Distributive lattices have the intersection property

• Main Author: Henri Mühle
• Format: Article
• Language: English
• Published: Institute of Mathematics of the Czech Academy of Science 2021-04-01
• Series: Mathematica Bohemica
• Subjects: distributive lattice; congruence-uniform lattice; canonical join complex; core label order; intersection property
• Online Access: http://mb.math.cas.cz/full/146/1/mb146_1_2.pdf
• ISSN: 0862-7959; 2464-7136

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.