Closures of fuzzy linguistic truth value with regards to the extension principle

We studied non-convex or sub-normal linguistic truth values that has not been considered so far. The logical operations based on the extension principle are closed into V/sub N/, V/sub C/ and V/sub R/ and arbitrary intersection of any of these is also closed. There are infinite number of subsets outside of the convex truth value, that are classified by the ranks. We show that a set of linguistic truth values of a common rank is not closed.