Decidability of Well Quasi-Order and Atomicity for Equivalence Relations Under Embedding Orderings

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 \({{\,\textrm{Av}\,}}(\rho _1,\dots ,\rho _k)\) 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?