Cartesian product of compressible effect algebras

In the paper, we prove that is compatible with p}, the set of commutant of p, and , the projection commutant of a, are all normal sub-effect algebras of a compressible effect algebra E, and 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. This work was supported by the SF of Education Department of Shaanxi Province (Grant No. 07JK267), P. R. China.