Oriented matroid systems

The domination invariant has played an important part in reliability theory. While most of the work in this field has been restricted to various types of network system models, many of the results can be generalized to much wider families of systems associated with matroids. A matroid is an ordered pair (F,M), where F is a nonempty finite set and M is a collection of incomparable subsets of F , called circuits, satisfying certain closure properties. For any given matroid (F,M) where F = (E ∪ x) and x / ∈ E we can associate a reliability system with component set E and with minimal path sets P = {(M \ x) : M ∈M, x ∈M}. Previous papers have explored the relation between undirected network systems and matroids. In this paper the main focus is on directed network systems and their relation to oriented matroids. Oriented matroids are a special type of matroids where the circuits are signed sets. Using these signed sets one can e.g., obtain a set theoretic representation of the direction of the edges of a directed network system. Classical results for directed network systems include the fact that the signed domination is either +1 or −1 if the network is acyclic, and zero otherwise. It turns out that these results can be generalized to systems derived from oriented matroids. Several classes of systems for which the generalized results hold will be discussed. These include oriented versions of k-out-of-n-systems and a certain class of systems associated with matrices.