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Cartesian closedness of a category of non-frame valued complete fuzzy orders
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ScienceDirect Fuzzy Sets and Systems 390 (2020) 96–104 www.elsevier.com/locate/fss
Cartesian closedness of a category of non-frame valued complete fuzzy orders Min Liu School of Sciences, Chang’an University, 710064, Xi’an, PR China Received 11 July 2019; received in revised form 7 February 2020; accepted 11 February 2020 Available online 14 February 2020
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. © 2020 Elsevier B.V. All rights reserved. Keywords: Fuzzy relations; L-order; Liminf complete L-ordered set; Complete L-ordered set; Cartesian closed category
1. 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 [9,10,13,25–27,29]. 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 Lvalued 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 E-mail address: [email protected]. https://doi.org/10.1016/j.fss.2020.02.004 0165-0114/© 2020 Elsevier B.V. All rights reserved.
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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 (Q, &) be the unit interval equipped with a continuous t-norm. In [14], it is proved that the category Liminf(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. 2. L-ordered sets In this paper, (L, ∧, ∨, ∗, →, 0, 1) always denotes a complete residuated lattice [3]. For concepts and results in domain theory and category theory, we refer to [2,8,12]. Definition 2.1 (Wagner [23], Fan [6], Bˇelohlávek [3]). An L-ordered set is a pair (A, e) such that A is a set and e : A × A −→ L is a map, called an L-order, that satisfies for every x, y, z ∈ A, (E1) e(x, x) = 1 (reflexivity); (E2) e(x, y) ∗ e(y, z) ≤ e(x, z) (transitivity); (E3) e(x, y) = e(y, z) = 1 implies x = y (antisymmetry). A complete residuated lattice is a commutative unital quantale [18]. Thus, an L-ordered set in the above sense is a quantale-enriched category [23,24,13]. Further more, by the fact that a commutative unital quantale is a quantaloid [19], a quantale-enriched category in the sense of [24] can be viewed as a quantaloid-enriched category [19]. For sake of simplicity, we often write A for an L-ordered set (A, e) and write A(x, y) for e(x, y) if no confusion would arise. A map f : A −→ B between two L-ordered sets is called L-order preserving (or order preserving for short) if A(x1 , x2 ) ≤ B(f (x1 ), f (x2 )) for all x1 , x2 ∈ A. Let (A, e) be an L-ordered set. The L-order e induces a partial order ≤e on A defined by x ≤e y iff e(x, y) = 1. We will denote ≤e simply by ≤, denote (A, ≤e ) by A0 and denote the join and meet operations in A0 by ∨ and ∧. Example 2.2. (1) Let {} be a single point set. Define e(, ) = 1. Then ({}, e) is an H -ordered set. ({}, e) is the terminal object in the category of L-ordered sets. (2) (Sub-L-ordered set) Let A be an L-ordered set and B ⊆ A. Let B(x, y) = A(x, y), for x, y ∈ B. Then B is an L-ordered set. (3) (Product L-ordered set) Let A, B be L-ordered sets. For (x1 , y1 ), (x2 , y2 ) ∈ A × B, let (A × B)((x1 , y1 ), (x2 , y2 )) = A(x1 , x2 ) ∧ B(y1 , y2 ). Then A × B is an L-ordered set.
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Definition 2.3 (Stubbe [20,21], Lai and Zhang [13]). Let A be an L-ordered set, μ ∈ LA , {xi } ⊆ A, x ∈ A, α ∈ L. (i) An element a ∈ A is called a join of μ, in symbols a = sup μ, if A(a, y) = x∈X μ(x) → A(x, y) for all y ∈ A. (ii) An element a ∈ A is called a conical colimit of {xi }, in symbols a = conicolimiti∈I xi , if A(a, y) = i A(xi , y) for all y ∈ A. (iii) An element a ∈ A is called a tensor of x and α, in symbols a = α ⊗ x, if A(a, y) = α → A(x, y) for all y ∈ A. An L-ordered set A is said to be complete if sup μ exits for all μ ∈ LA ; A is said to be conically cocomplete if conicolimiti∈I xi exists for all {xi } ⊆ A; A is said to be tensored if α ⊗ x exists for all α ∈ L, x ∈ A; A is said to be order complete if A0 is complete. Theorem 2.4 (Stubbe [20,21]). Let A be an L-ordered set. (1) A is complete if and only if A is tensored and conically cocomplete. (2) If the conical colimit of a family of elements {xi } ⊆ A exists, then conicolimiti∈I xi = i xi . (3) If A is conically cocomplete, then A is order complete. Recall that a net {xλ }λ∈D in an L-ordered set A is forward Cauchy [14], if 1= A(xμ , xσ ). λ∈D λ≤μ≤σ
A forward Cauchy net {xλ }λ∈D in an L-ordered set A converges to a ∈ A, or a is a liminf of {xλ }λ∈D , if for all x ∈ A, A(a, x) = λ∈D λ≤μ A(xμ , x). We denote the liminf of {xλ }λ∈D by lim infλ∈D xλ if it exists. An L-ordered set A is liminf complete [14] if each forward Cauchy net has a liminf. An order preserving function f : A → B is liminf continuous [14] if it preserves the liminf of forward Cauchy nets in A. Denote the set of all liminf continuous functions from A to B by [A → B]. Define P : [A → B] × [A → B] −→ L as follows: P (f, g) = {p ∈ L|∀x, y ∈ A, p ∧ A(x, y) ≤ B(f (x), g(y))}. Suppose (L, ∧) is a frame. Then P is an L-order on [A → B] (see [5]). Denote the L-ordered set ([A → B], P ) by [A → B]P . Define ev : [A → B]P × A −→ B by ev(f, x) = f (x). Then ev is order preserving (see [5]). 3. Cartesian closedness of the category of complete H -ordered sets The complete residuated lattice considered in this note is the MV-algebra H with three elements. Precisely, H = {0, 12 , 1} and p&q = max{p + q − 1, 0} for all p, q ∈ H . For more information about H , we refer to [4,7,17]. Clearly, (H, ∧) is a frame [11]. We note that connectives on three values and their relations are extensively exposed in [4]. By Table 1 in [4], we know & is the unique conjunction on H such that (H, &) is a complete residuated lattice and & = ∧. &
∧
→− , respectively. Let Abe an H -ordered set The right adjoint for p&− and p∧− will be denoted by p →− and p and {xλ }λ∈D a monotone net in the underlying ordered set A0 . Then λ∈D λ≤μ A(xμ , x) = i∈D A(xi , x) for all x ∈ A [16]. This shows that lim infλ∈D xλ exists if and only if the conical colimit of {xλ }λ∈D exists, in such case they are equal. Let Liminf(H ) denote the category of liminf complete H -ordered sets and liminf continuous functions. In [16], it is proved that Liminf(H ) is Cartesian closed. The exponentiation B A of objects A, B is the function space [A → B]P . Proposition 3.1. If A is a conically cocomplete H -ordered set, then A is liminf complete. Lemma 3.2 ([16]). Let A, B be two H -ordered sets, f : A → B an order preserving map. Then (1) A is liminf complete if and only if every monotone net in A0 converges in A. (2) If A is liminf complete then f is liminf continuous if and only if f preserves liminfs of monotone nets if and only if f : A0 → B0 is Scott continuous.
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Lemma 3.3. Let A be an H -ordered set, B a conically cocomplete H -ordered set. Then [A → B]P is conically cocomplete. Moreover, the conical colimit f of {fi }i∈I ⊆ [A → B]P is given by f (x) = i∈I fi (x) for every x ∈ A. Proof. Let {fi }i∈I ⊆ [A → B]P , and assume that f (x) = net in A0 , y ∈ B. Then B(f (lim inf xλ ), y) = B( fi (lim inf xλ ), y) λ∈D
=
(By conically cocompleteness of B)
B(lim inf fi (xλ ), y)
(By liminf continuity of fi )
λ∈D
λ∈D
i∈I
=
fi (x) for every x ∈ A. Let {xλ }λ∈D be a monotone
B(fi (lim inf xλ ), y)
i∈I
=
i∈I
λ∈D
i∈I
B(fi (xλ ), y)
(By monotonicity of {fi (xλ )}λ∈D )
B( fi (xλ ), y)
(By conically cocompleteness of B)
i∈I λ∈D
=
λ∈D
=
i∈I
B(f (xλ ), y)
λ∈D
Thus, f (lim infλ∈D xλ ) = lim infλ∈D f (xλ ), i.e., f is liminf continuous. For every g ∈ [A → B]P , we have ∧ P (f, g) = A(x, y) → B(f (x), g(y)) x,y∈A
=
∧ A(x, y) → B( fi (x), g(y))
x,y∈A
=
i∈I
∧
A(x, y) →
x,y∈A
=
B(fi (x), g(y))
i∈I ∧
A(x, y) → B(fi (x), g(y))
x,y∈A i∈I
=
P (fi , g).
i∈I
Thus, f is the conical colimit of {fi }i∈I , i.e., [A → B]P is conically cocomplete. Lemma 3.4. If A and B are conically cocomplete H -ordered sets, then so is A × B. Proof. Let {(xi , yi )}i∈I ⊆ A × B. Define (x, y) = ( i∈I xi , i∈I yi ). Then, for every (a, b) ∈ A × B, we have A × B((xi , yi ), (a, b)) = (A(xi , a) ∧ B(yi , b)) i∈I
i∈I
=
A(xi , a) ∧
i∈I
B(yi , b)
i∈I
= A( xi , a) ∧ B( yi , b) i∈I
i∈I
= A(x, b) ∧ B(y, b) = A × B((x, y), (a, b)). Thus, (x, y) is the conical colimit of {(xi , yi )}i∈I , i.e., A × B is conically cocomplete.
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Let CSupliminf(H ) denotes the full subcategory of Liminf(H ) consisting of conically cocomplete H -ordered sets. By Lemma 3.3, Lemma 3.4 and the Cartesian closedness of Liminf(H ), we can obtain the following result: Theorem 3.5. CSupliminf(H ) is Cartesian closed. Let Supliminf(H ) denotes the full subcategory of Liminf(H ) consisting of complete H -ordered sets. Lemma 3.6. Let A, B be two complete H -ordered sets. (i) The single point H -ordered set {} is the terminal object of Supliminf(H ). (ii) The product A × B of two complete H -ordered set is complete. Thus, A × B is the categorical product of A, B in Supliminf(H ). (iii) If the exponentiation B A of objects A, B exists, then B A is the function space [A → B]P . Proof. (i) is trivial. (ii) By Theorem 2.4 and Lemma 3.4, we only need to show that A × B is tensored. Let (x, y) ∈ A × B, α ∈ L. Then, for every (a, b) ∈ A × B, we have &
&
α → A × B((x, y), (a, b)) = α → (A(x, a) ∧ B(y, b)) &
&
= (α → A(x, a)) ∧ (α → B(y, b)) = A(α ⊗ x, a) ∧ B(α ⊗ y, b) = A × B((α ⊗ x, α ⊗ y), (a, b)). Thus, the tensor of α and (x, y) exists and equal to (α ⊗ x, α ⊗ y). (iii) The proof mimics the techniques used by Smyth (see [22] Lemma 5).
Example 3.7. Let A = {ai : i ∈ N} ∪ {a, b}, where ai , a, b are mutually distinct, N is the set of natural numbers. Define an H -valued relation e : A × A → H as follows (Table 1): Table 1 The H -valued relation on A. e
a1
a2
a3
···
a
b
a1
1
1
1
···
1
1
a2
1
1
···
1
1
a3 .. .
1 2 1 2
1 2
···
.. .
.. .
1 .. .
1 .. .
1 .. .
a
1 2
1 2
1 2
···
1
1
b
0
0
0
···
1 2
1
Then (A, e) is a complete H -ordered set, but [A → A]P is not complete. Proof. Step (1): We claim that (A, e) is a complete H -ordered set. (a) It is routine to check that e is an H -order. (b) A is tensored: We can check that ∀x ∈ A, 0 ⊗ x = 0, 1 ⊗ x = 1. By the fact that 1 & 1, y = a or b; → e(b, y) = 1 2 2 , y = ai , i ∈ N, we have
1 & 2 → e(b, y) = e(a, y)
for all y ∈ A. Thus,
1 2
⊗ b = a. Similarly,
1 2
⊗ a = a1 ,
1 2
⊗ ai = a1 for all i ∈ N.
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(c) A is conically cocomplete: Let {xi : i ∈ I } ⊆ A. If {xi : i ∈ I } has a maximal element x, then i∈I e(xi , y) = e(x, y) for all y ∈ A. In this case, the conical colimit of {xi : i ∈ I } is x. If {xi : i ∈ I } does not have a maximal element, then {x i : i ∈ I } is cofinal in {ai : i ∈ N}, that is {xi : i ∈ I } ⊆ {ai : i ∈ N} and ∀i ∈ N, ∃j ∈ I such that xj ≥ ai . In this case, i∈I xi = a, 1, y = a or b; e(xi , y) = 1 2 , y = ai , i ∈ N. i∈I
Thus, i∈I e(xi , y) = e(a, y) for every y ∈ A. In this case, the conical colimit of {xi : i ∈ I } is a. So we can conclude that A is conically cocomplete. By (a), (b) and (c), we deduce that (A, e) is a complete H -ordered set. Step (2): We shall characterize all liminf continuous functions from A to A. Suppose f : A0 → A0 is monotone. Then the equality e(x, y) ≤ e(f (x), f (y)) holds for x, y ∈ A with e(f (x), f (y)) = 1 or e(f (x), f (y)) = 12 . Thus, f : A → A is H -order preserving if and only if (1) f : A0 → A0 is monotone; and (2) e(x, y) ≤ e(f (x), f (y)) for x, y ∈ A with e(f (x), f (y)) = 0. Suppose f : A0 → A0 is monotone. Then f : A0 → A0 is Scott continuous if and only if f ( i∈I xi ) = i∈I f (xi ) for any {xi : i ∈ I } ⊆ {ai : i ∈ N} which is cofinal in {ai : i ∈ N} if and only if f ( i∈N ai ) = i∈N f (ai ). By Lemma 3.2, we know that a map f : A → A is liminf continuous if and only if (1)f : A0 → A0 is monotone; (2) e(x, y) ≤ e(f (x), f (y)) for x, y ∈ A with e(f (x), f (y)) = 0 and (3) f ( i∈N ai ) = i∈N f (ai ). Now suppose that f : A → A is liminf continuous. Let F be the set of all functions from A to A. Case (1): f (a) = b. a ≤ b implies that ≤ f (b). Thus, f (b) = b. b = f (a) b = f (a) = f ( i∈N ai ) = i∈N f (ai ) implies that there is a smallest i0 ∈ N such that f (ai0 ) = b. If i0 = 1, then f is the constant function valued at b. Let I = {f ∈ F |f (x) = b, ∀x ∈ A}. Suppose i0 = 1. Then, f (aj ) = b for all j ∈ N with j ≥ i0 , and f (a) = f (b) = b. For every i ∈ N with i < i0 , we have f (ai ) < b and 12 = e(ai0 , ai ) ≤ e(f (ai0 ), f (ai )) = e(b, f (ai )). Thus, f (ai ) = a for every i ∈ N with i < i0 . Let II = {f ∈ F |f (a) = f (b) = b, ∃i0 ∈ N, i0 = 1, s.t. f (ai ) = a for all i ∈ N with i < i0 , f (aj ) = b for all j ∈ N with j ≥ i0 }. Case (2): f (a) = a. a = f (a) = f ( i∈N ai ) = i∈N f (ai ) implies that there is a smallest i0 ∈ N such that f (ai0 ) = a or {f (ai )}i∈N is cofinal in {ai }i∈N . Case (2)-1: f (b) = b. By the value of f (ai ), we have three situations. Case (2)-1-1: i0 = 1. Then f (ai ) = a for every i ∈ N. Let III = {f ∈ F |f (b) = b, f (a) = a, f (ai ) = a for every i ∈ N}. Case (2)-1-2: i0 = 1. By monotonicity of f , f (ai ) = a for every i ∈ N with i ≥ i0 , and ∀i, j ∈ N with i ≤ j < i0 , ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj . Let IV = {f ∈ F |f (b) = b, f (a) = a, ∃i0 ∈ N, i0 = 1, s.t. f (ai ) = a for every i ∈ N with i ≥ i0 , ∀i, j ∈ N with i ≤ j < i0 , ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj }. Case (2)-1-3: {f (ai )}i∈N is cofinal in {ai }i∈N . Let V = {f ∈ F |f (b) = b, f (a) = a, ∀i, j ∈ N with i ≤ j, ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj and {f (ai )}i∈N is cofinal in {ai }i∈N }. Case (2)-2: f (b) = a. Similar to Case (2)-1, f must belong to one of the following three sets.
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Let VI = {f ∈ F |f (x) = a for every x ∈ A}. Let VII = {f ∈ F |f (b) = f (a) = a, ∃i0 ∈ N, i0 = 1, s.t. f (ai ) = a for every i ∈ N with i ≥ i0 , ∀i, j ∈ N with i ≤ j < i0 , ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj }. Let VIII = {f ∈ F |f (b) = f (a) = a, ∀i, j ∈ N with i ≤ j, ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj , {f (ai )}i∈N is cofinal in {ai }i∈N }. Case (3): f (a) = ai0 for some i0 ∈ N. By the inequality 21 = e(b, a) ≤ e(f (b), f (a)) = e(f (b), ai0 ), we deduce that f (b) = b. Clearly, f (b) ≥ f (a) = ai 0 . By the requirement that ai0 = f (a) = f ( i∈N ai ) = i∈N f (ai ), there exists a smallest j0 ∈ N such that f (aj0 ) = ai0 . By the monotonicity of f , we know ∀i ≥ j0 , f (ai ) = ai0 . Case (3)-1: f (b) = a. Let IX = {f ∈ F |f (b) = a, ∃i0 , j0 ∈ N, s.t. f (a) = ai0 , ∀i ∈ N with i ≥ j0 , f (ai ) = ai0 , ∀i, j ∈ N with i ≤ j ≤ j0 , ∃i , j ∈ N with i ≤ j ≤ i0 , s.t. f (ai ) = ai , f (aj ) = aj }. Case (3)-2: f (b) = ak for some k ≥ i0 . Let X = {f ∈ F |∃k, i0 , j0 ∈ N with k ≥ i0 , s.t. f (b) = ak , f (a) = ai0 , ∀i ∈ N with i ≥ j0 , f (ai ) = ai0 , ∀i, j ∈ N with i ≤ j ≤ j0 , ∃i , j ∈ N with i ≤ j ≤ i0 , s.t. f (ai ) = ai , f (aj ) = aj }. We can check that all elements of (I)-(X) are liminf continuous. Thus [A → A] = I ∪ II ∪ · · · ∪ X. Let id : A → A be the identity function. Then id ∈ V. Step (3): We claim that the tensor of 12 and id does not exist in [A → A]P . For g ∈ [A → A], ∧ P (id, g) = 0 ⇔ A(x, y) → A(x, g(y)) = 0 x,y∈A ∧
⇔ ∃x, y ∈ A, s.t. A(x, y) → A(x, g(y)) = 0 1 ⇔ ∃x, y ∈ A, s.t. A(x, y) ≥ and A(x, g(y)) = 0 2 ⇔ ∃x, y ∈ A, s.t. (x, y) ∈ (A × A) \ {(b, ai )|i ∈ N}, (x, g(y)) ∈ {(b, ai )|i ∈ N} ⇔ ∃x, y ∈ A, i0 ∈ N, s.t. x = b, g(y) = ai0 , y ∈ {a, b} ⇔ ∃i ∈ N, s.t. g(a) = ai or ∃j ∈ N, s.t. g(b) = aj ⇔ g ∈ IX ∪ X. Thus,
M. Liu / Fuzzy Sets and Systems 390 (2020) 96–104
1 & → P (id, g) = 2
1 2,
1,
103
g ∈ IX ∪ X; g ∈ I ∪ II ∪ · · · ∪ VIII. &
Suppose that h is the tensor of 12 and id. Then, by definition, P (h, g) = 12 → P (id, g). Since ∀g ∈ IX ∪ X, P (h, g) = 1 2 , h does not belong to IX ∪ X. Since ∀g ∈ I ∪ II ∪ · · · ∪ VIII, P (h, g) = 1, h should be the smallest element in I ∪ II ∪ · · · ∪ VIII. This contradicts the fact that I ∪ II ∪ · · · ∪ VIII does not have a smallest element. Thus, the claim holds. So we can conclude that [A → A]P is not complete. By Lemma 3.6 and Example 3.7, we can obtain that Theorem 3.8. Supliminf(H ) is not Cartesian closed. 4. Conclusion Since the category of liminf complete H -ordered sets is Cartesian closed, it was natural to believe that the category of complete H -ordered sets should be Cartesian closed as well. But, in this paper, we obtain that this category is not Cartesian closed. This seems counter-intuitive. Main reason behind it is that liminf completeness does not require tensor completeness. Similarly, conical cocompleteness does not require tensor completeness, thus it is easy to show that the category of conically complete H -ordered sets is Cartesian closed. But, the result of this paper is not sufficient to show that the category of continuous complete H -ordered sets is not Cartesian closed. Thus, the problem about the Cartesian closedness of the category of continuous complete H -ordered sets remains open. Based on the Cartesian closedness of Ord(H ), we obtained that Liminf(H ) and CSupliminf(H ) are Cartesian closed. A main contribution of these results is the confirmation of the existence of a complete residuated lattice that is not a frame such that these categories are Cartesian closed. So, it is natural to ask that if these results extend to more than three truth-values. Furthermore, the characterization of the truth-value lattices such that these categories are Cartesian closed is a tricky problem. Acknowledgements The author would like to express his sincere gratitude to the editors and reviewers. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11501048, 11871320) and the Fundamental Research Funds for the Central Universities (Grant No. 300102128102). References [1] S. Abramsky, A. Jung, Domain theory, in: S. Abramsky, D.M. Gabbay, T.S.E. Maibaum (Eds.), Semantic Structures, in: Handbook of Logic in Computer Science, vol. 3, Clarendon Press, 1994, pp. 1–168. [2] J. Adámek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990. [3] R. Bˇelohlávek, Fuzzy Relational Systems, Foundations and Principles, Kluwer Academic Publishers, Plenum Publishers, New York, 2002. [4] D. Ciucci, D. Dubois, A map of dependencies among three-valued logics, Inf. Sci. 250 (2013) 162–177. [5] M.M. Clementino, D. Hofmann, Exponentiation in V-categories, Topol. Appl. 153 (2006) 3113–3128. [6] L. Fan, A new approach to quantitative domain theory, Electron. Notes Theor. Comput. Sci. 45 (2001) 77–87. [7] J.C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets Syst. 69 (1995) 141–156. [8] G. Gierz, K.H. Hofmann, K. Keimel, et al., Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003. [9] S.E. Han, L.X. Lu, W. Yao, Quantale-valued fuzzy Scott topology, Iran. J. Fuzzy Syst. 16 (3) (2019) 175–188. [10] D. Hofmann, P.L. Waszkiewicz, Approximation in quantale-enriched categories, Topol. Appl. 158 (2011) 963–977. [11] P.T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982. [12] G.M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes Series, vol. 64, Cambridge University Press, 1982. [13] H. Lai, D. Zhang, Complete and directed complete -categories, Theor. Comput. Sci. 388 (2007) 1–25. [14] H. Lai, D. Zhang, Closedness of the category of liminf complete fuzzy orders, Fuzzy Sets Syst. 282 (2016) 86–98. [15] M. Liu, B. Zhao, Two Cartesian closed subcategories of fuzzy domains, Fuzzy Sets Syst. 238 (2014) 102–112. [16] M. Liu, B. Zhao, A non-frame valued Cartesian closed category of liminf complete fuzzy orders, Fuzzy Sets Syst. 321 (2017) 50–54. [17] P. Perny, B. Roy, The use of fuzzy outranking relations in preference modelling, Fuzzy Sets Syst. 49 (1992) 33–53.
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K.I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics Series, vol. 234, Longman, Essex, 1990. K.I. Rosenthal, The Theory of Quantaloids, Pitman Research Notes in Mathematics Series, vol. 348, Longman, Essex, 1996. I. Stubbe, Categorical structures enriched in a quantaloid: categories, distributors, functors, Theory Appl. Categ. 14 (2005) 1–45. I. Stubbe, Categorical structures enriched in a quantaloid: tensored and cotensored categories, Theory Appl. Categ. 16 (2005) 283–306. M.B. Smyth, The largest Cartesian closed category of domains, Theor. Comput. Sci. 27 (1983) 109–119. K.R. Wagner, Solving Recursive Domain Equations with Enriched Categories, Ph.D. Thesis, School of Computer Science, Carnegie-Mellon University, Pittsburgh, 1994. K.R. Wagner, Liminf convergence in -categories, Theor. Comput. Sci. 184 (1997) 61–104. W.X. Xie, Q.Y. Zhang, L. Fan, The Dedekind-MacNeille completions for fuzzy posets, Fuzzy Sets Syst. 160 (2009) 2292–2316. W. Yao, Quantitative domains via fuzzy sets: Part I: continuity of fuzzy directed-complete poset, Fuzzy Sets Syst. 161 (2010) 983–987. W. Yao, Quantitative domains via fuzzy sets: Part II: fuzzy Scott topology on fuzzy directed-complete posets, Fuzzy Sets Syst. 173 (2011) 60–80. L.A. Zadeh, Similarity relations and fuzzy orderings, Inf. Sci. 3 (1971) 177–200. Q.Y. Zhang, L. Fan, Continuity in quantitative domains, Fuzzy Sets Syst. 154 (2005) 118–131.
ScienceDirect Fuzzy Sets and Systems 390 (2020) 96–104 www.elsevier.com/locate/fss
Cartesian closedness of a category of non-frame valued complete fuzzy orders Min Liu School of Sciences, Chang’an University, 710064, Xi’an, PR China Received 11 July 2019; received in revised form 7 February 2020; accepted 11 February 2020 Available online 14 February 2020
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. © 2020 Elsevier B.V. All rights reserved. Keywords: Fuzzy relations; L-order; Liminf complete L-ordered set; Complete L-ordered set; Cartesian closed category
1. 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 [9,10,13,25–27,29]. 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 Lvalued 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 E-mail address: [email protected]. https://doi.org/10.1016/j.fss.2020.02.004 0165-0114/© 2020 Elsevier B.V. All rights reserved.
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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 (Q, &) be the unit interval equipped with a continuous t-norm. In [14], it is proved that the category Liminf(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. 2. L-ordered sets In this paper, (L, ∧, ∨, ∗, →, 0, 1) always denotes a complete residuated lattice [3]. For concepts and results in domain theory and category theory, we refer to [2,8,12]. Definition 2.1 (Wagner [23], Fan [6], Bˇelohlávek [3]). An L-ordered set is a pair (A, e) such that A is a set and e : A × A −→ L is a map, called an L-order, that satisfies for every x, y, z ∈ A, (E1) e(x, x) = 1 (reflexivity); (E2) e(x, y) ∗ e(y, z) ≤ e(x, z) (transitivity); (E3) e(x, y) = e(y, z) = 1 implies x = y (antisymmetry). A complete residuated lattice is a commutative unital quantale [18]. Thus, an L-ordered set in the above sense is a quantale-enriched category [23,24,13]. Further more, by the fact that a commutative unital quantale is a quantaloid [19], a quantale-enriched category in the sense of [24] can be viewed as a quantaloid-enriched category [19]. For sake of simplicity, we often write A for an L-ordered set (A, e) and write A(x, y) for e(x, y) if no confusion would arise. A map f : A −→ B between two L-ordered sets is called L-order preserving (or order preserving for short) if A(x1 , x2 ) ≤ B(f (x1 ), f (x2 )) for all x1 , x2 ∈ A. Let (A, e) be an L-ordered set. The L-order e induces a partial order ≤e on A defined by x ≤e y iff e(x, y) = 1. We will denote ≤e simply by ≤, denote (A, ≤e ) by A0 and denote the join and meet operations in A0 by ∨ and ∧. Example 2.2. (1) Let {} be a single point set. Define e(, ) = 1. Then ({}, e) is an H -ordered set. ({}, e) is the terminal object in the category of L-ordered sets. (2) (Sub-L-ordered set) Let A be an L-ordered set and B ⊆ A. Let B(x, y) = A(x, y), for x, y ∈ B. Then B is an L-ordered set. (3) (Product L-ordered set) Let A, B be L-ordered sets. For (x1 , y1 ), (x2 , y2 ) ∈ A × B, let (A × B)((x1 , y1 ), (x2 , y2 )) = A(x1 , x2 ) ∧ B(y1 , y2 ). Then A × B is an L-ordered set.
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Definition 2.3 (Stubbe [20,21], Lai and Zhang [13]). Let A be an L-ordered set, μ ∈ LA , {xi } ⊆ A, x ∈ A, α ∈ L. (i) An element a ∈ A is called a join of μ, in symbols a = sup μ, if A(a, y) = x∈X μ(x) → A(x, y) for all y ∈ A. (ii) An element a ∈ A is called a conical colimit of {xi }, in symbols a = conicolimiti∈I xi , if A(a, y) = i A(xi , y) for all y ∈ A. (iii) An element a ∈ A is called a tensor of x and α, in symbols a = α ⊗ x, if A(a, y) = α → A(x, y) for all y ∈ A. An L-ordered set A is said to be complete if sup μ exits for all μ ∈ LA ; A is said to be conically cocomplete if conicolimiti∈I xi exists for all {xi } ⊆ A; A is said to be tensored if α ⊗ x exists for all α ∈ L, x ∈ A; A is said to be order complete if A0 is complete. Theorem 2.4 (Stubbe [20,21]). Let A be an L-ordered set. (1) A is complete if and only if A is tensored and conically cocomplete. (2) If the conical colimit of a family of elements {xi } ⊆ A exists, then conicolimiti∈I xi = i xi . (3) If A is conically cocomplete, then A is order complete. Recall that a net {xλ }λ∈D in an L-ordered set A is forward Cauchy [14], if 1= A(xμ , xσ ). λ∈D λ≤μ≤σ
A forward Cauchy net {xλ }λ∈D in an L-ordered set A converges to a ∈ A, or a is a liminf of {xλ }λ∈D , if for all x ∈ A, A(a, x) = λ∈D λ≤μ A(xμ , x). We denote the liminf of {xλ }λ∈D by lim infλ∈D xλ if it exists. An L-ordered set A is liminf complete [14] if each forward Cauchy net has a liminf. An order preserving function f : A → B is liminf continuous [14] if it preserves the liminf of forward Cauchy nets in A. Denote the set of all liminf continuous functions from A to B by [A → B]. Define P : [A → B] × [A → B] −→ L as follows: P (f, g) = {p ∈ L|∀x, y ∈ A, p ∧ A(x, y) ≤ B(f (x), g(y))}. Suppose (L, ∧) is a frame. Then P is an L-order on [A → B] (see [5]). Denote the L-ordered set ([A → B], P ) by [A → B]P . Define ev : [A → B]P × A −→ B by ev(f, x) = f (x). Then ev is order preserving (see [5]). 3. Cartesian closedness of the category of complete H -ordered sets The complete residuated lattice considered in this note is the MV-algebra H with three elements. Precisely, H = {0, 12 , 1} and p&q = max{p + q − 1, 0} for all p, q ∈ H . For more information about H , we refer to [4,7,17]. Clearly, (H, ∧) is a frame [11]. We note that connectives on three values and their relations are extensively exposed in [4]. By Table 1 in [4], we know & is the unique conjunction on H such that (H, &) is a complete residuated lattice and & = ∧. &
∧
→− , respectively. Let Abe an H -ordered set The right adjoint for p&− and p∧− will be denoted by p →− and p and {xλ }λ∈D a monotone net in the underlying ordered set A0 . Then λ∈D λ≤μ A(xμ , x) = i∈D A(xi , x) for all x ∈ A [16]. This shows that lim infλ∈D xλ exists if and only if the conical colimit of {xλ }λ∈D exists, in such case they are equal. Let Liminf(H ) denote the category of liminf complete H -ordered sets and liminf continuous functions. In [16], it is proved that Liminf(H ) is Cartesian closed. The exponentiation B A of objects A, B is the function space [A → B]P . Proposition 3.1. If A is a conically cocomplete H -ordered set, then A is liminf complete. Lemma 3.2 ([16]). Let A, B be two H -ordered sets, f : A → B an order preserving map. Then (1) A is liminf complete if and only if every monotone net in A0 converges in A. (2) If A is liminf complete then f is liminf continuous if and only if f preserves liminfs of monotone nets if and only if f : A0 → B0 is Scott continuous.
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Lemma 3.3. Let A be an H -ordered set, B a conically cocomplete H -ordered set. Then [A → B]P is conically cocomplete. Moreover, the conical colimit f of {fi }i∈I ⊆ [A → B]P is given by f (x) = i∈I fi (x) for every x ∈ A. Proof. Let {fi }i∈I ⊆ [A → B]P , and assume that f (x) = net in A0 , y ∈ B. Then B(f (lim inf xλ ), y) = B( fi (lim inf xλ ), y) λ∈D
=
(By conically cocompleteness of B)
B(lim inf fi (xλ ), y)
(By liminf continuity of fi )
λ∈D
λ∈D
i∈I
=
fi (x) for every x ∈ A. Let {xλ }λ∈D be a monotone
B(fi (lim inf xλ ), y)
i∈I
=
i∈I
λ∈D
i∈I
B(fi (xλ ), y)
(By monotonicity of {fi (xλ )}λ∈D )
B( fi (xλ ), y)
(By conically cocompleteness of B)
i∈I λ∈D
=
λ∈D
=
i∈I
B(f (xλ ), y)
λ∈D
Thus, f (lim infλ∈D xλ ) = lim infλ∈D f (xλ ), i.e., f is liminf continuous. For every g ∈ [A → B]P , we have ∧ P (f, g) = A(x, y) → B(f (x), g(y)) x,y∈A
=
∧ A(x, y) → B( fi (x), g(y))
x,y∈A
=
i∈I
∧
A(x, y) →
x,y∈A
=
B(fi (x), g(y))
i∈I ∧
A(x, y) → B(fi (x), g(y))
x,y∈A i∈I
=
P (fi , g).
i∈I
Thus, f is the conical colimit of {fi }i∈I , i.e., [A → B]P is conically cocomplete. Lemma 3.4. If A and B are conically cocomplete H -ordered sets, then so is A × B. Proof. Let {(xi , yi )}i∈I ⊆ A × B. Define (x, y) = ( i∈I xi , i∈I yi ). Then, for every (a, b) ∈ A × B, we have A × B((xi , yi ), (a, b)) = (A(xi , a) ∧ B(yi , b)) i∈I
i∈I
=
A(xi , a) ∧
i∈I
B(yi , b)
i∈I
= A( xi , a) ∧ B( yi , b) i∈I
i∈I
= A(x, b) ∧ B(y, b) = A × B((x, y), (a, b)). Thus, (x, y) is the conical colimit of {(xi , yi )}i∈I , i.e., A × B is conically cocomplete.
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Let CSupliminf(H ) denotes the full subcategory of Liminf(H ) consisting of conically cocomplete H -ordered sets. By Lemma 3.3, Lemma 3.4 and the Cartesian closedness of Liminf(H ), we can obtain the following result: Theorem 3.5. CSupliminf(H ) is Cartesian closed. Let Supliminf(H ) denotes the full subcategory of Liminf(H ) consisting of complete H -ordered sets. Lemma 3.6. Let A, B be two complete H -ordered sets. (i) The single point H -ordered set {} is the terminal object of Supliminf(H ). (ii) The product A × B of two complete H -ordered set is complete. Thus, A × B is the categorical product of A, B in Supliminf(H ). (iii) If the exponentiation B A of objects A, B exists, then B A is the function space [A → B]P . Proof. (i) is trivial. (ii) By Theorem 2.4 and Lemma 3.4, we only need to show that A × B is tensored. Let (x, y) ∈ A × B, α ∈ L. Then, for every (a, b) ∈ A × B, we have &
&
α → A × B((x, y), (a, b)) = α → (A(x, a) ∧ B(y, b)) &
&
= (α → A(x, a)) ∧ (α → B(y, b)) = A(α ⊗ x, a) ∧ B(α ⊗ y, b) = A × B((α ⊗ x, α ⊗ y), (a, b)). Thus, the tensor of α and (x, y) exists and equal to (α ⊗ x, α ⊗ y). (iii) The proof mimics the techniques used by Smyth (see [22] Lemma 5).
Example 3.7. Let A = {ai : i ∈ N} ∪ {a, b}, where ai , a, b are mutually distinct, N is the set of natural numbers. Define an H -valued relation e : A × A → H as follows (Table 1): Table 1 The H -valued relation on A. e
a1
a2
a3
···
a
b
a1
1
1
1
···
1
1
a2
1
1
···
1
1
a3 .. .
1 2 1 2
1 2
···
.. .
.. .
1 .. .
1 .. .
1 .. .
a
1 2
1 2
1 2
···
1
1
b
0
0
0
···
1 2
1
Then (A, e) is a complete H -ordered set, but [A → A]P is not complete. Proof. Step (1): We claim that (A, e) is a complete H -ordered set. (a) It is routine to check that e is an H -order. (b) A is tensored: We can check that ∀x ∈ A, 0 ⊗ x = 0, 1 ⊗ x = 1. By the fact that 1 & 1, y = a or b; → e(b, y) = 1 2 2 , y = ai , i ∈ N, we have
1 & 2 → e(b, y) = e(a, y)
for all y ∈ A. Thus,
1 2
⊗ b = a. Similarly,
1 2
⊗ a = a1 ,
1 2
⊗ ai = a1 for all i ∈ N.
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(c) A is conically cocomplete: Let {xi : i ∈ I } ⊆ A. If {xi : i ∈ I } has a maximal element x, then i∈I e(xi , y) = e(x, y) for all y ∈ A. In this case, the conical colimit of {xi : i ∈ I } is x. If {xi : i ∈ I } does not have a maximal element, then {x i : i ∈ I } is cofinal in {ai : i ∈ N}, that is {xi : i ∈ I } ⊆ {ai : i ∈ N} and ∀i ∈ N, ∃j ∈ I such that xj ≥ ai . In this case, i∈I xi = a, 1, y = a or b; e(xi , y) = 1 2 , y = ai , i ∈ N. i∈I
Thus, i∈I e(xi , y) = e(a, y) for every y ∈ A. In this case, the conical colimit of {xi : i ∈ I } is a. So we can conclude that A is conically cocomplete. By (a), (b) and (c), we deduce that (A, e) is a complete H -ordered set. Step (2): We shall characterize all liminf continuous functions from A to A. Suppose f : A0 → A0 is monotone. Then the equality e(x, y) ≤ e(f (x), f (y)) holds for x, y ∈ A with e(f (x), f (y)) = 1 or e(f (x), f (y)) = 12 . Thus, f : A → A is H -order preserving if and only if (1) f : A0 → A0 is monotone; and (2) e(x, y) ≤ e(f (x), f (y)) for x, y ∈ A with e(f (x), f (y)) = 0. Suppose f : A0 → A0 is monotone. Then f : A0 → A0 is Scott continuous if and only if f ( i∈I xi ) = i∈I f (xi ) for any {xi : i ∈ I } ⊆ {ai : i ∈ N} which is cofinal in {ai : i ∈ N} if and only if f ( i∈N ai ) = i∈N f (ai ). By Lemma 3.2, we know that a map f : A → A is liminf continuous if and only if (1)f : A0 → A0 is monotone; (2) e(x, y) ≤ e(f (x), f (y)) for x, y ∈ A with e(f (x), f (y)) = 0 and (3) f ( i∈N ai ) = i∈N f (ai ). Now suppose that f : A → A is liminf continuous. Let F be the set of all functions from A to A. Case (1): f (a) = b. a ≤ b implies that ≤ f (b). Thus, f (b) = b. b = f (a) b = f (a) = f ( i∈N ai ) = i∈N f (ai ) implies that there is a smallest i0 ∈ N such that f (ai0 ) = b. If i0 = 1, then f is the constant function valued at b. Let I = {f ∈ F |f (x) = b, ∀x ∈ A}. Suppose i0 = 1. Then, f (aj ) = b for all j ∈ N with j ≥ i0 , and f (a) = f (b) = b. For every i ∈ N with i < i0 , we have f (ai ) < b and 12 = e(ai0 , ai ) ≤ e(f (ai0 ), f (ai )) = e(b, f (ai )). Thus, f (ai ) = a for every i ∈ N with i < i0 . Let II = {f ∈ F |f (a) = f (b) = b, ∃i0 ∈ N, i0 = 1, s.t. f (ai ) = a for all i ∈ N with i < i0 , f (aj ) = b for all j ∈ N with j ≥ i0 }. Case (2): f (a) = a. a = f (a) = f ( i∈N ai ) = i∈N f (ai ) implies that there is a smallest i0 ∈ N such that f (ai0 ) = a or {f (ai )}i∈N is cofinal in {ai }i∈N . Case (2)-1: f (b) = b. By the value of f (ai ), we have three situations. Case (2)-1-1: i0 = 1. Then f (ai ) = a for every i ∈ N. Let III = {f ∈ F |f (b) = b, f (a) = a, f (ai ) = a for every i ∈ N}. Case (2)-1-2: i0 = 1. By monotonicity of f , f (ai ) = a for every i ∈ N with i ≥ i0 , and ∀i, j ∈ N with i ≤ j < i0 , ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj . Let IV = {f ∈ F |f (b) = b, f (a) = a, ∃i0 ∈ N, i0 = 1, s.t. f (ai ) = a for every i ∈ N with i ≥ i0 , ∀i, j ∈ N with i ≤ j < i0 , ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj }. Case (2)-1-3: {f (ai )}i∈N is cofinal in {ai }i∈N . Let V = {f ∈ F |f (b) = b, f (a) = a, ∀i, j ∈ N with i ≤ j, ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj and {f (ai )}i∈N is cofinal in {ai }i∈N }. Case (2)-2: f (b) = a. Similar to Case (2)-1, f must belong to one of the following three sets.
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Let VI = {f ∈ F |f (x) = a for every x ∈ A}. Let VII = {f ∈ F |f (b) = f (a) = a, ∃i0 ∈ N, i0 = 1, s.t. f (ai ) = a for every i ∈ N with i ≥ i0 , ∀i, j ∈ N with i ≤ j < i0 , ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj }. Let VIII = {f ∈ F |f (b) = f (a) = a, ∀i, j ∈ N with i ≤ j, ∃i , j ∈ N with i ≤ j , s.t. f (ai ) = ai , f (aj ) = aj , {f (ai )}i∈N is cofinal in {ai }i∈N }. Case (3): f (a) = ai0 for some i0 ∈ N. By the inequality 21 = e(b, a) ≤ e(f (b), f (a)) = e(f (b), ai0 ), we deduce that f (b) = b. Clearly, f (b) ≥ f (a) = ai 0 . By the requirement that ai0 = f (a) = f ( i∈N ai ) = i∈N f (ai ), there exists a smallest j0 ∈ N such that f (aj0 ) = ai0 . By the monotonicity of f , we know ∀i ≥ j0 , f (ai ) = ai0 . Case (3)-1: f (b) = a. Let IX = {f ∈ F |f (b) = a, ∃i0 , j0 ∈ N, s.t. f (a) = ai0 , ∀i ∈ N with i ≥ j0 , f (ai ) = ai0 , ∀i, j ∈ N with i ≤ j ≤ j0 , ∃i , j ∈ N with i ≤ j ≤ i0 , s.t. f (ai ) = ai , f (aj ) = aj }. Case (3)-2: f (b) = ak for some k ≥ i0 . Let X = {f ∈ F |∃k, i0 , j0 ∈ N with k ≥ i0 , s.t. f (b) = ak , f (a) = ai0 , ∀i ∈ N with i ≥ j0 , f (ai ) = ai0 , ∀i, j ∈ N with i ≤ j ≤ j0 , ∃i , j ∈ N with i ≤ j ≤ i0 , s.t. f (ai ) = ai , f (aj ) = aj }. We can check that all elements of (I)-(X) are liminf continuous. Thus [A → A] = I ∪ II ∪ · · · ∪ X. Let id : A → A be the identity function. Then id ∈ V. Step (3): We claim that the tensor of 12 and id does not exist in [A → A]P . For g ∈ [A → A], ∧ P (id, g) = 0 ⇔ A(x, y) → A(x, g(y)) = 0 x,y∈A ∧
⇔ ∃x, y ∈ A, s.t. A(x, y) → A(x, g(y)) = 0 1 ⇔ ∃x, y ∈ A, s.t. A(x, y) ≥ and A(x, g(y)) = 0 2 ⇔ ∃x, y ∈ A, s.t. (x, y) ∈ (A × A) \ {(b, ai )|i ∈ N}, (x, g(y)) ∈ {(b, ai )|i ∈ N} ⇔ ∃x, y ∈ A, i0 ∈ N, s.t. x = b, g(y) = ai0 , y ∈ {a, b} ⇔ ∃i ∈ N, s.t. g(a) = ai or ∃j ∈ N, s.t. g(b) = aj ⇔ g ∈ IX ∪ X. Thus,
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1 & → P (id, g) = 2
1 2,
1,
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g ∈ IX ∪ X; g ∈ I ∪ II ∪ · · · ∪ VIII. &
Suppose that h is the tensor of 12 and id. Then, by definition, P (h, g) = 12 → P (id, g). Since ∀g ∈ IX ∪ X, P (h, g) = 1 2 , h does not belong to IX ∪ X. Since ∀g ∈ I ∪ II ∪ · · · ∪ VIII, P (h, g) = 1, h should be the smallest element in I ∪ II ∪ · · · ∪ VIII. This contradicts the fact that I ∪ II ∪ · · · ∪ VIII does not have a smallest element. Thus, the claim holds. So we can conclude that [A → A]P is not complete. By Lemma 3.6 and Example 3.7, we can obtain that Theorem 3.8. Supliminf(H ) is not Cartesian closed. 4. Conclusion Since the category of liminf complete H -ordered sets is Cartesian closed, it was natural to believe that the category of complete H -ordered sets should be Cartesian closed as well. But, in this paper, we obtain that this category is not Cartesian closed. This seems counter-intuitive. Main reason behind it is that liminf completeness does not require tensor completeness. Similarly, conical cocompleteness does not require tensor completeness, thus it is easy to show that the category of conically complete H -ordered sets is Cartesian closed. But, the result of this paper is not sufficient to show that the category of continuous complete H -ordered sets is not Cartesian closed. Thus, the problem about the Cartesian closedness of the category of continuous complete H -ordered sets remains open. Based on the Cartesian closedness of Ord(H ), we obtained that Liminf(H ) and CSupliminf(H ) are Cartesian closed. A main contribution of these results is the confirmation of the existence of a complete residuated lattice that is not a frame such that these categories are Cartesian closed. So, it is natural to ask that if these results extend to more than three truth-values. Furthermore, the characterization of the truth-value lattices such that these categories are Cartesian closed is a tricky problem. Acknowledgements The author would like to express his sincere gratitude to the editors and reviewers. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11501048, 11871320) and the Fundamental Research Funds for the Central Universities (Grant No. 300102128102). References [1] S. Abramsky, A. Jung, Domain theory, in: S. Abramsky, D.M. Gabbay, T.S.E. Maibaum (Eds.), Semantic Structures, in: Handbook of Logic in Computer Science, vol. 3, Clarendon Press, 1994, pp. 1–168. [2] J. Adámek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990. [3] R. Bˇelohlávek, Fuzzy Relational Systems, Foundations and Principles, Kluwer Academic Publishers, Plenum Publishers, New York, 2002. [4] D. Ciucci, D. Dubois, A map of dependencies among three-valued logics, Inf. Sci. 250 (2013) 162–177. [5] M.M. Clementino, D. Hofmann, Exponentiation in V-categories, Topol. Appl. 153 (2006) 3113–3128. [6] L. Fan, A new approach to quantitative domain theory, Electron. Notes Theor. Comput. Sci. 45 (2001) 77–87. [7] J.C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets Syst. 69 (1995) 141–156. [8] G. Gierz, K.H. Hofmann, K. Keimel, et al., Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003. [9] S.E. Han, L.X. Lu, W. Yao, Quantale-valued fuzzy Scott topology, Iran. J. Fuzzy Syst. 16 (3) (2019) 175–188. [10] D. Hofmann, P.L. Waszkiewicz, Approximation in quantale-enriched categories, Topol. Appl. 158 (2011) 963–977. [11] P.T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982. [12] G.M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes Series, vol. 64, Cambridge University Press, 1982. [13] H. Lai, D. Zhang, Complete and directed complete -categories, Theor. Comput. Sci. 388 (2007) 1–25. [14] H. Lai, D. Zhang, Closedness of the category of liminf complete fuzzy orders, Fuzzy Sets Syst. 282 (2016) 86–98. [15] M. Liu, B. Zhao, Two Cartesian closed subcategories of fuzzy domains, Fuzzy Sets Syst. 238 (2014) 102–112. [16] M. Liu, B. Zhao, A non-frame valued Cartesian closed category of liminf complete fuzzy orders, Fuzzy Sets Syst. 321 (2017) 50–54. [17] P. Perny, B. Roy, The use of fuzzy outranking relations in preference modelling, Fuzzy Sets Syst. 49 (1992) 33–53.
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