Common ordering extensions

This article provides necessary and sufficient conditions for a collection of binary relations to have a common ordering extension. We also characterize the quasi-ordering that is obtained by taking the intersection over all these ordering extensions. Next, we consider the special case where the collection contains only two relations. In this special case, our necessary and sufficient conditions can be reformulated to include solely binary relations that are defined on a certain subset of the universal domain. The usefullness of our results are illustrated with several examples and we relate our findings to the results in the literature.