×

Irredundant sets in Boolean algebras. (English) Zbl 0781.06010

A subset \(X\) of a Boolean algebra is called irredundant if no proper subset of \(X\) generates the same subalgebra as \(X\). In any Boolean algebra both chains and infinite disjoint sets are irredundant. E. K. van Douwen, J. D. Monk and M. Rubin [Algebra Univers. 11, 220-243 (1980; Zbl 0451.06014)] asked whether all uncountable Boolean algebras contain uncountable irredundant sets. In the paper under review there is proved that under the Proper Forcing Axiom the answer is affirmative. The very deep Theorem 1 says that every uncountable Boolean algebra has an uncountable irredundant set whenever PFA holds true, whereas Theorem 2 says that in fact the axiomatic approach to the problem was necessary. Theorem 2 says that the irredundant sets have an influence on the real line: if every uncountable Boolean algebra has an uncountable irredundant set, then every subset of \(\omega^ \omega\) of size \(\aleph_ 1\) has an upper bound in the ordering of eventual dominance. Another yet interesting result of the paper says that for every Boolean algebra \(B\), \(| B|\leq\text{irr}(B)\cdot\text{ig}(B)^ +\), where \(\text{irr}(B)\) is the supremum of cardinalities of irredundant sets in \(B\) and \(\text{ig}(B)\) is the minimal cardinal \(\tau\) such that every ideal in \(B\) has a set of generators of size not greater than \(\tau\). In particular, under Martin’s Axiom every Boolean algebra of countable irredundance is of size at most \(\aleph_ 1\).

MSC:

06E05 Structure theory of Boolean algebras
03E50 Continuum hypothesis and Martin’s axiom

Citations:

Zbl 0451.06014
PDFBibTeX XMLCite
Full Text: DOI