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We consider the posets of equivalence relations on finite sets under thestandard embedding ordering and under the consecutive embedding ordering. Inthe latter case, the relations are also assumed to have an underlying linearorder, which governs consecutive embeddings. For each poset we ask the wellquasi-order and atomicity decidability questions: Given finitely manyequivalence relations $\rho_1,\dots,\rho_k$, is the downward closed setAv$(\rho_1,\dots,\rho_k)$ consisting of all equivalence relations which do notcontain any of $\rho_1,\dots,\rho_k$: (a) well-quasi-ordered, meaning that itcontains no infinite antichains? and (b) atomic, meaning that it is not a unionof two proper downward closed subsets, or, equivalently, that it satisfies thejoint embedding property?

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