Two categories $\mathbb{Set}(\Omega )$ and $\mathbb{SetF}(\Omega )$ of fuzzy sets over an $MV$-algebra $\Omega $ are investigated. Full subcategories of these categories are introduced consisting of objects $(\mathop {{\mathrm sub}}(A,\delta )$, $\sigma )$, where $\mathop {{\mathrm sub}}(A,\delta )$ 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.
In this paper we apply the notion of the product $MV$-algebra in accordance with the definition given by B. Riečan. We investigate the convex embeddability of an $MV$-algebra into a product $MV$-algebra. We found sufficient conditions under which any two direct product decompositions of a product $MV$-algebra have isomorphic refinements.