The universal homogeneous binary tree

TL;DR

This paper classifies the model-complete cores and reducts of a unique countable existentially closed semilinear order, linking these structures to permutation groups and their definability properties.

Contribution

It provides the first classification of the model-complete cores of reducts of the universal homogeneous semilinear order S2.

Findings

Classification of model-complete cores of reducts of S2

Identification of all reducts up to first-order interdefinability

Connection between reducts and closed permutation groups

Abstract

A partial order is called semilinear iff the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which we denote by S2. We study the reducts of S2, that is, the relational structures with the same domain as S2 all of whose relations are first-order definable in S2. Our main result is a classification of the model-complete cores of the reducts of S2. From this, we also obtain a classification of reducts up to first-order interdefinability, which is equivalent to a classification of all closed permutation groups that contain the automorphism group of S2.