Congruence lattices of small planar lattices

Grätzer, G.; Lakser, H.; Schmidt, E. T.

doi:10.1090/S0002-9939-1995-1301498-5

Congruence lattices of small planar lattices
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by G. Grätzer, H. Lakser and E. T. Schmidt PDF

Proc. Amer. Math. Soc. 123 (1995), 2619-2623 Request permission

Abstract:

For a finite distributive lattice D with n join-irreducible elements, we construct a finite (planar) lattice L with $O({n^2})$ elements such that the congruence lattice of L is isomorphic to D. This improves on an early result of R. P. Dilworth (around 1940) and G. Grätzer and E. T. Schmidt (1962) constructing such a (nonplanar) lattice L with $O({2^{2n}})$ elements, and on a recent construction of G. Grätzer and H. Lakser which yields a finite (planar) lattice L with $O({n^3})$ elements.

References

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