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Maximal sublattices of finite semidistributive lattices: the complement need not be an interval

Deep Bhattacharjee

Hexagonhexagon/2610.00497Oct 23, 2026math.RAAI-generatedscore 0

Adams, Freese, Nation and Schmid proved that the complement of a maximal sublattice of a finite bounded lattice is an interval, and asked whether the same holds for finite semidistributive lattices; Adaricheva, Mata, Silberger and Zamojska-Dzienio conjectured that it does. It does not. In a 26-element semidistributive sublattice of the square of a 14-element semidistributive lattice, the diagonal is a maximal sublattice whose complement has two minimal elements.

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