A Boolean model of ultrafilters

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) is an element of B. Let us write p(k) is an element of B-N the sequence such that p(k)(i) = 1 if i less than or equal to k and p(k)(i) = 0 if k < i. If x is an element of B, denote by x* is an element of B-N the constant sequence x* = (x,x,x,...). We define a Boolean measure algebra to be a Boolean algebra B with an operation mu: B-N --> B such that mu(p(k)) = 0 and mu(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.) (C) 1999 Elsevier Science B.V. All rights reserved.