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\(\Lambda \)-Ultrametric spaces and lattices of equivalence relations

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Abstract

For a finite lattice \(\Lambda \), \(\Lambda \)-ultrametric spaces have, among other reasons, appeared as a means of constructing structures with lattices of equivalence relations embedding \(\Lambda \). This makes use of an isomorphism of categories between \(\Lambda \)-ultrametric spaces and structures equipped with certain families of equivalence relations. We extend this isomorphism to the case of infinite lattices. We also pose questions about representing a given finite lattice as the lattice of \(\emptyset \)-definable equivalence relations of structures with model-theoretic symmetry properties.

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References

  1. Bergman, G.: An Invitation to General Algebra and Universal Constructions. Springer, Cham (2015)

    Book  Google Scholar 

  2. Bodirsky, M.: Complexity classification in infinite-domain constraint satisfaction (2012). arXiv:1201.0856

  3. Braunfeld, S.: The lattice of definable equivalence relations in homogeneous \(n\)-dimensional permutation structures. Electron. J. Combin. 23, #44 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Braunfeld, S.: Infinite limits of finite-dimensional permutation structures, and their automorphism groups: between model theory and combinatorics. PhD thesis, Rutgers University, New Brunswick (2018)

  5. Priess-Crampe, S., Ribenboim, P.: Equivalence relations and spherically complete ultrametric spaces. C. R. Acad. Sci. Paris Sér. I Math. 320, 1187–1192 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Priess-Crampe, S., Ribenboim, P.: Generalized ultrametric spaces I. Abh. Math. Semin. Univ. Hambg. 66, 55–73 (1996)

    Article  MathSciNet  Google Scholar 

  7. Pudlák, P., Tůma, J.: Every finite lattice can be embedded in a finite partition lattice. Algebra Univ. 10, 74–95 (1980)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I thank Gregory Cherlin for helpful discussions.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Samuel Braunfeld

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  1. Samuel Braunfeld
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Correspondence to Samuel Braunfeld.

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Dedicated to Ralph Freese, Bill Lampe, and J.B. Nation.

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This article is part of the topical collection “Algebras and Lattices in Hawaii” edited by W. DeMeo.

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Braunfeld, S. \(\Lambda \)-Ultrametric spaces and lattices of equivalence relations. Algebra Univers. 80, 33 (2019). https://doi.org/10.1007/s00012-019-0606-4

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  • DOI: https://doi.org/10.1007/s00012-019-0606-4

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