One of the ways to describe a matroid M(E) in Chapter 2 was by means of a closure operator cl on the set of subsets of E. This closure operator distinguishes a closed set or flat of the matroid M(E) as a set T ⊂ E with the property T = cl(T). In this chapter we want to study the collection L(M) of flats of M(E) and find out how much of the structure of M(E) is reflected in the structure of L(M).

L(M) is (partially) ordered by set-theoretic inclusion. Furthermore, there are two natural binary operations on L(M): For every pair T1, T2 of flats of M(E) define T1 ∨ T2 as the smallest flat containing both T1 and T2, and T1 ∧ T2 as the largest flat contained in both T1 and T2. [It is not difficult to see that these operations are well defined. In fact, T1 ∨ T2 = cl(T1 ∪ T2), and T1 ∧ T2 = T1 ∩ T2.]

The appropriate setting for study of this algebraic structure on L(M) is lattice theory. So we shall briefly introduce the concept of a lattice and then proceed to investigate special classes of lattices. The guiding example thereby is the lattice of subspaces of a finite-dimensional vector space. Such a lattice is, in particular, modular. Hence, we shall look at the concept of modularity in a lattice, with special emphasis on the notion of a modular pair of elements.