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 language | English |
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Number of pages | 26 |
Journal | Order |
Early online date | 14 Feb 2024 |
DOIs | |
Publication status | E-pub ahead of print - 14 Feb 2024 |
Keywords
- Equivalence relation
- Embedding
- Poset
- Well quasi-order
- Antichain
- Atomic
- Joint embedding property
- Graph
- Path
- Subpath
- Decidability