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.
