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

Order Pub Date : 2024-02-14 DOI:10.1007/s11083-024-09659-9

引用次数: 0

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?

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嵌入排序下等价关系的井式准排序可判性与原子性

摘要我们考虑了在标准嵌入排序和连续嵌入排序下有限集上等价关系的集合。在后一种情况下，我们还假定这些关系有一个基本的线性秩，它支配着连续嵌入。对于每一个正集，我们都会提出准有序性和原子性的可解性问题：给定有限多个等价关系 \(\rho _1,\dots ,\rho _k\)，向下闭集 \({{\,\textrm{Av}\,}}(\rho _1,\dots ,\rho _k)\)是否由所有不包含 \(\rho _1,\dots ,\rho _k\)的等价关系组成：(a) 准有序，即它不包含无限反链？(b) 原子性，即它不是两个适当的向下封闭子集的联合，或者，等价地，它满足联合嵌入属性？

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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