Complete subobjects of fuzzy sets over $MV$-algebras

summary: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