On Cellularity In Homorphic Images of Boolean Algebras
by
J. Donald Monk
University of Colorado, Boulder

For any infinite Boolean algebra A, let c_Hr A be the set of all pairs (\kappa, \lambda) such that A has an infinite homomorphic image B of size \lambda and cellularity \kappa (the cellularity of B is the supremum of the cardinalities of disjoint subsets of B). In topological terms, for any compact totally disconnected Hausdorff space X we are dealing with the set of all pairs (\kappa, \lambda) such that \lambda is the weight of an infinite closed subspace Y of X and \kappa is the cellularity of Y. After a brief survey of previously known results on this notion, we mention some new facts and give a complete proof for one of them.

Special cases of the new facts give, under GCH, BA's A and B such that c_Hr A = { (\omega, \omega₁), (\omega₁, \omega₁), (\omega₂, \omega₂) } and c_Hr B = { (\omega, \omega₁), (\omega₁, \omega₁), (\omega₁, \omega₂), (\omega₂, \omega₂) }.

Also, we have one result on the analogous notion c_Sr A for subalgebras: If c(A) >= \omega₂ and (\omega, \omega₂) in c_Sr A, then (\omega₁, \omega₂) in c_Sr A.

These results solve problems stated in Monk, Cardinal invariants on Boolean algebras, Birkhäuser 1996.

Date received: June 25, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-35.