  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.
