Weak maps and stabilizers of classes of matroids
Let \(N\) be a matroid belonging to a class \({\mathcal N}\) of matroids representable over a field \(F\), and assume that \({\mathcal N}\) is closed under minors and duals. \(N\) is an \(F\)-stabilizer of \({\mathcal N}\) if every representation of any 3-connected member of \({\mathcal N}\) is determined (up to elementary row and column operations) by any \(N\)-minor. A universal stabilizer is an \(F\)-stabilizer for every \(F\) over which the class \({\mathcal N}\) is respresentable. The authors examine the basic properties of universal stabilizers and prove the following theorem. (Note that an element is fixed (cofixed) in \(M\) if there is no elementary extension (coextension) of \(M\) by an element \(x'\) which is equivalent to \(x\) but not dependent on \(x\).) Theorem 6.1: Let \(N\) be a 3-connected matroid in a well-closed class \({\mathcal N}\) and suppose that \(| E(N)| \geq 2\). Then \(N\) is a universal stabilizer for \({\mathcal N}\) if and only if the following three conditions hold: (i) If \(M\) is a 3-connected member of \({\mathcal N}\) with an element \(x\) such that \(M \setminus x = N\), then \(x\) is fixed in \(M\). (ii) If \(M\) is a 3-connected member of \({\mathcal N}\) with an element \(y\) such that \(M/y = N\), then \(y\) is cofixed in \(M\). (iii) If \(M\) is a 3-connected member of \({\mathcal N}\) with a pair of elements \(x\) and \(y\) such that \(M \setminus x / y = N\) and \(M \setminus x\) is 3-connected, then \(x\) is fixed in \(M\).
scientific article; zbMATH DE number 1221598