A subobjects structure of the category $\Omega $- of $\Omega $-fuzzy sets over a complete $MV$-algebra $\Omega =(L,\wedge ,\vee ,\otimes ,\rightarrow )$ is investigated, where an $\Omega $-fuzzy set is a pair ${\mathbf A}=(A,\delta )$ such that $A$ is a set and $\delta \:A\times A\rightarrow \Omega $ is a special map. Special subobjects (called complete) of an $\Omega $-fuzzy set ${\mathbf A}$ which can be identified with some characteristic morphisms ${\mathbf A}\rightarrow \Omega ^*=(L\times L,\mu )$ are then investigated. It is proved that some truth-valued morphisms $\lnot _{\Omega }\:\Omega ^*\rightarrow \Omega ^*,\cap _{\Omega }$, $\cup _{\Omega } \:\Omega ^*\times \Omega ^*\rightarrow \Omega ^*$ are characteristic morphisms of complete subobjects.
A method is presented making it possible to construct $po$-groups with a strong theory of quasi-divisors of finite character and with some prescribed properties as subgroups of restricted Hahn groups $H(\Delta ,\mathbb{Z})$, where $\Delta $ are finitely atomic root systems. Some examples of these constructions are presented.
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.
For an order embedding $G\overset{h}{\rightarrow }{\rightarrow }\Gamma $ of a partly ordered group $G$ into an $l$-group $\Gamma $ a topology $\mathcal T_{\widehat{W}}$ is introduced on $\Gamma $ which is defined by a family of valuations $W$ on $G$. Some density properties of sets $h(G)$, $h(X_t)$ and $(h(X_t)\setminus \lbrace h(g_1),\dots ,h(g_n)\rbrace )$ ($X_t$ being $t$-ideals in $G$) in the topological space $(\Gamma ,\mathcal T_{\widehat{W}})$ are then investigated, each of them being equivalent to the statement that $h$ is a strong theory of quasi-divisors.