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Keywords:
free sequences; Boolean algebras; cardinal functions; 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.
References:
[BK81] Baumgartner J.E., Komjáth P.: Boolean algebras in which every chain and antichain is countable. Fund. Math. 111 (1981), 125–133. MR 0609428 | Zbl 0452.03044
[KMB89] Koppelberg S., Monk J.D., Bonnet R.: Handbook of Boolean Algebras. vol. 1989, North-Holland, Amsterdam, 1989.
[Kos99] Koszmider P.: Forcing minimal extensions of Boolean algebras. Trans. Amer. Math. Soc. 351 (1999), no. 8, 3073–3117. MR 1467471 | Zbl 0922.03071
[Mon08] Monk J.D.: Maximal irredundance and maximal ideal independence in Boolean algebras. J. Symbolic Logic 73 (2008), no. 1, 261–275. DOI 10.2178/jsl/1208358753 | MR 2387943 | Zbl 1141.06011
[Mon11] Monk J.D.: Maximal free sequences in a Boolean algebra. Comment. Math. Univ. Carolin. 52 (2011), no. 4, 593–610. MR 2864001 | Zbl 1249.06034
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