On $\lam$-existence over a predicate

TL;DR

This paper proves that in fully stable theories over a predicate, any $ am$-complete set can be extended to a $ am$-saturated model without altering the predicate part, generalizing previous results in difference fields.

Contribution

It introduces the notion of $ am$-completeness and shows that $ am$-existence always holds in fully stable theories, extending prior work to a broader context.

Findings

Any $ am$-complete set can be extended to a $ am$-saturated model in fully stable theories.

$ am$-existence only fails for trivial reasons in such theories.

Generalizes results from difference fields of characteristic 0.

Abstract

We prove that in a countable theory $T$ fully stable over a predicate $P$, any $\lam$-complete set $A$ has the $\lam$-existence property. This means that $A$ can be extended to a $\lam$-saturated model of $T$ without changing the $P$-part. The notion of $\lam$-completeness, introduced in this paper, captures some obvious necessary conditions for such an extension to be possible (for example, the $P$-part of $A$ has to be a $\lam$-saturated model of the appropriate theory). So in a fully stable theory $T$, $\lam$-existence can only fail for trivial reasons. This generalizes results of Chatzidakis in the context of difference fields of characteristic 0.