Variations of Independence in Boolean Algebras

TL;DR

This paper introduces new classes of boolean algebras with generalized independence properties, explores their combinatorial and cardinal invariants, and investigates their connections to hypergraph theory and set theory.

Contribution

It defines n-independent classes of boolean algebras, studies their properties, and examines related cardinal invariants, extending the understanding of independence in algebraic and set-theoretic contexts.

Findings

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

Cardinal functions p and s_mm satisfy p <= s_mm for infinite boolean algebras.

Moderately generated boolean algebras are characterized by finitely splitting generating sets.

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. Ideal-independence is also considered, and it is shown that the cardinal function p <= s_mm for infinite boolean algebras. We also define and consider moderately generated boolean algebras; that is, those boolean algebras that have a generating set consisting of elements that split finitely many elements of the boolean algebra.