Craig, Andrew, P. K.

Dual digraphs of finite meet-distributive and modular lattices / Andrew Craig ... [et al.]. -- 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 four open problems.

In Cubo : a mathematical journal. -- Temuco : Department of mathematics and statistics of the Universidad de La Frontera, 2024. -- ISSN 0716-7776. -- ISSN 0719-0646. -- Vol. 26, no. 2 (2024), pp. 279-302