Quotients of Boolean algebras and regular subalgebras

0342828 - MÚ 2011 RIV DE eng J - Journal Article
Balcar, Bohuslav - Pazák, Tomáš
Quotients of Boolean algebras and regular subalgebras.
Archive for Mathematical Logic. Roč. 49, č. 3 (2010), s. 329-342. ISSN 0933-5846. E-ISSN 1432-0665
R&D Projects: GA AV ČR IAA100190509; GA MŠMT MEB060909
Institutional research plan: CEZ:AV0Z10190503; CEZ:AV0Z10750506
Keywords : Boolean algebra * sequential topology * ZFC extension * ideal
Subject RIV: BA - General Mathematics
Impact factor: 0.414, year: 2010
http://link.springer.com/article/10.1007%2Fs00153-010-0174-y

Let B and C be Boolean algebras and e : B -> C an embedding. We examine the hierarchy of ideals on C for which (e) over bar : B -> C/I is a regular (i.e. complete) embedding. As an application we deal with the interrelationship between P(omega)/fin in the ground model and in its extension. If M is an extension of V containing a new subset of omega, then in M there is an almost disjoint refinement of the family ([omega](omega))(V). Moreover, there is, in M, exactly one ideal I on omega such that (P(omega)/fin)(V) is a dense subalgebra of (P(omega)/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(omega)/fin hooked right arrow P(omega)/(U(Os)(B))(G) is a regular one, where U(Os)(B) is the Urysohn closure of the zero-convergence structure on B.
Permanent Link: http://hdl.handle.net/11104/0185452