The category of algebraic fuzzy closure L -systems on fuzzy complete lattices

Abstract

Based on a complete residuated lattice, algebraic fuzzy closure operators and algebraic fuzzy closure L-systems on a fuzzy complete lattice are defined and investigated. We establish a “one-to-one” correspondence between algebraic fuzzy closure operators and algebraic fuzzy closure L-systems under a condition on fuzzy order. Moreover, it is shown that the category of (algebraic) fuzzy closure operator spaces is isomorphic to the category of (algebraic) fuzzy closure L-system spaces.

Keywords

(Algebraic) fuzzy closure operators fuzzy complete lattices category

1 Introduction

The partially ordered set (poset, for short) is an important mathematical structure which attracts attention in many areas (see [5]). In order to study the poset in the general framework, many authors studied it from the fuzzy point of view [1 , 30]. One of them is the fuzzy order originally proposed by Bělohlávek [1, 2], where the fundamental theorems of concept lattice were fuzzified successfully. Another kind of fuzzy order was defined by Fan and Zhang in [6, 30] and used to introduce a new approach to quantitative domain theory and continuity in quantitative domains, respectively. Later on, Yao proved that these two kinds of fuzzy order are equivalent to each other in [26].

Closure operators and closure systems have been studied in several areas of classical mathematics, e.g., topology, algebra, logic, data mining, knowledge representation. Particularly, they play a significant role in theoretical computer science and have a close relation with Galois connections [5, 8]. The notions of fuzzy L_K-closure operator and fuzzy L_K-closure system were firstly given in [3] within the framework of fuzzy powerset, where the truth value L is a complete residuated lattice. In addition, several particular fuzzy closure (resp., interior) operators and fuzzy closure systems have been considered in fuzzy subalgebras, fuzzy congruences, fuzzy topologies, etc. [4 , 15].

Moreover, fuzzy closure operators and fuzzy interior operators based on fuzzy order were firstly provided in [27] to study the relations between them and fuzzy Galois connections. Go on this work, Yao and Zhao [28] presented fuzzy kernel systems and fuzzy kernel L-systems on fuzzy poset and established a “one-to-one” correspondence between fuzzy kernel (or interior) operators and kernel (L-)systems. However, in their paper, L must be a complete Heyting algebra and the condition “with ⊔· existing” was required in order to establish the “one-to-one” correspondence between fuzzy kernel operators and fuzzy kernel L-systems. For the purpose of expressing conveniently, we assume that (X, e) is a fuzzy complete lattice throughout this paper.

The query that arise from the previous works can be stated as follows:

How to make the assertion: “on a crisp complete lattice, every order-preserving map can induce a closure system related to it (see [25])” hold in the fuzzy setting underlying a more general structure of truth values, such as the complete residuated lattice.

In the classical setting, algebraic closure operators and algebraic closure systems on dcpo have been discussed in [8]. Therefore, the problem how to define the proper fuzzy algebraic closure operators and systems emerges naturally.

One aim of this paper is to study the relationship between fuzzy closure operators and fuzzy closure L-systems using the complete residuated lattice as the structure of truth values. But, we proposed the Condition (∗) on fuzzy order, because the assertion in (1) above does not hold in the fuzzy setting.

Another aim is to define the appropriate concepts of algebraic fuzzy closure operators and algebraic fuzzy closure L-systems on fuzzy complete lattices. Moreover, we established a relationship between them. Of course, we did all these work based on the complete residuated lattice and our given condition on fuzzy order.

The paper is organized as follows: In Section 2, we recall necessary basic concepts and propositions. In Section 3, fuzzy closure operators and fuzzy closure L-systems are introduced. After we propose the Condition (∗) on fuzzy order, a “one-to-one” correspondence between fuzzy closure operators and fuzzy closure L-systems is established under this condition. In Sections 4, the concepts of algebraic fuzzy closure operators and algebraic fuzzy closure L-systems on fuzzy complete lattices are proposed. Moreover, the relationship between them is obtained under the Condition (∗) . In Sections 5, from the category view, we further analyze the connections between the (algebraic) fuzzy closure operators and the (algebraic) fuzzy closure L-systems. Conclusions are given in Section 6.

2 Preliminaries

For the convenience of the readers, in this section, some basic knowledge are reviewed. More detailed information can be found in |nces.

Definition 2.1. [20] A residuated lattice L is a structure (L, ∨ , ∧ , ⊗ , → , ⇒ , 0, 1, T), where (L, ∨ , ∧ , 0, 1) is a bounded lattice, (L, ⊗ , T) a monoid which is an algebraic structure with a single associative binary operation and an identity element, ⊗ a binary operation on L, → and ⇒ are left and right residuum of ⊗ by residuation conditions (Rc . 1) a ⊗ b ≤ c ⇔ a ≤ b → c and (Rc . 2) a ⊗ b ≤ c ⇔ b ≤ a ⇒ c for any a, b, c ∈ L, respectively. It is easy to check that if the respective residua of ⊗ exist, then a → b = ∨ {c ∈ L : a ⊗ c ≤ b} and a ⇒ b = ∨ {c ∈ L : c ⊗ a ≤ b} for any a, b, c ∈ L (see [24]).

A residuated lattice is called commutative iff ⊗ is commutative (clearly, if ⊗ is commutative then → =⇒), integral iff T = 1, complete iff the underlying lattice (L, ∨ , ∧ , 0, 1) is complete. The commutative integral complete residuated lattice (L, ∨ , ∧ , ⊗ , → , ⇒ , 0, 1, T) is short for (L, ∨ , ∧ , ⊗ , → , 0, 1).

Given a residuated lattice (L, ∨ , ∧ , ⊗ , → , 0, 1), a unary operator ¬ is referred as the precomplement operator, by ¬a = a → 0.

A (complete) residuated lattice is called (complete) Heyting algebra if ⊗ =∧. In a residuated lattice (L, ⊗), if ⊗ is idempotent, i.e., a ⊗ a = a for all a ∈ L, then ⊗ =∧ and consequently L is a Heyting algebra. Properties of complete residuated lattices and complete Heyting algebras are well-known and can be found in many papers, e.g. [10, 11].

In what follows, ⊗ is sometimes called generalized triangular norm and → the residuum of ⊗.

Proposition 2.1. [16 , 23] In any commutative integral complete residuated lattice (L, ∨ , ∧ , ⊗ , → , 0, 1), the following properties hold for all a, b, a_i, b_i, c ∈ L (i ∈ I):

⊗ is isotone in both arguments, → antitone in the 1st argument and isotone in the 2nd argument.

a ≤ (b → a ⊗ b).

a ⊗ (∨ _i∈Ib_i) = ∨ _i∈I (a ⊗ b_i).

a → ∧ _i∈Ib_i = ∧ _i∈I (a → b_i),

∨ _i∈Ia_i → b = ∧ _i∈I (a_i → b).

a ⊗ b ≤ a ∧ b.

(a ⊗ b) → c = b → (a → c).

b → c ≤ ((a → b) → (a → c)),

b → c ≤ ((c → a) → (b → a)).

1 → a = a.

Because of the outstanding properties of the complete residuated lattice above, throughout this paper, unless other wise stated, L always denotes a complete residuated lattice which is both commutative and integral.

Theorem 2.1. [18] (L, ∨ , ∧ , ⊗ , → , 0, 1) is a residuated lattice if and only if for any a, b, c ∈ L:

a ⊗ b ≤ c ⇔ a ≤ b → c.

1 ⊗ a = a (resp . , a ≤ b ⇔ a → b = 1).

a → (b → c) = b → (a → c).

Definition 2.2. [22] Let X be a nonempty set. A map A : X → L is called an L-fuzzy set in X and A_α = {x ∈ X : A (x) ≥ α} is called α-level subset of A for any α ∈ L. Furthermore, A ⊆ B means A (x) ≤ B (x) for every x ∈ X.

Obviously, for any X₁ ⊆ X, χ_{X
₁} ∈ L^X, where χ_{X
₁} is the characteristic function of X₁.

We abbreviate all “L-fuzzy” to “fuzzy” in the rest of this paper.

Definition 2.3. [1, 26] A fuzzy poset is a pair (X, e) such that X is a set and e : X × X → L is a map, called a fuzzy order, which satisfies for any x, y ∈ X,

e (x, x) =1 (reflexivity),

e (x, y) ⊗ e (y, z) ≤ e (x, z) (transitivity),

e (x, y) = e (y, x) =1 implies x = y (antisymmetry).

Given two fuzzy posets (X, e_X) and (Y, e_Y), a map f : X → Y is said to be a fuzzy order-preserving map if e_X (x, y) ≤ e_Y (f (x) , f (y)) for any x, y ∈ X.

Let (X, e) be a fuzzy poset. ≤_e ∈ X × X is defined by “x ≤ _ey ⇔ e (x, y) =1 for any x, y ∈ X”.

Remark 2.1. It is a direct conclusion that if (X, e) is a fuzzy poset, then (X, ≤ _e) is a crisp poset.

Proposition 2.2. [26] Let (X, e) be a fuzzy poset. Then for all x, y ∈ X,

$\begin{matrix} e (x, y) & = & ⋀_{z \in X} (e (z, x) \to e (z, y)) \\ = & ⋀_{z \in X} (e (y, z) \to e (x, z)) . \end{matrix}$

Definition 2.4. [26] Let (X, e) be a fuzzy poset, x₀ ∈ X, A ∈ L^X. The element x₀ is called a join (resp., meet) of A, in symbols x₀ = ⊔ A (resp., x₀ = ⊓ A), if

∀x ∈ X, A (x) ≤ e (x, x₀)

(resp . , A (x) ≤ e (x₀, x)),

∀y ∈ X, ⋀ _x∈X (A (x) → e (x, y)) ≤ e (x₀, y)

(resp . , ⋀ _x∈X (A (x) → e (y, x)) ≤ e (y, x₀)).

It is easy to verify that if x₁, x₂ are two joins (or meets) of A, then x₁ = x₂ by the antisymmetry of fuzzy order e. That is, each A ∈ L^X has at most one join (or meet).

Proposition 2.3. [26] (1) x₀ = ⊔ A iff ∀y ∈ X, $e (x_{0}, y) = ⋀_{x \in X} (A (x) \to e (x, y)) .$

(2) x₀ = ⊓ A iff ∀y ∈ X, $e (y, x_{0}) = ⋀_{x \in X} (A (x) \to e (y, x)) .$

Definition 2.5. [26] A fuzzy poset (X, e) is called a fuzzy complete lattice if for all A ∈ L^X, ⊔A and ⊓A exist.

Proposition 2.4. [26] Let (X, e) be a fuzzy poset. The following statements are equivalent:

(X, e) is a fuzzy complete lattice.

For any A ∈ L^X, ⊔ A exists.

For any A ∈ L^X, ⊓ A exists.

Remark 2.2. Let (X, e) be a fuzzy complete lattice. Then (X, ≤ _e) is a crisp complete lattice.

Proof. Suppose (X, e) is a fuzzy complete lattice. Then we proof ∧X₁ exists for any X₁ ⊆ X here.

Obviously, χ_{X
₁} ∈ L^X, so ⊓χ_{X
₁} exists. We claim ⊓χ_{X
₁} = ∧ X₁. In fact,

(i) For any x ∈ X₁, e (⊓ χ_{X
₁}, x) ≥ χ_{X
₁} (x) =1 implies e (⊓ χ_{X
₁}, x) =1, i . e . , ⊓ χ_{X
₁} ≤ _ex.

(ii) If for any y ∈ X such that y ≤ _ex for any x ∈ X₁, then $\begin{matrix} e (y, ⊓ χ_{X_{1}}) \geq ⋀_{x \in X} (χ_{X_{1}} (x) \to e (y, x)) \\ = (⋀_{x \in X_{1}} (χ_{X_{1}} (x) \to e (y, x))) \\ \land (⋀_{x \in X \ X_{1}} (χ_{X_{1}} (x) \to e (y, x))) \\ = (⋀_{x \in X_{1}} (1 \to 1)) \land (⋀_{x \in X \ X_{1}} (0 \to e (y, x))) \\ = 1, \end{matrix}$ that is, y ≤ _e ⊓ χ_{X
₁}. □

Definition 2.6. [19, 27] For each map f : X → Y, the L-forward powerset operator f^→ : L^X → L^Y is defined by $(\forall y \in X) (\forall A \in L^{X}) (f^{\to} (A) (y) = ⋁_{f (x) = y} A (x)) .$

Definition 2.7. [7, 28] On a set X, a map Sub : L^X → L^X is defined by $\forall A, B \in L^{X}, Sub (A, B) = ⋀_{x \in X} (A (x) \to B (x)) .$ Then Sub is a fuzzy order on L^X, called the subsethood operator on (X, e). Clearly, $A \subseteq B \Leftrightarrow Sub (A, B) = 1 .$

Remark 2.3. [32] Let (X, e) be a fuzzy complete lattice. Then for any A, B ∈ L^X,

Sub (A, B) ≤ e (⊔ A, ⊔ B),

Sub (A, B) ≤ e (⊓ B, ⊓ A).

3 Fuzzy closure L-systems and their correspondence to fuzzy closure operators

In this section, we discuss fuzzy closure operators and fuzzy closure L-systems on fuzzy complete lattices and establish a “one-to-one” correspondence between them under our given condition.

Definition 3.1. [1] Let (X, e) be a fuzzy poset. A map c : X → X is called a fuzzy closure operator on (X, e) if,

e (x, c (x)) =1 for all x ∈ X,

c is fuzzy order-preserving, i.e. e (x, y) ≤ e (c (x) , c (y)) for all x, y ∈ X,

e (cc (x) , c (x)) =1 for all x ∈ X.

Definition 3.2. Let (X, e) be a fuzzy complete lattice. Then Σ ∈ L^X is called a fuzzy closure L-system on (X, e) if

Sub (S, Σ) ≤ Σ (⊓ S) for all S ∈ L^X,

Σ (x) ⊗ e (x, y) ⊗ e (y, x) ≤ Σ (y) for all x, y ∈ X.

Remark 3.1. In the Definition 3.2, if L is replaced by a complete Heyting algebra and (X, e) is replaced by a fuzzy poset, respectively, then it is exactly the dual of the fuzzy kernel L-system defined in [28]. Furthermore, this definition degrades into the dual of the fuzzy kernel system in [28] if Σ is a crisp subset of X, i.e., “Σ ∈ L^X”, the item (2) is removed and the item (1) is replaced by “for any S ⊆ χ_Σ, ⊓S ∈ Σ”.

Remark 3.2. Let (X, e) be a fuzzy complete lattice and Σ a fuzzy closure L-system on (X, e). Then for any α ∈ L, Σ_α is a crisp closure system on the crisp complete lattice (X, ≤ _e) defined in the usual sense.

Proof. Suppose Σ is a fuzzy closure L-system on (X, e). We need to prove that ∧H ∈ Σ_α, i . e . , Σ (∧ H) ≥ α for any H ⊆ Σ_α. In fact, from Remark 2.2, we have $\begin{matrix} Σ (\land H) = Σ (⊓ χ_{H}) \geq sub (χ_{H}, Σ) \\ = ⋀_{x \in X} (χ_{H} (x) \to Σ (x)) \\ = (⋀_{x \in H} (χ_{H} (x) \to Σ (x))) \\ \land (⋀_{x \in X \ H} (χ_{H} (x) \to Σ (x))) \\ = (⋀_{x \in H} (1 \to Σ (x))) \land (⋀_{x \in X \ H} (0 \to Σ (x))) \\ = ⋀_{x \in H} Σ (x) \geq α . \end{matrix}$ □

Proposition 3.1. Let (X, e) be a fuzzy complete lattice and Σ a fuzzy closure L-system on (X, e). Then the map c_Σ : X → X (x ↦ ⊓ Σ_x) is a fuzzy closure operator on (X, e), where Σ_x (z) = Σ (z) ⊗ e (x, z) for any z ∈ X.

Proof. It is similar to the proof of Proposition 4.9 in [28]. □

In order to prove that every fuzzy order-preserving map can induce a fuzzy closure L-system, the authors request L to be a complete Heyting algebra in [28]. It is well known that the complete residuated lattice is a generalization of the complete Heyting algebra. In this paper, the structure of truth values is a complete residuated lattice. Thus, we need to add a condition for the purpose of making the above result hold, that is, the fuzzy order e satisfies $\begin{matrix} Condition (*) e (x, y) \land e (y, z) \\ \leq e (x, z) (\forall x, y, z \in X) . \end{matrix}$ Obviously, if the structure of truth values is a complete Heyting algebra, then e satisfies Condition (∗) . The following Example 3.1 illustrates the fact that if L is a complete residuated lattice but not a complete Heyting algebra, there exists a fuzzy order satisfying Condition (∗) .

Example 3.1. Let L = {0, a, b, c, d, 1} be a complete lattice depicted in Fig. 1.

The precomplement operator ¬ is given in Table 1.

The generalized triangular norm ⊗ and the implication operator →in L are defined as follows: for any x, y ∈ L, $\begin{matrix} x \otimes y & = & {\begin{matrix} 0, & x \leq \neg y, \\ x \land y, & x ≰ \neg y . \end{matrix} \\ x \to y & = & {\begin{matrix} 1, & x \leq y, \\ \neg x \lor y, & x ≰ y . \end{matrix} \end{matrix}$ It can be easily verified that (L, ∨ , ∧ , ⊗ , → , 0, 1) is a complete residuated lattice.

Pick X = {x₁, x₂, x₃} and define e : X × X → L as following matrix: $(\begin{matrix} 1 & a & b \\ a & 1 & c \\ b & c & 1 \end{matrix}) .$ Then we can verify that e is a fuzzy order on X. Actually, we only need to verify transitivity here: $\begin{matrix} e (x_{1}, x_{2}) \otimes e (x_{2}, x_{3}) = a \otimes c = 0 \leq e (x_{1}, x_{3}), \\ e (x_{1}, x_{3}) \otimes e (x_{3}, x_{2}) = b \otimes c = 0 \leq e (x_{1}, x_{2}), \\ e (x_{2}, x_{3}) \otimes e (x_{3}, x_{1}) = c \otimes b = 0 \leq e (x_{2}, x_{1}), \\ e (x_{2}, x_{1}) \otimes e (x_{1}, x_{3}) = a \otimes b = 0 \leq e (x_{2}, x_{3}), \\ e (x_{3}, x_{1}) \otimes e (x_{1}, x_{2}) = b \otimes a = 0 \leq e (x_{3}, x_{2}), \\ e (x_{3}, x_{2}) \otimes e (x_{2}, x_{1}) = c \otimes a = 0 \leq e (x_{3}, x_{1}) . \end{matrix}$

Apparently, c ⊗ c = 0 implies that L is not a complete Heyting algebra, however, we claim that e satisfies Condition (∗) , i . e . , e (x, y) ∧ e (y, z) ≤ e (x, z) for any x, y, z ∈ X. In fact, $\begin{matrix} e (x_{1}, x_{2}) \land e (x_{2}, x_{3}) = a \land c = c \leq b = e (x_{1}, x_{3}), \\ e (x_{1}, x_{3}) \land e (x_{3}, x_{2}) = b \land c = c \leq a = e (x_{1}, x_{2}), \\ e (x_{2}, x_{3}) \land e (x_{3}, x_{1}) = c \land b = c \leq a = e (x_{2}, x_{1}), \\ e (x_{2}, x_{1}) \land e (x_{1}, x_{3}) = a \land b = 0 \leq c = e (x_{2}, x_{3}), \\ e (x_{3}, x_{1}) \land e (x_{1}, x_{2}) = b \land a = 0 \leq c = e (x_{3}, x_{2}), \\ e (x_{3}, x_{2}) \land e (x_{2}, x_{1}) = c \land a = c \leq b = e (x_{3}, x_{1}) . \end{matrix}$

Proposition 3.2. Let (X, e) be a fuzzy complete lattice with e satisfying Condition (∗) and h : X → X a fuzzy order-preserving map. Then N_h ∈ L^X is a fuzzy closure L-system on (X, e), where N_h (x) = e (h (x) , x).

Proof. (1) By Condition (∗) and Proposition 2.1 (4), for any S ∈ L^X, we have $\begin{matrix} e (h (⊓ S), ⊓ S) \\ = ⋀_{x \in X} (S (x) \to e (h (⊓ S), x)) \\ \geq ⋀_{x \in X} (S (x) \to (e (h (⊓ S), h (x)) \land e (h (x), x))) \\ \geq ⋀_{x \in X} (S (x) \to (e (⊓ S, x) \land N_{h} (x))) \\ = ⋀_{x \in X} ((S (x) \to e (⊓ S, x)) \land (S (x) \to N_{h} (x))) \\ = ⋀_{x \in X} (1 \land (S (x) \to N_{h} (x))) \\ = ⋀_{x \in X} (S (x) \to N_{h} (x)) = Sub (S, N_{h}) . \end{matrix}$

(2) h is fuzzy order-preserving implies $\begin{matrix} N_{h} (x) \otimes e (x, y) \otimes e (y, x) \\ = e (h (x), x) \otimes e (x, y) \otimes e (y, x) \\ \leq e (h (x), x) \otimes e (x, y) \otimes e (h (y), h (x)) \\ \leq e (h (y), x) \otimes e (x, y) \leq e (h (y), y) = N_{h} (y) . □ \end{matrix}$

The following Example 3.2 expresses that the Condition (∗) is indispensable in the Proposition 3.2.

Example 3.2. Let L be a complete residuated lattice defined in Example 3.1.

Define e_L (x, y) = x → y for any x, y ∈ L, then (L, e_L) is a fuzzy complete lattice where $⊔ A = ⋁_{x \in L} (A (x) \otimes x), ⊓ A = ⋀_{x \in L} (A (x) \to x)$ for all A ∈ L^L (see [26]).

Directly, $\begin{matrix} e_{L} (1, d) \land e_{L} (d, 0) = (1 \to d) \land (d \to 0) \\ = d \land c = c ≰ e_{L} (1, 0) = 1 \to 0 = 0 \end{matrix}$ implies that e_L does not satisfy Condition (∗) .

We consider the map h^a : L → L by x ↦ a → x, obviously, h^a is a fuzzy order-preserving map from Proposition 2.1 (7). We claim that N_{h
^a} is not a fuzzy closure L-system, i.e., Sub (S, N_{h
^a}) ≰ N_{h
^a} (⊓ S). In fact, for any x ∈ L, $\begin{matrix} N_{h^{a}} (x) & = & e_{L} (h^{a} (x), x) = e_{L} (a \to x, x) \\ = & (a \to x) \to x, \end{matrix}$ then we have, $N_{h^{a}} = {\frac{a}{0}, \frac{a}{a}, \frac{1}{b}, \frac{a}{c}, \frac{d}{d}, \frac{1}{1}} .$ Pick $S = {\frac{c}{0}, \frac{c}{a}, \frac{0}{b}, \frac{c}{c}, \frac{b}{d}, \frac{d}{1}} \subseteq N_{h^{a}},$ so, $Sub (S, N_{h^{a}}) = 1 .$ However, $\begin{matrix} ⊓ S = ⋀_{x \in L} (S (x) \to x) \\ = (c \to 0) \land (c \to a) \land (0 \to b) \\ \land (c \to c) \land (b \to d) \land (d \to 1) \\ = d \land 1 \land 1 \land 1 \land 1 \land 1 = d . \end{matrix}$

That is, $\begin{matrix} N_{h^{a}} (⊓ S) = e_{L} (h^{a} (⊓ S), ⊓ S) \\ = e_{L} (a \to ⊓ S, ⊓ S) \\ = e_{L} (a \to d, d) = e_{L} (1, d) \\ = d ≱ 1 = Sub (S, N_{h^{a}}) . \end{matrix}$

The following Theorem 3.1 is similar to Theorem 4.11 in [28], we give a different proof here.

Theorem 3.1. Let (X, e) be a fuzzy complete lattice with e satisfying Condition (∗) , c : X → X a fuzzy closure operator and Σ a fuzzy closure L-system on (X, e). Then c_{Σ
_c} = c and Σ_{c
_Σ} = Σ.

Proof. (1) For any x ∈ X, we have c_{Σ
_c} (x) = ⊓ (Σ_c) _x, where (Σ_c) _x (z) = Σ_c (z) ⊗ e (x, z) = e (c (z) , z) ⊗ e (x, z) for any z ∈ X, then c is a fuzzy closure operator implies that $\begin{matrix} e (c (x), c_{Σ_{c}} (x)) = e (c (x), ⊓ (Σ_{c})_{x}) \\ = ⋀_{z \in X} ((Σ_{c})_{x} (z) \to e (c (x), z)) \\ \geq ⋀_{z \in X} ((Σ_{c})_{x} (z) \to (e (c (x), c (z)) \otimes e (c (z), z))) \\ \geq ⋀_{z \in X} ((Σ_{c})_{x} (z) \to (e (x, z) \otimes e (c (z), z))) = 1 . \end{matrix}$ $\begin{matrix} e (c_{Σ_{c}} (x), c (x)) = e (⊓ (Σ_{c})_{x}, c (x)) \geq (Σ_{c})_{x} (c (x)) \\ = e (cc (x), c (x)) \otimes e (x, c (x)) = 1 . \end{matrix}$ Thus, for any x ∈ X, c_{Σ
_c} (x) = c (x) , i . e . , c_{Σ
_c} = c.

(2) The proof of Σ_{c
_Σ} = Σ is similar to the proof of Theorem 4.11 in [28]. □

4 Algebraic fuzzy closure L-systems and algebraic fuzzy closure operators

At the beginning of this section, we introduce some preliminary knowledge firstly.

Definition 4.1. Let (X, e) be a fuzzy poset. Then for each A ∈ L^X, ↓A ∈ L^X (resp., ↑A ∈ L^X) is defined by $\begin{matrix} (\forall x \in X) (↓ A (x) & = & ⋁_{y \in X} (A (y) \otimes e (x, y))) \\ (resp ., (↑ A (x) & = & ⋁_{y \in X} (A (y) \otimes e (y, x)))) . \end{matrix}$

Definition 4.2. [26] A ∈ L^X is called a fuzzy lower set (resp., a fuzzy upper set) of X if A (x) ⊗ e (y, x) ≤ A (y) (resp., A (x) ⊗ e (x, y) ≤ A (y)) for any x, y ∈ X.

Proposition 4.1. Let (X, e) be a fuzzy complete lattice. Then for any A ∈ L^X,

A ⊆ ↓ A (resp., A ⊆ ↑ A).

↓A (resp., ↑A) is a fuzzy lower (resp., upper) set.

A is a fuzzy lower set (resp., a fuzzy upper set) iff ↓A = A (resp., ↑A = A).

⊔A = ⊔ ↓ A (resp., ⊓A = ⊓ ↑ A).

Proof. (1) For any x ∈ X, ↓A (x) = ⋁ _y∈X (A (y) ⊗ e (x, y)) ≥ A (x) ⊗ e (x, x) = A (x), whence, A ⊆ ↓ A.

(2) By proposition 2.1 (3), for any x, y ∈ X, $\begin{matrix} ↓ A (x) \otimes e (y, x) \\ = (⋁_{m \in X} (A (m) \otimes e (x, m))) \otimes e (y, x) \\ = ⋁_{m \in X} (A (m) \otimes e (x, m) \otimes e (y, x)) \\ \leq ⋁_{m \in X} (A (m) \otimes e (y, m)) = ↓ A (y), \end{matrix}$ so ↓A is a fuzzy lower set.

(3) (⇒) A is a fuzzy lower set implies that A (y) ⊗ e (x, y) ≤ A (x) for any x, y ∈ X, $↓ A (x) = ⋁_{y \in X} (A (y) \otimes e (x, y)) \leq ⋁_{y \in X} A (x) = A (x),$ that is, ↓A ⊆ A. Thus, ↓A = A by (1).

(⟸) This is a direct conclusion by (2).

(4) From Remark 2.3 (1) and (1) in this Proposition, we have e (⊔ A, ⊔ ↓ A) ≥ Sub (A, ↓ A) =1, i.e., e (⊔ A, ⊔ ↓ A) =1.

On the other hand, by Proposition 2.1 (4) and (6), we have $\begin{matrix} e (⊔ ↓ A, ⊔ A) \\ = ⋀_{x \in X} (↓ A (x) \to e (x, ⊔ A)) \\ = ⋀_{x \in X} ((⋁_{y \in X} (A (y) \otimes e (x, y))) \to e (x, ⊔ A)) \\ = ⋀_{x \in X} ⋀_{y \in X} ((A (y) \otimes e (x, y)) \to e (x, ⊔ A)) \\ = ⋀_{x \in X} ⋀_{y \in X} (A (y) \to (e (x, y) \to e (x, ⊔ A))) \\ = ⋀_{y \in X} (A (y) \to ⋀_{x \in X} (e (x, y) \to e (x, ⊔ A))) \\ = ⋀_{y \in X} (A (y) \to e (y, ⊔ A)) = 1 . \end{matrix}$

Thus, ⊔A = ⊔ ↓ A.

The case of ↑A can be proved analogously. □

Definition 4.3. [13 , 26] Let (X, e) be a fuzzy poset. Then D ∈ L^X is called a fuzzy directed subset of X if

⋁_x∈XD (x) =1,

D (x) ⊗ D (y) ≤ ⋁ _z∈XD (z) ⊗ e (x, z) ⊗ e (y, z) for any x, y ∈ X.

Denote $D (X) = {D : D is a fuzzy directedsubset of X}$ .

Definition 4.4. [13, 26] Let (X, e) be a fuzzy poset. Then a fuzzy lower set I ∈ L^X is called a fuzzy ideal of X if it is a fuzzy directed subset of X.

Denote $I (X) = {I : I is a fuzzy ideal of X}$ .

Proposition 4.2. Let (X, e) be a fuzzy complete lattice and h : X → X a fuzzy order-preserving map. Then for any $D \in D (X)$ ,

$↓ D \in I (D)$ .

$h^{\to} (D) \in D (X)$ .

⊔h^→ (↓ D) = ⊔ h^→ (D).

Proof. (1) It can be found in [21], but the authors did not prove it, so we verify it here.

By Proposition 4.1 (2), ↓D is a fuzzy lower set, we claim that it is a fuzzy directed subset. In fact,

⋁_x∈X ↓ D (x) ≥ ⋁ _x∈XD (x) =1 by Proposition 4.1 (1), whence, ⋁_x∈X ↓ D (x) =1.

From Propositions 4.1 (1) and 2.1 (3), we have

\begin{matrix} ↓ D (x) \otimes ↓ D (y) \\ = (⋁_{m \in X} (D (m) \otimes e (x, m))) \\ \otimes (⋁_{n \in X} (D (n) \otimes e (y, n))) \\ = ⋁_{m \in X} ⋁_{n \in X} ((D (m) \otimes D (n)) \otimes e (x, m) \otimes e (y, n)) \\ \leq ⋁_{m \in X} ⋁_{n \in X} ((⋁_{h \in X} D (h) \otimes e (m, h) \otimes e (n, h)) \\ \otimes e (x, m) \otimes e (y, n)) \\ = ⋁_{m \in X} ⋁_{n \in X} ⋁_{h \in X} (D (h) \otimes e (m, h) \otimes e (n, h) \\ \otimes e (x, m) \otimes e (y, n)) \\ \leq ⋁_{h \in X} (D (h) \otimes e (x, h) \otimes e (y, h)) \\ \leq ⋁_{h \in X} (↓ D (h) \otimes e (x, h) \otimes e (y, h)), \end{matrix}

thus,

↓ D \in I (D)

(2) This is a special case of Proposition 5.3 in [26].

(3) For any x ∈ X, we have

(i) h^→ (↓ D) (x) = ⋁ _h(y)=x ↓ D (y) ≥ ⋁ _h(y)=xD (y) = h^→ (D) (x) by Proposition 4.1 (1). This implies h^→ (D) ⊆ h^→ (↓ D).

So, from Remark 2.3 (1), $e (⊔ h^{\to} (D), ⊔ h^{\to} (↓ D)) \geq Sub (h^{\to} (D), h^{\to} (↓ D)) = 1,$ that is e (⊔ h^→ (D) , ⊔ h^→ (↓ D)) =1 .

(ii) Proposition 2.1 (3) and h : X → X is a fuzzy order-preserving map imply $\begin{matrix} h^{\to} (↓ D) (x) = ⋁_{h (y) = x} ↓ D (y) \\ = ⋁_{h (y) = x} (⋁_{m \in X} (D (m) \otimes e (y, m))) \\ \leq ⋁_{h (y) = x} ⋁_{m \in X} (D (m) \otimes e (h (y), h (m))) \\ = ⋁_{m \in X} (D (m) \otimes ⋁_{h (y) = x} e (h (y), h (m))) \\ = ⋁_{m \in X} (D (m) \otimes e (x, h (m))) \\ = ⋁_{l \in X} ((⋁_{h (m) = l} D (m)) \otimes e (x, l)) \\ = ⋁_{l \in X} (h^{\to} (D) (l) \otimes e (x, l)) = ↓ h^{\to} (D) (x), \end{matrix}$ so, by Proposition 4.1 (4) and Remark 2.3 (1), $\begin{matrix} e (⊔ h^{\to} (↓ D), ⊔ h^{\to} (D)) & = & e (⊔ h^{\to} (↓ D), ⊔ ↓ h^{\to} (D)) \\ \geq & Sub (h^{\to} (↓ D), ↓ h^{\to} (D)) \\ = & 1, \end{matrix}$ i.e., e (⊔ h^→ (↓ D) , ⊔ h^→ (D)) =1 . Thus, ⊔h^→ (↓ D) = ⊔ h^→ (D). □

In the following, based on above mentioned results, we investigated algebraic fuzzy closure operators and algebraic fuzzy closure L-systems on fuzzy complete lattices.

Definition 4.5. Let (X, e) be a fuzzy complete lattice and c a fuzzy closure operator. Then c is called an algebraic fuzzy closure operator on (X, e) if for any $D \in D (X)$ , c (⊔ D) = ⊔ c^→ (D).

Lemma 4.1. A fuzzy closure operator is algebraic iff for any $I \in I (X)$ , c (⊔ I) = ⊔ c^→ (I).

Proof. (⇒) It is obvious.

(⟸) By Propositions 4.1 (4) and 4.2 (3), we have c (⊔ D) = c (⊔ ↓ D) = ⊔ c^→ (↓ D) = ⊔ c^→ (D) for any $D \in D (X)$ . □

Definition 4.6. Let (X, e) be a fuzzy complete lattice and Σ a fuzzy closure L-system on (X, e). Then Σ is called an algebraic fuzzy closure L-system on (X, e) if for any $D \in D (X)$ , Sub (D, Σ) ≤ Σ (⊔ D).

The following Remark 4.1 is similar to Remark 3.2, so we omit the proof here.

Remark 4.1. Let (X, e) be a fuzzy complete lattice, Σ an algebraic fuzzy closure L-system on (X, e), then for any α ∈ L, Σ_α is an algebraic crisp closure system on the crisp complete lattice (X, ≤ _e) defined in the usual sense.

Theorem 4.1. Let (X, e) be a fuzzy complete lattice and Σ an algebraic fuzzy closure L-system on (X, e). Then c_Σ is an algebraic fuzzy closure operator on (X, e), where c_Σ is defined as in Proposition 3.1.

Proof. From Proposition 3.1, c_Σ is a fuzzy closure operator. Thus, we only need to prove that c_Σ is algebraic here, that is for any $I \in I (X)$ , $c_{Σ} (⊔ I) = ⊔ c_{Σ}^{\to} (I)$ by Lemma 4.1.

(1) c_Σ is a fuzzy closure operator implies $\begin{matrix} e (⊔ c_{Σ}^{\to} (I), c_{Σ} (⊔ I)) \\ = ⋀_{x \in X} (c_{Σ}^{\to} (I) (x) \to e (x, c_{Σ} (⊔ I))) \\ = ⋀_{x \in X} ((⋁_{c_{Σ} (m) = x} I (m)) \to e (x, c_{Σ} (⊔ I))) \\ = ⋀_{x \in X} ⋀_{c_{Σ} (m) = x} ((I (m) \to e (x, c_{Σ} (⊔ I))) \\ = ⋀_{m \in X} (I (m) \to e (c_{Σ} (m), c_{Σ} (⊔ I))) \\ \geq ⋀_{m \in X} (I (m) \to e (m, ⊔ I)) = 1, \end{matrix}$ that is, $e (⊔ c_{Σ}^{\to} (I), c_{Σ} (⊔ I)) = 1$ .

(2) $\begin{matrix} e (c_{Σ} (⊔ I), ⊔ c_{Σ}^{\to} (I)) = e (⊓ Σ_{⊔ I}, ⊔ c_{Σ}^{\to} (I)) \\ \geq Σ_{⊔ I} (⊔ c_{Σ}^{\to} (I)) = Σ (⊔ c_{Σ}^{\to} (I)) \otimes e (⊔ I, ⊔ c_{Σ}^{\to} (I)) . \end{matrix}$

In fact, (i) By Proposition 4.2 (2), $c_{Σ}^{\to} (I) \in D (X)$ , so that Σ is an algebraic fuzzy closure L-system reveals that $\begin{matrix} Σ (⊔ c_{Σ}^{\to} (I)) \geq Sub (c_{Σ}^{\to} (I), Σ) \\ = ⋀_{x \in X} (c_{Σ}^{\to} (I) (x) \to Σ (x)) \\ = ⋀_{x \in X} (⋁_{c_{Σ} (y) = x} I (y) \to Σ (x)) \\ = ⋀_{x \in X} ⋀_{c_{Σ} (y) = x} (I (y) \to Σ (x)) \\ = ⋀_{y \in X} (I (y) \to Σ (c_{Σ} (y))) \\ = ⋀_{y \in X} (I (y) \to Σ (⊓ Σ_{y})) \\ = ⋀_{y \in X} (I (y) \to 1) = 1, \end{matrix}$ that is, $Σ (⊔ c_{Σ}^{\to} (I)) = 1$ .

(ii) I is a fuzzy lower set implies that I (x) ⊗ e (y, x) ≤ I (y) for any x, y ∈ X, that is, I (x) → I (y) ≥ e (y, x) for any x, y ∈ X by Theorem 2.1 (1), thus $\begin{matrix} e (⊔ I, ⊔ c_{Σ}^{\to} (I)) = ⋀_{x \in X} (I (x) \to e (x, ⊔ c_{Σ}^{\to} (I))) \\ \geq ⋀_{x \in X} (I (x) \to c_{Σ}^{\to} (I) (x)) \\ = ⋀_{x \in X} (I (x) \to ⋁_{c_{Σ} (y) = x} I (y)) \\ = ⋀_{x \in X} ⋁_{c_{Σ} (y) = x} (I (x) \to I (y)) \\ = ⋀_{y \in X} (I (c_{Σ} (y)) \to I (y)) \\ \geq ⋀_{y \in X} e (y, c_{Σ} (y)) = 1, \end{matrix}$ that is, $e (⊔ I, ⊔ c_{Σ}^{\to} (I)) = 1$ .

Hence, $e (c_{Σ} (⊔ I), ⊔ c_{Σ}^{\to} (I)) = 1$ .

Therefore, $c_{Σ} (⊔ I) = ⊔ c_{Σ}^{\to} (I)$ , that is, c_Σ is algebraic. □

Theorem 4.2. Let (X, e) be a fuzzy complete lattice with e satisfying Condition (∗) and c an algebraic fuzzy closure operator on (X, e). Then Σ_c is an algebraic fuzzy closure L-system on (X, e), where Σ_c is defined as in Proposition 3.2.

Proof. From Proposition 3.2, Σ_c is a fuzzy closure L-system on (X, e). Thus, we only need to prove that Σ_c is algebraic here. Since c is algebraic and e satisfies Condition (∗) , for any $D \in D (X)$ , $\begin{matrix} Σ_{c} (⊔ D) & = & e (c (⊔ D), ⊔ D) = e (⊔ c^{\to} (D), ⊔ D) \\ = & ⋀_{x \in X} (c^{\to} (D) (x) \to e (x, ⊔ D)) \\ = & ⋀_{x \in X} ((⋁_{c (m) = x} D (m)) \to e (x, ⊔ D)) \\ = & ⋀_{m \in X} (D (m) \to e (c (m), ⊔ D)) \\ \geq & ⋀_{m \in X} (D (m) \to (e (c (m), m) \land e (m, ⊔ D))) \\ = & ⋀_{m \in X} ((D (m) \to Σ_{c} (m)) \\ \land (D (m) \to e (m, ⊔ D))) \\ = & ⋀_{m \in X} ((D (m) \to Σ_{c} (m)) \land 1) \\ = & ⋀_{m \in X} (D (m) \to Σ_{c} (m)) = Sub (D, Σ_{c}) . \end{matrix}$ Thus, Σ_c is algebraic. □

5 The category of algebraic fuzzy closure L-systems

In this section, two categories are introduced, called the category of (algebraic) fuzzy closure operator spaces and the category of (algebraic) fuzzy closure L-system paces, respectively. A(n) (algebraic) fuzzy closure operator space is a triple (X, e, c) consisting of a set X, a fuzzy order e satisfying Condition (∗) and a(n) (algebraic) fuzzy closure operator c on (X, e). A map f : X → Y is called a(n) (algebraic) fuzzy continuous map between (algebraic) fuzzy closure operator spaces (X, e_X, c^X) and (Y, e_Y, c^Y) if e_X (x₁, x₂) = e_Y (f (x₁) , f (x₂)) for any x₁, x₂ ∈ X and f (c^X (x)) = c^Y (f (x)) for any x ∈ X. Alternatively, one may introduce (algebraic) fuzzy closure L-system spaces as triples (X, e, Σ) where e is a fuzzy order and Σ a(n) (algebraic) fuzzy closure L-system on (X, e). A map g : X → Y is called a(n) (algebraic) fuzzy continuous map between (algebraic) fuzzy closure L-system spaces (X, e_X, Σ^X) and (Y, e_Y, Σ^Y) if e_X (x₁, x₂) = e_Y (g (x₁) , g (x₂)) for any x₁, x₂ ∈ X and Σ^Y (y) = Σ^X (g^-1 (y)) for any y ∈ Y.

Then we can obtain the following conclusions.

Proposition 5.1. (1) (Algebraic) fuzzy closure operator spaces together with (algebraic) fuzzy continuous maps as morphisms form a category (AL -) COS.

(2) (Algebraic) fuzzy closure L-system spaces together with (algebraic) fuzzy continuous maps as morphisms form a category (AL -) CLSS.

Proof. (1) Let f₁ ∈ Hom_COS (X, Y), f₂ ∈ Hom_COS (Y, Z), and Id_X : X → X an identity map. Then, for any x ∈ X, $\begin{matrix} (f_{2} \circ f_{1}) (c^{X} (x)) & = & f_{2} (f_{1} (c^{X} (x))) \\ = & f_{2} (c^{Y} (f_{1} (x))) \\ = & c^{Z} (f_{2} (f_{1} (x))) \\ = & c^{Z} ((f_{2} \circ f_{1}) (x)), \end{matrix}$ which implies f₂ ∘ f₁ ∈ Hom_COS (X, Z). Obviously, Id_X ∈ Hom_COS (X, X).

(2) Let g₁ ∈ Hom_CLSS (X, Y), g₂ ∈ Hom_CLSS (Y, Z), and Id_X : X → X an identity map. Then, for any y ∈ Y, $Σ^{X} ((g_{2} \circ g_{1})^{- 1} (z)) = Σ^{X} (g_{1}^{- 1} (g_{2}^{- 1} (z))) = Σ^{Y} (g_{2}^{- 1} (z)) = Σ^{Z} (z),$ which implies g₂ ∘ g₁ ∈ Hom_CLSS (X, Z). Obviously, Id_X ∈ Hom_CLSS (X, X). □

Proposition 5.2. (1) For all (X, e_X, c^X), (Y, e_Y, c^Y) ∈ ob (COS) and f ∈ Hom_COS (X, Y), let $𝕂 ((X, e_{X}, c^{X})) = (X, e_{X}, Σ_{c^{X}})$ and $𝕂 (f) = f$ . Then $𝕂$ is a functor from the category COS to the category CLSS, i.e., the following diagram is commutative. $\begin{array}{l} (X, e_{X}, c^{X}) \overset{f}{\to} (Y, e_{Y}, c^{Y}) \\ K ↓ K ↓ \\ (X, e_{X}, Σ_{c^{X}}) \overset{K (f) = f}{\to} (Y, e_{Y}, Σ_{c^{Y}}) \end{array}$

(2) For all (X, e_X, Σ^X), (Y, e_Y, Σ^Y) ∈ ob (CLSS) and g ∈ Hom_CLSS (X, Y), let $ℍ ((X, e_{X}, Σ^{X})) = (X, e_{X}, c_{Σ^{X}})$ and $ℍ (f) = f$ . Then $ℍ$ is a functor from the category CLSS to the category COS, i.e., the following diagram is commutative. $\begin{array}{l} (X, e_{X}, Σ^{X}) \overset{g}{\to} (Y, e_{Y}, Σ^{Y}) \\ ℍ ↓ ℍ ↓ \\ (X, e_{X}, c_{Σ^{X}}) \overset{ℍ (g) = g}{\to} (Y, e_{Y}, c_{Σ^{Y}}) \end{array}$

Proof. (1) (i) By Proposition 3.2, we have (X, e_X, Σ_{c
^X}) ∈ob (CLSS).

(ii) For f ∈ Hom_COS (X, Y), we have $\begin{matrix} Σ_{c}^{X} (f^{- 1} (y)) \\ = e_{X} (c^{X} (f^{- 1} (y)), f^{- 1} (y)) \\ = e_{Y} (f (c^{X} (f^{- 1} (y))), f (f^{- 1} (y))) \\ = e_{Y} (c^{Y} (f (f^{- 1} (y))), f (f^{- 1} (y))) \\ = e_{Y} (c^{Y} (y), y) = Σ_{c}^{Y} (y) (\forall y \in Y), \end{matrix}$ that is, $𝕂 (f) = f \in {Hom}_{CLSS} (X, Y)$ .

(2) Similar to (1), we only need to show that for any g ∈ Hom_CLSS (X, Y), $ℍ (g) = g \in {Hom}_{COS} (X, Y)$ , $i . e ., g (c_{Σ^{X}} (x)) = c_{Σ^{Y}} (g (x)) = ⊓ Σ_{g (x)}^{Y}$ for any x ∈ X. In fact, $\begin{matrix} e (⊓ Σ_{g (x)}^{Y}, g (c_{Σ^{X}} (x))) \geq Σ_{g (x)}^{Y} (g (c_{Σ^{X}} (x))) \\ = Σ^{Y} (g (c_{Σ^{X}} (x))) \otimes e_{Y} (g (x), g (c_{Σ^{X}} (x))) \\ = Σ^{X} (c_{Σ^{X}} (x)) \otimes e_{X} (x, c_{Σ^{X}} (x)) = 1 . \end{matrix}$ On the other hand, by Proposition 2.1 (6) and (2), we have $\begin{matrix} e_{Y} (g (c_{Σ^{X}} (x)), ⊓ Σ_{g (x)}^{Y}) \\ = ⋀_{q \in Y} (Σ_{g (x)}^{Y} (q) \to e_{Y} (g (c_{Σ^{X}} (x)), q)) \\ = ⋀_{q \in Y} ((Σ^{Y} (q) \otimes e_{Y} (g (x), q)) \\ \to e_{Y} (g (c_{Σ^{X}} (x)), q)) \\ = ⋀_{q \in Y} (e_{Y} (g (x), q) \to (Σ^{Y} (q) \\ \to e_{Y} (g (c_{Σ^{X}} (x)), q))) \\ = ⋀_{q \in Y} (e_{Y} (g (x), q) \to (Σ^{X} (g^{- 1} (q)) \\ \to e_{X} (c_{Σ^{X}} (x), g^{- 1} (q)))) \\ = ⋀_{q \in Y} (e_{Y} (g (x), q) \to (Σ^{X} (g^{- 1} (q)) \\ \to e_{X} (⊓ Σ_{x}^{X}, g^{- 1} (q)))) \\ \geq ⋀_{q \in Y} (e_{Y} (g (x), q) \to (Σ^{X} (g^{- 1} (q)) \\ \to Σ_{x}^{X} (g^{- 1} (q)))) \\ = ⋀_{q \in Y} (e_{Y} (g (x), q) \to (Σ^{X} (g^{- 1} (q)) \\ \to (Σ^{X} (g^{- 1} (q)) \otimes e_{X} (x, g^{- 1} (q))))) \\ \geq ⋀_{q \in Y} (e_{Y} (g (x), q) \to e_{X} (x, g^{- 1} (q))) \\ = ⋀_{q \in Y} (e_{X} (x, g^{- 1} (q)) \to e_{X} (x, g^{- 1} (q))) = 1 . \end{matrix}$

We conclude that g (c_{Σ
^X} (x)) = c_{Σ
^Y} (g (x)). □

By Propositions 5.1 and 5.2 above, we obtain the following Theorem 5.1.

Theorem 5.1. The categories COS and CLSS are isomorphic.

Proof. From Theorem 3.1 and Propositions 5.1 and 5.2 above, they are isomorphic, i.e., the following diagram is commutative. $\begin{array}{l} (X, e_{X}, c^{X}) \overset{f}{\to} (Y, e_{Y}, c^{Y}) \\ I d_{X} ↓ I d_{Y} ↓ \\ ℍ (K (X, e_{X}, c^{X})) \overset{ℍ (K (f)) = f}{\to} ℍ (K (Y, e_{Y}, c^{Y})) \\ = (X, e_{X}, c^{X}) = (Y, e_{Y}, c^{Y}) \end{array}$

□

AL - COS and AL - CLSS are full categories of COS and CLSS, respectively. Thus, we have the following Corollary 5.1.

Corollary 5.1. The categories AL - COS and AL - CLSS are isomorphic.

6 Conclusions

In this work, we discuss the relationship between fuzzy closure operators and fuzzy closure L-systems based on complete residuated lattice, establish a “one-to-one” correspondence between fuzzy closure operators and fuzzy closure L-systems under our given condition on fuzzy order. We also propose the notions of the algebraic fuzzy closure operators and the algebraic fuzzy closure L-systems on fuzzy complete lattices, which generalizes algebraic closure operators and the algebraic closure systems on classical setting. Moreover, in order to further exploit the relationship between (algebraic) fuzzy closure operators and (algebraic) fuzzy closure L-systems from the category view, we obtain that the category of (algebraic) fuzzy closure operators is isomorphic to the category of (algebraic) fuzzy closure L-systems.

Footnotes

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (No. 11371130) and Research Fund for the Doctoral Program of Higher Education of China (No. 20120161110017).

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