Abstract:We make use of a forcing technique for extending Boolean algebras. The same type of forcing was employed in Baumgartner J.E., Komj\'ath P., {\it Boolean algebras in which every chain and antichain is countable\/}, Fund. Math. {\bf 111} (1981), 125--133, Koszmider P., {\it Forcing minimal extensions of Boolean algebras\/}, Trans. Amer. Math. Soc. {\bf 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., {\it Maximal irredundance and maximal ideal independence in Boolean algebras\/}, J. Symbolic Logic {\bf 73} (2008), no.~1, 261--275, and Monk J.D., {\it Maximal free sequences in a Boolean algebra\/}, Comment. Math. Univ. Carolin. {\bf 52} (2011), no.~4, 593--610.
Keywords: free sequences; Boolean algebras; cardinal functions; ultrafilter number
DOI: DOI 10.14712/1213-7243.015.110
AMS Subject Classification: 06E05 54A25