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Partial Fields and Matroid Representation

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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 varphi: P 1 -< P 2 is a non-trivial partial-field homomorphism, then every matroid representable over P 1 is representable over P 2 . 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.

Document Type: Research Article

Affiliations: Department of Mathematics, Victoria University, Wellington, New Zealand

Publication date: June 1, 1996

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