Meet-Distributive Lattices have the Intersection Property
arXiv:1810.01528 · doi:10.21136/MB.2022.0072-21
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.
8 pages, 4 figures. I have discovered an error in the proof of Proposition 3.5 of version 3. This has the effect that the main result extends to a larger class of lattices and the proof is much simpler. Comments are very welcome