Atomic effect algebras with compression bases

Abstract

Compression base effect algebras were recently introduced by Gudder [6]. They generalize sequential effect algebras [7] and compressible effect algebras [5]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [11]. The notion of projection-atomicity is introduced and studied and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.