Binary simple homogeneous structures

TL;DR

This paper characterizes all binary simple homogeneous structures using definable equivalence relations that organize the structure and influence its dividing and extension properties.

Contribution

It provides a complete description of binary simple homogeneous structures through 0-definable equivalence relations and their role in the structure's properties.

Findings

All such structures are described by specific equivalence relations.

These relations determine dividing and extension behaviors.

The paper offers a classification framework for these structures.

Abstract

We describe all binary simple homogeneous structures M in terms of 0-definable equivalence relations on M, which "coordinatize" M and control dividing, and extension properties that respect these equivalence relations.