Local stability in structures with a standard sort

TL;DR

This paper explores local stability in structures with a standard sort, extending classical approaches and analyzing variants of the order property to understand definability in these structures.

Contribution

It introduces a detailed analysis of local stability in structures with a standard sort and compares different variants of the order property.

Findings

Sets externally definable by stable formulas are definable in an appropriate sense

Three variants of the order property are examined and shown to be non-equivalent

Extensions of classical stability results to structures with a standard sort

Abstract

Recently, a classical approach to continuous structures has been proposed in [ABBMZ] and [Z] that extends the class of structures falling under the scope of [HI] or [BBHU]. These articles introduce the notion of structures with a standard sort. We discuss local stability in this context. We examine three variants of the order property which are prima facie non equivalent. For each variant we show that sets externally definable by stable formulas are definable in some appropriate sense.