LAUSR

Abstract We demonstrate a one-to-one correspondence between idempotent closure operators and the so-called saturated quasi-uniform structures on a category C . Not only this result allows to obtain a categorical counterpart P of the Csaszar-Pervin quasi-uniformity P , that we characterize as a transitive quasi-uniformity compatible with an idempotent interior operator, but also permits to describe those topogenous orders that are induced by a transitive quasi-uniformity on C . The categorical counterpart P ⁎ of P − 1 is characterized as a transitive quasi-uniformity compatible with an idempotent closure operator. When applied to other categories outside topology P allows, among other things, to generate a family of idempotent closure operators on Grp, the category of groups and group homomorphisms, determined by the normal closure.