Matroid representation over partial fields

Abstract

A partial field P is an algebraic structure that behaves very much like a field except that addition is a partial binary operation, that is, for some a,b Є P, a + b may not be defined. In the first half of this thesis we axiomatise the notion of a partial field, and then develop a theory of matroid representation over partial fields. It is shown that many important classes of matroids arise as the classes of matroids representable over a partial field. For example, the class of regular matroids is a class of matroids representable over a partial field. The matroids representable over a partial field are closed under standard matroid operations such as the taking of minors, duals, direct sums, and 2-sums. We define homomorphisms of partial fields. It is shown that if φ : P1→P2 is a non-trivial partial field homomorphism, then every matroid representable over P1 is also representable over P2. In the second half we make observations of three classes of matroids that would arise in attempting to characterise when a matroid is representable over GF(4) and other fields. Each class is representable over some partial field with the property that the class is contained in the class of matroids representable over GF(4).