A simultaneous generalization of independence and disjointness in boolean algebras

TL;DR

This paper introduces a new class of boolean algebras that generalize free boolean algebras, exploring their properties, combinatorial structures, and connections to hypergraph theory and cardinal invariants.

Contribution

It defines n-independent sets in boolean algebras, investigates n-free and omega-free classes, and studies related cardinal functions, including their independence from ZFC.

Findings

Properties of n-free and omega-free boolean algebras are characterized.

Cardinal invariants like nInd, i_n, and i_omega are introduced and analyzed.

Values of i_n and i_omega on P(omega)/fin are shown to be independent of ZFC.

Abstract

We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n-independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, $n$Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n-independent subset; and i_omega, the minimum size of an omega-independent subset, are introduced and investigated. The values of i_n and i_omega on P(omega)/fin are shown to be independent of ZFC.