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Maximal irredundance and maximal ideal independence in Boolean algebras. (English) Zbl 1141.06011

A subset \(X\) of a Boolean algebra \(A\) is irredundant iff \(x \notin \langle X \setminus \{ x \} \rangle\) for all \(x \in X\), where \(\langle X \setminus \{ x \} \rangle\) denotes the subalgebra generated by \(X \setminus \{ x \}\). \(X\) is ideal independent if the above property holds for the ideals instead of subalgebras. Let \(S\) be the set of all maximal irredundant subsets of \(A\). Then Irr\(_{mm}\) denotes the minimum of \(\{ | X| : X \in S \}\).
The author shows that there is an atomless Boolean algebra \(A\) of size \(2^{\omega}\) with Irr\(_{mm}(A) = \omega\).
Let \(T\) denote the set of all maximal ideal independent subsets of \(A\). Then \(s_{\text{spect}}(A)\) denotes the set \(\{ | X| : X \in T \}\) and \(s_{mm}(A)\) denotes the minimum of \(s_{\text{spect}}(A)\). The author shows that many sets of infinite cardinals can appear as \(s_{\text{spect}}(A)\). Further on he shows that it is relatively consistent that \(s_{mm}({\mathcal{P}}(\omega)/\text{fin}) < 2^{\omega}\).

MSC:

06E05 Structure theory of Boolean algebras
06E10 Chain conditions, complete algebras
03E35 Consistency and independence results
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References:

[1] Continuum cardinals generalized to Boolean algebras 66 pp 1928– (2001)
[2] Cardinal invariants on Boolean algebras (1996) · Zbl 0849.03038
[3] Handbook of set theory
[4] Set theory (1980)
[5] Handbook on Boolean algebras 1 (1989)
[6] On some small cardinals for Boolean algebras 69 pp 674– (2004)
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