The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous ndimensional permutation structures (homogeneous structures with n linear orders) for all finite n. We prove here that the lattice of ∅-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of ∅-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of ∅-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2.