Partial Fields and Matroid Representation

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 ∈ P, a + b may not be defined. We develop a theory of matroid representation over partial fields. It is shown that many important classes of matroids arise as the 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. Homomorphisms of partial fields are defined. It is shown that if φ : P1 → P2 is a non-trivial partial field homomorphism, then every matroid representable over P1 is representable over P2. The connection with Dowling group geometries is examined. It is shown that if G is a finite abelian group, and r > 2, then there exists a partial field over which the rank–r Dowling group geometry is representable if and only if G has at most one element of order 2, that is, if G is a group in which the identity has at most two square roots.