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A Note on the Modularization of Lattices

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Abstract

Valuations on finite lattices have been known for a long time. In this paper, we present a combinatorial procedure called modularization that associates a modular lattice to any given finite lattice such that they have the same valuation polytopes.

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References

  1. Birkhoff, G.: Lattice Theory, vol. 25. American Mathematical Soc., Providence (1967)

    Google Scholar 

  2. Chappell, T., Friedl, T., Sanyal, R.: Two double poset polytopes. SIAM J. Discret. Math. 31(4), 2378–2413 (2017)

    Article  MathSciNet  Google Scholar 

  3. Dobbertin, H.: About polytopes of valuations on finite distributive lattices. Order 2(2), 193–198 (1985)

    Article  MathSciNet  Google Scholar 

  4. Geissinger, L.: The face structure of a poset polytope. In: Proceedings of the Third Caribbean Conference on Combinatorics and Computing, vol. 59. Univ. West Indies, Barbados (1981)

  5. Grätzer, G.: General Lattice Theory. Springer Science & Business Media (2002)

  6. Grätzer, G., Wehrung, F.: Tensor products of semilattices with zero, revisited. J. Pure Appl. Algebra 147(3), 273–301 (2000)

    Article  MathSciNet  Google Scholar 

  7. Liu, F.: On positivity of Ehrhart polynomials. arXiv:1711.09962 (2017)

  8. Stanley, R: Two poset polytopes. Discret. Comput. Geom. 1(1), 9–23 (1986)

    Article  MathSciNet  Google Scholar 

  9. Stanley, R.: Enumerative Combinatorics, vol. 1. Cambridge University Press, Cambridge (2012)

    MATH  Google Scholar 

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Acknowledgements

The author thanks Prof. Richard Stanley for supervising this project.

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Correspondence to Yibo Gao.

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Gao, Y. A Note on the Modularization of Lattices. Order 37, 311–318 (2020). https://doi.org/10.1007/s11083-019-09507-1

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  • DOI: https://doi.org/10.1007/s11083-019-09507-1

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