Partial Fields and Matroid Representation

✍ Scribed by Charles Semple; Geoff Whittle

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 228 KB
• Volume: 17
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1996.0010

✦ Synopsis

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 g P, a q 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 : P ª P is a non-trivial partial-field homomorphism, 1 2 then every matroid representable over P is representable over P . The connec-1 2 tion 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.