Abstract Let H = { 0 , 1 2 , 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.

中文翻译：

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一类非帧值完全模糊阶的笛卡尔封闭性

摘要 令 H = { 0 , 1 2 , 1 } 具有自然顺序且 p & q = max ⁡ { p + q − 1 , 0 } 对所有 p , q ∈ H 。我们知道 liminf 完全 H 序集的范畴是笛卡尔闭的。本文证明了以liminf连续函数为态射的圆锥共完备H序集的范畴是笛卡尔闭的。更重要的是，给出了一个反例，表明由完全H序集的liminf连续函数组成的函数空间不需要是完备的。因此，具有 liminf 连续函数作为态射的完全 H 序集的范畴不是笛卡尔封闭的。