On a Conjecture Concerning Dyadic Oriented Matroids

A rational matrix is totally dyadic if all of its nonzero subdeterminants are in f\Sigma2 k : k 2 Zg. An oriented matriod is dyadic if it has a totally dyadic representation A. A dyadic oriented matriod is dyadic of order k if it has a totally dyadic representation A with full row rank and with the property that for each pair of adjacent bases A 1 and A 2 2 \Gammak fi fi fi fi det(A 1 ) det(A 2 ) fi fi fi fi 2 k : In this note we present a counterexample to a conjecture on the relationship between the order of a dyadic oriented matroid and the ratio of agreement to disagreement in sign of its signed circuits and cocircuits (Conjecture 5.2, Lee (1990)). A rational matrix is totally dyadic if all of its nonzero subdeterminants are in f\Sigma2 k : k 2 Zg. An oriented matriod is dyadic if it has a totally dyadic representation A. A dyadic oriented matriod is dyadic of order k if it has a totally dyadic representation A with full row rank and with the property that for each ..

A Characterization of the Orientations of Ternary Matroids

A matroid or oriented matroid is dyadic if it has a rational representation with all nonzero subdeterminants in f\Sigma2 k : k 2 Zg. Our main theorem is that an oriented matroid is dyadic if and only if the underlying matroid is ternary. A consequence of our theorem is the recent result of G. Whittle that a rational matroid is dyadic if and only if it is ternary. Along the way, we establish that each whirl has three inequivalent orientations. Furthermore, except for the rank-3 whirl, no pair of these are isomorphically equivalent. A rational matrix is totally dyadic if all of its nonzero subdeterminants are in D := f\Sigma2 k : k 2 Zg. A matroid or oriented matroid is dyadic if it can be represented over Q by a totally--dyadic matrix. It is easy to see that dyadic matroids are ternary, since elements of D map to nonzeros of GF(3) when viewed modulo 3 (see Lee (1990), for example). Hence the matroids that underlie dyadic oriented-matroids are ternary. Our main result is the "if" pa..