Defining matroids through sequential selection

Let E be a finite set and S be a collection of subsets of E . For each x ∈ E let S x = { S ∈ S ∣ x ∈ S } . Suppose we choose elements x 1 , … , x n in such a way that we first choose x 1 belonging to some set of S x 1 . For i = 2 , … , n we choose x i belonging to some set of S x i ∖ ( S x 1 ∪ ⋯ ∪ S x i − 1 ) . We call the set { x 1 , … , x n } a sequential transversal of S , and we let T S be the set of all sequential transversals of S , which includes 0̸ as well. We examine conditions under which the pair ( E , T S ) is a matroid. We show that ( E , T S ) is a matroid iff T S = T b ( max ( T S ) ) where b ( max ( T S ) ) denotes the blocker of the maximal sets of T S . It is also shown that every matroid on a set E can be defined as a pair ( E , T S ) where T S is order-independent; that is, the elements in any sequential transversal can be picked in any order. Various conditions and examples are provided in which ( E , T S ) is a matroid.