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:我们认为在标准嵌入顺序和连续嵌入顺序下定定定定定定定定定定定定定定定定定定定定定定定定定定定定定定定的等同关系。在后一种情况下,假设这些关系也有一个基本线性顺序,它制约着连续嵌入;对于每个摆定,我们问的是准顺序和原子性可变性的问题:鉴于相当数量的等同关系,$\rho_1,\dots,\rho_k$是下向封闭定定定定定定定定定定定定定定定定定定的Av$(rho_1,\dots,\rho_k)$,包括所有不包含任何$(rho_1,\dots,\k$)的等同关系:(a) 合理顺序,这意味着它没有无限的反链?和(b)原子,意味着它不是由两个正常的向下封闭子组合的组合,或者同样,它满足联合嵌入的属性?