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Cartesian closedness of a category of non-frame valued complete fuzzy orders. (English) Zbl 1452.18006

Summary: Let \(H = \{0, \frac{ 1}{ 2}, 1 \}\) with the natural order and \(p \operatorname{\&} q = \max \{p + q - 1, 0 \}\) for all \(p, q \in H\). 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.

MSC:

18B35 Preorders, orders, domains and lattices (viewed as categories)
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
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