Homogeneous 3-dimensional permutation structures

TL;DR

This paper classifies all homogeneous structures built from three linear orders, advancing the understanding of finite-dimensional permutation structures and completing a broader classification effort.

Contribution

It provides a complete classification of homogeneous 3-dimensional permutation structures and describes all known finite-dimensional cases.

Findings

Classification of all homogeneous 3D permutation structures

Natural description of known finite-dimensional structures

Completion of the existing classification census

Abstract

We provide a classification of the homogeneous 3-dimensional permutation structures, i.e. homogeneous structures in a language of 3 linear orders, partially answering a question of Cameron. We also arrive at a natural description of all known homogeneous finite-dimensional permutation structures by modifying the language used in the construction from a previous paper, completing the "census" begun there.