Lattice of Algebraically Closed Sets in One-Based Theories

Abstract

Let $T$ be a one-based theory. We define a notion of width, in the case of $T$ having the finiteness property, for the lattice of finitely generated algebraically closed sets and prove Theorem. Let $T$ be one-based with the finiteness property. If $T$ is of bounded width, then every type in $T$ is nonorthogonal to a weight one type. If $T$ is countable, the converse is true.