Noncommutative lattices

Lattices appear throughout mathematics: from logics and set theory, to algebra (congruence lattice and lattice of subvarieties), discrete mathematics (face lattice of a polytope and matroids) and topology. Noncommutative lattices can be studied either as generalizations of lattices or as double semigroups of idempotents. We shall describe both approaches and present results that reveal an interesting geometrical structure that allows several combinatorial observations.