Complements and quasicomplements in the lattice of subalgebras of P(ω)

AbstractIn the lattice of subalgebras of a Boolean algebra D call Ba complement of A if A ∩ B = {0,1} and {A ∪ B} generates D. B is called a quasicomplement of A if it is maximal w.r.t. the property A ∩ B = {0, 1}. We characterize those countable subalgebras of P(ω) which have a complement, and, assuming Martin's Axiom, describe the isomorphism types of some quasicomplements of the finite-cofinite subalgebra of P(ω)