The Reducts of the Homogeneous Binary Branching C-relation

TL;DR

This paper classifies the reducts of a unique countable homogeneous structure with a binary branching C-relation, confirming a conjecture and analyzing their endomorphism monoids.

Contribution

It proves the finiteness of reducts up to interdefinability for the structure (L;C) and categorizes their endomorphism monoids.

Findings

Finitely many reducts up to existential interdefinability.

Finitely many reducts up to first-order interdefinability.

Endomorphism monoids fall into four categories.

Abstract

Let (L;C) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of (L;C), i.e., the structures with domain L that are first-order definable in (L;C). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of (L;C). We also study the endomorphism monoids of such reducts and show that they fall into four categories.