algebras. (English) Zbl 0932.03072

Gudder, Stanley

Int. J. Theor. Phys. 37, No. 3, 915-923 (1998).

Summary: A special type of effect algebra called an \(S\)-dominating effect algebra is introduced. It is shown that an \(S\)-dominating effect algebra \(P\) has a naturally defined Brouwer-complementation that gives \(P\) the structure of a Brouwer-Zadeh poset. This enables us to prove that the sharp elements of \(P\) form an orthomodular lattice. We then show that a standard Hilbert space effect algebra is \(S\)-dominating. We conclude that \(S\)-dominating effect algebras may be useful abstract models for sets of quantum effects in physical systems.

MSC:

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

Keywords:

\(D\)-posets; \(S\)-dominating effect algebra; Brouwer-complementation; Brouwer-Zadeh poset; sharp elements; orthomodular lattice; Hilbert space effect algebra