A Boolean model of ultrafilters Original Research Article

We introduce the notion of Boolean measure algebra. It can be described shortly using some standard notations and terminology. If B is any Boolean algebra, let BN denote the algebra of sequences (xn), xn ∈ B. Let us write pk ∈ BN the sequence such that pk(i) = 1 if i ⩽ K and Pk(i) = 0 if k < i. If x ∈ B, denote by x∗ ∈ BN the constant sequence x∗ = (x,x,x,…). We define a Boolean measure algebra to be a Boolean algebra B with an operation μ:BN → B such that μ(pk) = 0 and μ(x∗) = x. Any Boolean measure algebra can be used to model non-principal ultrafilters in a suitable sense. Also, we can build effectively the initial Boolean measure algebra. This construction is related to the closed open Ramsey Theorem (J. Symbolic Logic 38 (1973) 193–198.)