Lattices of Cyclic Flats of Matroids: from Finite Rank to Finitary and Beyond

Abstract

In the finite case, there are close connections between lattices and matroids. More notably, every lattice can be realised as the lattice of cyclic flats of some matroid. The main concern of this paper is finding analogous connections in the infinite case. In particular, we introduce a class of set systems (which forms a superclass of matroids) whose lattices of cyclic flats model all (possibly infinite) complete lattices.