Maximal free sequences in a Boolean algebra. (Q2897377)

The results of the author are related, in particular, to his book [Cardinal invariants on Boolean algebras. Basel: Birkhäuser Verlag (1996; Zbl 0849.03038); reprint (2010; Zbl 1206.03042)]. A free sequence in a Boolean algebra \(B\) is a sequence \(\left \langle a_{\xi }\: \xi < \alpha \right \rangle \) of elements of \(B\) indexed by an ordinal \(\alpha \) such that for all finite subsets \(F\) and \(G\) of \(\alpha \), if \(\xi <\zeta \) for every \(\xi \in F\) and \(\zeta \in G\) then \(\prod _{\xi \in F} a_\xi \cdot \prod _{\zeta \in G} -a_{\zeta }\neq 0\). The notion of maximal free sequence is defined in a natural way. A free sequence of length \(\alpha \) exists iff in the Stone space \(\text{Ult}(B)\) of \(B\) there is a free sequence of length \(\alpha \). That is, there is a sequence \(\left \langle x_{\xi }\: \xi < \alpha \right \rangle \) of elements of \(\text{Ult}(B)\) such that \(\overline { \{ x_{\xi }\: \xi < \zeta \}} \cap \overline { \{ x_{\xi }\: \xi \geq \beta \}} = \emptyset \). So different cardinal functions \(\mathfrak {f}_{\mathrm{sp}}(B)\) related to this notion can be defined:NEWLINENEWLINE\(\mathfrak {f}_{\mathrm{sp}}(B):= \{| \alpha | \: \text{ there is an infinite maximal free sequence of length }\alpha \}\) and \(\mathfrak {f}(B) = \min (\mathfrak {f}_{\mathrm{sp}}(B))\). For instance, assume that \(\kappa \) is an infinite cardinal. Then, (1) \(\mathfrak {f}_{\mathrm{sp}}(FC(\kappa )) = \{\omega \}\) where \(FC(\kappa )\) is the algebra of finite/cofinite subsets of \(\kappa \) (Theorem 1.2); (2) \(\mathfrak {f}_{\mathrm{sp}}(\text{Fr}(\kappa )) = \{\kappa \}\) where \(\text{Fr}(\kappa )\) is the free algebra over \(\kappa \) (Theorem 1.3); (3) \(\mathfrak {f}_{\mathrm{sp}}(\mathcal {P}(\kappa )) = [\kappa ,2^{\kappa }]\) (Proposition 1.11). These cardinal functions are intensively developed in \S 1 and a lot of results are established, for instance for the weak product of Boolean algebras (Proposition 1.15).NEWLINENEWLINEIn the other sections, other cardinal functions, replacing in \(\mathfrak {f}_{\mathrm{sp}}(B)\), ``maximal free sequences'' by the minimal cardinal \(| X | \) such that \(X \subseteq B\) is a ``maximal independent set'', or by the minimal cardinal \(| X | \) such that \(X \subseteq B\) is a ``maximal chain sequence''. In the case of an interval algebra \(B(C)\) generated by a chain \(C\), the relationship between ``maximal chain sequences'' and the chain \(C\) are developed.