Residuated mappings
Jean-Claude Derderian · Pacific Journal of Mathematics · 1967
In a series of papers, D. J. Foulis developed a theory in the course of which he obtained analogues of Yon 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. Nδbeling.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.Preliminaries.Unless further restricted, P, Q, R denote arbitrary posets whose order relations are all written <;.DEFINITION 2. If 9? e S(P, Q) satisfies any of the conditions of the proposition it is said to be range-closed.The set of all such