Freese, Ralph; Hyndman, Jennifer; Nation, J. B. Whaley’s theorem for finite lattices. (English) Zbl 1045.06002 Order 20, No. 3, 223-228 (2003). 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 \(|\mathbf S| \geqslant \frac{3}{4}|\mathbf L|\). Cited in 1 Document MSC: 06B05 Structure theory of lattices 06A06 Partial orders, general Keywords:maximal sublattice; ordered sets; finite lattices PDFBibTeX XMLCite \textit{R. Freese} et al., Order 20, No. 3, 223--228 (2003; Zbl 1045.06002) Full Text: DOI