Decidability of well quasi-order and atomicity for equivalence relations under embedding orderings

TL;DR

This paper investigates the decidability of well quasi-ordering and atomicity in posets of finite equivalence relations under embedding and consecutive embedding orders, providing insights into their structural properties.

Contribution

It establishes decidability results for well quasi-order and atomicity in posets of equivalence relations under specific embedding orders, advancing understanding of their structural complexity.

Findings

Decidability of well quasi-ordering for certain equivalence relation posets.

Decidability of atomicity in these posets.

Characterization of conditions under which these properties hold.

Abstract

We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations $\rho_1,\dots,\rho_k$, is the downward closed set Av$(\rho_1,\dots,\rho_k)$ consisting of all equivalence relations which do not contain any of $\rho_1,\dots,\rho_k$: (a) well-quasi-ordered, meaning that it contains no infinite antichains? and (b) atomic, meaning that it is not a union of two proper downward closed subsets, or, equivalently, that it satisfies the joint embedding property?