The spectrum of maximal independent subsets of a Boolean algebra (Q598298)

An independent subset of a Boolean algebra is a set \(X\) so that if \(F, G\) are disjoint finite subsets of \(X\) then \(\prod_{x \in F}x \cdot \prod_{y \in G}-y \neq 0\). Spind\((B) = \{| X| : X\) is a maximal independent subset of \(B\}\). \(\mathfrak i(B) = \inf\) Spind\((B)\); Ind\((B) = \sup\) Spind\((B)\). The main result of this paper is that any set of infinite cardinals is Spind\((B)\) for some \(B\). It also investigates Spind\((B)\) and \(\mathfrak i (B)\) where \(B\) is defined via weak product, product, sum, or some combination of these, applied to free algebras.