The (λ,κ)-Freese-Nation Property for Boolean Algebras and Compacta

Abstract

We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite κ, the (κ,κ)-FN, the (κ + ,κ)-FN, and the κ-FN are known to be equivalent; we show that the family of properties (λ, μ)-FN for λ > μ form a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, and the exponential operation. The \((\kappa,\aleph_0)\)-FN in particular has strong consequences for base properties of compacta (stronger still for homogeneous compacta), and these consequences have natural duals in terms of special subsets of boolean algebras. We show that the \((\kappa,\aleph_0)\)-FN also leads to a generalization of the equality of weight and π-character in dyadic compacta. Elementary subalgebras and their duals, elementary quotient spaces, were originally used to define the (λ,κ)-FN and its topological dual, which naturally generalized from Stone spaces to all compacta, thereby generalizing Ščepin’s notion of openly generated compacta. We introduce a simple combinatorial definition of the (λ,κ)-FN that is equivalent to the original for regular infinite cardinals λ > κ.