Quasi-uniform structures determined by closure operators

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 Császár-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.