ELEMENTARY PROPERTIES OF LIMIT REDUCED POWERS WITH APPLICATIONS TO BOOLEAN POWERS

Abstract

This chapter discusses the elementary properties of limit reduced powers with applications to Boolean powers. Limit reduced powers were introduced by Keis1er and are so named because every limit reduced power of A is, in a natural way, a direct limit of reduced powers of A. For algebras A and B, A = B means that A is elementarily equivalent with B. For a filter F on I × I and an algebra A, the limit power AI | F is the subalgebra of AI consisting of all functions f: I → A with Ker f ∈ ∈ F. For any topological space X and any algebra A, C(X, A) is the algebra of all continuous functions from X into A (with the discrete topology) with the operations defined component-wise.