Dual digraphs of finite meet-distributive and modular lattices

TL;DR

This paper characterizes dual digraph representations of finite lattices with meet-distributivity and modularity, linking these properties to convex geometries and providing conditions for modularity in TiRS digraphs.

Contribution

It introduces a dual digraph framework for finite meet-distributive and modular lattices, extending previous dual representations and connecting to convex geometries.

Findings

Dual digraphs of finite lattices can describe meet-distributivity and modularity.

Properties of lattices have simple descriptions in dual digraphs.

Conditions on TiRS digraphs imply modularity of the dual lattice.

Abstract

We describe the digraphs that are dual representations of finite lattices satisfying conditions related to meet-distributivity and modularity. This is done using the dual digraph representation of finite lattices by Craig, Gouveia and Haviar (2015). These digraphs, known as TiRS digraphs, have their origins in the dual representations of lattices by Urquhart (1978) and Plo\v{s}\v{c}ica (1995). We describe two properties of finite lattices which are weakenings of (upper) semimodularity and lower semimodularity respectively, and then show how these properties have a simple description in the dual digraphs. Combined with previous work on dual digraphs of semidistributive lattices (2022), it leads to a dual representation of finite meet-distributive lattices. This provides a natural link to finite convex geometries. In addition, we present two sufficient conditions on a finite TiRS digraph for its dual lattice to be modular. We close by posing four open problems.