Weak maps and stabilizers of classes of matroids

Let F be a field and let N be a matroid in a class N of F-representable matroids that is closed under miners and the taking of duals. Then N is an F-stabilizer for N if every representation of a 3-connected member of Ar is determined up to elementary row operations and column scaling by a representation of any one of its N-minors. The study of stabilizers was initiated by Whittle. This paper extends that study by examining certain types of stabilizers and considering the connection with weak maps. The notion of a universal stabilizer is introduced to identify the underlying matroid structure that guarantees that N will be an F'-stabilizer for Jy for every field F' over which members of JY are representable. It is shown that, just as with F-stabilizers, one can establish whether or not N is a universal stabilizer for N by an elementary finite check. If N is a universal stabilizer for N we determine additional conditions on N and N that ensure that if N is not a strict rank-preserving weak-map image of any matroid in N, then no connected matroid in N with an N-minor is a strict rank-preserving weak-map image of any 3-connected matroid in N. Applications of the theory are given for quaternary matroids. For example, it is shown that Ut, is a universal stabilizer for the class of quaternary matroids with no U-3,U-6-minor. Moreover, if M-1 and M-2 are distinct quaternary matroids with U-2,U-5-minors but no U-3,U-6-minors and M-1 is connected while M-2 is 3-connected, then M-1 is not a rank-preserving weak-map image of M-2. (C) 1998 Academic Press.