Cartesian-closedness and subcategories of (L, M)-fuzzy Q-convergence spaces

In this paper, we first construct the function space of (L, M)-fuzzy Q-convergence spaces to show the Cartesian-closedness of the category (L, M)-QC of (L, M)-fuzzy Q-convergence spaces. Secondly, we introduce several subcategories of (L, M)-QC, including the category (L, M)-KQC of (L, M)-fuzzy Kent Q-convergence spaces, the category (L, M)-LQC of (L, M)-fuzzy Q-limit spaces and the category (L, M)-PQC of (L, M)-fuzzy pretopological Q-convergence spaces, and investigate their relationships.


Introduction
In general topology, function spaces of topological spaces cannot be constructed in a satisfactory way. This means the category of topological spaces with continuous mappings as morphisms is not Cartesian-closed. In order to overcome this deficiency, the concept of filter convergence spaces (convergence spaces in short) was proposed and discussed (Choquet 1948;Fischer 1959;Kent 1964;Kowalsky 1954). In Preuss (2002), Preuss gave a systematical collection of convergence structures, including function spaces and subcategories of convergence spaces as well as their connections with topological spaces.
With the development of fuzzy set theory, many mathematical structures have been generalized to the fuzzy case (Arqub and Al-Smadi 2020;Arqub et al. 2016Arqub et al. , 2017Li and Wang 2020;Xiu 2020;Zhang and Pang 2020). In the theory of fuzzy topology (Chang 1968;Kubiak 1985;Šostak 1985), many types of fuzzy convergence structures have been proposed, such as stratified L-generalized convergence structure (Jäger 2001(Jäger , 2016bJin 2012, 2014) (Pang 2018;Xu 2001;Yao 2009), L-convergence tower structure (Flores et al. 2006;Jäger 2016a;Pang 2019), L-ordered convergence structure (Fang 2010a, b), (Enriched) (L, M)-fuzzy (Q-)convergence structure (Pang 2014a, b;Zhao 2016, 2017),convergence structure Yue 2017, 2021;Jin et al. 2019;Yu and Fang 2017;Yue and Fang 2020) and so forth. Fuzzy convergence structures are usually discussed from two aspects. On the one hand, the categorical relationship between fuzzy convergence structures and fuzzy topologies is discussed. For example, Yu and Fang (Yu and Fang 2017) showed that the category of strong L-topological spaces can be embedded in the category of -convergence spaces as a reflective subcategory and the category of topological -convergence spaces is isomorphic to that of strong Ltopological spaces. On the other hand, the categorical properties of fuzzy convergence spaces are investigated. Zhang et al. (2019) showed the monoidal closedness of the category of L-generalized convergence spaces. Pang and Zhao (2017) established the categorical properties among subcategories of enriched (L, M)-fuzzy convergence spaces. Recently, Pang (Pang 2018(Pang , 2019 discussed the Cartesian-closedness, extensionality and productivity of quotient mappings of subcategories of L-fuzzifying convergence spaces and stratified L-generalized convergence tower spaces. In the theory of fuzzy convergence spaces, many researchers usually show the Cartesian-closedness of fuzzy convergence spaces by constructing the corresponding function space, i.e., the power object in the category of fuzzy convergence spaces. Actually, there are different approaches to show the Cartesian-closedness of a category (Preuss 2002). For example, a topological category A is Cartesian-closed if and only if the functor A × − : A −→ A : B −→ A × B preserves final epi-sinks for each object A in A. In this approach, Pang and Li showed the Cartesian-closedness of the categories of (L, M)-fuzzy convergence spaces (Pang 2014b) and L-fuzzy Q-convergence spaces (Li 2016), respectively. Later, (Pang and Zhao 2016) introduced the concept of stratified (L, M)fuzzy Q-convergence spaces and proved that the resulting category is Cartesian-closed. From a theoretical aspect, Cartesian-closedness of a category ensures the existence of its corresponding function space. However, the researchers failed to construct the corresponding function space although they showed the Cartesian-closedness of the categories of their corresponding fuzzy convergence spaces (Li 2016;Pang 2014b;Pang and Zhao 2016). By this motivation, we will focus on the function space of (L, M)-fuzzy Q-convergence spaces (called stratified (L, M)-fuzzy Q-convergence spaces in Pang and Zhao (2016)), which is an essential part of the theory of (L, M)-fuzzy Q-convergence spaces. Concretely, we will construct the concrete form of the corresponding function space of (L, M)-fuzzy Q-convergence spaces. Moreover, as generalizations of Kent convergence spaces, limit spaces and pretopological convergence spaces, we will introduce several types of (L, M)-fuzzy Q-convergence spaces, including (L, M)-fuzzy Kent Q-convergence spaces, (L, M)-fuzzy Q-limit spaces, and (L, M)-fuzzy pretopological Q-convergence spaces, and then study their mutual relationships from a categorical aspect. This paper is organized as follows. In Sect. 2, we recall some necessary concepts and notations. In Sect. 3, we construct the function space of (L, M)-fuzzy Q-convergence structures to show the Cartesian-closedness of the resulting category. In Sects. 4-6, we propose the concepts of (L, M)fuzzy Kent Q-convergence spaces, (L, M)-fuzzy Q-limit spaces, and (L, M)-fuzzy pretopological Q-convergence spaces and investigate their categorical relationships.

Preliminaries
Throughout this paper, both L and M denote completely distributive lattices and is an order-reversing involution on L. The smallest element and the largest element in L (M) are denoted by ⊥ L (⊥ M ) and L ( M ), respectively. For a, b ∈ L, we say that a is wedge below For a nonempty set X , L X denotes the set of all L-subsets on X . L X is also a complete lattice when it inherits the structure of the lattice L in a natural way, by defining ∨, ∧ and ≤ pointwisely. The smallest element and the largest element in L X are denoted by ⊥ X L and X L , respectively. For each x ∈ X and a ∈ L, the L-subset x a , defined by x a (y) = a if y = x, and x a (y) = ⊥ L if y = x, is called a fuzzy point. The set of nonzero coprime elements in L X is denoted by J (L X ). It is easy to see that J (L X ) = {x λ | x ∈ X , λ ∈ J (L)}. We say that a fuzzy point x λ quasi-coincides with A, denoted by The family of all (L, M)-fuzzy filters on X is denoted by F L M (X ). (Pang 2014a) For each x λ ∈ J (L X ), we definê q(x λ ) : L X −→ M as follows:

Example 2.2
Then,q(x λ ) is an (L, M)-fuzzy filter on X .
On the set F L M (X ) of all (L, M)-fuzzy filters on X , we define an order by F ≤ G if F(A) ≤ G(A) for all A ∈ L X . Then for a family of (L, M)-fuzzy filters {F j | j ∈ J }, the infimum is given by For an (L, M)-fuzzy Q-convergence structure q on X , the pair (X , q) is called an (L, M)-fuzzy Q-convergence space.
A continuous mapping between (L, M)-fuzzy Qconvergence spaces (X , q X ) and (Y , q Y ) is a mapping f : It is easy to check that (L, M)-fuzzy Q-convergence spaces and their continuous mappings form a category, denoted by (L, M)-QC.
Example 2.4 (Pang and Zhao 2016) Let X be a nonempty set.
(1) Define qc * : F L M (X ) −→ L X as follows: It is easy to verify that qc * is an (L, M)-fuzzy Qconvergence structure on X .
(2) Define qc * : F L M (X ) −→ L X as follows: It is easy to check that qc * is an (L, M)-fuzzy Qconvergence structure on X .
In order to provide an example from the aspect of fuzzy topology, we first recall the following definition.
Definition 2.5 (Höhle and Šostak 1999) A stratified (L, M)fuzzy topology on X is a mapping τ : L X −→ M which satisfies: For a stratified (L, M)-fuzzy topology τ on X , the pair (X , τ ) is called a stratified (L, M)-fuzzy topological space.
Example 2.6 (Pang and Zhao 2016) Let (X , τ ) be a stratified (L, M)-fuzzy topological space and define qc τ : F L M (X ) −→ L X as follows: Notice that (L, M)-fuzzy Q-convergence structures in Definition 2.3 are exactly stratified (L, M)-fuzzy Qconvergence structures in Pang and Zhao (2016). In this paper, we will focus on this kind of fuzzy convergence structures and explore the concrete form of its function spaces as well as its subcategories.
Definition 2.7 (Pang and Zhao 2016) Let {(X j , q j )} j∈J be a family of (L, M)-fuzzy Q-convergence spaces and { p k :

Function space of (L, M)-fuzzy Q-convergence spaces
In this section, we will construct the function space of (L, M)-fuzzy Q-convergence spaces. By means of the constructed function space, we will show the Cartesianclosedness of (L, M)-QC.
In order to guarantee the existence of the product of (L, M)-fuzzy filters, we assume that ⊥ L is prime in this section. Let , we denote two subsets of L as follows: In order to show q [X ,Y ] is an (L, M)-fuzzy Q-convergence structure on [X , Y ], the following lemma is necessary.
For (1), take each μ ∈ J (L) with μ ≤ λ and a ∈ L with μ a . Then it follows that λ a , which means f λq a, By (1) and (2), we have Then, there exists ν a ∈ J (L) such that ν a a and for each This shows the continuity of ev.
where the third equality holds since p Y •x = id Y . Now for each A ∈ L X with x μq A, i.e., μ A (x), it follows from y μ ≤ q Y (G) and (LMQC3) that G(A(x)) = M . Then, where the second quality holds since , we obtain f x = f •x (as the composition of two continuous mappingŝ x and f ) is continuous, as desired.

Theorem 3.7 The category (L, M)-QC is Cartesian-closed.
Actually, Pang and Zhao (2016) showed the Cartesianclosedness of the category of (L, M)-fuzzy Q-convergence spaces (which is called stratified (L, M)-fuzzy Qconvergence space in Pang and Zhao (2016)). However, they failed to construct the corresponding function spaces. In this section, we provide the concrete form of the corresponding function spaces, which gives an answer to the question proposed in Pang and Zhao (2016).

(L, M)-fuzzy Kent Q-convergence spaces
In this section, we will generalize the notion of Kent convergence spaces to the (L, M)-fuzzy case and study its relationship with (L, M)-fuzzy Q-convergence spaces.

Then q r is an (L, M)-fuzzy Kent Q-convergence structure on X .
Proof It is enough to show that q r satisfies (LMQC1)-(LMQC3) and (LMKQC). Indeed, (LMQC1) and (LMQC2) are straightforward.
(LMQC3) Take each x λ ∈ J (L X ), F ∈ F L M (X ) and a ∈ L such that x λ ≤ q r (F) and λ a . This implies Then there exists λ a ∈ J (L) such that λ a a and there exists G ∈ F L M (X ) such that x λ a ≤ q(G) and G ∧q(x λ a ) ≤ F. Since q satisfies (LMQC3), it follows from x λ a ≤ q(G) and λ a a that G(a) = M , and further F(a) ≥ G(a) ∧ q(x λ a )(a) = M .
(2) For each (L, M)-fuzzy Kent Q-convergence space (Y , q Y ) and each mapping f : X −→ Y , the continuity of f : For (1), it is easy to verify that q(F) ≤ q r (F) for each F ∈ F L M (X ).
For (2), take each x λ ∈ J (L X ) and F ∈ F L M (X ) such that This implies f (x) μ ≤ q Y ( f ⇒ (F)). By the arbitrariness of μ, we obtain f (x) λ ≤ q Y ( f ⇒ (F)). This proves the continuity of f : (X , q r ) −→ (Y , q Y ).
Then, q c is an (L, M)-fuzzy Kent Q-convergence structure on X .
Proof (LMQC1) and (LMQC2) are easy to be verified and omitted.
(LMQC3) Take each x λ ∈ J (L X ), F ∈ F L M (X ) and a ∈ L such that x λ ≤ q c (F) and λ a . It follows that Then, there exists λ a ∈ J (L) such that λ a a and for each μ ≺ λ a , x μ ≤ q(F ∧q(x μ )). Since λ a a , there exists μ a ≺ λ a such that μ a a . This implies x μ a ≤ q(F ∧q(x μ a )) and μ a a . Since q satisfies (LMQC3), we Then, there exists λ 1 ∈ J (L) such that ν ≤ λ 1 and for each μ ≺ λ 1 , x μ ≤ q(F ∧q(x μ )). Thus, for each μ ∈ J (L) with μ ≺ ν, it follows that This implies By the arbitrariness of ν, we obtain λ ≤ q c (F ∧q(x λ ))(x), that is, x λ ≤ q c (F ∧q(x λ )), as desired. (1) id X : (X , q c ) −→ (X , q) is continuous.
(2) For each (L, M)-fuzzy Kent Q-convergence space (Y , q Y ) and each mapping f : Y −→ X , the continu- For (1), it is easy to show q c (F) ≤ q(F) for each F ∈ F L M (X ).
For (2), take each G ∈ F L M (Y ) and y λ ∈ J (L Y ) such that y λ ≤ q Y (G). Then, for each μ ≺ λ, it follows that ( f (y) μ )). From the definition of q c , we get This shows f (y) λ ≤ q c ( f ⇒ (G)), as desired.

Lemma 4.6 (Preuss 2002) Suppose that A is a topological category. If B is a bicoreflective (full and isomorphic closed) subcategory of A which is closed under formation of finite products in A, then B is Cartesian-closed whenever A is
Cartesian-closed.

(L, M)-fuzzy Q-limit spaces
In this section, we will propose the concept of (L, M)-fuzzy Q-limit spaces, which is a generalization of limit spaces in general topology. Then, we will study its relationship with (L, M)-fuzzy Kent Q-convergence spaces from a categorical aspect.
Definition 5.1 An (L, M)-fuzzy Q-convergence structure q on X is called an (L, M)-fuzzy Q-limit structure if it satisfies For an (L, M)-fuzzy Q-limit structure q on X , the pair (X , q) is called an (L, M)-fuzzy Q-limit space.
The full subcategory of (L, M)-QC, consisting of (L, M)fuzzy Q-limit spaces, is denoted by (L, M)-LQC.
In order to show the further relationship between (L, M)fuzzy Kent Q-convergence spaces and (L, M)-fuzzy Q-limit spaces, we first give the following lemma.
Lemma 5.2 Let (X , q) be an (L, M)-fuzzy Kent Qconvergence space and define q l : Then, q l is an (L, M)-fuzzy Q-limit structure on X .
(LMQC3) Take each F ∈ F L M (X ), x λ ∈ J (L X ) and a ∈ L such that x λ ≤ q l (F) and λ a . Then, q l (F)(x) a . By the definition of q l (F), there exists λ ∈ J (L) such that λ a and there exist F 1 , it follows that μ ≺ q l (F)(x) and μ ≺ q l (G)(x). Then, there exist λ 1 , λ 2 ∈ J (L) and F 1 , F 2 , . . . , By the arbitrariness of μ, we obtain λ ≤ q l (F ∧ G)(x), that is, x λ ≤ q l (F ∧ G), as desired. Proof Let (X , q) be an (L, M)-fuzzy Kent Q-convergence space. By Lemma 5.2, we know q l is an (L, M)-fuzzy Qlimit structure on X . Next we claim that id X : (X , q) −→ (X , q l ) is the (L, M)-LQC-bireflector. For this, it suffices to verify (1) id X : (X , q) −→ (X , q l ) is continuous.
(2) For each (L, M)-fuzzy Q-limit space (Y , q Y ) and each mapping f : X −→ Y , the continuity of f : For (1), it follows immediately from q(F) ≤ q l (F) for each F ∈ F L M (X ).
For (2), take each F ∈ F L M (X ) and x λ ∈ J (L X ) such that x λ ≤ q l (F). Then, for each μ ≺ λ, there exists λ μ ∈ J (L) such that μ ≤ λ μ and there exist F 1 , . . . , Next we discuss the Cartesian-closedness of (L, M)-LQC. To this end, the following two lemmas are necessary.

Lemma 5.7 (Preuss 2002) Suppose that A is a topological category. If B is a bireflective (full and isomorphic closed) subcategory of A which is closed under formation of power objects in A, then B is Cartesian-closed whenever A is
Cartesian-closed.
Proof It follows immediately from Theorems 2.8, 5.3 and 5.6, and Lemma 5.7.

Remark 5.9
It is required that ⊥ L should be prime in several conclusions. This requirement seems to be strong. However, the real unit interval I = [0, 1] at least fulfils this requirement. Moreover, I fulfills the assumption of being completely distributive lattice with an order reversing involution.

(L, M)-fuzzy pretopological and topological Q-convergence spaces
In this section, we will introduce the concept of (L, M)fuzzy pretopological Q-convergence spaces and discuss its relationship with (L, M)-fuzzy Q-limit spaces and (L, M)fuzzy topological Q-convergence spaces (Pang and Zhao 2016). For this, we first recall the following notation.
Then F q x λ is an (L, M)-fuzzy filter on X satisfying F q x λ ≤ q(x λ ). Definition 6.1 An (L, M)-fuzzy Q-convergence structure q on X is called pretopological if it satisfies For an (L, M)-fuzzy pretopological Q-convergence structure q on X , the pair (X , q) is called an (L, M)-fuzzy pretopological Q-convergence space.
Lemma 6.3 Let (X , q) be an (L, M)-fuzzy Q-limit space and define q p : F L M (X ) −→ L X by Then q p is an (L, M)-fuzzy pretopological Q-convergence structure on X .
Proof (LMQC1) and (LMQC2) are straightforward. (LMQC3) Take each F ∈ F L M (X ), x λ ∈ J (L X ) and a ∈ L such that x λ ≤ q p (F) and λ a . It follows that Then, there exists λ a ∈ J (L) such that F q x λa ≤ F and λ a a . This implies (LMPQC) For each x λ ∈ J (L X ) and F ∈ F L M (X ) with x λ ≤ q p (F), take each μ ∈ J (L) such that μ ≺ λ. It follows that Then, there exists ν ∈ J (L) such that μ ≤ ν and F q x ν ≤ F. This implies F q By the arbitrariness of μ, we get λ ≤ q p (F q p x λ )(x), i.e., x λ ≤ q p (F q p x λ ), as desired. Theorem 6.4 (L, M)-PQC is a bireflective subcategory of (L, M)-LQC.
Proof Let (X , q) be an (L, M)-fuzzy Q-limit convergence space. By Lemma 6.3, we know q p is an (L, M)-fuzzy pretopological Q-convergence structure on X . Next we claim that id X : (X , q) −→ (X , q p ) is the (L, M)-PQCbireflector. For this, it suffices to verify (1) id X : (X , q) −→ (X , q p ) is continuous.
For (1), take each x λ ∈ J (L X ) and F ∈ F L M (X ) such that x λ ≤ q(F). Then it follows that F q x λ ≤ F, which means x λ ≤ q p (F). This shows q(F) ≤ q p (F).
For (2), take each F ∈ F L M (X ) and x λ ∈ J (L X ) such that x λ ≤ q p (F). Then, for each μ ∈ J (L) with μ ≺ λ, it follows that This means there exists ν ∈ J (L) such that F q x ν ≤ F and μ ≤ ν. Then it follows that By the arbitrariness of μ, we obtain f (x) λ ≤ q Y ( f ⇒ (F)). This proves f : (X , q p ) −→ (Y , q Y ) is continuous.
Next let us recall the definition of (L, M)-fuzzy topological Q-convergence structures in Pang (2014b). For an (L, M)-fuzzy topological Q-convergence structure q on X , the pair (X , q) is called an (L, M)-fuzzy topological Q-convergence space.