Transversal and function matroidal structures of covering-based rough sets
RSKT'11 Proceedings of the 6th international conference on Rough sets and knowledge technology
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Let E be a finite set and S be a collection of subsets of E. For each x ∈ E let Sx = {S∈S|x∈S}. Suppose we choose elements x1,...,xn in such a way that we first choose x1 belonging to some set of Sx1. For i=2,...,n we choose xi belonging to some set of Sxi\(Sx1∪...∪Sxi-1). We call the set {x1,...,xn} a sequential transversal of S, and we let TS be the set of all sequential transversals of S, which includes 0 as well. We examine conditions under which the pair (E, TS) is a matroid. We show that (E, TS) is a matroid iff TS = Tb(max(TS)) where b(max(TS)) denotes the blocker of the maximal sets of TS. It is also shown that every matroid on a set E can be defined as a pair (E, TS) where TS 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, TS) is a matroid.