Abstract
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 L with |L|≥3 contains a proper sublattice S with |S|≥|L|1/3. It is also shown that that every finite modular lattice L with |L|≥3 contains a proper sublattice S with |S|≥|L|1/2, and every finite distributive lattice L with |L|≥4 contains a proper sublattice S with |S|≥3/4|L|.
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Freese, R., Hyndman, J. & Nation, J.B. Whaley's Theorem for Finite Lattices. Order 20, 223–228 (2003). https://doi.org/10.1023/B:ORDE.0000026464.36426.09
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DOI: https://doi.org/10.1023/B:ORDE.0000026464.36426.09