On weak maps of ternary matroids (Q1266395)

Let \(M\) and \(N\) be ternary matroids on a common ground set \(E\). The identity on \(E\) is a weak map from \(M\) to \(N\) if every independent set in \(N\) is also independent in \(M\). In this case, \(N\) is called a weak-map image of \(M\). If, moreover, \(M\) and \(N\) have the same rank, \(N\) is rank-preserving weak-map image of \(M\). The main result of this paper proves that, if \(M\) is a 3-connected ternary matroid on \(E\), \(N\) is a matroid on \(E\), a rank-preserving weak image of \(M\), connected and non-binary, then \(M= N\). Examples are given to show that this result, proved first for \(N 3\)-connected and then in the general case, is the best possible.