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?

翻译：我们研究在标准嵌入序和连续嵌入序下有限集上等价关系构成的偏序集。在后一种情况下，假设这些关系具有一个基础线性序，该线性序控制连续嵌入。对于每个偏序集，我们提出良拟序和原子性可判定性问题：给定有限个等价关系 $\rho_1,\dots,\rho_k$，由所有不包含任何 $\rho_1,\dots,\rho_k$ 的等价关系构成的下向闭集 Av$(\rho_1,\dots,\rho_k)$ 是否：(a) 为良拟序，即它不包含无穷反链？(b) 为原子，即它不能表示为两个真下向闭子集的并集，或者等价地，它满足联合嵌入性质？