Remarks on continuum cardinals on Boolean algebras

For set-theoretical notation we follow [3]. We follow [2] for Boolean algebraic notation, and Monk [5] for more specialized notation concerning cardinal functions on Boolean algebras. Fr(κ) is the free Boolean algebra on κ generators. A is the completion of A. If L is a linear order, then Intalg(L) is the interval algebra over L (perhaps after adjoining a first element to L). Any element x of Intalg(L) has the form [a0 , b0) ∪ · · · ∪ [am−1 , bm−1), with a0 < b0 < · · · < bm−1 ≤ ∞. (Here ∞ is not in L.) The intervals [ai, bi) are called the components of x. Now we define some notions entering into the definitions of our cardinal functions: A tower in A is a subset T of A well-ordered by the Boolean ordering in a limit ordinal type and with sum 1. We say that X is weakly dense in A if for all a ∈ A there is an x ∈ X+ such that x ≤ a or x ≤ −a. A weak partition of A is a system 〈bξ : ξ < α〉 of pairwise disjoint elements with sum 1; it is not assumed that all bξ are nonzero. We say that X ⊆ A is independent if for all F,G ∈ [X] we have [ F ∩G = ∅ → ∏ x∈F x · ∏ x∈G −x = 0 ] ; we say that it is ideal independent if for all x ∈ X and all F ∈ [X\{x}]<ω we have that x ≤ ∑ y∈F y. A set X ⊆ A+ is dense in an ultrafilter D of A if for all a ∈ D there is an x ∈ X such that x ≤ a. It is not assumed that X ⊆ D. A subset X of A+ splits A if for every a ∈ A for which A a is infinite there is an x ∈ X such that a ·x = 0 = a ·−x. A free sequence in a Boolean algebra A is a sequence 〈aξ : ξ < α〉 of elements of A such that