A general categorical reflection for various completions of Q-closure spaces and Q-ordered sets

In this paper, we consider a full subcategory S of the category CLQ of stratified Q-closure spaces, satisfying some conditions. We construct a reflection from the category CLQ onto S, called the S-completion, which encompasses a variety of completions for Q-closure spaces, Q-cotopological spaces, and Q-ordered sets. The notion of a fuzzy subset system Z on a category C of structured sets is introduced. By considering fuzzy subset systems Z on the category CLQ, we introduce the concept of a Z-convergence Q-closure space, as a generalization of sober/bounded sober/monotone convergence Q-cotopological spaces. We show that the category S is exactly a category of Z-convergence spaces for some fuzzy subset system Z. And, in the case that Z consists of irreducible fuzzy sets, the S-completion can be restricted to the setting of Q-cotopological spaces. The sobrification, bounded sobrification, D-completion of Q-cotopological spaces are special cases of the S-completion. After establishing the connections between fuzzy subset systems on CLQ and those on the category ORDQ of Q-ordered sets, we provide a general approach to generating various completions of Q-ordered sets from the S-completion of corresponding Q-closure spaces. Completions of Q-ordered sets with respect to different types of fuzzy directedness can be obtained directly in this useful framework.