The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. This paper considers a generalisation of these classes to the so called k-regular matroids. The main result of the paper determines the automorphisms of the algebraic structure associated with the class of k-regular matroids. This result is the first step in establishing a unique representation property for k-regular matroids.