Extremality in semidistributive lattices

Abstract

We establish several independent results concerning extremal, left modular, congruence uniform, and semidistributive lattices. An equivalent characterization of left modular lattices is obtained in terms of edge-labellings, together with necessary and sufficient conditions on the doubling steps in the construction of congruence normal lattices that ensure left modularity or extremality. We prove that a congruence uniform lattice is shellable if and only if it is extremal. We answer a question of Barnard by constructing a counterexample showing that an induced subcomplex of a canonical join complex need not itself be such a complex. Finally, we show that the order dimension of a semidistributive extremal lattice equals the chromatic number of the complement of its Galois graph, generalizing a theorem of Dilworth for distributive lattices. As an application, we determine the dimensions of generalizations of the Hochschild lattice, of the parabolic Tamari lattice, and of some lattices of torsion classes.