Matroids enter combinatorial optimization problems at various levels. Whitney's (1935) motivation to introduce matroids as combinatorial objects in their own right stemmed from his interest in approaching the Four-Color Problem algebraically and combining the combinatorial and algebraic-geometric aspects of graphs into the notion of a matroid.

Graphs furnish the most important models for combinatorial optimization problems. Thus it is natural to ask to what extent graph properties actually are properties of the underlying matroid and to study more general classes of matroids that enjoy, for example, the ‘max-flow-min-cut’ property of network flows (cf. Seymour 1977). This approach leads to fundamental structural questions about matroids per se which, nevertheless, have many practical implications. One of the foremost results in this area is Seymour's (1980) decomposition theory for regular matroids exhibiting regular matroids as being essentially built up by graphic and cographic matroids. As a consequence, efficient procedures can be developed to test whether a matrix is totally unimodular or whether certain linear programs actually are (better tractable) network problems (see, e.g., Welsh 1982 and Bixby 1982 for an introduction into this aspect of matroid theory).

Matroids also compose the combinatorial structure of linear programming (Minty 1966, Rockafellar 1969). Indeed, pivoting in linear programming may be carried out purely ‘combinatorially’ (Bland 1977).

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.