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arxiv: 1807.08384 · v1 · pith:TYAEGFEXnew · submitted 2018-07-22 · 🧮 math.RA · math.CO

Lattices with many congruences are planar

classification 🧮 math.RA math.CO
keywords congruenceslatticeplanarelementexactlyexistsfinitelattices
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Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly $2^{n-5}$ congruences.

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  1. One hundred twenty-seven subsemilattices and planarity

    math.RA 2019-06 unverdicted novelty 6.0

    A finite n-element semilattice is planar if it has at least 127 * 2^(n-8) subsemilattices, and this bound is sharp for n > 8 via an explicit non-planar counterexample with one fewer subsemilattice.