A note on Maximality of ideal-independent sets

In this note we derive a property of maximal ideal-independent subsets of boolean algebras which has corollaries regarding the continuum cardinals p and s_mm(P(omega)/fin)

Ideal Independence, Free Sequences, and the Ultrafilter Number

We make use of a forcing technique for extending Boolean algebras. The same type of forcing was employed in [BK81], [Kos99], and elsewhere. Using and modifying a lemma of Koszmider, and using CH, we obtain an atomless BA, A such that f(A) = smm(A) < u(A), answering questions raised by [Mon08] and [Mon11]

Irredundant Sets in Atomic Boolean Algebras

Assuming GCH, we construct an atomic boolean algebra whose pi-weight is strictly less than the least size of a maximal irredundant family.Comment: This version corrects some errors in the original arXiv versio

Ideal independence, free sequences, and the ultrafilter number

summary:We make use of a forcing technique for extending Boolean algebras. The same type of forcing was employed in Baumgartner J.E., Komjáth P., Boolean algebras in which every chain and antichain is countable, Fund. Math. 111 (1981), 125–133, Koszmider P., Forcing minimal extensions of Boolean algebras, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3073–3117, and elsewhere. Using and modifying a lemma of Koszmider, and using CH, we obtain an atomless BA, $A$ such that $\mathfrak{f}(A) = \text{s}_{\text{mm}}(A) <\frak{u}(A)$, answering questions raised by Monk J.D., Maximal irredundance and maximal ideal independence in Boolean algebras, J. Symbolic Logic 73 (2008), no. 1, 261–275, and Monk J.D., Maximal free sequences in a Boolean algebra, Comment. Math. Univ. Carolin. 52 (2011), no. 4, 593–610