Cartesian product of compressible effect algebras

Abstract

In the paper, we prove that $C(p)=\{a\in E\mid a$ is compatible with p}, the set of commutant of p, and $C_p(a)=\{p\in P(E)\mid a=J_p(a)\oplus J_{p'}(a)\}=\{p\in P(E)\mid a\in C(p)\}$ , the projection commutant of a, are all normal sub-effect algebras of a compressible effect algebra E, and $Q = \{q\mid J_q$ is a direct retraction on E} is a normal sub-effect algebra of an effect algebra E. Moreover, we answer an open question in Gudder’s (Rep Math Phys 54:93–114, 2004), Compressible effect algebras, Rep Math Phys, by showing that the cartesian product of an infinite number of E _i is a compressible effect algebra if and only if each E _i is a compressible effect algebra.