Two categories $\mathrm{𝕊𝕖𝕥}\left(\Omega \right)$ and $\mathrm{𝕊𝕖𝕥𝔽}\left(\Omega \right)$ of fuzzy sets over an $MV$-algebra $\Omega$ are investigated. Full subcategories of these categories are introduced consisting of objects $\left(\mathrm{s}ub\right(A,\delta )$, $\sigma )$, where $\mathrm{s}ub(A,\delta )$ is a subset of all extensional subobjects of an object $(A,\delta )$. It is proved that all these subcategories are quasi-reflective subcategories in the corresponding categories.

The main goal of this paper is to construct fuzzy connectives on algebraic completely distributive lattice(ACDL) by means of extending fuzzy connectives on the set of completely join-prime elements or on the set of completely meet-prime elements, and discuss some properties of the new fuzzy connectives. Firstly, we present the methods to construct t-norms, t-conorms, fuzzy negations valued on ACDL and discuss whether De Morgan triple will be kept. Then we put forward two ways to extend fuzzy implications...