A Basis for the Theory of Residuated Groupoids

The main part of the theory of residuated groupoids has been technical and the results apparently diverse in nature. However P. Dubreil and R. Croisot [5], using the concepts of residuated and residual maps, were able to present the basic properties of residuals in a simple unified manner. R. Croisot [3] carried on this study. In this paper we build a theory of residuated and residual maps which serves as a unifying basis for much of the theory of residuated groupoids. First (§1, (c)) we elucidate the structure of residuated and residual maps, and so examine their interaction. This investigation enables us to interpret a closure on a partially ordered set as a residuated map (see Proposition 3, Corollary). Moreover we deduce the fundamental Inverting Theorems which enable certain conditions on residuated maps to be presented in terms of equivalent conditions on residual maps (Theorems 1, 3 and 5). Residuated groupoids are those partially ordered groupoids in which left and right multiplication by any element is a residuated map. Hence we can apply the Inverting Theorems to present certain conditions on the multiplication in terms of equivalent conditions on the residuation, and also to investigate the interaction of closures and anti-closures with the multiplication and residuation. This theory motivates and unifies results due to I. Molinaro, R. McFadden and J. Querre [7, 9, 10, 11, 12]. Moreover, the specified restrictions on the application of the Inverting Theorems explains the pattern of these results (especially those of J. Querr6 [12]). Finally (§3) we show how R. McFadden's m-congruence theory [10] provides an elegant method presenting and motivating J. Querre's non-commutative >l-nomal theory. The various generalisations of this theory will be dealt with in another paper.