Classification properties for some ternary structures

We provide a model-theoretic classification of the countable homogeneous ${\mathbf{H}}_4$-free 3-hypertournament studied by Cherlin, Hubička, Konečný, and Nešetřil. Our main result is that the theory of this structure is ${\mathrm{S}\mathrm{O}\mathrm{P}}_3$, ${\mathrm{T}\mathrm{P}}_2$, and ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_4$. We offer two proofs of this fact: one is a direct proof, and the other employs part of the abstract machinery recently developed by Mutchnik.