On fuzzy complements

In [1] Bellman and Giertz showed that under reasonable restrictions the generalized operations of union and intersection for fuzzy sets have to be l.u.b, and g.l.b, in IX endowed with the usual order, where I denotes the unit interval and X is an arbitrary set. It is the purpose of this paper to characterize those operations which are acceptable as complementation. We demand that the operation would be idempotent, orderreversing and that it would be a productmap. We see that these conditions are minimal and not sufficient to guarantee the preservation of important properties of complementation with regard to sets and functions. We then define a category of fuzzy complemented spaces and give necessary and sufficient conditions for subcategories to have properties analogous to those of the category of sets and functions.