  In this article we investigate the relations between three classes of
lattices each extending the class of distributive lattices in a different way.
In particular, we consider join-semidistributive, join-extremal and
left-modular lattices, respectively. Our main motivation is a recent result by
Thomas and Williams proving that every semidistributive, extremal lattice is
left modular. We prove the converse of this on a slightly more general level.
Our main result asserts that every join-semidistributive, left-modular lattice
is join extremal. We also relate these properties to the topological notion of
lexicographic shellability.
