In a series of papers, D. J.
Foulis developed a theory in the course of which he obtained analogues of Von
Neumann’s Coordinatization Theorem by making use of ∗-monotone mappings. A
generalization of these mappings, residuated mappings, leads to extensions of
his results. Residuated mappings also arise independently in studies of R.
Croisot and G. Nöbeling. The purpose of this paper is to develop their
properties systematically. Of particular help is the link established with
the basic properties of M-homomorphisms between groups with operators
yielding analogues of the Fundamental Theorem of Homomorphisms and
Fitting’s Lemma, and with the study of residuation especially in Noetherian
rings.
