A Boolean model of ultrafilters

Abstract

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 B N denote the algebra of sequences ( x n ), x n ∈ B. Let us write p k ∈ B N the sequence such that p k ( i) = 1 if i ⩽ K and P k ( i) = 0 if k < i. If x ∈ B, denote by x ∗ ∈ B N the constant sequence x ∗ = (x,x,x,…) . We define a Boolean measure algebra to be a Boolean algebra B with an operation μ: B N → B such that μ( p k ) = 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.)