Quotients of Boolean algebras and regular subalgebras

Let 𝔹 and ℂ be Boolean algebras and e: 𝔹→ℂ an embedding. We examine the hierarchy of ideals on ℂ for which e̅: 𝔹→ℂ / I is a regular (i.e. complete) embedding. As an application we deal with the interrelationship between P(ω)/ fin in the ground model and in its extension. If M is an extension of V containing a new subset of ω, then in M there is an almost disjoint refinement of the family ([ω] ω ) V . Moreover, there is, in M , exactly one ideal I on ω such that (P(ω)/ fin)^V is a dense subalgebra of (P(ω)/I)^M if and only if M does not contain an independent (splitting) real. We show that for a generic extension V [ G ], the canonical embedding P^V(ω)/ fin↪P(ω)/(U(Os)(𝔹))^G is a regular one, where U(Os)(𝔹) is the Urysohn closure of the zero-convergence structure on 𝔹 .