On ordering free groups

The positive cones of the left orders on a free group can be described by their finite subsets. An algorithm is given for recognizing when a finite subset of a free group lies in a positive cone. This is used to show how one can construct a sequence of finite subsets of a positive cone whose union is the positive cone. Moreover, the method gives an overview of the positive cones of a free group. It is still an open problem whether there are positive cones which can be generated as a subsemigroup by a finite subset.