The (\lambda, \kappa)-Freese-Nation property for boolean algebras and compacta

TL;DR

This paper introduces a two-parameter hierarchy of the Freese-Nation property for boolean algebras and compacta, revealing new structural insights and extending classical notions to a broader, more robust framework.

Contribution

It defines a two-dimensional hierarchy of the (,)-FN properties, demonstrating their stability and implications for compacta and boolean algebras, generalizing previous concepts.

Findings

The (,)-FN hierarchy is robust under coproducts, retracts, and exponentials.

The (,\u2060)-FN influences base properties of compacta, especially homogeneous ones.

Generalizes the equality of weight and -character in dyadic 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 \kappa, the (\kappa,\kappa)-FN, the (\kappa^+,\kappa)-FN, and the \kappa-FN are known to be equivalent; we show that the family of properties (\lambda,\mu)-FN for \lambda>\mu 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 \pi-character in dyadic compacta. Elementary subalgebras and their duals, elementary quotient spaces, were originally used to define the (\lambda, \kappa)-FN and its topological dual, which naturally generalized from Stone spaces to all compacta, thereby generalizing Shchepin's notion of openly generated compacta. We introduce a simple combinatorial definition of the (\lambda, \kappa)-FN that is equivalent to the original for regular infinite cardinals \lambda>\kappa.