Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On the Representations of Finite Distributive Lattices

  • Siggers, Mark (Department of Mathematics, Kyungpook National University)
  • Received : 2017.01.24
  • Accepted : 2020.01.16
  • Published : 2020.03.31
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Abstract

A simple but elegant result of Rival states that every sublattice L of a finite distributive lattice 𝒫 can be constructed from 𝒫 by removing a particular family 𝒥L of its irreducible intervals. Applying this in the case that 𝒫 is a product of a finite set 𝒞 of chains, we get a one-to-one correspondence L ↦ 𝒟𝒫(L) between the sublattices of 𝒫 and the preorders spanned by a canonical sublattice 𝒞⋄ of 𝒫. We then show that L is a tight sublattice of the product of chains 𝒫 if and only if 𝒟𝒫(L) is asymmetric. This yields a one-to-one correspondence between the tight sublattices of 𝒫 and the posets spanned by its poset J(𝒫) of non-zero join-irreducible elements. With this we recover and extend, among other classical results, the correspondence derived from results of Birkhoff and Dilworth, between the tight embeddings of a finite distributive lattice L into products of chains, and the chain decompositions of its poset J(L) of non-zero join-irreducible elements.

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Acknowledgement

Supported by : National Research Foundation (NRF)

I thank the anonymous referees for their patience and their constructive comments.

References

  1. G. Birkhoff, Rings of sets, Duke Math. J., 3(3)(1937), 443-454. https://doi.org/10.1215/S0012-7094-37-00334-X
  2. R. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math., 51(1950), 161-166. https://doi.org/10.2307/1969503
  3. R. Dilworth, Some combinatorial problems on partially ordered sets, Proc. Sympos. Appl. Math., 10(1960), 85-90.(American Mathematical Society, Providence, R. I.)
  4. G. Gratzer, Lattice theory: foundation, Birkhauser/Springer Basel AG, Basel, 2011.
  5. K. M. Koh, On the lattice of maximum-sized antichains of a finite poset, Alg. Universalis, 17(1983), 73-86. https://doi.org/10.1007/BF01194515
  6. R. N. Larson, Embeddings of finite distributive lattices into products of chains, Semigroup Forum, 56(1998), 70-77. https://doi.org/10.1007/s00233-002-7005-3
  7. V. Retakh and M. Saks, On the rational relationships among pseudo-roots of a non-commutative polynomial, Preprint.
  8. I. Rival, Maximal sublattices of finite distributive lattices II, Proc. Amer. Math. Soc., 44(1974), 263-268. https://doi.org/10.1090/S0002-9939-1974-0360393-5
  9. M. Siggers, Distributive lattice polymorphisms on reflexive graphs, Bull. Korean Math. Soc., 55(2018), 81-105. https://doi.org/10.4134/BKMS.b160853