Semitopological groups, semiclosure semigroups and quantales

Abstract

As a valued domain, quantales have been applied to the study of enriched category, fuzzy algebra, fuzzy topology and fuzzy domain. One purpose of this paper is to investigate semitopological groups and semiclosure semigroups from the viewpoint of quantales. We first show that a semitopological group is completely determined by the topological ideal conuclei of the power-set quantale over a group. Next, we prove that the concepts of semitopological groups and semiclosure groups are dually equivalent. Another purpose of this paper is to investigate ordered semigroups, conditionally complete quantales and quantales from the viewpoint of semiclosure semigroups. As an important result of semiclosure semigroups, we prove that the category of complete semiclosure semigroups is isomorphic to the category of quantales.