Binary matroids play an important theoretical role, partly because they were the first class of coordinatizable matroids to be completely characterized, but also because the class of binary matroids contains the unimodular matroids and the graphic matroids, two classes fundamental to matroid theory. There are numerous characterizations of binary matroids, very different in nature, and expressive of the richness of the concept.

Definition and Basic Properties

Definition. A matroid is binary if it is representable (coordinatizable) over the two-element field GF(2).

According to the general definition of representable matroids, (see Chapter 1), a matroid M(E) on a finite set E is binary if there is a mapping α of E into a GF(2)-vector space V such that a subset X ⊆ E is independent in M(E) if and only if the restriction of α to X is injective and the set {α(x)|x ∈ X} of vectors in V is linearly independent. The mapping a is then called a binary representation of the matroid M(E).

Example. Denote by Ur,n up to isomorphism, the matroid on a set of n elements, in which the bases are those subsets which have r elements. Then U2,3 is binary. (This matroid is identified with the projective line over the field GF(2).) On the other hand, U2,4. is not binary; this matroid, which consists of four geometric points on a line, is a typical non-binary matroid, and serves to characterize the binary matroids, as we shall see later.