T. Bisztriczky - A Signature Theorem for uniform oriented matroids

Let M be an uniform oriented matroid over a set E of n elements endowed with the linear order . We say that e(i) and e(i+1) are successive elements of E. The Signature Lemma of Cordovil and Duchet states that if for any circuit C, (*) successive elements of Ethat are in C have alternating signs in C, then M is alternating with respect to the linear order. In the Signature Theorem, the condition (*) is weakened.