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

Abstract

Let $H=\{0,\frac{1}{2},1\}$ with the natural order and $p\mathrm{\&}q=\max \{p+q-1,0\}$ 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.

Introduction

The notion of fuzzy order [28] was introduced by Zadeh in 1971. Since then, theoretical and applicational aspects of fuzzy orders have been developed rapidly. Along this line, the concept of L-ordered sets was introduced in [3]. L-ordered sets in the sense of [3] not only extend the truth-value set to be a complete residuated lattice but also modify the original antisymmetry condition. Categorically speaking, an L-ordered set is just an L-enriched category [12], [13], [23], [24].

In 1994, quantale-enriched categories were introduced to the study of domain theory [1], [8] as a unification of metric and order theoretical approach to domain theory [23], [24]. As a particular quantale-enriched category, Fan introduced the notion of L-Fuzzy poset [6] as a fuzzy approach to study quantitative domain theory. At present, many basic concepts such as ideal, continuity and Scott topology in domain theory have been extended to the framework of quantale-enriched categories [9], [10], [13], [25], [26], [27], [29].

Due to the strong application background, Cartesian closed categories of domains play an important role in domain theory. Although concepts such as ideal and continuity and some important results have been extended to the L-valued setting with L being a complete residuated lattice, the fruitful results about Cartesian closed categories in domain theory can hardly be extended to the L-valued setting with L being a complete residuated lattice. In [27], it is proved that the category of fuzzy dcpos (a fuzzy dcpo is a fuzzy poset in which every fuzzy directed set has a join [27]) is Cartesian closed. Later, it is proved that the category of fuzzy continuous lattices and the category of fuzzy algebraic lattices are Cartesian closed [15]. We note that all these results are obtained under the assumption that the truth-value lattice is a frame. Recall that a frame is just a complete Heyting algebra [11]. In the fuzzy community, one can argue that this assumption is too limited. Thus, can we extend those results to a more general setting is an interesting work. In other words, the problem is whether we can extend the truth-value lattice to be a proper complete residuated lattice. In fact, exponentiation in quantale-enriched categories have been characterized by Clementino and Hofmann in [5]. Generally speaking, quantale-enriched categories are not Cartesian closed. We know the unit interval play an important in the study of fuzzy mathematics. Let $(\mathcal{Q},\mathrm{\&})$ be the unit interval equipped with a continuous t-norm. In [14], it is proved that the category Liminf$(\mathcal{Q})$ of liminf complete fuzzy orders and liminf continuous maps is Cartesian closed if and only if & is the t-norm min.

Let H be the three-valued MV-algebra. In [14], it is proved that the category of H-ordered sets Ord(H) is Cartesian closed. As a continuation, in [16] it is further proved that the category of liminf complete H-ordered sets is Cartesian closed. These results seem to be promising to provide us a platform to construct Cartesian closed categories of quantitative domains in the framework of non-frame valued setting. Naturally, we can ask whether the category of complete H-ordered sets is Cartesian closed.

In this paper, a counterexample is given which shows that the function space of two complete H-ordered sets may not be complete. Thus, perhaps disappointing, we obtain that the category of complete H-ordered sets is not Cartesian closed. Additionally, we show that the category of conically cocomplete H-ordered sets is Cartesian closed.