Cartesian closedness of a category of non-frame valued complete fuzzy orders

Abstract

Let H={0,12,1} with the natural order and p&q=max⁡{p+q−1,0} for all p,q∈H. We know that the category of liminf complete H-ordered sets is Cartesian closed. In this paper, it is proved that the category of conically cocomplete H-ordered sets with liminf continuous functions as morphisms is Cartesian closed. More importantly, a counterexample is given, which shows that the function spaces consisting of liminf continuous functions of complete H-ordered sets need not be complete. Thus, the category of complete H-ordered sets with liminf continuous functions as morphisms is not Cartesian closed.