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

Abstract

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 $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-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 $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-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.

Homogeneous 3-Dimensional Permutation Structures

“…The following classification in the primitive case was conjectured in [1]. We show in Proposition 2.9 that this would follow from a positive answer to the above question.…”

“…This section is not strictly needed for the classification of the 3-dimensional case, but does provide context by giving the necessary background for Question 1. Because we hewed to the language of linear orders, we were unable to provide a satisfactory "census" in [1] of homogeneous finite-dimensional permutation structures, since some known examples were not produced by the construction. After modifying the construction to work with subquotient orders, we show such examples, for which we can now give a straightforward description.…”

“…Conjecture 1 (Primitivity Conjecture, [1]). Every primitive homogeneous finite dimensional permutation structure can be constructed by the following procedure.…”

The Classification of Homogeneous Finite-Dimensional Permutation Structures

“…A construction producing many new imprimitive examples of such structures was introduced in [1]. The structures produced by a slight generalization of that construction, making use of the subquotient orders introduced in §2 in place of linear orders, were put forward as a conjecturally complete catalog in [2], which confirmed the case of 3 linear orders.…”

“…The primitive case, in which there is no ∅-definable equivalence relation, is foundational for proving the completeness of the catalog. The Primitivity Conjecture of [1] conjectured that, modulo the agreement of certain orders up to reversal, a primitive homogeneous finite-dimensional permutation structure is the Fraïssé limit of all finite n-dimensional permutation structures, for some n. In the case of 2 [4] and 3 [2] linear orders, the conjecture was proven by increasingly involved direct amalgamation arguments. A description of the ways linear orders can interact in certain ω-categorical structures, as well as of the closed sets ∅-definable in products of such structures, was given in [11], and as an application of these model-theoretic results, the Primitivity Conjecture was confirmed.…”