  We consider a school choice matching model where the priorities for schools
are represented by binary relations that may not be weak order. We focus on the
(total order) extensions of the binary relations. We introduce a class of
algorithms to derive one of the extensions of a binary relation and
characterize them by using the class. We show that if the binary relations are
the partial orders, then for each stable matching for the profile of the binary
relations, there is an extension for which it is also stable. Moreover, if
there are multiple stable matchings for the profile of the binary relations
that are ranked by Pareto dominance, there is an extension for which all of
those matchings are stable. We provide several applications of these results.
