States on pseudo-effect algebras with general comparability. (English) Zbl 1249.03117

Dvurečenskij, Anatolij

Kybernetika 40, No. 4, 397-419 (2004).

Summary: Pseudo-effect algebras are partial algebras \((E;+,0,1)\) with a partially defined addition \(+\) which is not necessarily commutative and therefore with two complements, left and right ones. General comparability allows one to compare elements of \(E\) in some intervals with Boolean ends. Such an algebra is always a pseudo-MV-algebra. We show that it admits a state, and we describe the state space from the topological point of view. We prove that every pseudo-effect algebra is in fact a pseudo-MV-algebra which is a subdirect product of linearly ordered pseudo-MV-algebras. In addition, we present many illustrating examples.

MSC:

03G12	Quantum logic
06D35	MV-algebras

Keywords:

pseudo-effect algebra; pseudo-MV-algebra; general comparability; state