Quotients of Boolean algebras and regular subalgebras

Abstract

Let ${\mathbb{B}}$ and ${\mathbb{C}}$ be Boolean algebras and ${e: \mathbb{B}\rightarrow \mathbb{C}}$ an embedding. We examine the hierarchy of ideals on ${\mathbb{C}}$ for which ${ \bar{e}: \mathbb{B}\rightarrow \mathbb{C} / \fancyscript{I}}$ is a regular (i.e. complete) embedding. As an application we deal with the interrelationship between ${\fancyscript{P}(\omega)/{{\rm fin}}}$ 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 ${\fancyscript{I}}$ on ω such that ${(\fancyscript{P}(\omega)/{{\rm fin}})^V}$ is a dense subalgebra of ${(\fancyscript{P}(\omega)/\fancyscript{I})^M}$ if and only if M does not contain an independent (splitting) real. We show that for a generic extension V[G], the canonical embedding

$$\fancyscript{P}^V(\omega){/}{\rm fin}\hookrightarrow \fancyscript{P}(\omega){/}(U(Os)(\mathbb{B}))^G$$

is a regular one, where ${U(Os)(\mathbb{B})}$ is the Urysohn closure of the zero-convergence structure on ${\mathbb{B}}$.