Graphs and Matroids - Kaioke Begay- Finitary and Cofinitary Oriented Matroids

Abstract: Much like matroids, oriented matroids can be defined on finite ground sets using a set of circuit axioms. Oriented matroid duality is an important property which is difficult to maintain in the infinite setting. One way to define infinite oriented matroids in a way that preserves duality is using signed set orthogonality. This allows for the notion of finitary and cofinitary oriented matroids. An oriented matroid is called finitary if all of its circuits have finite support, and cofinitary if it is the dual of a finitary oriented matroid. It has recently been shown that cofinitary oriented matroids do not necessarily satisfy the circuit axioms. In the finite setting, there are many ways to define oriented matroids which are equivalent to the circuit axioms. Here, we prove that some of these definitions still hold for cofinitary oriented matroids, even when the circuit axioms fail.