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

Victoria Louise Ironmonger*, Nik Ruskuc*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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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 ρ1,...,ρk, is the downward closed set Av(ρ1,...,ρk) consisting of all equivalence relations which do not contain any of ρ1,...,ρk (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?
Original languageEnglish
Number of pages26
JournalOrder
Early online date14 Feb 2024
DOIs
Publication statusE-pub ahead of print - 14 Feb 2024

Keywords

  • Equivalence relation
  • Embedding
  • Poset
  • Well quasi-order
  • Antichain
  • Atomic
  • Joint embedding property
  • Graph
  • Path
  • Subpath
  • Decidability

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