Local stability in structures with a standard sort

TL;DR

The paper develops a two-sorted framework for structures with a standard sort and investigates local stability via three variants of the order property: finitary stability, non-finitary stability, and $\varepsilon$-stability. It proves that externally definable sets and, more generally, types, are internally definable in each variant by finite combinatorial data, and it introduces a $K(S)$-duality and an auxiliary sort to manage external information. The results unify definability of external sets with internal descriptions in both classical and continuous-like logics, notably through finite matrices that realize global types and stable definable functions. Collectively, these findings provide a robust transfer principle for external definability in the framework of standard-sort structures, with potential applications to continuous logic and related formalisms.

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.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Theorem 3: (Compactness)
• Definition 4
• Theorem 14
• Definition 15
• Definition 17
• Definition 18
• Theorem 19
• Definition 20