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The Best Extending Cover-preserving Geometric Lattices of Semimodular Lattices

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Abstract

In 2010, Gábor Czédli and E. Tamás Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices. Advances in Mathematics, 225, 2455–2463 (2010)]. That is to say: What are the geometric lattices G such that a given finite semimodular lattice L has a cover-preserving embedding into G with the smallest ∣G∣? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices G of a given semimodular lattice L and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice G equal the length of L and the number of non-zero join-irreducible elements of L, respectively. Therefore, we solve the problem on the best cover-preserving embedding of a given semimodular lattice raised by Gábor Czédli and E. Tamás Schmidt.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, China

    Peng He

  2. School of Mathematical Sciences, Sichuan Normal University, Chengdu, 610066, China

    Xue Ping Wang

Authors
  1. Peng He
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  2. Xue Ping Wang
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Corresponding author

Correspondence to Xue Ping Wang.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 11901064 and 12071325)

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He, P., Wang, X.P. The Best Extending Cover-preserving Geometric Lattices of Semimodular Lattices. Acta. Math. Sin.-English Ser. 39, 1369–1388 (2023). https://doi.org/10.1007/s10114-023-1531-1

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  • DOI: https://doi.org/10.1007/s10114-023-1531-1

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