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Journal Article

Quotients of Boolean algebras and regular subalgebras

Balcar Bohuslav, Pazák Tomáš

: Archive for Mathematical Logic vol.49, 3 (2010), p. 329-342

: CEZ:AV0Z10190503

: CEZ:AV0Z10750506

: IAA100190509, GA AV ČR, MEB060909, GA MŠk

: Boolean algebra, sequential topology, ZFC extension, ideal

: 10.1007/s00153-010-0174-y

: http://link.springer.com/article/10.1007%2Fs00153-010-0174-y

(eng): 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.

: BA

07.01.2019 - 08:39