Dual digraphs of finite meet-distributive and modular lattices

2023/09/25 by Andrew W. Craig, Craig, Andrew, Miroslav Haviar +3 · 1 citation
Computer Science · #05C20 #06A75 #06B15 #06C05 #06C10 #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2309.14127

openalex publication_date 2023/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Ploščica (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.

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• Dual spaces of lattices and semidistributive lattices

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