Secondary basis unique augmentation matroids and union minimal matroids

摘要

By the augmentation axiom of matroids, given two independent sets whose cardinalities are not equal, there exists some other independent set of which the independent set with smaller cardinality is a proper subset. The augmentation axiom requires only the existence of independent set augmentation, not its uniqueness. This causes that the collections of bases of some matroids can be reduced while the unions of the collections of bases of these matroids remain unchanged. We may say these matroids are not minimal. This paper studies a type of matroids whose augmentations of independent sets have some degree of uniqueness and whose collections of bases are minimal in a sense. First, we propose the concept of secondary basis unique augmentation matroid, and prove a matroid is a secondary basis unique augmentation matroid iff the collection of the circuits of its dual matroid is a partition. Then we propose the concept of union minimal matroid based on rank-preserving weak-map, and prove that secondary basis unique augmentation matroids are union minimal matroids. Finally, we propose the concept of basis unique exchange matroid and study its properties.