We discuss the compatibilities between approach limit structures and group structures; present some basic facts, provide several natural examples, and include some characterization theorems. Introducing the notions of approach pre-Cauchy group and approach Cauchy group, we look at the relationship of these notions with approach limit group. In particular, we consider uniformization of approach limit groups.

We introduce probabilistic limit groups under a t-norm and study their basic properties. We show that for the classes of strict t-norms, all categories of probabilistic limit groups under such t-norms are isomorphic. The same is true for nilpotent t-norms. We further show that for each probabilistic limit group under a t-norm there is a natural probabilistic uniform limit structure which has the same underlying probabilistic Cauchy structure as the probabilistic limit group.

In this paper, we focus on enriched cl-premonoid-valued topological groups, and their so-called change-of-basis lattice. In so doing, we take L as an enrichedcl-premonoid and present a category SL-NTopGrp, of stratified enriched cl-premonoid-valued neighborhood topological groups. We produce some characterization theorems, and prove that every stratified L-neighborhood topological group is uniformizable.