Effect algebras with compressions *

The notion of a CB-effect algebra as an effect algebra equipped with a compression base was recently introduced by S. Gudder as an analogue of the notion of a unital group with a compression base (CB-group) introduced by D. Foulis. The present paper extends the investigation of CB-effect algebras with the projection cover property, the Rickart projection property, and introduces the so-called b-general comparability, which is an effect algebra version of general comparability in CB-groups. Commutativity properties, blocks and C-blocks are studied, and it is shown that a CB-effect algebra with b-general comparability can be covered by its C-blocks, which are maximal sets of commuting elements, and can be organized into MV algebras. Connections with sequential effect algebras (SEAS) are studied.