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The Number of Subuniverses, Congruences, Weak Congruences of Semilattices Defined by Trees

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Abstract

We determine the number of subuniverses of semilattices defined by arbitrary and special kinds of trees via combinatorial considerations. Using a result of Freese and Nation, we give a formula for the number of congruences of semilattices defined by arbitrary and special kinds of trees. Using both results, we prove a formula for the number of weak congruences of semilattices defined by a binary tree; we discuss some special cases. We solve two apparently nontrivial recurrences applying the method of Aho and Sloane.

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Acknowledgements

The authors thank the anonymous referee for his/her valuable comments, especially those that led to the current form of Theorem 3.2. We are indebted to this reviewer for calling our attention to the fact mentioned in Remark 4.5.

This research of the second author was supported by the National Research, Development and Innovation Fund of Hungary under funding scheme K 138892. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    Delbrin Ahmed

  2. Department of Algebra and Number Theory, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    Eszter K. Horváth

  3. Department of Analysis, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    Zoltán Németh

Authors
  1. Delbrin Ahmed
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  2. Eszter K. Horváth
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  3. Zoltán Németh
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Correspondence to Eszter K. Horváth.

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Ahmed, D., Horváth, E.K. & Németh, Z. The Number of Subuniverses, Congruences, Weak Congruences of Semilattices Defined by Trees. Order 40, 335–348 (2023). https://doi.org/10.1007/s11083-022-09611-9

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