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We introduce and investigate the resolvent order, which is a binary relationon the set of firmly nonexpansive mappings. It unifies well-known ordersintroduced by Loewner (for positive semidefinite matrices) and by Zarantonello(for projectors onto convex cones). A connection with Moreau's order of convexfunctions is also presented. We also construct partial orders on (quotient setsof) proximal mappings and convex functions. Various examples illustrate ourresults.

The resolvent order: a unification of the orders by Zarantonello, by Loewner, and by Moreau