A Simultaneous Generalization of Independence and Disjointness in Boolean Algebras

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 ω-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, $\mathfrak{i}_{n}$ , the minimum size of a maximal n-independent subset and $\mathfrak{i}_{\omega}$ , the minimum size of an ω-independent subset, are introduced and investigated. The values of $\mathfrak {i}_{n}$ and $\mathfrak {i}_{\omega}$ on are shown to be independent of ZFC.