Meet-Distributive Lattices have the Intersection Property

TL;DR

This paper characterizes finite meet-distributive lattices using the core label order and canonical join complex, revealing that the core label order always forms a meet-semilattice, thus deepening understanding of their structure.

Contribution

It provides a new characterization of finite meet-distributive lattices through their core label order and canonical join complex, highlighting their structural properties.

Findings

Core label order of finite meet-distributive lattices is always a meet-semilattice.

Characterization of these lattices via secondary structures.

Insight into the relationship between closure operators and lattice properties.

Abstract

Meet-distributive lattices form an intriguing class of lattices, because they are precisely the lattices obtainable from a closure operator with the so-called anti-exchange property. Moreover, meet-distributive lattices are join semidistributive. Therefore, they admit two natural, secondary structures: the core label order is an alternative order on the lattice elements and the canonical join complex is the flag-simplicial complex on canonical join representations. In this article we present a characterization of finite meet-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 meet-distributive lattice is always a meet-semilattice.