STRATIFIED (L, M)-FUZZY Q-CONVERGENCE SPACES

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1 Iranian Journal of Fuzzy Systems Vol. 13, No. 4, (2016) pp STRATIFIED (L, M)-FUZZY Q-CONVERGENCE SPACES B. PANG AND Y. ZHAO Abstract. This paper presents the concepts of (L, M)-fuzzy Q-convergence spaces and stratified (L, M)-fuzzy Q-convergence spaces. It is shown that the category of stratified (L, M)-fuzzy Q-convergence spaces is a bireflective subcategory of the category of (L, M)-fuzzy Q-convergence spaces, and the former is a Cartesian-closed topological category. Also, it is proved that the category of stratified (L, M)-fuzzy topological spaces can be embedded in the category of stratified (L, M)-fuzzy Q-convergence spaces as a reflective subcategory, and the former is isomorphic to the category of topological stratified (L, M)-fuzzy Q-convergence spaces. 1. Introduction Fuzzy convergence theory takes an important part in the development of fuzzy topology. In recent years, fuzzy convergence theory deserves wide attention. With respect to different kinds of fuzzy topological spaces, there have been different kinds of fuzzy convergence structures. In the situation of L-topology, Lowen [17] defined the concept of a prefilter as a subset of I X in order to study the theory of fuzzy topology. Min [18] proposed fuzzy limit structures by means of prefilters. Using stratified L-filters, Jäger [8] introduced stratified L-fuzzy convergence spaces (which are called stratified L-generalized convergence spaces in [9]). There are some following works to study the properties of this kind of fuzzy convergence structures [4, 5, 10, 11, 12, 15, 16, 27]. In the framework of L-fuzzifying topology, Xu [26] introduced fuzzifying convergence structures by means of classical filters. Yao [28] introduced L-fuzzifying convergence structures using L-filters of ordinary subsets. In (L, M)-fuzzy setting, Güloǧu et al. [6] introduced the concept of I-fuzzy convergence structures by means of I-filters. Pang and Fang [19, 20] used L-filters to define L-fuzzy Q-convergence structures and studied its separation properties. Later, Pang proposed the concepts of (L, M)-fuzzy convergence spaces [21, 22] and enriched (L, M)-fuzzy convergence spaces [23], and studied their categorical properties and their categorical relations with (L, M)-fuzzy topological spaces. Compared with L-topological spaces [2], stratified L-topological spaces [7] have better properties in some aspects. In a natural way, the stratification condition is generalized to (L, M)-fuzzy topological spaces, generating stratified (L, M)-fuzzy topological spaces (called weakly enriched (L, M)-fuzzy topological spaces in [7]). From the viewpoint of convergence theory, stratified L-generalized convergence Received: October 2015; Revised: March 2016; Accepted: April 2016 Key words and phrases: (Stratified) (L, M)-fuzzy topology, (Stratified) (L, M)-fuzzy Q- convergence structure, Topological category, Cartesian-closedness.

2 96 B. Pang and Y. Zhao structures in the sense of Jäger provide a good tool for interpreting stratified L- topology. In [13] introduced stratified LM N-convergence tower spaces, which can induce stratified (L, M)-fuzzy convergence spaces. Actually, the stratification of L-topology and (L, M)-fuzzy topology depends on the stratification of stratified L-filters. This motivates us to how to define stratified (L, M)-fuzzy convergence structures by using L-filters or (L, M)-fuzzy filters without the stratification condition. In this paper, we will focus on fuzzy convergence structures in the framework of stratified (L, M)-fuzzy topological spaces. We will first generalize L-fuzzy Q- convergence structures to (L, M)-fuzzy Q-convergence structures. Based on this new concept, we will introduce stratified (L, M)-fuzzy Q-convergence structures and study its relations with (L, M)-fuzzy Q-convergence structures. Further, we will investigate the categorical properties of stratified (L, M)-fuzzy Q-convergence structures and study its relations with stratified (L, M)-fuzzy topologies in a categorical sense. 2. 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 b in L, in symbols a b, if for every subset D L, D b implies d a for some d D. A complete lattice L is completely distributive if and only if b = {a L a b} for each b L. An element a in L is called coprime if a b c implies a b or a c. The set of nonzero coprime elements in L (M) is denoted by J(L) (J(M)), respectively. For a nonempty set X, L X denotes the set of all L-subsets on X. The smallest element and the largest element in L X are denoted by X L and X L, respectively. LX is also a completely distributive De Morgan algebra when it inherits the structure of the lattice L in a natural way, by defining,, and pointwisely. 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 X, J(L)}. We say that a fuzzy point J(L X ) quasi-coincides with A, denoted by ˆqA, if A (x) or equivalently A. The relation does not quasi-coincide with is denoted by ˆq. For a L, a denotes the constant mapping X L, x a. Let ϕ : X Y be a mapping. Define ϕ : L X L Y and ϕ : L Y L X by ϕ (A)(y) = ϕ(x)=y A(x) for A LX and y Y, and ϕ (B) = B ϕ for B L Y, respectively. Definition 2.1. [29] A mapping F : L X M is called an (L, M)-fuzzy filter on X if it satisfies: (LMF1) F( X L ) = M, F( X L ) = M ; (LMF2) F(A B) = F(A) F(B). The family of all (L, M)-fuzzy filters on X is denoted by F LM (X). On the set F LM (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. Obviously, (F LM (X), ) is a partially ordered set.

3 Stratified (L, M)-fuzzy Q-convergence Spaces 97 Example 2.2. [21] For each J(L X ), we define ˆq( ) : L X M as follows: { A L X M, x, ˆq( )(A) = ˆqA, M, ˆqA. Then ˆq( ) is an (L, M)-fuzzy filter. Definition 2.3. [29] Let F F LM (X) and ϕ : X Y be a mapping. Then ϕ (F) : L Y M, A F(ϕ (A)) is an (L, M)-fuzzy filter on Y and is called the image of F under ϕ. Definition 2.4. [14, 25] An (L, M)-fuzzy topology on X is a mapping τ : L X M which satisfies: (LFT1) τ( X L ) = τ( X L ) = M ; (LFT2) τ(a B) τ(a) τ(b); (LFT3) τ( j J A j) j J τ(a j). For an (L, M)-fuzzy topology τ on X, the pair (X, τ) is called an (L, M)-fuzzy topological space. A continuous mapping between (L, M)-fuzzy topological spaces (X, τ X ) and (Y, τ Y ) is a mapping ϕ : X Y such that τ X (ϕ (A)) τ Y (A) for each A L Y. The category of (L, M)-fuzzy topological spaces with continuous mappings as morphisms will be denoted by (L, M)-FTop. Definition 2.5. [7] An (L, M)-fuzzy topology τ on X is called stratified if it satisfies: (SLFT) a L, τ(a) = M. For a stratified (L, M)-fuzzy topology τ on X, the pair (X, τ) is called a stratified (L, M)-fuzzy topological space. The full subcategory of (L, M)-FTop, consisting of stratified (L, M)-fuzzy topological spaces, is denoted by S(L, M)-FTop. Definition 2.6. [3] An (L, M)-fuzzy quasi-coincident neighborhood system on X is defined to be a set Q = {Q x J(L X )} of mappings Q x : L X M satisfying the following conditions: (LFQ1) Q x ( X L ) = M, Q x ( X L ) = M ; (LFQ2) Q x (A) M implies ˆqA; (LFQ3) Q x (A B) = Q x (A) Q x (B). The pair (X, Q) is called an (L, M)-fuzzy quasi-coincident neighborhood space, and it will be called topological if it satisfies moreover, (LFQ4) Q x (A) = ˆqB A y µ ˆqB Q y µ (B). A continuous mapping between two (L, M)-fuzzy quasi-coincident neighborhood spaces (X, Q X ) and (Y, Q Y ) is a mapping ϕ : X Y such that Q X (ϕ (A)) Q Y ϕ(x) (A) for each J(L X ) and A L Y. Proposition 2.7. [3] Let Q = {Q x J(L X )} be an (L, M)-fuzzy quasicoincident neighborhood system on X and define τ Q : L X M by A L X, τ Q (A) = Q x (A). ˆqA

4 98 B. Pang and Y. Zhao Then τ Q is an (L, M)-fuzzy topology on X. Proposition 2.8. [3] Let τ be an (L, M)-fuzzy topology on X and define Q τ : L X M by A L X, Q τ (A) = τ(b). ˆqB A Then Q τ = {Q τ J(L X )} is an (L, M)-fuzzy quasi-coincident neighborhood system on X. Moreover, τ(a) = ˆqA Qτ (A). Definition 2.9 (Adámek et al. [1]). A category A is called a topological category over Set provided that for any set X, any class J, any family ((X j, ξ j )) j J of A- objects and any family (ϕ j : X X j ) j J of mappings, there exists a unique A- structure ξ on X which is initial with respect to the source (ϕ j : X (X j, ξ j )) j J. This means that for an A-object (Y, η), a mapping ψ : (Y, η) (X, ξ) is an A- morphism iff for all j J, ϕ j ψ : (Y, η) (X j, ξ j ) is an A-morphism. The class of objects of a category A is denoted by A. For more notions related to category theory we refer to [1, 24]. 3. Stratified (L, M)-fuzzy Q-convergence Spaces In this section, we will introduce the concepts of (L, M)-fuzzy Q-convergence spaces and stratified (L, M)-fuzzy Q-convergence spaces. Also, we will study their relations in a categorical sense. Definition 3.1. An (L, M)-fuzzy Q-convergence structure on X is a mapping qc : F LM (X) L X which satisfies: (LFQC1) qc(ˆq( )); (LFQC2) F G implies qc(f) qc(g). For an (L, M)-fuzzy Q-convergence structure qc on X, the pair (X, qc) is called an (L, M)-fuzzy Q-convergence space. Definition 3.2. A mapping ϕ : (X, qc X ) (Y, qc Y ) between (L, M)-fuzzy Q- convergence spaces is called continuous provided that qc X (F) implies ϕ(x) qc Y (ϕ (F)) for each F F LM (X) and J(L X ). That is, ϕ (qc X (F)) qc Y (ϕ (F)) for each F F LM (X). It is easy to check that all (L, M)-fuzzy Q-convergence spaces and their continuous mappings form a category, denoted by (L, M)-FQC. Remark 3.3. When L = M, the concept of (L, M)-fuzzy Q-convergence structures in Definition 3.1 is exactly the concept of L-fuzzy Q-convergence structures in [19]. Sometimes, we need to compare more than one (L, M)-fuzzy Q-convergence structure on a common domain X, so we define as follows: Definition 3.4. For two (L, M)-fuzzy Q-convergence spaces (X, qc 1 ) and (X, qc 2 ), we say that (X, qc 2 ) is finer than (X, qc 1 ), or (X, qc 1 ) is coarser than (X, qc 2 ), denoted by (X, qc 2 ) (X, qc 1 ), if the identity mapping id X : (X, qc 2 ) (X, qc 1 ) is continuous, that is, we have qc 2 (F) qc 1 (F) for each F F LM (X). We also write qc 2 qc 1 in this case.

5 Stratified (L, M)-fuzzy Q-convergence Spaces 99 Example 3.5. Let X be a nonempty set. (1) Define qc : F LM (X) L X as follows: { X F F LM (X), qc (F) = L, J(L X ) s.t. ˆq( ) F; X L, otherwise. It is easy to verify that qc is the finest (L, M)-fuzzy Q-convergence structure on X. (2) Define qc : F LM (X) L X as follows: F F LM (X), qc (F) = X L. It is easy to check that qc is the coarsest (L, M)-fuzzy Q-convergence structure on X. Next we introduce the notion of stratified (L, M)-fuzzy Q-convergence spaces and study its relations with (L, M)-fuzzy Q-convergence spaces. Definition 3.6. An (L, M)-fuzzy Q-convergence structure qc on X is called stratified provided that (SLFQC) If qc(f) and a, then F(a) = M. The pair (X, qc) is called a stratified (L, M)-fuzzy Q-convergence space. The full subcategory of (L, M)-FQC, consisting of stratified (L, M)-fuzzy Q- convergence spaces, is denoted by S(L, M)-FQC. Remark 3.7. In the axiom (SLFQC), we call an (L, M)-fuzzy filter F -stratified provided that a implies F(a) = M. In this case, (SLFQC) can be interpreted as qc(f) implies that F is -stratified. This means that only the -stratified (L, M)-fuzzy filter can converge to. Theorem 3.8. The category S(L, M)-FQC is a topological category over Set. Proof. We need only prove the existence of initial structures. Let {(X j, qc j )} j J be a family of stratified (L, M)-fuzzy Q-convergence spaces and X be a nonempty set. If {ϕ j : X (X j, qc j )} j J is a source, then qc X : F LM (X) L X defined by F F LM (X), qc X (F) = j J ϕ j (qc j (ϕ j (F))) is a stratified (L, M)-fuzzy Q-convergence structure on X. In fact, (LFQC1) it suffices to remark that for a mapping ϕ : X Y and the fuzzy point filter ˆq( ), we have for A L Y, ϕ (ˆq( ))(A) = ˆq( )(ϕ (A)) = ˆq(ϕ(x) )(A), i.e., ϕ (ˆq( )) = ˆq(ϕ(x) ). This implies that qc X (ˆq( )) = j J ϕ j (qc j (ˆq(ϕ j (x) )) j J ϕ j (ϕ j (x) ). (LFQC2) Obviously. (SLFQC) If qc X (F) and a, then ϕ j (x) qc j (ϕ j Since qc j satisfies (SLFQC), we know F(a) = F(ϕ j (a)) = ϕ j (F)(a) = M. (F)) for each j J.

6 100 B. Pang and Y. Zhao Let further (Y, qc Y ) S(L, M)-FQC and ϕ : Y X be a mapping. Assume that ϕ j ϕ is continuous for every j J. Then for each G F LM (Y ), we have y µ qc Y (G) = ϕ j ϕ(y) µ qc j ((ϕ j ϕ) (G)) for each j J = ϕ j (ϕ(y)) µ qc j (ϕ j (ϕ (G))) for each j J = ϕ(y) µ j J ϕ j (qc j (ϕ j (ϕ (G)))) = ϕ(y) µ qc X (ϕ (G)). This shows ϕ : (Y, qc Y ) (X, qc X ) is continuous. Therefore, ϕ is continuous if and only if ϕ j ϕ is continuous for each j J, as desired. Example 3.9 (Product space). Let {(X j, qc j )} j J be a family of stratified (L, M)- fuzzy Q-convergence spaces and {p k : X := j J X j (X k, qc k )} k J be the source formed by the family of the projection mappings {p k : j J X j X k } k J. The stratified (L, M)-fuzzy Q-convergence structure on X, denoted by j J qc j, that is initial with respect to {p j : X (X j, c j )} j J, is called the product stratified (L, M)-fuzzy Q-convergence structure and the pair (X, j J qc j) is called the product space. For the product of two stratified (L, M)-fuzzy Q-convergence spaces (X, qc X ) and (Y, qc Y ), we usually write (X Y, qc X qc Y ). For a nonempty set X, let QC s (X) denote the fibre {(X, qc) : qc is a stratified (L, M)-fuzzy Q-convergence structure on X} of X. By Theorem 3.8, we obtain Corollary (QC s (X), ) is a complete lattice. Corollary Let (Y, qc Y ) be a stratified (L, M)-fuzzy Q-convergence space and ϕ : X Y be a mapping. Define qc X : F LM (X) L X by F F LM (X), qc X (F) = ϕ (qc Y (ϕ (F))). Then qc X is a stratified (L, M)-fuzzy Q-convergence structure on X. Theorem S(L, M)-FQC is a bireflective subcategory of (L, M)-FQC. Proof. Suppose that (X, qc) is an (L, M)-fuzzy Q-convergence space and E qc = {c (X, c) S(L, M)-FQC and qc c}. Let qc r = E qc. By Corollary 3.10, we know qc r is a stratified (L, M)-fuzzy Q-convergence structure on X. Then id X : (X, qc) (X, qc r ) is the S(L, M)- FQC-bireflection. Next we prove it by two steps. Step 1: id X : (X, qc) (X, qc r ) is continuous. By the definition of qc r, it is easy to see that qc qc r, which shows the continuity of id X : (X, qc) (X, qc r ). Step 2: For a stratified (L, M)-fuzzy Q-convergence space (Y, qc Y ) and a mapping ϕ : X Y, the continuity of ϕ : (X, qc) (Y, qc Y ) implies the continuity of ϕ : (X, qc r ) (Y, qc Y ). Take any J(L X ) and F F LM (X) such that qc r (F). Then for each stratified (L, M)-fuzzy Q-convergence space (X, c) satisfying qc c, it follows that qc(f). By Corollary 3.11, we know

7 Stratified (L, M)-fuzzy Q-convergence Spaces 101 that qc X is a stratified (L, M)-fuzzy Q-convergence structure on X. Since ϕ : (X, qc) (Y, qc Y ) is continuous, it is easy to verify that qc qc X. Hence, we obtain qc X (F) = ϕ (qc Y (ϕ (F))), that is, ϕ(x) qc Y (ϕ (F)). This implies the continuity of ϕ : (X, qc r ) (Y, qc Y ). 4. Relations Between S(L, M)-FQC and S(L, M)-FTop In this section, we will study the relations between stratified (L, M)-fuzzy Q- convergence spaces and stratified (L, M)-fuzzy topological spaces from a categorical aspect. Definition 4.1. Let (X, qc) be a stratified (L, M)-fuzzy Q-convergence space and J(L X ). Define Fx qc : L X M by Fx qc (A) = qc(f) F(A) for each A L X. Then Fx qc is an (L, M)-fuzzy filter, which is called the quasi-coincident neighborhood filter at. Proposition 4.2. Let (X, qc) be a stratified (L, M)-fuzzy Q-convergence space and define τ qc : L X M by A L X, τ qc (A) = Fx qc (A). ˆqA Then τ qc is a stratified (L, M)-fuzzy topology on X. Proof. It suffices to verify that τ qc satisfies (LFT1) (LFT3) and (SLFT). In fact, (LFT1) τ qc ( X L ) = ˆq X L τ qc ( X L ) = ˆq X L (LFT2) Take any A, B L X. Then τ qc (A) τ qc (B) = ( X L ) = = M, ( X L ) = M = M. = = ˆqA (A) ˆqA yµ ˆqB yµ ˆqB yµ (B) (A) yµ (B) ˆq(AB) yµ ˆq(AB) x (A) x (B) ˆq(AB) x (A B) ˆq(AB) = τ qc (A B). (A) yµ (B) (LFT3) For {A j j J} L X, it follows that τ qc A j = j J ˆq x A j j J A j j J = x A j j J ˆqA j j J x (A j ) j J ˆqA j = τ qc (A j ). j J

8 102 B. Pang and Y. Zhao (SLFT) For each a L, it follows that τ qc (a) = ˆqa qc(f) = a qc(f) F(a) F(a) = M. (by (SLFQC)) This shows that τ qc is a stratified (L, M)-fuzzy topology on X. Proposition 4.3. If ϕ : (X, qc X ) (Y, qc Y ) is continuous, then so is ϕ : (X, τ qc X ) (Y, τ qc Y ). Proof. Since ϕ : (X, qc X ) (X, qc X ) is continuous, it follows that for each F F LM (X), ϕ (qc X (F)) qc Y (ϕ (F)). Then for each B L Y, we have τ qc X (ϕ (B)) = = = ˆqϕ (B) ˆqϕ (B) X (ϕ (B)) ϕ ( X )(B) ϕ(x) ˆqB qc X (F) ϕ (F)(B) ϕ(x) ˆqB ϕ(x) qc Y (ϕ (F)) y µ ˆqB y µ qc Y (G) = y µ ˆqB = τ qc Y (B). Y y µ (B) G(B) ϕ (F)(B) Thus, ϕ : (X, τ qc X ) (Y, τ qc Y ) is continuous. By Propositions 4.2 and 4.3, we obtain a functor F : S(L, M)-FTop S(L, M)- FQC, defined by F(X, τ) = (X, qc τ ) and F(ϕ) = ϕ. Proposition 4.4. Let (X, τ) be an (L, M)-fuzzy topological space and define qc τ : F LM (X) L X as follows: F F LM (X), qc τ (F) = y µ. Q τ yµ F Then qc τ is an (L, M)-fuzzy Q-convergence structure on X.

9 Stratified (L, M)-fuzzy Q-convergence Spaces 103 Proof. It is enough to show that qc τ satisfies (LFQC1) and (LFQC2). (LFQC1) Take any J(L X ). Then Q τ F LM (X) and Q τ ˆq( ). Further, it follows that qc τ (ˆq( )) = y µ. (LFQC2) Obviously. Q τ yµ ˆq() Lemma 4.5. Let (X, τ) be an (L, M)-fuzzy topological space. Then for each J(L X ) and F F LM (X), qc τ (F) if and only if Q τ F. Proof. The sufficiency is obvious. It suffices to show the necessity. Suppose that qc τ (F). Take any A L X and α J(M) such that α Q τ (A) = Q τ y µ (B). ˆqB A y µ ˆqB Then there exists B α L X such that ˆqB α A and for each y µˆqb α, it follows that α Q τ y µ (B α ). Then we have qc τ (F) B α. By the definition of qc τ, there exists z ν J(L X ) such that Q τ z ν F and z ν B α. Thus, By the arbitrariness of α, we get α Q τ z ν (B α) F(B α) F(A). F(A) {α J(M) α Q τ (A)} = Q τ (A), as desired. Proposition 4.6. Let (X, τ) be a stratified (L, M)-fuzzy topological space. Then qc τ is a stratified (L, M)-fuzzy Q-convergence structure on X. Proof. By Proposition 4.4, it is enough to show that qc τ satisfies (SLFQC). For each qc τ (F) and a, by Lemma 4.5, we know Q τ F. Then it follows that F(a) Q τ (a) = τ(b) τ(a) = M. ˆqB a Proposition 4.7. If ϕ : (X, τ X) (Y, τ Y ) is continuous, then so is ϕ : (X, qc τ X ) (Y, qc τ Y ). Proof. Since ϕ : (X, τ X ) (Y, τ Y ) is continuous, it follows that ϕ : (X, Q τ X ) (Y, Q τ Y ) is continuous, that is, Then for each F F LM (X), we have J(L X ), ϕ (Q τ X x ) Q τ Y ϕ(x). qc τ X (F) Q τ X x F (by Lemma 4.5) = ϕ (Q τ X x ) ϕ (F) = Q τ Y ϕ(x) ϕ (F) ϕ(x) qc τ Y (ϕ (F)) (by Lemma 4.5) ϕ (qc τ Y (ϕ (F))).

10 104 B. Pang and Y. Zhao By the arbitrariness of, we obtain qc τ X (F) ϕ (qc τ Y (ϕ (F))), that is, ϕ (qc τ X (F)) qc τ Y (ϕ (F)). This shows that ϕ : (X, qc τ X ) (Y, qc τ Y ) is continuous. By Propositions 4.6 and 4.7, we obtain a functor G :S(L, M)-FQC S(L, M)- FTop, defined by G(X, qc) = (X, τ qc ) and G(ϕ) = ϕ. Theorem 4.8. (G, F) is a Galois correspondence. Moreover, G is a left inverse of F. Proof. We need only show that F G I S(L,M) FQC and G F = I S(L,M) FTop. That is, for a stratified (L, M)-fuzzy Q-convergence structure qc on X and a stratified (L, M)-fuzzy topology τ on X, it follows that qc τ qc qc qc and τ qcτ = τ. For qc τ qc qc qc, take any J(L X ) and A L X. It follows that Then Q τ qc Q τ qc (A) = ˆqB A y µ ˆqB y µ (B) (A). Fx qc. Further, take any F F LM (X). Then qc(f) = Fx qc F = Q τ qc F qc τ qc (F). (by Lemma 4.5) By the arbitrariness of, we obtain qc qc τ qc. For τ qcτ = τ, by Propositions 2.7 and 4.2, it suffices to show that Q τ = Fx qcτ for each J(L X ). By Lemma 4.5, for each A L X, we have Fx qcτ (A) = F(A) = F(A) = Q τ (A), as desired. qc τ (F) Q τ F Corollary 4.9. The category S(L, M)-FTop can be embedded in the category S(L, M)-FQC as a reflective subcategory. In general topology, we all know that the category of topological convergence spaces, which is a special kind of convergence spaces, is isomorphic to the category of topological spaces. In the framework of stratified (L, M)-fuzzy topology, we have the similar conclusion. Definition A stratified (L, M)-fuzzy Q-convergence structure on X is called topological if it satisfies (LFPQC) qc(fx qc ); (LFTQC) Fx qc (A) = ˆqB A y µ ˆqB F y qc µ (B). For a topological stratified (L, M)-fuzzy Q-convergence structure c on X, the pair (X, c) is called a topological stratified (L, M)-fuzzy Q-convergence space. The full subcategory of S(L, M)-FQC, consisting of topological stratified (L, M)- fuzzy Q-convergence spaces, is denoted by S(L, M)-FTQC.

11 Stratified (L, M)-fuzzy Q-convergence Spaces 105 Theorem S(L, M)-FTQC is isomorphic to S(L, M)-FTop. Proof. By Theorem 4.8, it suffices to show that for each topological stratified (L, M)-fuzzy Q-convergence space (X, qc), qc τ qc = qc. Take any J(L X ) and A L X. By (LFTQC), Q τ qc (A) = τ qc (B) = ˆqB A ˆqB A y µ ˆqB = (A). Then for each F F LM (X), by (LFPQC), qc(f) Fx qc F Q τ qc F y µ (B) qc τ qc (F). (by Lemma 4.5) By the arbitrariness of, we get qc = qc τ qc, as desired. 5. Cartesian-closedness of S(L, M)-FQC Compared with topological spaces, convergence spaces possess better categorical properties, such as Cartesian-closedness. In the section, we will discuss the Cartesian-closedness of S(L, M)-FQC. There are several ways to show that a category is Cartesian closed. Here we adopt the following way. Lemma 5.1. [1] Let A be a well-fibred topological category. Then the following statements are equivalent. (1) A is Cartesian closed. (2) For each A A, the functor (A ) preserves final epi-sinks. In Theorem 3.8, we showed that S(L, M)-FQC is a topological category. Next we show it is well-fibred. Proposition 5.2. S(L, M)-FQC is fibre small. Proof. For a fixed set X, the class of all stratified (L, M)-fuzzy Q-convergence structures on X is a subset of (L X ) F LM (X). This shows the fibre-smallness of S(L, M)-FQC. Lemma 5.3. Let X = {x} and let qc be an (L, M)-fuzzy Q-convergence structure on X. Then the following statements are equivalent. (1) qc is stratified, that is, qc satisfies (SLFQC). (2) J(L X ), F F LM (X), qc(f) iff F ˆq( ). (3) F F LM (X), qc(f) = F ˆq( ).

12 106 B. Pang and Y. Zhao Proof. Since X is a singleton, each L-subset on X is actually a constant L-subset and J(L X ) = { J(L)}. Next we show the equivalence of these conditions. (1)= (2): Sufficiency. If F ˆq( ), then by (LFQC1) and (LFQC2), we have qc(ˆq( )) qc(f). Necessity. Take any a L X with ˆqa, i.e., a. Since qc(f), by (1), we obtain that F(a) = M. Thus, F(a) ˆq( )(a). If ˆqa, then F(a) M = ˆq( )(a). This shows F ˆq( ). (2)= (3): Straightforward. (3)= (1): Take any J(L) and a L such that a and qc(f). Then qc(f)(x) = µ. F ˆq(x µ) Since a, we know that F ˆq(x µ) µ a. Then there exists µ J(L) such that F ˆq(x µ ) and µ a, i.e., x µˆqa. This implies that as desired. F(a) ˆq(x µ )(a) = M, By Lemma 5.3, we can obtain the following result. Proposition 5.4. For a singleton X = {x}, there is a unique stratified (L, M)- fuzzy Q-convergence structure on X. That is, S(L, M)-FQC satisfies the terminal separator property. By Theorem 3.8, Propositions 5.2 and 5.4, we have Theorem 5.5. S(L, M)-FQC is a well-fibred topological category. In order to show the Cartesian-closedness of S(L, M)-FQC, we will list some conclusions with respect to (L, M)-fuzzy filters in [21]. Lemma 5.6. [21] For a nonempty family {F j } j J of (L, M)-fuzzy filters, the following statements are equivalent: (1) There exists an (L, M)-fuzzy filter F such that F F j for all j J. (2) F j1 (A 1 ) F jn (A n ) = M if A 1 A n = X L (n N, A 1,, A n L X, {j 1,, j n } J). In the case of existence, we find F j (A) = {Fj1 (A 1 ) F jn (A n ) A 1 A n A} j J n N as the supremum of {F j } j J in (F LM (X), ). We denote it by j J F j. For two (L, M)-fuzzy filters F and G, we write F G F LM (X) if F G exists. Lemma 5.7. [21] Let F F LM (Y ) and ϕ : X Y be a mapping. Then the following statements are equivalent: (1) ϕ (F) : L X M defined by ϕ (F)(A) = ϕ (B) A F(B) is an (L, M)- fuzzy filter on X. (2) B L Y, ϕ (B) = X L implies F(B) = M.

13 Stratified (L, M)-fuzzy Q-convergence Spaces 107 In case ϕ (F) F LM (X), we call ϕ (F) the inverse image of F under ϕ. Obviously, ϕ (F) exists if ϕ is surjective. Lemma 5.8. [21] Suppose that L is prime. Let {X j } j J be a family of nonempty sets, p k : j J X j X k the projection mapping and F j F LM (X j ) ( j J). Then ( ) j J p j (F j) F LM j J X j, which is called the product (L, M)-fuzzy filter and denoted by j J F j. For two (L, M)-fuzzy filters F and G, we write F G. Lemma 5.9. [21] Suppose that L is prime. Let {X j } j J be a family of nonempty sets, p k : j J X j X k the projection mapping, F j F LM (X j ) ( j J) and ( ) F F LM j J X j. Then the following statements hold: (1) j J p j (F) F. ( (2) p k j J j) F F k, k J. Lemma [21] Suppose that L is prime. Let F F LM (X 1 ), G F LM (X 2 ) and let ϕ : X 1 Y 1 and ψ : X 2 Y 2 be mappings. Then (ϕ ψ) (F G) = ϕ (F) ψ (G). Definition [1] Let A be a category. A family {ϕ j : A j A} j J of A- morphisms indexed by some class J, is called an epi-sink in A provided that for any pair α, β : A D of A-morphisms such that α ϕ j = β ϕ j for each j J, it follows that α = β. Lemma Let {(X j, qc j )} j J and (X, qc X ) be a family of stratified (L, M)- fuzzy Q-convergence spaces. Then {ϕ j : (X j, qc j ) (X, qc X )} j J is an epi-sink if and only if X = j J ϕ j[x j ] (ϕ j [X j ] = {ϕ j (x j ) x j X j }). Proof. The proof of Lemma 6.8 in [21] can be adopted. Since the category S(L, M)-FQC is a topological category, there exists a unique final structure with respect to a sink {ϕ j : (X j, qc j ) X} j J in S(L, M)-FQC. However, here we only explore the concrete form of the final structure with respect to the epi-sink. Theorem Let {(X j, qc j)} j J be a family of stratified (L, M)-fuzzy Q-convergence spaces and X be a nonempty set. If {ϕ j : (X j, qc j ) X} j J is an epi-sink, then qc X : F LM (X) L X defined by F F LM (X), qc X (F) = { J(L X ) x j X j, F j F LM (X j ) s.t. ϕ j (x j ) = x, ϕ j (F j ) F and x j qc j(f j )} is the unique final structure with respect to the epi-sink {ϕ j : (X j, qc j ) X} j J. Proof. Firstly, we show that qc X is a stratified (L, M)-fuzzy Q-convergence structure on X. That is, qc X satisfies (LFQC1), (LFQC2) and (SLFQC). In fact, (LFQC1) Take any J(L X ). By Lemma 5.12, we obtain x ϕ j [X j ]. Then there exist j J and x j X j such that ϕ j (x j ) = x. Obviously, ˆq(x j ) F LM (X j ),

14 108 B. Pang and Y. Zhao ϕ j (ˆq(xj )) = ˆq(ϕ j(x j ) ) = ˆq( ) and x j qc j(ˆq(x j )). This implies that qc X (ˆq( )). (LFQC2) Straightforward. (SLFQC) Take any J(M) and a L such that a and qc X (F). Then qc X (F)(x) = {µ J(L) x j X j, F j F LM (X j ) s.t. ϕ j (x j ) = x, ϕ j (F j ) F and x j µ qc j (F j )} Since a, it follows that qc X (F)(x) a. This means that there exist µ J(L), j J, x j X j and F j F LM (X j ) such that ϕ j (x j ) = x, ϕ j (F j) F, x j µ qc j (F j ) and µ a. Since qc j satisfies (SLFQC), we obtain F j (a) = M. This implies that F(a) ϕ j (F j )(a) = F j (ϕ j (a)) = F j (a) = M. Secondly, let (Y, qc Y ) be a stratified (L, M)-fuzzy Q-convergence space and ϕ : X Y be a mapping satisfying that ϕ ϕ j : (X j, qc j ) (Y, qc Y ) is continuous for all j J. Take any F F LM (X) and J(L X ) such that qc X (F), that is, qc X (F)(x). Let µ J(L) such that µ. Then there exist ν J(L), j J, x j X j, F j F LM (X j ) such that ϕ j (x j ) = x, ϕ j (F j) F, x j ν qc j (F j ) and µ ν. Obviously, x j µ qc j (F j ). By the continuity of ϕ ϕ j, we obtain ϕ(ϕ j (x j )) µ qc Y (ϕ (ϕ j (F j))). This implies ϕ(x) µ qc Y (ϕ (F)). By the arbitrariness of µ, we obtain ϕ(x) qc Y (ϕ (F)). This proves the continuity of ϕ : (X, qc X ) (Y, qc Y ). As a consequence, the continuity of ϕ ϕ j for all j J implies the continuity of ϕ. Hence, we obtain that qc X is the final structure on X with respect to the sink {ϕ j : (X j, qc j ) X} j J. Theorem Suppose that L is prime. Then S(L, M)-FQC is Cartesian closed. Proof. Suppose that {ϕ j : (X j, qc j ) (X, qc X )} j J is a final epi-sink in S(L, M)- FQC. By Theorem 5.13, we know that F F LM (X), qc X (F) = { J(L X ) x j X j, F j F LM (X j ) s.t. ϕ j (x j ) = x, ϕ j (F j ) F and x j qc j(f j )} Next we show that for each stratified (L, M)-fuzzy Q-convergence space (Y, qc Y ), the family {1 Y ϕ j : (Y X j, qc Y qc j ) (Y X, qc Y qc X )} j J is a final epi-sink in S(L, M)-FQC. Let (Z, qc Z ) be a stratified (L, M)-fuzzy Q-convergence space and α, β : (Y X, qc Y qc X ) (Z, qc Z ) be continuous such that α (1 Y ϕ j ) = β (1 Y ϕ j ) for each j J. Take each (y, x) Y X. Since X = j J ϕ j[x j ], there exist j J and x j X j such that ϕ j (x j ) = x. Hence, α((y, x)) = α((y, ϕ j (x j ))) = α((1 Y ϕ j )((y, x j ))) = β((1 Y ϕ j )((y, x j ))) = β((y, ϕ j (x j ))) = β((y, x)), i.e., α = β. This means {1 Y ϕ j : (Y X j, qc Y qc j ) Y X} j J is an epi-sink in S(L, M)-FQC.

15 Stratified (L, M)-fuzzy Q-convergence Spaces 109 In order to show that qc Y qc X is the final structure with respect to the sink {1 Y ϕ j : (Y X j, qc Y qc j ) Y X} j J, we need only show that for each stratified (L, M)-fuzzy Q-convergence space (Z, qc Z ) and each mapping ϕ : Y X Z, the continuity of ϕ (1 Y ϕ j ) for each j J implies the continuity of ϕ. Take any H F LM (Y X) and (y, x) J(L Y X ) such that (y, x) (qc Y qc X )(H ). By Example 3.10, it follows that qc Y (p Y (H ))(y) and qc X(p X (H ))(x). Take any µ J(L) such that µ qc X (p X (H ))(x). Then there exist ν J(L), j J, x j X j, F j F LM (X j ) such that ϕ j (x j ) = x, ϕ j (F j) p X (H ), xj ν qc j (F j ) and µ ν. Thus, we get µ qc Y (p Y (H ))(y) and µ ν qc j(f j )(x j ). By Lemma 5.9 (2), we obtain µ qc Y (p Y (H ))(y) qc j (F j )(x j ) (qc Y qc j )(p Y (H ) F j )((y, x j )). That is, (y, x j ) µ (qc Y qc j )(p Y (H ) F j). By the continuity of ϕ (1 Y ϕ j ), we obtain ϕ(y, x) µ = (ϕ (1 Y ϕ j ))(y, x j ) µ qc Z ((ϕ (1 Y ϕ j )) (p Y (H ) F j )) = qc Z (ϕ (p Y (H ) ϕ j (F j ))) (by Lemma 5.10) qc Z (ϕ (p Y (H ) p X (H ))) qc Z (ϕ (H )). (by Lemma 5.9 (2)) By the arbitrariness of µ, we obtain ϕ(y, x) qc Z (ϕ (H )). This proves that ϕ : (Y X, c Y c X ) (Z, c Z ) is continuous. As a consequence, we obtain that the family {1 Y ϕ j : (Y X j, qc Y qc j ) (Y X, qc Y qc X )} j J is a final episink in S(L, M)-FQC. By Lemma 5.1, we know that the category S(L, M)-FQC is Cartesian closed. 6. Conclusions In this paper, we generalized L-fuzzy Q-convergence structures to (L, M)-fuzzy Q-convergence structures and proposed the concept of stratified (L, M)-fuzzy Q- convergence structures. This new concept not only provides a good approach to describe stratified (L, M)-fuzzy topology, but also possesses nice categorical properties. For example, the resulting category of stratified (L, M)-fuzzy Q-convergence spaces is a Cartesian closed topological category. As we all know, stratified (L, M)- fuzzy topological spaces have better properties than (L, M)-fuzzy topological spaces in some aspects. This motivates us to consider topological properties that stratified (L, M)-fuzzy Q-convergence spaces possess. In the future, we will consider topological properties of stratified (L, M)-fuzzy topological spaces, such as compactness and separation properties. Acknowledgements. The authors would like to express their sincere thanks to the anonymous reviewers for their careful reading and constructive comments. This work is supported by National Nature Science Foundation Committee (NSFC) of China (No ), China Postdoctoral Science Foundation (No. 2015M581434), Fundamental Research Project of Shenzhen (No. JCYJ and No. JCYJ ).

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17 Stratified (L, M)-fuzzy Q-convergence Spaces 111 Bin Pang, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, P.R. China address: Yi Zhao, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, P.R. China address: *Corresponding author