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?
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Atminas, A., Lozin, V., Moshkov, M.: WQO is decidable for factorial languages. Inform. Comput. 256, 321–333 (2017)
Braunfeld, S.: The undecidability of joint embedding and joint homomorphism for hereditary graph classes. Discrete Math. Theor. Comput. Sci. 21(9), 17 (2019)
Braunfeld, S.: The undecidability of joint embedding for 3-dimensional permutation classes. Discrete Math. Theor. Comput. Sci. 22(10), 20 (2021)
Cherlin, G.: Forbidden substructures and combinatorial dichotomies: WQO and universality. Discrete Math. 311, 1543–1584 (2011)
Ding, G.: Subgraphs and well quasi-ordering. J. Graph Theory 16, 489–502 (1992)
Elizalde, S.: A survey of consecutive patterns in permutations. Recent trends in combinatorics, 601–618, IMA Vol. Math. Appl. 159, Springer, (2016)
Elizalde, S., McNamara, P.R.W.: The structure of the consecutive pattern poset. Int. Math. Res. Not. IMRN, 2099–2134 (2018)
Fraïssé, R.: Theory of relations. Stud. Logic Found. Math. 145, North-Holland Publishing Co., Amsterdam (2000)
Higman, G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2, 326–336 (1952)
Hodges, W.: Model theory, Encyclopedia of Mathematics and its Applications 42. CUP, Cambridge (1993)
Huczynska, S., Ruškuc, N.: Well quasi-order in combinatorics: embeddings and homomorphisms, Surveys in combinatorics 2015, 261–293, LMS Lecture Note Ser. 424, CUP, Cambridge (2015)
Liu, C-H.: Recent progress on well-quasi-ordering graph, Well-quasi orders in computation, logic, language and reasoning—a unifying concept of proof theory, automata theory, formal languages and descriptive set theory, 161–188, Trends Log. Stud. Log. Libr., 53, Springer, Cham (2020)
McDevitt, M., Ruškuc, N.: Atomicity and well quasi-order for consecutive orderings on words and permutations. SIAM J. Discrete Math. 35(1), 495–520 (2021)
Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. J. Combin. Theory Ser. B 92, 325–357 (2004)
Vatter, V.: Permutation classes, Handbook of enumerative combinatorics, 753–833, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL (2015)
Acknowledgements
The authors would like to thank the anonymous referee for their helpful suggestions to improve the paper.
Funding
None.
Author information
Authors and Affiliations
Contributions
The aricle in its entirety is the result of equal contributions of the two co-authors. Both authors reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ironmonger, V., Ruškuc, N. Decidability of Well Quasi-Order and Atomicity for Equivalence Relations Under Embedding Orderings. Order (2024). https://doi.org/10.1007/s11083-024-09659-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11083-024-09659-9