The spectrum of maximal independent subsets of a Boolean algebra Original Research Article

Recall that a subset X of a Boolean algebra (BA) A is independent if for any two finite disjoint subsets F,G of X we have View the MathML source The independence of a BA A, denoted by Ind(A), is the supremum of cardinalities of its independent subsets. We can also consider the maximal independent subsets. The smallest size of an infinite maximal independent subset is the cardinal invariant View the MathML source, well known in the case View the MathML source. In this article we consider the collection of all cardinalities of infinite maximal independent subsets of a BA A; we call this set the spectrum of infinite maximal independent subsets, denoted by Spind(A). Note that infinite maximal independent subsets exist in any BA which is not superatomic. The main result is that any set of infinite cardinals can occur as Spind(A) for some infinite BA A. Beyond this we give results concerning the way that Spind(A) changes under various algebraic operations. However, the basic components of most algebras that we deal with are free algebras.