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.
Keywords:  distributive lattice; congruence-uniform lattice; canonical join complex; core label order; intersection property
Classification MSC:  06D05


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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


 
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