On the completeness of localic groups via generators and relations

Abstract In the late ‘90s, Banaschewski and Vermeulen showed that any localic group is complete in its two-sided uniformity. In this paper, we provide a method of constructing extensions of localic groups. The raison d’être of this paper, however, is to show using generators and relations that if the left and the right uniformity coincide, then the localic group must be complete.