Lascar Strong Types in Some Simple Theories

In this paper a class of simple theories, called the low theories is developed, and the following is proved. Theorem Let T be a low theory, A a set and a; b elements realizing the same strong type over A. Then, a and b realize the same Lascar strong type over A. The reader is expected to be familiar with forking in simple theories, as developed in Kim's thesis Kim]. The Lascar strong type of a over A is denoted lstp(a=A). Unless stated otherwise, we work in the context of a simple theory in this paper. 1 Amalgamation properties Type amalgamation (the Independence Theorem) is perhaps the most useful property of forking dependence in a simple theory. First, we stress an important fact from Kim]. Lemma 1.1 Let A be a set, a; b elements such that lstp(a=A) = lstp(b=A) and a j ^ A b. Then there is an innnite sequence (I; <) , with a; b 2 I and a < b , such that (I; <) is indiscernible over A. There are several closely related results that can be called \type amalgamation". The two we use here are: Proposition 1.2 Let A B , A C be sets, B j ^ A C , and b; c elements such that lstp(b=A) = lstp(c=A) , b j ^ A B and c j ^ A C. Then, tp(b=B) tp(c=C) is consistent and does not fork over A .