Around Rubin's "Theories of linear order"

TL;DR

This paper explores conditions under which the theories of linearly ordered structures are as simple as colored orders, extending Rubin's ideas to characterize their definable sets and automorphism properties.

Contribution

It introduces three conditions inspired by Rubin's work that ensure theories are geometrically simple and characterizes theories of colored orders with additional structure.

Findings

Identifies conditions guaranteeing theories are similar to colored orders.

Proves the strongest condition characterizes theories of colored orders with convex equivalence classes.

Shows models under these conditions have simple definable sets.

Abstract

Let $\mathcal M=(M,<,...)$ be a linearly ordered first-order structure and $T$ its complete theory. We investigate conditions for $T$ that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders with added unary predicates). Motivated by Rubin's work, we label three conditions expressing properties of types of $T$ and/or automorphisms of models of $T$. We prove several results which indicate the "geometric" simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest condition characterizes, up to definitional equivalence (inter-definability), theories of colored orders expanded by equivalence relations with convex classes.