Mathematica Bohemica, Vol. 146, No. 1, pp. 7-17, 2021
Distributive lattices have the intersection property
Henri Mühle
Received November 27, 2018. Published online December 9, 2019.
Abstract: Distributive lattices form an important, well-behaved class of lattices. They are instances of two larger classes of lattices: congruence-uniform and semidistributive lattices. Congruence-uniform lattices allow for a remarkable second order of their elements: the core label order; semidistributive lattices naturally possess an associated flag simplicial complex: the canonical join complex. In this article we present a characterization of finite distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite distributive lattice is always a meet-semilattice.
Affiliations: Author's address: Henri Mühle, Technische Universität Dresden, Institut für Algebra, Zellescher Weg 12-14, 01069 Dresden, Germany, e-mail: henri.muehle@tu-dresden.de