∆-matroid and Jump System

In [7, 10], the concept of pseudomatroid was developed as a proper generalization of the concept of matroid. The same concept was independently developed as ∆-matroid in [4, 5]. Throughout the paper, we use the more popular name ∆-matroid for this structure. In [6], the concept of ∆-matroid was further generalized to jump system. Further interesting results on jump system are reported in [1, 3, 12, 14, 15]. In this paper, we show that jump systems are, in some sense, equivalent to ∆-matroids. Using this equivalence and the ∆-matroid theory, we give simple proofs and extensions of many of the results on jump systems in [3, 6, 12]. In Section 2, we introduce notations and basic definitions. In Section 3, we give known and some new results on ∆-matroid. In Section 4, we prove equivalence between jump systems and ∆-matroids. We use this equivalence to give simple proofs of some of the known results on jump systems in Section 5.