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.
Eighty-three sublattices and planarity
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abstract
Let $L$ be a finite $n$-element lattice. We prove that if $L$ has at least $83\cdot 2^{n-8}$ sublattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar lattice with exactly $83\cdot 2^{n-8}-1$ sublattices.
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math.RA 1years
2019 1verdicts
UNVERDICTED 1representative citing papers
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One hundred twenty-seven subsemilattices and planarity
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.