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Whaley’s theorem for finite lattices. (English) Zbl 1045.06002

Summary: Whaley’s Theorem on the existence of large proper sublattices of infinite lattices is extended to ordered sets and finite lattices. As a corollary it is shown that every finite lattice \(\mathbf L\) with \(|\mathbf L| \geqslant 3\) contains a proper sublattice \(\mathbf S\) with \(|\mathbf S|\geqslant |\mathbf L|^{1/3}\). It is also shown that every finite modular lattice \(\mathbf L\) with \(|\mathbf L| \geqslant 3\) contains a proper sublattice \(\mathbf S\) with \(|\mathbf S|\geqslant |\mathbf L|^{1/2}\), and every finite distributive lattice \(\mathbf L\) with \(|\mathbf L|\geqslant 4\) contains a proper sublattice \(\mathbf S\) with \(|\mathbf S| \geqslant \frac{3}{4}|\mathbf L|\).

MSC:

06B05 Structure theory of lattices
06A06 Partial orders, general
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