Dual digraphs of finite meet-distributive and modular lattices

Andrew Craig; Miroslav Haviar; Klarise Marais

doi:10.56754/0719-0646.2602.279

Authors

Andrew Craig Department of Mathematics and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park, 2006 South Africa – National Institute for Theoretical and Computational Sciences (NITheCS), South Africa. https://orcid.org/0000-0002-4787-3760
Miroslav Haviar Department of Mathematics and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park, 2006 South Africa – Department of Mathematics, Faculty of Natural Sciences, M. Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia. https://orcid.org/0000-0002-9721-152X
Klarise Marais Department of Mathematics and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park, 2006 South Africa. https://orcid.org/0000-0003-1164-8825

DOI:

https://doi.org/10.56754/0719-0646.2602.279

Keywords:

Semimodular lattice, lower semimodular lattice, modular lattice, TiRS digraph, meet- distributive lattice, finite convex geometry

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 in this journal 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 three open problems.

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