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Generalization of L-closure spaces
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Fuzzy Sets and Systems 149 (2005) 415 – 432 www.elsevier.com/locate/fss
Generalization of L-closure spaces Wu-Neng Zhou Institute of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, PR China Received 5 January 2001; received in revised form 8 March 2004; accepted 29 March 2004
Abstract This paper introduces the concepts of L-closure spaces and the convergence in L-closure spaces. In the variable-basis setting of such closure spaces, continuous generalized order homomorphisms are characterized by the convergence theory of nets, 2lters and ideals. Furthermore, in the 2xed-basis setting of such closure spaces, some constructions on L-closure spaces, such as the subspace of an L-cs, the sum L-cs, the product of L-cs’ and the induced L-cs of a crisp closure space, are investigated. We obtain an important result that the category L-CLOSURE is a topological category over SET w.r.t. the forgetful functor from L-CLOSURE to SET. We obtain another important result that there is an adjunction between the category of crisp closure spaces and the full subcategory of those L-closure spaces which are induced by some crisp closure space. c 2004 Published by Elsevier B.V. Keywords: L-closure space; L-convergence; L-continuous mapping; L-closure subspaces; L-closure sum spaces; L-closure-induced spaces
1. Introduction The theory of fuzzy closure spaces has been established by Mashhour and Ghanim [6] and = Srivastava et al. [8,9]. The de2nition of Mashhour and Ghanim is an analogue of Cech closure spaces and Srivastava et al. have introduced it as an analogue of Birkho@ closure spaces in [8]. Based on [8], Rekha Srivastava and Manjari Srivastava studied the subspace of a fuzzy closure space, the sum of a family of pairwise disjoint fuzzy closure spaces and the product of a family of fuzzy closure spaces in [9]. The notion of T1 fuzzy closure space was also introduced in [9]. However, all the above works are in the 2xed-basis setting [0; 1], and there are no ideals of topological category in the above obtained results. In this paper, the aim is to introduce the concept of L-closure spaces which is a generalization of the concept of fuzzy closure spaces and to study many important properties for L-closure spaces, such as category property, especially.
Supported by the National Natural Science Foundation of PR China (NSFC: 10371106). E-mail address: [email protected] (W.-N. Zhou).
c 2004 Published by Elsevier B.V. 0165-0114/$ - see front matter doi:10.1016/j.fss.2004.03.026
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This paper is organized as follows. Section 2 introduces the concepts of L-closure operator and L-closure spaces and discusses the relation between L-closure spaces and L-topological spaces. Two L-closure operators on L-unit interval are constructed. They are important for the subsequent discussions. Section 3 introduces the concepts of convergence for net and ideal in L-closure spaces and gets the characteristic properties. The concepts of continuous generalized order homomorphism and L-continuous mapping are introduced and many characteristic properties for generalized order homomorphism are discussed in Section 4. Section 5 contains one of the most important results, namely that the category whose objects are all L-closure spaces and arrows all L-continuous mappings between two L-closure spaces is a topological category over the category SET. Section 6 constructs many L-closure spaces from some known L-closure spaces. Another important result is obtained in this section, namely that there is an adjunction between the category of crisp closure spaces and the full subcategory of those L-closure spaces which are induced by some crisp closure spaces. In the conclusion section, we summarize the results of this paper and point out some problems to be further studied. 2. Technical preliminaries Throughout this paper, X; Y; Xt (t ∈ T , T is an indexing set) denote nonempty crisp sets, L; L1 ; L2 denote completely distributive lattices with an order-reversing involution “” (i.e. ∀a; b∈L; a6b⇒a ¿ b and for every a ∈ L, a = a). LX = {A | A : X → L is a mapping}. 0 and 1 denote the smallest and the largest element of L, respectively. ∈ L is called a union-irreducible element, if = a ∨ b implies
= a or = b for arbitrary a; b ∈ L. M (L) denotes the set of all nonzero union-irreducible elements of L. For each x ∈ X; ∈ M (L); x ∈ LX is called a molecule which satis2es x (y) = for y = x and x (y) = 0 for y = x. Let 0X ; 1X ∈ LX be L-sets de2ned by 0X (x) = 0; 1X (x) = 1 for all x ∈ X . Let M ∗ (LX ) be the set {x ∈ LX : x ∈ X; ∈ M (L)}. For the sake of convenience, we recall the following concepts and results used later in this paper. De nition 2.1 (Liu and Luo [4], Wang [10]). Let be a nonempty subset of LX . We call an L-topology on X if satis2es the following conditions: (1) 0X ; 1X ∈ . (2) If U; V ∈ , thenU ∧ V ∈ . (3) If 1 ⊆ , then U ∈1 U ∈ . The pair (LX ; ) is called an L-topological space (L-ts for short). Each element of is called an open set in (LX ; ). The complement of an open set is called a closed set in (LX ; ). In an L-ts (LX ; ), c∗ (A) = ∧ {F ∈ LX : F is a closed set and F¿A} is called the closure of A for A ∈ LX . Lemma 2.1 (Liu and Luo [4]). Let (LX ; ) be an L-ts. Then the following hold for all A; B ∈ LX . (1) c∗ (0X ) = 0X ; c∗ (1X ) = 1X ; (2) A6c∗ (A);
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(3) A6B ⇒ c∗ (A)6c∗ (B); (4) c∗ (A ∨ B) = c∗ (A) ∨ c∗ (B); (5) c∗ (c∗ (A)) = c∗ (A). De nition 2.2 (Liu and Luo [4]). Let A; B be posets and f : A → B, g : B → A be order preserving mappings. f is called a left adjoint of g and g is called a right adjoint of f, denoted by f g, if ∀a ∈ A; ∀b ∈ B;
f(a) 6 b ⇔ a 6 g(b):
Equivalently, f g if and only if ∀a ∈ A; ∀b ∈ B; a6gf(a) and fg(b)6b. 3. L-closure spaces In this section, we shall establish the concepts of L-closure spaces and L-closed sets and discuss the relation between L-closure spaces and L-topological spaces. Firstly, we introduce the concept of L-closure spaces which is a generalization of fuzzy closure spaces in [6,8,9]. De nition 3.1. A mapping c : LX →; LX is called an L-closure operator (L-co for short) or an L-closure, if it satis2es the following conditions for any A; B ∈ LX : (cl1) (cl2) (cl3) (cl4)
c(0X ) = 0X . A6c(A). A6B implies c(A)6c(B). c(c(A)) = c(A).
The pair (LX ; c) is called an L-closure space (L-cs for short) and F ∈ LX is called an L-closed set in (LX ; c) if c(F) = F. Let c = {F ∈ LX : c(F) = F}. Remark 3.1. From Lemma 2.1, we know that the closure operator c∗ : LX → LX in an L-topological space satis2es the conditions mentioned in De2nition 3.1. So an L-topological space (LX ; ) can induce an L-cs (LX ; c∗ ) where c∗ is the closure operator in L-ts (LX ; ). But the converse proposition is not true, which is shown in the following Example 3.1. Therefore the L-cs’ are more than L-ts’. Example 3.1. Let X = {a; b; c}; = {0X ; {a}; {b}; {a; c}; 1X }; L = I = [0; 1]. De2ne c : I X → I X as following: c(A) = ∧{F ∈ : F ¿ A}
for A ∈ I X :
Then c is an L-co by the de2nition of c(A). So (LX ; c) is an L-cs. But c is not a closure operator in any L-topological space. In fact, if c is a closure operator in an L-topological space (LX ; ), then c({a} ∨ {b}) = c({a}) ∨ c({b}). Now c({a} ∨ {b}) = c({a; b}) = 1X , and c({a}) ∨ c({b}) = {a} ∨ {b} = {a; b}. Hence c({a} ∨ {b}) = c({a}) ∨ c({b}). In the following example, we shall construct two L-co’s on I (L), the L-unit interval, which will be also used in Example 5.1 in the sequel.
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Example 3.2. Let R be the real line, : R → L be an order-reversing mapping (i.e. ∀s; t ∈ R; s6t ⇒
(s)¿ (t)) that satis2es the following condition: ∀t ¡ 0; (t) = 1;
∀t ¿ 1; (t) = 0:
denotes the set of this kind of mappings. For every ∈ and every t ∈ R, de2ne
(s); (t−) =
(s):
(t+) = s¿t
s¡t
De2ne an equivalence relation “∼” on as follows:
∼ ⇔ ∀t ∈ R; (t−) = (t−); (t+) = (t+): For every ∈ , let [ ] be the equivalence class in with respect to ∼, i.e. [ ] = { ∈ : ∼ }: Denote the family of all the equivalence classes in with respect to ∼ by I (L). For convenience, for every ∈ , if no confusion will be caused, we identify with every such that ∼ , and still use to denote [ ]. For every t ∈ R, de2ne Mt ; Nt ∈ LI (L) as follows:
(s); Nt ( ) = ( (t+)) = ( (s)) : ∀ ∈ I (L); Mt ( ) = (t−) = s¡t
s¿t
De2ne c : LI (L) → LI (L) such that Mt : ∀A ∈ LI (L) ; c(A) = M t ¿A
Then we have the following results:
c(0I (L) ) = 0I (L) . In fact, Mt ( ) = 0 whenever t¿1. So c(0I (L) ) = Mt ¿0I (L) Mt = 0I (L) . ∀A ∈ LI (L) ; A6c(A). This is obvious from the de2nition of c(A). ∀A; B ∈ LI (L) ; A6B ⇒ c(A)6c(B). This is obvious from the de2nition of c(A) and c(B). ∀t ∈ R; c(Mt ) = Mt . ∀A ∈ LI (L) ; c(c(A)) = c(A). In fact, c(c(A)) = c( Mt ¿A Mt )6 Mt ¿A c(M t ) = Mt ¿A Mt = c(A). ∀R1 ⊂ R with R1 = ∅and sup R1 = r, t ∈R1 Mt = Mr . In fact, ∀ ∈ ; ( t ∈R1 Mt )( ) = {
t ∈ R1 (v) : v¡t}; Mr ( ) = { (u) : u¡r}. ∀t ∈ R and v¡t, we have v¡t6r. So
(v)¿ { (u) : 1 u¡r} and hence t ∈R1 { (v) : v¡t}6 { (u) : u¡r}. Conversely, ∀u¡r; ∃t ∈ R1 such that u¡t6r. Take v ∈R with u¡v¡t. Then
(u)¿ (v)¿ { (v) : v¡t} by the decreasing t ∈ R1 property of . So t ∈R1 { (v) : v¡t}¿ { (u) : u¡r}. Therefore ( t ∈R1 Mt )( ) = (Mr )( ) for every ∈ . (7) If A ∈ LI (L) ; A = 0I (L) and c(A) = A, then there exists t0 ∈R such that A = Mt0 . In fact, c(A) = { (s) : s¡0} = 1; M0 ¿A. So T = ∅. M t ¿ A Mt = t ∈T {Mt : Mt ¿A}. Because ∀ ∈ ; M0 ( ) = On the other hand, if t¿1, then Mt = 0I (L) . So sup T exists and let r = sup T . Therefore c(A) = Mr by (6).
(1) (2) (3) (4) (5) (6)
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So c is an L-co on LI (L) , (LI (L) ; c) is an L-cs and every L-closed set in (LI (L) ; c) possesses form Mt . Similarly, de2ne c1 : LI (L) → LI (L) such that ∀A ∈ LI (L) ;
c1 (A) =
Nt :
Nt ¿ A
Then we can prove that c1 is another L-co on LI (L) , (LI (L) ; c1 ) is another L-cs and every L-closed set in (LI (L) ; c1 ) possesses form Nt .
4. L-convergence In this section, we shall introduce the concepts of convergence for net, ideal in L-cs and discuss their properties. The concept of remote neighborhood of a molecule plays an important part in discussing convergence. So we introduce the concept of remote neighborhood of a molecule, 2rstly. De nition 4.1. Let (LX ; c) be an L-cs, x ∈ M ∗ (L) and F ∈ c . If x 6 = F, then we call F an X L-closed remote neighborhood of x . If E ∈ L ; E6F ∈ c and x 6 = F, then we call E an L-remote neighborhood of x . &(x ) = {E ∈ LX : E is an L-remote neighborhood of x }. Applying the concept of remote neighborhood of a molecule, we can characterize the c(A) in L-cs (LX ; c) for A ∈ LX as follows. Theorem 4.1. Let (LX ; c) be an L-cs, x ∈ M ∗ (LX ) and A ∈ LX . Then x 6c(A) if and only if A6 = E for any E ∈ &(x ). This kind of x will be called an L-accumulation point of A and hence c(A) = {x ∈ M ∗ (LX ) : x is an L-accumulation point of A}. Proof. If x 6 = c(A), then c(A) ∈ &(x ). But A6c(A). So x is an L-accumulation point of A. Conversely, if there is P ∈ &(x ) such that A6P, then there is Q ∈ &(x ) ∩ c such that P6Q. Thus A6Q and c(A)6Q as Q is an L-closed set. Hence x 6 = c(A) since x 6 = Q. The molecular net is one of the most useful tools in studying the convergence in L-cs’. In the following, we introduce the concepts of limit point and cluster point of a molecular net in L-cs’ and discuss their properties, respectively. De nition 4.2. Let D be a directed set, S = {(xn ) n : n ∈ D; xn ∈ X; n ∈ M (L)} be a molecular net in (LX ; c) and x, ∈ M ∗ (LX ). (1) x, is called an L-limit point of S or S L-converges to x, , if for each P ∈ &(x, ); S is eventually not in P (i.e. ∃n0 ∈ D such that ∀n ∈ D and n¿n0 ; (xn ) n 6 = P), in symbol S →c x, . (2) x, is called an L-cluster point of S, if for each P ∈ &(x, ); S is frequently not in P (i.e. ∀n ∈ D; ∃m ∈ D and m¿n such that (xm ) m 6 = P), in symbol S ∝c x, .
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The union of all L-limit points of S is denoted as L-lim S. The union of all L-cluster points of S is denoted as L-clu S. Theorem 4.2. Let S be a molecular net in (LX ; c) and e ∈ M ∗ (LX ). Then e6L-lim S if and only if S →c e. e6L-clu S if and only if S ∝c e. Proof. It is obvious that if S →c e, then e6L-lim S by De2nition 4.2. Conversely, if e6L-lim S, then for every P ∈ &(e), there is F ∈ &(e) ∩ c such that P6F. Thus L-lim S6 = F. So there is an L-limit point , of S such that ,6 = F. Since S →c ,, S is eventually not in F. Thus S is eventually not in P. So S →c e. Similarly, it can be proved that e6L-clu S if and only if S ∝c e. Theorem 4.2 shows us that the molecule in L-set L-lim S (L-clu S, respectively) is equivalent to the L-limit point (L-cluster point, respectively) of S. Moreover, we can characterize the L-closure c(A) of an L-set A in (LX ; c) in terms of a molecular net as follows. Theorem 4.3. Let (LX ; c) be an L-cs, A ∈ LX and e ∈ M ∗ (LX ). (1) If there is a molecular net S in A and S ∝c e, then e6c(A). (2) If e6c(A) and &(e) is a directed set, then there is a molecular net in A such that S →c e. = P and e6c(A). Proof. (1) For every P ∈ &(e); S is frequently not in P by S ∝c e. So A6 (2) Let e6c(A). Then for every P ∈ &(e); A6 = P. Thus there is s(P)6A such that s(P)6 = P. So S = {s(P) : P ∈ &(e)} is a molecular net in A as &(e) is directed. Therefore for arbitrary P; Q ∈ &(e); s(Q)6 = P whenever Q¿P. So S →c e. The ideal is another useful tool in studying the convergence in L-cs’. Similarly, we introduce the concepts of limit point and cluster point of an ideal net in L-cs’ and discuss their properties, respectively. De nition 4.3. Let I be an ideal (i.e. I is an up directed set, a lower set and 1X ∈= I ) in (LX ; c) and x, ∈ M ∗ (LX ). (1) If &(x, ) ⊆ I , then x, is called an L-limit point of I or I L-converges to x, , written as I →c x, . (2) If for each A ∈ I and P ∈ &(x, ); A ∨ P = 1, then x, is called an L-cluster point of I , in symbol I ∝c x, . The union of all L-limit points of I is denoted as L-lim I . The union of all L-cluster points of I is denoted as L-clu I . In discussing the properties of convergence for ideals, the concepts of the minimal set of an , (, ∈ L) will be used which be quoted in the following de2nition. De nition 4.4 (Zhou andChen [11], Zhou and Meng [12]). Let A be a subset of L. A is called a minimal set of , ∈ L, if A = , and for each subset B of L with B¿,, and for each d ∈ A, there exists b ∈ B such that d6b. The union of all the minimal sets of , is denoted by 0(,), and let 0∗ (,) = 0(,) ∩ M (L).
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Theorem 4.4. Let I be an ideal in an L-cs (LX ; c); e ∈ M ∗ (LX ). Then e6L-lim I if and only if I →c e. e6L-clu I if and only if I ∝c e. Proof. It is obvious that if I →c e, then e6L-lim I by De2nition 4.3. Conversely, if e6L-lim I but I →c e is not true, then there is F ∈ &(e) such that F ∈= I . From F ∈ &(e), there is d ∈ 0∗ (e) such that F ∈ &(d). By d6e6L-lim I , there is an L-limit point u of I such that d6u. Thus I →c d. So F ∈ I . This is a contradiction. Therefore I →c e. Similarly, it can be proved that e6L-clu I if and only if I ∝c e. Theorem 4.4 shows us that there is no di@erence between the L-points in L-set L-lim I and the L-limit points of I . So does the L-points in L-set L-clu I and the L-cluster points of I . Similar to Theorem 4.3, we can characterize c(A) by ideal in L-cs’ as following theorem. Theorem 4.5. Let (LX ; c) be an L-cs, A ∈ LX ; e ∈ M ∗ (LX ) and &(e) a directed set. Then e6c(A) if and only if there is an ideal I of LX such that A ∈= I and I →c e. Proof. Let e6c(A). Then for every P ∈ &(e); A6 = P. Let I = &(e). Then I is an ideal such that =P I →c e and A ∈= I . Conversely, if an ideal I →c e and A ∈= I , then P ∈ I for every P ∈ &(e). Thus A6 and therefore e6c(A). 5. Continuous generalized order homomorphism In this section, we shall introduce the concepts of continuous generalized order homomorphism and L-continuous mapping and discuss their characterizations. De nition 5.1. Let f : LX1 → LY2 be a mapping. Call f a generalized order homomorphism (GOH for short) if it satis2es the following conditions: (1) f is -preserving. i.e. f( s∈S As ) = s∈S f(As ) for As ∈ LX1 , s ∈ S, S is an indexing set. (2) f (B ) = (f (B)) for every B ∈ LY2 where f (B) = {A ∈ LX1 : f(A)6B}. There is a relation between mapping f and mapping f in De2nition 5.1 which is shown in the following proposition. Proposition 5.1. Let f : LX1 → LY2 be a GOH. Then f f . Proof. This is an immediate corollary of the Adjoint Functor Theorem of [3]. Similar to the notion of the continuity in topological spaces, we can de2ne the continuity of GOH in L-cs as follows. De nition 5.2. Let (LX1 ; c1 ) be an L1 -cs, (LY2 ; c2 ) be an L2 -cs and f : LX1 → LY2 be a GOH. Call f a continuous GOH if f (B) is an L1 -closed set in (LX1 ; c1 ) for every L2 -closed set B in (LY2 ; c2 ). Specially, if L1 = L2 = L and f is Zadeh-type powerset operator, then we call a continuous GOH an
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L-continuous mapping between (LX ; c1 ) and (LY ; c2 ). Namely, f is L-continuous between (LX ; c1 ) and (LY ; c2 ) if and only if {B ◦ f : B ∈ c2 } ⊆ c1 . Because an L-ts can induce an L-cs as shown in Section 2, it is natural to consider the relation between the two notions of continuity in L-ts and in L-cs. Next remark illustrates that every L-continuous mapping between two L-ts’ can induce a continuous GOH between the L-cs’ induced by the two L-ts’. Remark 5.1. Let (LX ; ), (LY ; ) be L-ts’ and f : X → Y an ordinary mapping. Using the arrow notation [7], we produce the corresponding L-mappings f→ : LX → LY and f← : LY → LX as follows: ∀A ∈ LX ; ∀y ∈ Y; f→ (A)(y) = {A(x) : x ∈ X; f(x) = y}; ∀B ∈ LY ; f← (B) = {A ∈ LX : f→ (A) 6 B} = B ◦ f: f is called an L-continuous mapping [4] between (LX ; ) and (LY ; ), if ∀B ∈ ; f← (B) ∈ . Equivalently, f is L-continuous between (LX ; ) and (LY ; ) if and only if ∀B ∈ ; f← (B) ∈ since f← = (f→ ) . Let (LX ; c1 ) and (LY ; c2 ) be L-cs’ induced by (LX ; ); (LY ; ), respectively. Then for every L-continuous mapping f : (LX ; ) → (LY ; ), f→ is an L-continuous mapping between (LX ; c1 ) and (LY ; c2 ). To make the concept of L-continuity in L-cs clear, it is necessary to introduce the following example in which a continuous GOH in I -unit interval I I (I ) will be constructed. Example 5.1. In Example 3.2, let L = I . We take 1 ∈ as follows: 1; s 6 1;
1 (s) = 0; s ¿ 1: De2ne an ordinary mapping f : I (I ) → I (I ) as follows: ∀ ∈ ;
f( ) = [ 1 ]:
Then we can obtain the Zadeh-type powerset operator f→ : I I (I ) → I I (I ) induced by f and its right adjoint f← : I I (I ) → I I (I ) . In this case, ∀A ∈ I I (I ) ; ∀ ∈ , one gets {A( ) : ∈ }; = 0 ; → f (A)( ) = {A( ): f( ) = } = 0; = 0 : Furthermore, we can know that ∀B ∈ I I (I ) , 0; B( 1 ) = 0; f← (B) = {A : f→ (A) 6 B} = A : {A( ) : ∈ I (I )} 6 B( 0 ) ; B( 1 ) ¿ 0: It is obvious that f→ (A) is a fuzzy point whose supporting point is [ 1 ] and the height is {A( ) : ∈ }.
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Because f→ is a Zadeh-type powerset operator, f→ is a GOH (Wang, Theorem 1.8.5 [10]). On the other hand, the set of all L-closed sets in (I I (I ) ; c1 ) is {Nt : t ∈ R} by Example 3.2. Now ∀t ∈ R, 1; t ¿ 1; Nt ( 1 ) = ( 1 (s)) = (1 − 1 (s)) = 0; t ¡ 1: s¿t
So ←
f (Nt ) =
s¿t
0I (I ) ; 1I (I ) ;
t ¡ 1; t ¿ 1:
Therefore ∀t ∈ R; f← (Nt ) is an L-closed set in (I I (I ) ; c) and hence f→ is an L-continuous mapping in (I I (I ) ; c). We shall study some characteristic properties of a continuous GOH in terms of the convergence of molecular nets, ideals and 2lters. First, we can characterize a continuous GOH by L-remote neighborhood of a molecule, closed sets and molecular nets, etc. as following theorem. Theorem 5.1. Let (LX1 ; c1 ) be an L1 -cs, (LY2 ; c2 ) an L2 -cs and f : LX1 → LY2 be a GOH. Then the following conditions are equivalent: (1) (2) (3) (4) (5)
f is a continuous GOH. f(c1 (A))6c2 (f(A)) for every A ∈ LX1 . c1 (f (B))6f (c2 (B)) for every B ∈ LY2 . For every e ∈ M ∗ (LX1 ) and B ∈ &(f(e)); f (B) ∈ &(e). For every molecular net S in (LX1 ; c1 ), if S →c e and &(e) is a directed set, then the molecular net f(S) →c ; f(e).
Proof. Condition (1) implies (2). For every A ∈ LX1 ; A6f (f(A))6f (c2 (f(A))). Thus c1 (A)6f (c2 (f(A))). So f(c1 (A))6c2 (f(A)). Condition (2) implies (3). For every B ∈ LY2 ; f (B) ∈ LX1 . From (2), f(c1 (f (B)))6c2 (f(f (B))) 6c2 (B). So c1 (f (B))6f (c2 (B)). Condition (3) implies (1). Let B be an L2 -closed set in (LY2 ; c2 ). By (3), c1 (f (B))6f (c2 (B)) = f (B). So c1 (f (B)) = f (B) and f (B) is an L1 -closed set in (LX1 ; c1 ). Condition (1) implies (4). If e ∈ M ∗ (LX ) and B ∈ &(f(e)), then there is F ∈ c2 ∩ &(f(e)) such that B6F. So f (F) ∈ c1 ∩ &(e) by (1) and hence f (B) ∈ &(e). Condition (4) implies (1). Suppose that B ∈ c2 while f (B) ∈= c1 . Then there is a molecular e6c1 (f (B)) but e6 = f (B). Thus f(e)6 = B, i.e. B ∈ &(f(e)). From (4), f (B) ∈ &(e), i.e. there is F ∈ c1 ∩ &(e) such that f (B)6F. Thus c1 (f (B))6F and hence e6 = c(f (B)). This is a contra diction. So f (B) ∈ c1 . Condition (4) implies (5). Let S = {s(n) : n ∈ D} be a molecular net in LX1 and e ∈ M ∗ (LX1 ). If S →c e, then f(e) ∈ M ∗ (LY2 ) and f(S) →c f(e). In fact, for every B ∈ &(f(e)), by (4), f (B) ∈ &(e). Since S →c e, there is n0 ∈ D such that s(n)6 = f (B) whenever n¿n0 . Thus f(s(n))6 = B whenever n¿n0 . Condition (5) implies (4). Assume that (4) is not true. Then there are e ∈ M ∗ (LX1 ) and B ∈ &(f(e)) such that f (B) ∈= &(e). Thus for every E ∈ &(e); f (B)6 = E. There is e(E) ∈ M ∗ (LX1 ) such that e(E)6 f (B) but e(E)6 = E. Therefore S = {e(E) : E ∈ &(e)} →c f(e) is not true.
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In general, the image of an ideal in LX1 under a GOH is not an ideal, even an ideal base, in LY2 . However, under some condition, the image of an ideal in LX1 under a GOH may be an ideal base, even an ideal, in LY2 , which is shown in the following lemma. Lemma 5.1. Let f : LX1 → LY2 be a GOH and f(A) = 1Y if and only if A = 1X . If I is an ideal in LX1 , then f(I ) = {f(A) : A ∈ I } is an ideal base in LY2 . f(I )∗ denotes the ideal generated by f(I ). If f is also a bijective mapping, then f(I ) is an ideal in LY2 . Proof. We know 1Y ∈= LY2 as f(A) = 1Y if and only if A = 1X . If f(A); f(B) ∈ f(I ), then f(A ∨ B) ∈ f(I ) and f(A)6f(A ∨ B); f(B)6f(A ∨ B). So f(I ) is an ideal base in LY2 . Suppose that f is also a bijective mapping. If f(A) ∈ f(I ) and there exists E ∈ LY2 such that E6f(A), then f (E)6A. Thus f (E) ∈ I and hence E = f(f (E)) ∈ f(I ). So f(I ) is an ideal in LY2 . Naturally, we can obtain an ideal from a molecule whose convergence is uniform with the convergence of the molecule net. This is shown in the following result. Lemma 5.2. Let S = {s(n) : n ∈ D} be a molecular net in (LX ; c) and I (S) = {A : S is eventually not in A}. Then I (S) is an ideal in LX and L-lim S = L-lim I (S). Proof. Let S →c e, then for every P ∈ &(e); S is eventually not in P. So P ∈ I (S) and hence I (S) →c e. Conversely, if I (S) →c e, then for every P ∈ &(e); P ∈ I (S). Thus S is eventually not in P. So S →c e. Now we characterize a continuous GOH by ideals. Theorem 5.2. Let f : (LX1 ; c1 ) → (LY2 ; c2 ) be a GOH, f(A) = 1Y if and only if A = 1X and f is a bijective mapping. Then the following conditions are equivalent: (4) For every e ∈ M ∗ (LX1 ) and B ∈ &(f(e)); f (B) ∈ &(e). (6) If I is an ideal in LX1 ; I →c e and &(e) is directed, then f(I ) →c f(e). Proof. Let B ∈ &(f(e)). Then f (B) ∈ &(e) by (4). Thus f (B) ∈ I by I →c e. So B = f(f (B)) ∈ f(I ) and hence f(I ) →c f(e). Conversely, assume that (4) is not true. Then there is e ∈ M ∗ (LX1 ) and B ∈ &(f(e)) such that f (B) ∈= &(e). Let I = &(e). Then I is an ideal in LX1 and I →c e. But f(I ) →c f(e) is not true by B ∈= f(I ); B ∈ &(f(e)) and f (B) ∈= &(e). As we know, the concept of 2lter and the concept of ideal are dual. Therefore, we can also characterize a continuous GOH by 2lter. To do so, we introduce the concept of an L-cluster point of a 2lter, 2rstly. De nition 5.3. Let (LX ; c) be an L-cs, 1 is a 2lter (or 2lter base) in LX and e ∈ M ∗ (LX ). e is called an L-cluster point of 1 (in symbol 1 ∝c e) if for every P ∈ &(e) and every F ∈ 1; F6 = P. Similar to Lemma 5.1, we have the following result for the image of a 2lter under a GOH.
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Lemma 5.3. Let f : (LX1 ; c1 ) → (LY2 ; c2 ) be a GOH and f(A) = 0Y if and only if A = 0X . If 1 is a :lter in LX1 , then f(1) = {f(P) : P ∈ 1} is a :lter base in LY2 . Proof. 0Y ∈= f(1) as 0X ∈= 1 and f(A) = 0Y if and only if A = 0X . If f(A); f(B) ∈ f(1), then f(A ∧ B) ∈ f(1) and f(A)¿f(A ∧ B); f(B)¿f(A ∧ B). So f(1) is a 2lter base in LY2 . In the following lemma, we want to state that the convergence of 2lter base is equivalent to the convergence of the 2lter generated by the above 2lter base in an L-cs. So when we study the convergence of a 2lter, we need only consider the convergence of a 2lter base which generates the above 2lter. Lemma 5.4. Let 1 be a :lter base in (LX ; c); 1∗ is a :lter generated by 1 and e ∈ M ∗ (LX ). Then 1∗ ∝c e if and only if 1 ∝c e. Proof. The necessity is obvious by De2nition 5.3. Conversely, suppose that 1 ∝c e. For every E ∈ 1∗ , there exists F ∈ 1 such that F6E. So for every P ∈ &(e); F6 = P by De2nition 5.3. Thus ∗ E6 = P. So 1 ∝c ; e by De2nition 5.3. This completes the proof of the suPciency of this lemma. In the last part of this section, we shall characterize a continuous GOH by 2lter. Theorem 5.3. Let f : (LX1 ; c1 ) → (LY2 ; c2 ) be a GOH and f(A) = 0Y if and only if A = 0X . Then the following conditions are equivalent: (4) For every e ∈ M ∗ (LX1 ) and B ∈ &(f(e)); f (B) ∈ &(e). (7) If 1 is a :lter in LX1 and 1 ∝c e, then f(1) ∝c f(e). Proof. Let B ∈ &(f(e)) and f(F) ∈ f(1) where F ∈ 1. Then f (B) ∈ &(e) by (4). Thus F6 = f (B) by 1 ∝c e. So f(F)6 = B and hence f(1) ∝c f(e). Conversely, assume that (4) is not true. Then there exist e ∈ M ∗ (LX1 ) and B ∈ &(f(e)) such that f (B) ∈= &(e). Let 1 = {f (B)}. Then 1 is a 2lter base in LX1 and 1 ∝c e. In fact, from f (B) ∈= &(e), we have f (B)6 = F for every F ∈ &(e). But by f(f (B))6B; f(e) is not a cluster point of f(1∗ ). 6. The topological property of the category L-CLOSURE The main goal of this section is to get the important result that the category L-CLOSURE is a topological category over SET. First, we explain what is the category L-CLOSURE. Proposition 6.1. Let L be a :xed basis. Then the class of L-cs’ and L-continuous mappings between two L-cs’ form a category called L-CLOSURE. Proof. In L-CLOSURE, the object is an L-cs, the morphism is an L-continuous mapping between two L-cs’ and the composition of two morphisms is the composition of two L-continuous mappings. It is easy to check that the axioms of a category are satis2ed.
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In order to prove the main result, we need to discuss the following properties of the family of L-closed sets in an L-cs. Proposition 6.2. Let (LX ; c) be an L-cs. Then c possesses the following properties: (c1) 0X ∈ c ; 1X ∈ c ; (c2) If Fj ∈ c ( j ∈ J; J is the indexing set), then j∈J Fj ∈ c . Proof. It is obvious from De2nition 3.1. Proposition 6.3. If ∗ ⊆ LX satis:es conditions (c1) and (c2), then there exists a unique L-cs (LX ; c) on X such that the family c of its L-closed sets is just ∗ . Proof. De2ne operator c : LX → LX such that for every A ∈ LX ; c(A) = {F ∈ ∗ : A6F}. Then it is easy to show that (cl1), (cl2) and (cl3) hold. Moreover, since for every F ∈ ∗ ; c(F) = F and (c2), c(A) ∈ ∗ and hence (cl4) holds. So c is an L-closure. For the later part, on the one hand, ∗ ⊆ c from the de2nition of c. On the other hand, if E ∈ LX with c(E) = E, then E ∈ c by the de2nition of c(E) and condition (c2). Therefore c = ∗ . Finally, if there exists another L-closure c1 such that the family c1 of its L-closed sets is just ∗ , then it is easy to show that for every A ∈ LX ; c(A) = c1 (A). So c = c1 . This completes the proof of uniqueness. Let : L-CLOSURE → SET be the forgetful functor. In order to prove that (L-CLOSURE, ) is 2bre-complete, a partial ordered on the set CL (X ) of all L-closures on X is needed to introduce and we need to prove that the partial ordered set is a complete lattice. The partial ordered on the set CL (X ) is de2ned as follows. For every pair of c1 ; c2 ∈ CL (X ), c1 4 c2 ⇔ c1 ⊆ c2 . For the partial ordered set (CL (X ); 4), we have the following result. Proposition 6.4. (CL (X ); 4) is complete lattice. Proof. Obviously the mapping c : LX → LX de2ned by c (A) = A; ∀A ∈ LX is an L-closure and the universal upper bound in CL (X ) w.r.t. 4. Therefore, it is suPcient to verify that the in2mum of any nonempty family {ci : i ∈ I } of L-closures ci on X exists. In fact, X ∗ let ci be the family of all L-closed sets in (L ; ci ) and = i∈I ci . Then ∗ satis2es conditions (c1) and (c2) as well. So by Proposition 6.3, there exists an L-closure c on X such that the family c of all L-closed sets equals ∗ . Therefore c 4 ci (∀i ∈ I ). It is easy to show that c is the in2mum of {ci : i ∈ I }. Let Q be a nonempty subset of LX such that 0X ∈ . Q Then Q generates an L-closure cQ in the following sense: cQ := {c ∈ CL (X ) : Q ⊆ c }:
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By Proposition 6.4, cQ is the smallest L-closure on X whose family of all L-closed sets contains . Q Furthermore, to prove that every co-structured map ’ : ((LX ; cX )) → Y (see the proof of Theorem 6.1 below) has a unique 2nal lift, we shall introduce the concept of subbase of L-closure and give a condition for L-continuity by means of subbase as follows. De nition 6.1. A nonempty subset Q of LX with 0X is called a subbase of an L-closure c on X if c = cQ. Proposition 6.5. Let (LXi ; ci ) be an L-cs (i = 1; 2), Q a subbase of c2 and f : X1 → X2 an L-value Zadeh-type powerset operator. Then the following assertions are equivalent: (1) f→ is L-continuous mapping between (LX1 ; c1 ) and (LX2 ; c2 ). (2) {B ◦ f : B ∈ } Q ⊆ c1 . Proof. Implication (1) ⇒ (2) is obvious. In order to show (2) ⇒ (1), we consider a subset ∗ of LX2 where ∗ = {B ∈ LX2 : B ◦ f ∈ c1 }. Then it is easy to show that ∗ satis2es conditions (c1) and (c2), as c1 satis2es (c1) and (c2). So by Proposition 6.3, there exists an L-closure c∗ on X2 such that c∗ = ∗ . From Q ⊆ ∗ and De2nition 6.1, we know that c2 4 c∗ or c2 ⊆ ∗ . Therefore {B ◦ f : B ∈ c2 } ⊆ c1 and hence f is L-continuous. Finally, we recall the concept of topological category and prove the main result of this section. De nition 6.2 (Long de2nition of topological category, HRohle and Rodabaugh [2]). Let categories A and X be given, as well as a functor V : A → X, and let X ∈ |X|; Aj ∈ |A|, and fj : X → V (Aj ) in X, where j ∈ J for some indexing set J . Then (X; fj : X → V (Aj ))J is a V -structured source. Category A is topological w.r.t. category X and functor V i@ for each such V -structured source all the following hold: (1) V -lift: ∃Xˆ ∈ |A|; ∃(fˆj : Xˆ → Aj )J in A, V (Xˆ ) = X; V (fˆj ) = fj . (2) Initial V-lift: Given (Xˆ ; fˆj : Xˆ → Aj )J in A from (1), then ∀(Yˆ ; gˆj : Yˆ → Aj )J in A with Y ≡ V (Yˆ ) ˆ and and gj ≡ V (gˆj ); ∀h : Y → X in X with gj = fj ◦ h in X, ∃!hˆ : Yˆ → Xˆ in A with h = V (h) gˆj = fˆ ◦ hˆ in A. (3) Unique initial V-lift: Given (Xˆ ; fˆj : Xˆ → Aj )J satisfying (1) and (2), and given (XQ ; fQj : XQ → Aj )J satisfying (1) and (2), Xˆ = XQ and fˆj = fQj . Theorem 6.1. Let : L-CLOSURE → SET be the forgetful functor. Then L-CLOSURE is a topological category over SET w.r.t. . Proof. Let Q = {0Y ; 1Y } and ’ : ((LX ; cX )) → Y be a co-structured map. Then 0Y ◦ ’ ∈ cX ; 1Y ◦ ’ ∈ cX . Denote (LY ; cY ) by the unique L-closure on Y generated by . Q Then ’→ : (LX ; cX ) → (LY ; cY ) is L-continuous by Proposition 6.5. So ’ has a unique 2nal lift ’→ . Further, (L-CLOSURE, ) is 2bre-complete from Proposition 6.4. Hence, it is suPcient to show that for every L-closure space
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(LY ; cY∗ ), a map ’ : X → Y has a unique initial lift ’→ : (LX ; cX∗ ) → (LY ; cY∗ ) (cf. 21.c in [1]). Let ∗ = {B ◦ ’ ∈ LX : B ∈ cY∗ }. Denote (LX ; cX∗ ) by the unique L-closure space on X generated by ∗ . From the de2nition of ∗ and Proposition 6.5, ’→ : (LX ; cX∗ ) → (LY ; cY∗ ) is L-continuous w.r.t. cX∗ and cY∗ . In order to show that ’ is an initial morphism we consider a further L-continuous map → : (LZ ; c ) → (LY ; c∗ ) and a map : : Z → X such that = ’ ◦ :. From the de2nition of ∗ and the Z Y L-continuity of , we have that (B ◦ ’) ◦ : = B ◦
∈ cZ :
Since ∗ is the subbase of (LX ; cX∗ ), : is L-continuous; i.e. ’ is an initial arrow. The uniqueness of the lift follows from the antisymmetry of the partial ordering 4 “coarser”. This completes the proof of this theorem. 7. Constructions in L-CLOSURE In this section, the subspace of an L-cs, the sum L-cs, the product of L-cs’ and the induced L-cs of a crisp closure space will be constructed. From Theorem 6.1, we know that the category L-CLOSURE is topological over SET. Since a topological category inherits whatever limits and colimits the ground category has and SET is complete and cocomplete, the category L-CLOSURE is complete and cocomplete. So the following Theorems 7.2 and 7.3 certainly hold, the proofs are omitted here. First, we construct the subspace of an L-cs. De nition 7.1. Let Y be a subset of X . For A ∈ LY ; A∗ ∈ LX is given as: for every x ∈ X; A∗ (x) = A(x) if x ∈ Y and A∗ (x) = 0 if x ∈ X \Y . For B ∈ LX , B|Y ∈ LY is de2ned by (B|Y )(y) = B(y) if y ∈ X and (B|Y )(y) = 0 if y ∈ X \Y . We can make use of De2nition 7.1 to construct the subspace of an L-cs as follows: Theorem 7.1. Let (LX ; c) be an L-cs and Y a nonempty subset of X. If cY : LY → LY is such that cY (A) = c(A∗ )|Y for every A ∈ LY , then cY is an L-co and hence (LY ; cY ) is an L-cs on Y. In this case, we call cY a relative L-co of c and (LY ; cY ) an L-closure subspace of (LX ; c). Proof. (1) cY (0Y ) = c(0∗Y )|Y = 0Y . (2) cY (A) = c(A∗ )|Y ¿A∗ |Y = A. (3) If A6B, then A∗ 6B∗ and c(A∗ )6c(B∗ ). So cY (A) = c(A∗ )|Y 6c(B∗ )|Y = cY (B). (4) cY (cY (A)) = cY (c(A∗ )|Y ) = c((c(A∗ )|Y )∗ )|Y = c(c(A∗ ))|Y = c(A∗ )|Y = cY (A). As the same as in topological spaces, the L-closed set in L-closure subspace (LY ; cY ) may be generated by an L-close set in (LX ; c), which is shown in the following proposition. Proposition 7.1. Let (LX ; c) be an L-cs and (LY ; cY ) an L-closure subspace of (LX ; c). If A is L-closed in (LX ; c), then A|Y is L-closed in (LY ; cY ).
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Proof. Since c(A) = A, cY (A|Y ) = (c(A|Y )∗ )|Y = c(A)|Y = A|Y . Second, we construct the sum L-cs of L-cs’.
Theorem 7.2. Let = {(LXt ;
ct ) : t ∈ T } be pairwise disjoint L-cs’ and X = t ∈T Xt . Then ⊕ ct : LX → LX de:ned by ⊕ ct (A) = t ∈T ct (A|Xt ) for A ∈ LX is an L-co on X and hence (LX ; ⊕ ct ) is an L-cs. In this case, ⊕ ct is called sum L-co of ct (t ∈ T ) and (LX ; ⊕ct ) is called the sum L-cs of . Next we construct the product L-cs of L-cs’. Theorem 7.3. Let = {(LXt ; ct ) : t ∈ T } be a family of L-cs’, X = t ∈T Xt and pt : LX→ LXt be the X X X Zadeh-type powerset ←operator (t ∈ T ). Let c : L → L be a GOH de:ned Xby c(A) = i∈I {Vi ∈ L : Vi ¿A and Vi = {pj (Aj ) : Aj is L-closed, j = 1; : : : ; ki ; ki ∈ N }} for A ∈ L , where N is the set of all natural numbers, then c is an L-co on X and hence (LX ; c) is an L-cs and called the product L-cs of . In the following, we consider a crisp closure space based on which an L-cs can be constructed. Let c be a crisp closure on X , i.e. c : p(X ) → p(X ) satis2es the following conditions. (ccl1) (ccl2) (ccl3) (ccl4)
c(∅) = ∅. For every A ⊆ X; A ⊆ c(A). For every A; B ⊆ X , if A ⊆ B, then c(A) ⊆ c(B). For every A ⊆ X; c(c(A)) = c(A). The set of all closed sets in (X; c) is =, where = = {F ∈ p(X ) : c(F) = F} and = satis2es the following conditions. (cc1) ∅ ∈ =. (cc2) If {Fi : i ∈ I } ⊆ =, then i∈I Fi ∈ =.
Let A ∈ LX . Call A an upper semi-continuous function on (X; c) if for every a ∈ L; >a (A) ∈ =, where >a (A) = {x ∈ X : A(x)¿a}. Denote !L (=) the set of all upper semi-continuous functions on (X; c). Then !L (=) satis2es conditions (c1) and (c2). In fact, 0X ∈ !L (=) is obvious from the de2nition of uppersemi-continuous function. For the rest, let ⊆ !L (=). It is no harm to assume that = ∅. Let A = {B : B ∈ }. Then for every a ∈ L, >a (A) = {x ∈ X : A(x)¿a} = B∈ >a (B) ∈ = by (cc2). So by Proposition 6.3, there exists a unique L-closure denoted by !L (c) on X such that !L (c) = !L (=). To sum up, we have obtained an L-closure from a crisp closure and the L-closure is called induced L-closure. Namely, let C(X ) be the set of all crisp closures on X . Then we have a map !L : C(X ) → CL (X ) de2ned by c → !L (c): Let CLOSURE be the category of all crisp closures spaces on X and all continuous mappings (i.e. the preimage of every closed set in (Y; cY ) is closed set in (X; cX )). Then !L : CLOSURE → L-CLOSURE is a functor. Now we consider the inverse problem. Let (LX ; c) be an L-closure space, =∗ = {>a (B) : a ∈ L; B ∈ c } and @L (c ) = {F : F = i∈IF >ai (B); ai ∈ L; Bi ∈ =∗ ; i ∈ IF ; IF is an indexing set}. Then @L (c ) satis2es
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(cc1) and (cc2). So by Proposition 6.3 (the special case L = {0; 1}), there exists a unique crisp closure denoted by @L (c) on X such that the family =@L (c) of all closed sets in (X; @L (c)) is @L (c ). So we have a map @L : CL (X ) → C(X ) de2ned by c → @L (c) and a functor @L : L-CLOSURE → CLOSURE: The functors !L and @L possess many interesting properties which is shown in the following proposition. Proposition 7.2. The mappings !L ; @L possess the following properties: (1) For every c ∈ C(X ); (@L ◦ !L )(c) = c. (2) For every c ∈ CL (X ); (!L ◦ @L )(c) ¡ c. Particularly, if c = !L (c∗ ), then (!L ◦ @L )(c) = c, where c∗ ∈ C(X ). Proof. (1) Because {>a (B) : a ∈ L; B ∈ !L (=)} is the subbase of @L (!L (c)), (@L ◦ !L )(F) = F for every F ∈ =. So (@L ◦ !L )(=) ⊆ = and hence (@L ◦ !L )(c) 4 c. Conversely, for every F ∈ =, let A = AF , the characteristic function of F. Then A ∈ !L (=). Let a = 1 ∈ L. Then F = >a (A) = {x ∈ X : AF (x)¿1} ∈ {>a (B) : a ∈ L; B ∈ !L (=)}. So (@L ◦ !L )(=) ⊇ = and hence (@L ◦ !L )(c) ¡ c. Therefore ∀c ∈ C(X ); (@L ◦ !L )(c) = c. (2) Let c ∈ CL (X ), A ∈ c . Then ∀a ∈ L; >a (A) ∈ @L (c ). This shows that A is an upper semicontinuous function on (X; @L (c)). So A ∈ (!L ◦ @L )(c ) and hence (!L ◦ @L )(c) ¡ c. Especially, if c = !L (c∗ ), then (!L ◦ @L )(c) = (!L ◦ @L )(!L (c∗ )) = !L ((@L ◦ !L )(c∗ )) = !L (c∗ ) = c. From the concept of full subcategory, one can easily obtain the following result whose proof is omitted. Proposition 7.3. Category L-INDCLOSURE is a full subcategory of L-CLOSURE where each object of L-INDCLOSURE is (LX ; !L (c)) on LX induced by some crisp closure space (X; c) on X and each morphism of L-INDCLOSURE is the L-continuous mapping. If we restrict @L on the category L-INDCLOSURE, then we can obtain another important result in this paper. For this, we introduce the concept of adjunction of categories, :rstly. De nition 7.2 (Adjunction of categories HRohle and Rodabaugh [2], Mac Lane [5]). Let F : C → D and G : D → C be functors. We say F is left-adjoint to G if the following two criteria are satis2ed: (1) Lifting/continuity criterion: ∀A ∈ |C|; ∃ : A → GF(A); ∀B ∈ |D|; ∀f : A → G(B); Q ◦ : ∃!fQ : F(A) → B; f = G(f)
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(2) Naturality criterion: ∀f : A1 → A2
in C; F(f) = (or ≡)A2 ◦ f;
where the “≡” option allows F to only be de2ned initially on objects, in which (1) and “≡” will stipulate an action of F on morphisms such that F G. We also say that G is right-adjoint to F or that (F; G) is an adjunction, or we may write F G. The map is the unit of the adjunction; and the dual D morphism C of in the duals of the above statements is the counit of the adjunction. Finally, we state and proof another important result in this paper which shows us the relationship between two functors !L and @L . Theorem 7.4. The functor !L : CLOSURE → L-INDCLOSURE is the left-adjoint of the functor @L : L-INDCLOSURE → CLOSURE. Proof. (1) For every (X; c1 ) ∈ |CLOSURE|, let = @L ◦ !L . Then is the identity mapping by Proposition 7.2. ∀(LX ; !L (c2 )) ∈ |L-INDCLOSURE| and every continuous mapping f : (X; c1 ) → (X; @L (!L Q = f by Proposition 7.2. Therefore the Lifting=Continuity cri(c2 ))), taking fQ = !L (f), then ◦ @L (f) terion is satis2ed. (2) For every (X; c1 ); (X; c2 ) ∈ |CLOSURE|, (@L ◦ !L )((X; c1 )) = (X; c1 ) and (@L ◦ !L )((X; c2 )) = (X; c2 ) by Proposition 7.2. Taking A1 = A2 = id|CLOSURE| , then ∀f : (X; c1 ) → (X; c2 ), A2 ◦ f = f. So taking A2 ◦ f = !L (f), then A1 ◦ (@L (A2 ◦ f)) = A2 ◦ f. Therefore the Naturality criterion is satis2ed. By (1) and (2), the proof of this theorem is completed.
8. Conclusion In this paper, we have introduced the concepts of L-closure space and continuous generation order homomorphism and studied many important properties of L-closure space, such as category properties, especially. On the one hand, the concept of L-closure space in this paper is a generalization of fuzzy closure space introduced in [6,8,9]. On the other hand, the results about the category properties of L-closure space in this paper are deeper than in [6,8,9]. However, separation and compactness properties in L-closure spaces have not been studied in this paper and remain for further research.
Acknowledgements The author wishes to thank the Main Editor, Prof. S.E. Rodabaugh and the referees for their valuable comments and helpful suggestions.
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References [1] J. Ad˙amek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990. [2] U. HRohle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers, Dordrecht, 1999. [3] P.T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982. [4] Y.M. Liu, M.K. Luo, Fuzzy Topology, World Scienti2c Publishers, Singapore, 1997. [5] S. Mac Lane, Categories for the Working Mathematician, Springer, Berlin, Heidelberg, New York, 1971. [6] A.S. Mashhour, M.H. Ghanim, Fuzzy closure space, J. Math. Anal. Appl. 106 (1985) 154–170. [7] S.E. Rodabaugh, Categorical frameworks for Stone Representation Theories, in: S.E. Rodabaugh, et al., (Eds.), Applications of Category Theory to Fuzzy Subsets, Kluwer Academic Publishers, Netherlands, 1992, pp. 177–231. [8] R. Srivastava, A.K. Srivastava, A. Choubey, Fuzzy closure spaces, J. Fuzzy Math. 2 (1994) 525–534. [9] R. Srivastava, M. Srivastava, On T0 and T1 -closure spaces, Fuzzy Sets and Systems 109 (2000) 263–269. [10] G.J. Wang, Theory of L-fuzzy Topological Spaces, Press of Shaanxi Normal University, Xian, 1988 (in Chinese). [11] W.N. Zhou, S.L. Chen, Some important applications of net of L-fuzzy subsets, Fuzzy Sets and Systems 95 (1998) 91–98. [12] W.N. Zhou, P.Y. Meng, Relative remote neighborhood family and the characterizations of ultra-fuzzy compactness, Fuzzy Sets and Systems 109 (2000) 217–222.
Fuzzy Sets and Systems 149 (2005) 415 – 432 www.elsevier.com/locate/fss
Generalization of L-closure spaces Wu-Neng Zhou Institute of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, PR China Received 5 January 2001; received in revised form 8 March 2004; accepted 29 March 2004
Abstract This paper introduces the concepts of L-closure spaces and the convergence in L-closure spaces. In the variable-basis setting of such closure spaces, continuous generalized order homomorphisms are characterized by the convergence theory of nets, 2lters and ideals. Furthermore, in the 2xed-basis setting of such closure spaces, some constructions on L-closure spaces, such as the subspace of an L-cs, the sum L-cs, the product of L-cs’ and the induced L-cs of a crisp closure space, are investigated. We obtain an important result that the category L-CLOSURE is a topological category over SET w.r.t. the forgetful functor from L-CLOSURE to SET. We obtain another important result that there is an adjunction between the category of crisp closure spaces and the full subcategory of those L-closure spaces which are induced by some crisp closure space. c 2004 Published by Elsevier B.V. Keywords: L-closure space; L-convergence; L-continuous mapping; L-closure subspaces; L-closure sum spaces; L-closure-induced spaces
1. Introduction The theory of fuzzy closure spaces has been established by Mashhour and Ghanim [6] and = Srivastava et al. [8,9]. The de2nition of Mashhour and Ghanim is an analogue of Cech closure spaces and Srivastava et al. have introduced it as an analogue of Birkho@ closure spaces in [8]. Based on [8], Rekha Srivastava and Manjari Srivastava studied the subspace of a fuzzy closure space, the sum of a family of pairwise disjoint fuzzy closure spaces and the product of a family of fuzzy closure spaces in [9]. The notion of T1 fuzzy closure space was also introduced in [9]. However, all the above works are in the 2xed-basis setting [0; 1], and there are no ideals of topological category in the above obtained results. In this paper, the aim is to introduce the concept of L-closure spaces which is a generalization of the concept of fuzzy closure spaces and to study many important properties for L-closure spaces, such as category property, especially.
Supported by the National Natural Science Foundation of PR China (NSFC: 10371106). E-mail address: [email protected] (W.-N. Zhou).
c 2004 Published by Elsevier B.V. 0165-0114/$ - see front matter doi:10.1016/j.fss.2004.03.026
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This paper is organized as follows. Section 2 introduces the concepts of L-closure operator and L-closure spaces and discusses the relation between L-closure spaces and L-topological spaces. Two L-closure operators on L-unit interval are constructed. They are important for the subsequent discussions. Section 3 introduces the concepts of convergence for net and ideal in L-closure spaces and gets the characteristic properties. The concepts of continuous generalized order homomorphism and L-continuous mapping are introduced and many characteristic properties for generalized order homomorphism are discussed in Section 4. Section 5 contains one of the most important results, namely that the category whose objects are all L-closure spaces and arrows all L-continuous mappings between two L-closure spaces is a topological category over the category SET. Section 6 constructs many L-closure spaces from some known L-closure spaces. Another important result is obtained in this section, namely that there is an adjunction between the category of crisp closure spaces and the full subcategory of those L-closure spaces which are induced by some crisp closure spaces. In the conclusion section, we summarize the results of this paper and point out some problems to be further studied. 2. Technical preliminaries Throughout this paper, X; Y; Xt (t ∈ T , T is an indexing set) denote nonempty crisp sets, L; L1 ; L2 denote completely distributive lattices with an order-reversing involution “” (i.e. ∀a; b∈L; a6b⇒a ¿ b and for every a ∈ L, a = a). LX = {A | A : X → L is a mapping}. 0 and 1 denote the smallest and the largest element of L, respectively. ∈ L is called a union-irreducible element, if = a ∨ b implies
= a or = b for arbitrary a; b ∈ L. M (L) denotes the set of all nonzero union-irreducible elements of L. For each x ∈ X; ∈ M (L); x ∈ LX is called a molecule which satis2es x (y) = for y = x and x (y) = 0 for y = x. Let 0X ; 1X ∈ LX be L-sets de2ned by 0X (x) = 0; 1X (x) = 1 for all x ∈ X . Let M ∗ (LX ) be the set {x ∈ LX : x ∈ X; ∈ M (L)}. For the sake of convenience, we recall the following concepts and results used later in this paper. De nition 2.1 (Liu and Luo [4], Wang [10]). Let be a nonempty subset of LX . We call an L-topology on X if satis2es the following conditions: (1) 0X ; 1X ∈ . (2) If U; V ∈ , thenU ∧ V ∈ . (3) If 1 ⊆ , then U ∈1 U ∈ . The pair (LX ; ) is called an L-topological space (L-ts for short). Each element of is called an open set in (LX ; ). The complement of an open set is called a closed set in (LX ; ). In an L-ts (LX ; ), c∗ (A) = ∧ {F ∈ LX : F is a closed set and F¿A} is called the closure of A for A ∈ LX . Lemma 2.1 (Liu and Luo [4]). Let (LX ; ) be an L-ts. Then the following hold for all A; B ∈ LX . (1) c∗ (0X ) = 0X ; c∗ (1X ) = 1X ; (2) A6c∗ (A);
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(3) A6B ⇒ c∗ (A)6c∗ (B); (4) c∗ (A ∨ B) = c∗ (A) ∨ c∗ (B); (5) c∗ (c∗ (A)) = c∗ (A). De nition 2.2 (Liu and Luo [4]). Let A; B be posets and f : A → B, g : B → A be order preserving mappings. f is called a left adjoint of g and g is called a right adjoint of f, denoted by f g, if ∀a ∈ A; ∀b ∈ B;
f(a) 6 b ⇔ a 6 g(b):
Equivalently, f g if and only if ∀a ∈ A; ∀b ∈ B; a6gf(a) and fg(b)6b. 3. L-closure spaces In this section, we shall establish the concepts of L-closure spaces and L-closed sets and discuss the relation between L-closure spaces and L-topological spaces. Firstly, we introduce the concept of L-closure spaces which is a generalization of fuzzy closure spaces in [6,8,9]. De nition 3.1. A mapping c : LX →; LX is called an L-closure operator (L-co for short) or an L-closure, if it satis2es the following conditions for any A; B ∈ LX : (cl1) (cl2) (cl3) (cl4)
c(0X ) = 0X . A6c(A). A6B implies c(A)6c(B). c(c(A)) = c(A).
The pair (LX ; c) is called an L-closure space (L-cs for short) and F ∈ LX is called an L-closed set in (LX ; c) if c(F) = F. Let c = {F ∈ LX : c(F) = F}. Remark 3.1. From Lemma 2.1, we know that the closure operator c∗ : LX → LX in an L-topological space satis2es the conditions mentioned in De2nition 3.1. So an L-topological space (LX ; ) can induce an L-cs (LX ; c∗ ) where c∗ is the closure operator in L-ts (LX ; ). But the converse proposition is not true, which is shown in the following Example 3.1. Therefore the L-cs’ are more than L-ts’. Example 3.1. Let X = {a; b; c}; = {0X ; {a}; {b}; {a; c}; 1X }; L = I = [0; 1]. De2ne c : I X → I X as following: c(A) = ∧{F ∈ : F ¿ A}
for A ∈ I X :
Then c is an L-co by the de2nition of c(A). So (LX ; c) is an L-cs. But c is not a closure operator in any L-topological space. In fact, if c is a closure operator in an L-topological space (LX ; ), then c({a} ∨ {b}) = c({a}) ∨ c({b}). Now c({a} ∨ {b}) = c({a; b}) = 1X , and c({a}) ∨ c({b}) = {a} ∨ {b} = {a; b}. Hence c({a} ∨ {b}) = c({a}) ∨ c({b}). In the following example, we shall construct two L-co’s on I (L), the L-unit interval, which will be also used in Example 5.1 in the sequel.
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Example 3.2. Let R be the real line, : R → L be an order-reversing mapping (i.e. ∀s; t ∈ R; s6t ⇒
(s)¿ (t)) that satis2es the following condition: ∀t ¡ 0; (t) = 1;
∀t ¿ 1; (t) = 0:
denotes the set of this kind of mappings. For every ∈ and every t ∈ R, de2ne
(s); (t−) =
(s):
(t+) = s¿t
s¡t
De2ne an equivalence relation “∼” on as follows:
∼ ⇔ ∀t ∈ R; (t−) = (t−); (t+) = (t+): For every ∈ , let [ ] be the equivalence class in with respect to ∼, i.e. [ ] = { ∈ : ∼ }: Denote the family of all the equivalence classes in with respect to ∼ by I (L). For convenience, for every ∈ , if no confusion will be caused, we identify with every such that ∼ , and still use to denote [ ]. For every t ∈ R, de2ne Mt ; Nt ∈ LI (L) as follows:
(s); Nt ( ) = ( (t+)) = ( (s)) : ∀ ∈ I (L); Mt ( ) = (t−) = s¡t
s¿t
De2ne c : LI (L) → LI (L) such that Mt : ∀A ∈ LI (L) ; c(A) = M t ¿A
Then we have the following results:
c(0I (L) ) = 0I (L) . In fact, Mt ( ) = 0 whenever t¿1. So c(0I (L) ) = Mt ¿0I (L) Mt = 0I (L) . ∀A ∈ LI (L) ; A6c(A). This is obvious from the de2nition of c(A). ∀A; B ∈ LI (L) ; A6B ⇒ c(A)6c(B). This is obvious from the de2nition of c(A) and c(B). ∀t ∈ R; c(Mt ) = Mt . ∀A ∈ LI (L) ; c(c(A)) = c(A). In fact, c(c(A)) = c( Mt ¿A Mt )6 Mt ¿A c(M t ) = Mt ¿A Mt = c(A). ∀R1 ⊂ R with R1 = ∅and sup R1 = r, t ∈R1 Mt = Mr . In fact, ∀ ∈ ; ( t ∈R1 Mt )( ) = {
t ∈ R1 (v) : v¡t}; Mr ( ) = { (u) : u¡r}. ∀t ∈ R and v¡t, we have v¡t6r. So
(v)¿ { (u) : 1 u¡r} and hence t ∈R1 { (v) : v¡t}6 { (u) : u¡r}. Conversely, ∀u¡r; ∃t ∈ R1 such that u¡t6r. Take v ∈R with u¡v¡t. Then
(u)¿ (v)¿ { (v) : v¡t} by the decreasing t ∈ R1 property of . So t ∈R1 { (v) : v¡t}¿ { (u) : u¡r}. Therefore ( t ∈R1 Mt )( ) = (Mr )( ) for every ∈ . (7) If A ∈ LI (L) ; A = 0I (L) and c(A) = A, then there exists t0 ∈R such that A = Mt0 . In fact, c(A) = { (s) : s¡0} = 1; M0 ¿A. So T = ∅. M t ¿ A Mt = t ∈T {Mt : Mt ¿A}. Because ∀ ∈ ; M0 ( ) = On the other hand, if t¿1, then Mt = 0I (L) . So sup T exists and let r = sup T . Therefore c(A) = Mr by (6).
(1) (2) (3) (4) (5) (6)
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So c is an L-co on LI (L) , (LI (L) ; c) is an L-cs and every L-closed set in (LI (L) ; c) possesses form Mt . Similarly, de2ne c1 : LI (L) → LI (L) such that ∀A ∈ LI (L) ;
c1 (A) =
Nt :
Nt ¿ A
Then we can prove that c1 is another L-co on LI (L) , (LI (L) ; c1 ) is another L-cs and every L-closed set in (LI (L) ; c1 ) possesses form Nt .
4. L-convergence In this section, we shall introduce the concepts of convergence for net, ideal in L-cs and discuss their properties. The concept of remote neighborhood of a molecule plays an important part in discussing convergence. So we introduce the concept of remote neighborhood of a molecule, 2rstly. De nition 4.1. Let (LX ; c) be an L-cs, x ∈ M ∗ (L) and F ∈ c . If x 6 = F, then we call F an X L-closed remote neighborhood of x . If E ∈ L ; E6F ∈ c and x 6 = F, then we call E an L-remote neighborhood of x . &(x ) = {E ∈ LX : E is an L-remote neighborhood of x }. Applying the concept of remote neighborhood of a molecule, we can characterize the c(A) in L-cs (LX ; c) for A ∈ LX as follows. Theorem 4.1. Let (LX ; c) be an L-cs, x ∈ M ∗ (LX ) and A ∈ LX . Then x 6c(A) if and only if A6 = E for any E ∈ &(x ). This kind of x will be called an L-accumulation point of A and hence c(A) = {x ∈ M ∗ (LX ) : x is an L-accumulation point of A}. Proof. If x 6 = c(A), then c(A) ∈ &(x ). But A6c(A). So x is an L-accumulation point of A. Conversely, if there is P ∈ &(x ) such that A6P, then there is Q ∈ &(x ) ∩ c such that P6Q. Thus A6Q and c(A)6Q as Q is an L-closed set. Hence x 6 = c(A) since x 6 = Q. The molecular net is one of the most useful tools in studying the convergence in L-cs’. In the following, we introduce the concepts of limit point and cluster point of a molecular net in L-cs’ and discuss their properties, respectively. De nition 4.2. Let D be a directed set, S = {(xn ) n : n ∈ D; xn ∈ X; n ∈ M (L)} be a molecular net in (LX ; c) and x, ∈ M ∗ (LX ). (1) x, is called an L-limit point of S or S L-converges to x, , if for each P ∈ &(x, ); S is eventually not in P (i.e. ∃n0 ∈ D such that ∀n ∈ D and n¿n0 ; (xn ) n 6 = P), in symbol S →c x, . (2) x, is called an L-cluster point of S, if for each P ∈ &(x, ); S is frequently not in P (i.e. ∀n ∈ D; ∃m ∈ D and m¿n such that (xm ) m 6 = P), in symbol S ∝c x, .
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The union of all L-limit points of S is denoted as L-lim S. The union of all L-cluster points of S is denoted as L-clu S. Theorem 4.2. Let S be a molecular net in (LX ; c) and e ∈ M ∗ (LX ). Then e6L-lim S if and only if S →c e. e6L-clu S if and only if S ∝c e. Proof. It is obvious that if S →c e, then e6L-lim S by De2nition 4.2. Conversely, if e6L-lim S, then for every P ∈ &(e), there is F ∈ &(e) ∩ c such that P6F. Thus L-lim S6 = F. So there is an L-limit point , of S such that ,6 = F. Since S →c ,, S is eventually not in F. Thus S is eventually not in P. So S →c e. Similarly, it can be proved that e6L-clu S if and only if S ∝c e. Theorem 4.2 shows us that the molecule in L-set L-lim S (L-clu S, respectively) is equivalent to the L-limit point (L-cluster point, respectively) of S. Moreover, we can characterize the L-closure c(A) of an L-set A in (LX ; c) in terms of a molecular net as follows. Theorem 4.3. Let (LX ; c) be an L-cs, A ∈ LX and e ∈ M ∗ (LX ). (1) If there is a molecular net S in A and S ∝c e, then e6c(A). (2) If e6c(A) and &(e) is a directed set, then there is a molecular net in A such that S →c e. = P and e6c(A). Proof. (1) For every P ∈ &(e); S is frequently not in P by S ∝c e. So A6 (2) Let e6c(A). Then for every P ∈ &(e); A6 = P. Thus there is s(P)6A such that s(P)6 = P. So S = {s(P) : P ∈ &(e)} is a molecular net in A as &(e) is directed. Therefore for arbitrary P; Q ∈ &(e); s(Q)6 = P whenever Q¿P. So S →c e. The ideal is another useful tool in studying the convergence in L-cs’. Similarly, we introduce the concepts of limit point and cluster point of an ideal net in L-cs’ and discuss their properties, respectively. De nition 4.3. Let I be an ideal (i.e. I is an up directed set, a lower set and 1X ∈= I ) in (LX ; c) and x, ∈ M ∗ (LX ). (1) If &(x, ) ⊆ I , then x, is called an L-limit point of I or I L-converges to x, , written as I →c x, . (2) If for each A ∈ I and P ∈ &(x, ); A ∨ P = 1, then x, is called an L-cluster point of I , in symbol I ∝c x, . The union of all L-limit points of I is denoted as L-lim I . The union of all L-cluster points of I is denoted as L-clu I . In discussing the properties of convergence for ideals, the concepts of the minimal set of an , (, ∈ L) will be used which be quoted in the following de2nition. De nition 4.4 (Zhou andChen [11], Zhou and Meng [12]). Let A be a subset of L. A is called a minimal set of , ∈ L, if A = , and for each subset B of L with B¿,, and for each d ∈ A, there exists b ∈ B such that d6b. The union of all the minimal sets of , is denoted by 0(,), and let 0∗ (,) = 0(,) ∩ M (L).
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Theorem 4.4. Let I be an ideal in an L-cs (LX ; c); e ∈ M ∗ (LX ). Then e6L-lim I if and only if I →c e. e6L-clu I if and only if I ∝c e. Proof. It is obvious that if I →c e, then e6L-lim I by De2nition 4.3. Conversely, if e6L-lim I but I →c e is not true, then there is F ∈ &(e) such that F ∈= I . From F ∈ &(e), there is d ∈ 0∗ (e) such that F ∈ &(d). By d6e6L-lim I , there is an L-limit point u of I such that d6u. Thus I →c d. So F ∈ I . This is a contradiction. Therefore I →c e. Similarly, it can be proved that e6L-clu I if and only if I ∝c e. Theorem 4.4 shows us that there is no di@erence between the L-points in L-set L-lim I and the L-limit points of I . So does the L-points in L-set L-clu I and the L-cluster points of I . Similar to Theorem 4.3, we can characterize c(A) by ideal in L-cs’ as following theorem. Theorem 4.5. Let (LX ; c) be an L-cs, A ∈ LX ; e ∈ M ∗ (LX ) and &(e) a directed set. Then e6c(A) if and only if there is an ideal I of LX such that A ∈= I and I →c e. Proof. Let e6c(A). Then for every P ∈ &(e); A6 = P. Let I = &(e). Then I is an ideal such that =P I →c e and A ∈= I . Conversely, if an ideal I →c e and A ∈= I , then P ∈ I for every P ∈ &(e). Thus A6 and therefore e6c(A). 5. Continuous generalized order homomorphism In this section, we shall introduce the concepts of continuous generalized order homomorphism and L-continuous mapping and discuss their characterizations. De nition 5.1. Let f : LX1 → LY2 be a mapping. Call f a generalized order homomorphism (GOH for short) if it satis2es the following conditions: (1) f is -preserving. i.e. f( s∈S As ) = s∈S f(As ) for As ∈ LX1 , s ∈ S, S is an indexing set. (2) f (B ) = (f (B)) for every B ∈ LY2 where f (B) = {A ∈ LX1 : f(A)6B}. There is a relation between mapping f and mapping f in De2nition 5.1 which is shown in the following proposition. Proposition 5.1. Let f : LX1 → LY2 be a GOH. Then f f . Proof. This is an immediate corollary of the Adjoint Functor Theorem of [3]. Similar to the notion of the continuity in topological spaces, we can de2ne the continuity of GOH in L-cs as follows. De nition 5.2. Let (LX1 ; c1 ) be an L1 -cs, (LY2 ; c2 ) be an L2 -cs and f : LX1 → LY2 be a GOH. Call f a continuous GOH if f (B) is an L1 -closed set in (LX1 ; c1 ) for every L2 -closed set B in (LY2 ; c2 ). Specially, if L1 = L2 = L and f is Zadeh-type powerset operator, then we call a continuous GOH an
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L-continuous mapping between (LX ; c1 ) and (LY ; c2 ). Namely, f is L-continuous between (LX ; c1 ) and (LY ; c2 ) if and only if {B ◦ f : B ∈ c2 } ⊆ c1 . Because an L-ts can induce an L-cs as shown in Section 2, it is natural to consider the relation between the two notions of continuity in L-ts and in L-cs. Next remark illustrates that every L-continuous mapping between two L-ts’ can induce a continuous GOH between the L-cs’ induced by the two L-ts’. Remark 5.1. Let (LX ; ), (LY ; ) be L-ts’ and f : X → Y an ordinary mapping. Using the arrow notation [7], we produce the corresponding L-mappings f→ : LX → LY and f← : LY → LX as follows: ∀A ∈ LX ; ∀y ∈ Y; f→ (A)(y) = {A(x) : x ∈ X; f(x) = y}; ∀B ∈ LY ; f← (B) = {A ∈ LX : f→ (A) 6 B} = B ◦ f: f is called an L-continuous mapping [4] between (LX ; ) and (LY ; ), if ∀B ∈ ; f← (B) ∈ . Equivalently, f is L-continuous between (LX ; ) and (LY ; ) if and only if ∀B ∈ ; f← (B) ∈ since f← = (f→ ) . Let (LX ; c1 ) and (LY ; c2 ) be L-cs’ induced by (LX ; ); (LY ; ), respectively. Then for every L-continuous mapping f : (LX ; ) → (LY ; ), f→ is an L-continuous mapping between (LX ; c1 ) and (LY ; c2 ). To make the concept of L-continuity in L-cs clear, it is necessary to introduce the following example in which a continuous GOH in I -unit interval I I (I ) will be constructed. Example 5.1. In Example 3.2, let L = I . We take 1 ∈ as follows: 1; s 6 1;
1 (s) = 0; s ¿ 1: De2ne an ordinary mapping f : I (I ) → I (I ) as follows: ∀ ∈ ;
f( ) = [ 1 ]:
Then we can obtain the Zadeh-type powerset operator f→ : I I (I ) → I I (I ) induced by f and its right adjoint f← : I I (I ) → I I (I ) . In this case, ∀A ∈ I I (I ) ; ∀ ∈ , one gets {A( ) : ∈ }; = 0 ; → f (A)( ) = {A( ): f( ) = } = 0; = 0 : Furthermore, we can know that ∀B ∈ I I (I ) , 0; B( 1 ) = 0; f← (B) = {A : f→ (A) 6 B} = A : {A( ) : ∈ I (I )} 6 B( 0 ) ; B( 1 ) ¿ 0: It is obvious that f→ (A) is a fuzzy point whose supporting point is [ 1 ] and the height is {A( ) : ∈ }.
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Because f→ is a Zadeh-type powerset operator, f→ is a GOH (Wang, Theorem 1.8.5 [10]). On the other hand, the set of all L-closed sets in (I I (I ) ; c1 ) is {Nt : t ∈ R} by Example 3.2. Now ∀t ∈ R, 1; t ¿ 1; Nt ( 1 ) = ( 1 (s)) = (1 − 1 (s)) = 0; t ¡ 1: s¿t
So ←
f (Nt ) =
s¿t
0I (I ) ; 1I (I ) ;
t ¡ 1; t ¿ 1:
Therefore ∀t ∈ R; f← (Nt ) is an L-closed set in (I I (I ) ; c) and hence f→ is an L-continuous mapping in (I I (I ) ; c). We shall study some characteristic properties of a continuous GOH in terms of the convergence of molecular nets, ideals and 2lters. First, we can characterize a continuous GOH by L-remote neighborhood of a molecule, closed sets and molecular nets, etc. as following theorem. Theorem 5.1. Let (LX1 ; c1 ) be an L1 -cs, (LY2 ; c2 ) an L2 -cs and f : LX1 → LY2 be a GOH. Then the following conditions are equivalent: (1) (2) (3) (4) (5)
f is a continuous GOH. f(c1 (A))6c2 (f(A)) for every A ∈ LX1 . c1 (f (B))6f (c2 (B)) for every B ∈ LY2 . For every e ∈ M ∗ (LX1 ) and B ∈ &(f(e)); f (B) ∈ &(e). For every molecular net S in (LX1 ; c1 ), if S →c e and &(e) is a directed set, then the molecular net f(S) →c ; f(e).
Proof. Condition (1) implies (2). For every A ∈ LX1 ; A6f (f(A))6f (c2 (f(A))). Thus c1 (A)6f (c2 (f(A))). So f(c1 (A))6c2 (f(A)). Condition (2) implies (3). For every B ∈ LY2 ; f (B) ∈ LX1 . From (2), f(c1 (f (B)))6c2 (f(f (B))) 6c2 (B). So c1 (f (B))6f (c2 (B)). Condition (3) implies (1). Let B be an L2 -closed set in (LY2 ; c2 ). By (3), c1 (f (B))6f (c2 (B)) = f (B). So c1 (f (B)) = f (B) and f (B) is an L1 -closed set in (LX1 ; c1 ). Condition (1) implies (4). If e ∈ M ∗ (LX ) and B ∈ &(f(e)), then there is F ∈ c2 ∩ &(f(e)) such that B6F. So f (F) ∈ c1 ∩ &(e) by (1) and hence f (B) ∈ &(e). Condition (4) implies (1). Suppose that B ∈ c2 while f (B) ∈= c1 . Then there is a molecular e6c1 (f (B)) but e6 = f (B). Thus f(e)6 = B, i.e. B ∈ &(f(e)). From (4), f (B) ∈ &(e), i.e. there is F ∈ c1 ∩ &(e) such that f (B)6F. Thus c1 (f (B))6F and hence e6 = c(f (B)). This is a contra diction. So f (B) ∈ c1 . Condition (4) implies (5). Let S = {s(n) : n ∈ D} be a molecular net in LX1 and e ∈ M ∗ (LX1 ). If S →c e, then f(e) ∈ M ∗ (LY2 ) and f(S) →c f(e). In fact, for every B ∈ &(f(e)), by (4), f (B) ∈ &(e). Since S →c e, there is n0 ∈ D such that s(n)6 = f (B) whenever n¿n0 . Thus f(s(n))6 = B whenever n¿n0 . Condition (5) implies (4). Assume that (4) is not true. Then there are e ∈ M ∗ (LX1 ) and B ∈ &(f(e)) such that f (B) ∈= &(e). Thus for every E ∈ &(e); f (B)6 = E. There is e(E) ∈ M ∗ (LX1 ) such that e(E)6 f (B) but e(E)6 = E. Therefore S = {e(E) : E ∈ &(e)} →c f(e) is not true.
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In general, the image of an ideal in LX1 under a GOH is not an ideal, even an ideal base, in LY2 . However, under some condition, the image of an ideal in LX1 under a GOH may be an ideal base, even an ideal, in LY2 , which is shown in the following lemma. Lemma 5.1. Let f : LX1 → LY2 be a GOH and f(A) = 1Y if and only if A = 1X . If I is an ideal in LX1 , then f(I ) = {f(A) : A ∈ I } is an ideal base in LY2 . f(I )∗ denotes the ideal generated by f(I ). If f is also a bijective mapping, then f(I ) is an ideal in LY2 . Proof. We know 1Y ∈= LY2 as f(A) = 1Y if and only if A = 1X . If f(A); f(B) ∈ f(I ), then f(A ∨ B) ∈ f(I ) and f(A)6f(A ∨ B); f(B)6f(A ∨ B). So f(I ) is an ideal base in LY2 . Suppose that f is also a bijective mapping. If f(A) ∈ f(I ) and there exists E ∈ LY2 such that E6f(A), then f (E)6A. Thus f (E) ∈ I and hence E = f(f (E)) ∈ f(I ). So f(I ) is an ideal in LY2 . Naturally, we can obtain an ideal from a molecule whose convergence is uniform with the convergence of the molecule net. This is shown in the following result. Lemma 5.2. Let S = {s(n) : n ∈ D} be a molecular net in (LX ; c) and I (S) = {A : S is eventually not in A}. Then I (S) is an ideal in LX and L-lim S = L-lim I (S). Proof. Let S →c e, then for every P ∈ &(e); S is eventually not in P. So P ∈ I (S) and hence I (S) →c e. Conversely, if I (S) →c e, then for every P ∈ &(e); P ∈ I (S). Thus S is eventually not in P. So S →c e. Now we characterize a continuous GOH by ideals. Theorem 5.2. Let f : (LX1 ; c1 ) → (LY2 ; c2 ) be a GOH, f(A) = 1Y if and only if A = 1X and f is a bijective mapping. Then the following conditions are equivalent: (4) For every e ∈ M ∗ (LX1 ) and B ∈ &(f(e)); f (B) ∈ &(e). (6) If I is an ideal in LX1 ; I →c e and &(e) is directed, then f(I ) →c f(e). Proof. Let B ∈ &(f(e)). Then f (B) ∈ &(e) by (4). Thus f (B) ∈ I by I →c e. So B = f(f (B)) ∈ f(I ) and hence f(I ) →c f(e). Conversely, assume that (4) is not true. Then there is e ∈ M ∗ (LX1 ) and B ∈ &(f(e)) such that f (B) ∈= &(e). Let I = &(e). Then I is an ideal in LX1 and I →c e. But f(I ) →c f(e) is not true by B ∈= f(I ); B ∈ &(f(e)) and f (B) ∈= &(e). As we know, the concept of 2lter and the concept of ideal are dual. Therefore, we can also characterize a continuous GOH by 2lter. To do so, we introduce the concept of an L-cluster point of a 2lter, 2rstly. De nition 5.3. Let (LX ; c) be an L-cs, 1 is a 2lter (or 2lter base) in LX and e ∈ M ∗ (LX ). e is called an L-cluster point of 1 (in symbol 1 ∝c e) if for every P ∈ &(e) and every F ∈ 1; F6 = P. Similar to Lemma 5.1, we have the following result for the image of a 2lter under a GOH.
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Lemma 5.3. Let f : (LX1 ; c1 ) → (LY2 ; c2 ) be a GOH and f(A) = 0Y if and only if A = 0X . If 1 is a :lter in LX1 , then f(1) = {f(P) : P ∈ 1} is a :lter base in LY2 . Proof. 0Y ∈= f(1) as 0X ∈= 1 and f(A) = 0Y if and only if A = 0X . If f(A); f(B) ∈ f(1), then f(A ∧ B) ∈ f(1) and f(A)¿f(A ∧ B); f(B)¿f(A ∧ B). So f(1) is a 2lter base in LY2 . In the following lemma, we want to state that the convergence of 2lter base is equivalent to the convergence of the 2lter generated by the above 2lter base in an L-cs. So when we study the convergence of a 2lter, we need only consider the convergence of a 2lter base which generates the above 2lter. Lemma 5.4. Let 1 be a :lter base in (LX ; c); 1∗ is a :lter generated by 1 and e ∈ M ∗ (LX ). Then 1∗ ∝c e if and only if 1 ∝c e. Proof. The necessity is obvious by De2nition 5.3. Conversely, suppose that 1 ∝c e. For every E ∈ 1∗ , there exists F ∈ 1 such that F6E. So for every P ∈ &(e); F6 = P by De2nition 5.3. Thus ∗ E6 = P. So 1 ∝c ; e by De2nition 5.3. This completes the proof of the suPciency of this lemma. In the last part of this section, we shall characterize a continuous GOH by 2lter. Theorem 5.3. Let f : (LX1 ; c1 ) → (LY2 ; c2 ) be a GOH and f(A) = 0Y if and only if A = 0X . Then the following conditions are equivalent: (4) For every e ∈ M ∗ (LX1 ) and B ∈ &(f(e)); f (B) ∈ &(e). (7) If 1 is a :lter in LX1 and 1 ∝c e, then f(1) ∝c f(e). Proof. Let B ∈ &(f(e)) and f(F) ∈ f(1) where F ∈ 1. Then f (B) ∈ &(e) by (4). Thus F6 = f (B) by 1 ∝c e. So f(F)6 = B and hence f(1) ∝c f(e). Conversely, assume that (4) is not true. Then there exist e ∈ M ∗ (LX1 ) and B ∈ &(f(e)) such that f (B) ∈= &(e). Let 1 = {f (B)}. Then 1 is a 2lter base in LX1 and 1 ∝c e. In fact, from f (B) ∈= &(e), we have f (B)6 = F for every F ∈ &(e). But by f(f (B))6B; f(e) is not a cluster point of f(1∗ ). 6. The topological property of the category L-CLOSURE The main goal of this section is to get the important result that the category L-CLOSURE is a topological category over SET. First, we explain what is the category L-CLOSURE. Proposition 6.1. Let L be a :xed basis. Then the class of L-cs’ and L-continuous mappings between two L-cs’ form a category called L-CLOSURE. Proof. In L-CLOSURE, the object is an L-cs, the morphism is an L-continuous mapping between two L-cs’ and the composition of two morphisms is the composition of two L-continuous mappings. It is easy to check that the axioms of a category are satis2ed.
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In order to prove the main result, we need to discuss the following properties of the family of L-closed sets in an L-cs. Proposition 6.2. Let (LX ; c) be an L-cs. Then c possesses the following properties: (c1) 0X ∈ c ; 1X ∈ c ; (c2) If Fj ∈ c ( j ∈ J; J is the indexing set), then j∈J Fj ∈ c . Proof. It is obvious from De2nition 3.1. Proposition 6.3. If ∗ ⊆ LX satis:es conditions (c1) and (c2), then there exists a unique L-cs (LX ; c) on X such that the family c of its L-closed sets is just ∗ . Proof. De2ne operator c : LX → LX such that for every A ∈ LX ; c(A) = {F ∈ ∗ : A6F}. Then it is easy to show that (cl1), (cl2) and (cl3) hold. Moreover, since for every F ∈ ∗ ; c(F) = F and (c2), c(A) ∈ ∗ and hence (cl4) holds. So c is an L-closure. For the later part, on the one hand, ∗ ⊆ c from the de2nition of c. On the other hand, if E ∈ LX with c(E) = E, then E ∈ c by the de2nition of c(E) and condition (c2). Therefore c = ∗ . Finally, if there exists another L-closure c1 such that the family c1 of its L-closed sets is just ∗ , then it is easy to show that for every A ∈ LX ; c(A) = c1 (A). So c = c1 . This completes the proof of uniqueness. Let : L-CLOSURE → SET be the forgetful functor. In order to prove that (L-CLOSURE, ) is 2bre-complete, a partial ordered on the set CL (X ) of all L-closures on X is needed to introduce and we need to prove that the partial ordered set is a complete lattice. The partial ordered on the set CL (X ) is de2ned as follows. For every pair of c1 ; c2 ∈ CL (X ), c1 4 c2 ⇔ c1 ⊆ c2 . For the partial ordered set (CL (X ); 4), we have the following result. Proposition 6.4. (CL (X ); 4) is complete lattice. Proof. Obviously the mapping c : LX → LX de2ned by c (A) = A; ∀A ∈ LX is an L-closure and the universal upper bound in CL (X ) w.r.t. 4. Therefore, it is suPcient to verify that the in2mum of any nonempty family {ci : i ∈ I } of L-closures ci on X exists. In fact, X ∗ let ci be the family of all L-closed sets in (L ; ci ) and = i∈I ci . Then ∗ satis2es conditions (c1) and (c2) as well. So by Proposition 6.3, there exists an L-closure c on X such that the family c of all L-closed sets equals ∗ . Therefore c 4 ci (∀i ∈ I ). It is easy to show that c is the in2mum of {ci : i ∈ I }. Let Q be a nonempty subset of LX such that 0X ∈ . Q Then Q generates an L-closure cQ in the following sense: cQ := {c ∈ CL (X ) : Q ⊆ c }:
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By Proposition 6.4, cQ is the smallest L-closure on X whose family of all L-closed sets contains . Q Furthermore, to prove that every co-structured map ’ : ((LX ; cX )) → Y (see the proof of Theorem 6.1 below) has a unique 2nal lift, we shall introduce the concept of subbase of L-closure and give a condition for L-continuity by means of subbase as follows. De nition 6.1. A nonempty subset Q of LX with 0X is called a subbase of an L-closure c on X if c = cQ. Proposition 6.5. Let (LXi ; ci ) be an L-cs (i = 1; 2), Q a subbase of c2 and f : X1 → X2 an L-value Zadeh-type powerset operator. Then the following assertions are equivalent: (1) f→ is L-continuous mapping between (LX1 ; c1 ) and (LX2 ; c2 ). (2) {B ◦ f : B ∈ } Q ⊆ c1 . Proof. Implication (1) ⇒ (2) is obvious. In order to show (2) ⇒ (1), we consider a subset ∗ of LX2 where ∗ = {B ∈ LX2 : B ◦ f ∈ c1 }. Then it is easy to show that ∗ satis2es conditions (c1) and (c2), as c1 satis2es (c1) and (c2). So by Proposition 6.3, there exists an L-closure c∗ on X2 such that c∗ = ∗ . From Q ⊆ ∗ and De2nition 6.1, we know that c2 4 c∗ or c2 ⊆ ∗ . Therefore {B ◦ f : B ∈ c2 } ⊆ c1 and hence f is L-continuous. Finally, we recall the concept of topological category and prove the main result of this section. De nition 6.2 (Long de2nition of topological category, HRohle and Rodabaugh [2]). Let categories A and X be given, as well as a functor V : A → X, and let X ∈ |X|; Aj ∈ |A|, and fj : X → V (Aj ) in X, where j ∈ J for some indexing set J . Then (X; fj : X → V (Aj ))J is a V -structured source. Category A is topological w.r.t. category X and functor V i@ for each such V -structured source all the following hold: (1) V -lift: ∃Xˆ ∈ |A|; ∃(fˆj : Xˆ → Aj )J in A, V (Xˆ ) = X; V (fˆj ) = fj . (2) Initial V-lift: Given (Xˆ ; fˆj : Xˆ → Aj )J in A from (1), then ∀(Yˆ ; gˆj : Yˆ → Aj )J in A with Y ≡ V (Yˆ ) ˆ and and gj ≡ V (gˆj ); ∀h : Y → X in X with gj = fj ◦ h in X, ∃!hˆ : Yˆ → Xˆ in A with h = V (h) gˆj = fˆ ◦ hˆ in A. (3) Unique initial V-lift: Given (Xˆ ; fˆj : Xˆ → Aj )J satisfying (1) and (2), and given (XQ ; fQj : XQ → Aj )J satisfying (1) and (2), Xˆ = XQ and fˆj = fQj . Theorem 6.1. Let : L-CLOSURE → SET be the forgetful functor. Then L-CLOSURE is a topological category over SET w.r.t. . Proof. Let Q = {0Y ; 1Y } and ’ : ((LX ; cX )) → Y be a co-structured map. Then 0Y ◦ ’ ∈ cX ; 1Y ◦ ’ ∈ cX . Denote (LY ; cY ) by the unique L-closure on Y generated by . Q Then ’→ : (LX ; cX ) → (LY ; cY ) is L-continuous by Proposition 6.5. So ’ has a unique 2nal lift ’→ . Further, (L-CLOSURE, ) is 2bre-complete from Proposition 6.4. Hence, it is suPcient to show that for every L-closure space
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(LY ; cY∗ ), a map ’ : X → Y has a unique initial lift ’→ : (LX ; cX∗ ) → (LY ; cY∗ ) (cf. 21.c in [1]). Let ∗ = {B ◦ ’ ∈ LX : B ∈ cY∗ }. Denote (LX ; cX∗ ) by the unique L-closure space on X generated by ∗ . From the de2nition of ∗ and Proposition 6.5, ’→ : (LX ; cX∗ ) → (LY ; cY∗ ) is L-continuous w.r.t. cX∗ and cY∗ . In order to show that ’ is an initial morphism we consider a further L-continuous map → : (LZ ; c ) → (LY ; c∗ ) and a map : : Z → X such that = ’ ◦ :. From the de2nition of ∗ and the Z Y L-continuity of , we have that (B ◦ ’) ◦ : = B ◦
∈ cZ :
Since ∗ is the subbase of (LX ; cX∗ ), : is L-continuous; i.e. ’ is an initial arrow. The uniqueness of the lift follows from the antisymmetry of the partial ordering 4 “coarser”. This completes the proof of this theorem. 7. Constructions in L-CLOSURE In this section, the subspace of an L-cs, the sum L-cs, the product of L-cs’ and the induced L-cs of a crisp closure space will be constructed. From Theorem 6.1, we know that the category L-CLOSURE is topological over SET. Since a topological category inherits whatever limits and colimits the ground category has and SET is complete and cocomplete, the category L-CLOSURE is complete and cocomplete. So the following Theorems 7.2 and 7.3 certainly hold, the proofs are omitted here. First, we construct the subspace of an L-cs. De nition 7.1. Let Y be a subset of X . For A ∈ LY ; A∗ ∈ LX is given as: for every x ∈ X; A∗ (x) = A(x) if x ∈ Y and A∗ (x) = 0 if x ∈ X \Y . For B ∈ LX , B|Y ∈ LY is de2ned by (B|Y )(y) = B(y) if y ∈ X and (B|Y )(y) = 0 if y ∈ X \Y . We can make use of De2nition 7.1 to construct the subspace of an L-cs as follows: Theorem 7.1. Let (LX ; c) be an L-cs and Y a nonempty subset of X. If cY : LY → LY is such that cY (A) = c(A∗ )|Y for every A ∈ LY , then cY is an L-co and hence (LY ; cY ) is an L-cs on Y. In this case, we call cY a relative L-co of c and (LY ; cY ) an L-closure subspace of (LX ; c). Proof. (1) cY (0Y ) = c(0∗Y )|Y = 0Y . (2) cY (A) = c(A∗ )|Y ¿A∗ |Y = A. (3) If A6B, then A∗ 6B∗ and c(A∗ )6c(B∗ ). So cY (A) = c(A∗ )|Y 6c(B∗ )|Y = cY (B). (4) cY (cY (A)) = cY (c(A∗ )|Y ) = c((c(A∗ )|Y )∗ )|Y = c(c(A∗ ))|Y = c(A∗ )|Y = cY (A). As the same as in topological spaces, the L-closed set in L-closure subspace (LY ; cY ) may be generated by an L-close set in (LX ; c), which is shown in the following proposition. Proposition 7.1. Let (LX ; c) be an L-cs and (LY ; cY ) an L-closure subspace of (LX ; c). If A is L-closed in (LX ; c), then A|Y is L-closed in (LY ; cY ).
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Proof. Since c(A) = A, cY (A|Y ) = (c(A|Y )∗ )|Y = c(A)|Y = A|Y . Second, we construct the sum L-cs of L-cs’.
Theorem 7.2. Let = {(LXt ;
ct ) : t ∈ T } be pairwise disjoint L-cs’ and X = t ∈T Xt . Then ⊕ ct : LX → LX de:ned by ⊕ ct (A) = t ∈T ct (A|Xt ) for A ∈ LX is an L-co on X and hence (LX ; ⊕ ct ) is an L-cs. In this case, ⊕ ct is called sum L-co of ct (t ∈ T ) and (LX ; ⊕ct ) is called the sum L-cs of . Next we construct the product L-cs of L-cs’. Theorem 7.3. Let = {(LXt ; ct ) : t ∈ T } be a family of L-cs’, X = t ∈T Xt and pt : LX→ LXt be the X X X Zadeh-type powerset ←operator (t ∈ T ). Let c : L → L be a GOH de:ned Xby c(A) = i∈I {Vi ∈ L : Vi ¿A and Vi = {pj (Aj ) : Aj is L-closed, j = 1; : : : ; ki ; ki ∈ N }} for A ∈ L , where N is the set of all natural numbers, then c is an L-co on X and hence (LX ; c) is an L-cs and called the product L-cs of . In the following, we consider a crisp closure space based on which an L-cs can be constructed. Let c be a crisp closure on X , i.e. c : p(X ) → p(X ) satis2es the following conditions. (ccl1) (ccl2) (ccl3) (ccl4)
c(∅) = ∅. For every A ⊆ X; A ⊆ c(A). For every A; B ⊆ X , if A ⊆ B, then c(A) ⊆ c(B). For every A ⊆ X; c(c(A)) = c(A). The set of all closed sets in (X; c) is =, where = = {F ∈ p(X ) : c(F) = F} and = satis2es the following conditions. (cc1) ∅ ∈ =. (cc2) If {Fi : i ∈ I } ⊆ =, then i∈I Fi ∈ =.
Let A ∈ LX . Call A an upper semi-continuous function on (X; c) if for every a ∈ L; >a (A) ∈ =, where >a (A) = {x ∈ X : A(x)¿a}. Denote !L (=) the set of all upper semi-continuous functions on (X; c). Then !L (=) satis2es conditions (c1) and (c2). In fact, 0X ∈ !L (=) is obvious from the de2nition of uppersemi-continuous function. For the rest, let ⊆ !L (=). It is no harm to assume that = ∅. Let A = {B : B ∈ }. Then for every a ∈ L, >a (A) = {x ∈ X : A(x)¿a} = B∈ >a (B) ∈ = by (cc2). So by Proposition 6.3, there exists a unique L-closure denoted by !L (c) on X such that !L (c) = !L (=). To sum up, we have obtained an L-closure from a crisp closure and the L-closure is called induced L-closure. Namely, let C(X ) be the set of all crisp closures on X . Then we have a map !L : C(X ) → CL (X ) de2ned by c → !L (c): Let CLOSURE be the category of all crisp closures spaces on X and all continuous mappings (i.e. the preimage of every closed set in (Y; cY ) is closed set in (X; cX )). Then !L : CLOSURE → L-CLOSURE is a functor. Now we consider the inverse problem. Let (LX ; c) be an L-closure space, =∗ = {>a (B) : a ∈ L; B ∈ c } and @L (c ) = {F : F = i∈IF >ai (B); ai ∈ L; Bi ∈ =∗ ; i ∈ IF ; IF is an indexing set}. Then @L (c ) satis2es
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(cc1) and (cc2). So by Proposition 6.3 (the special case L = {0; 1}), there exists a unique crisp closure denoted by @L (c) on X such that the family =@L (c) of all closed sets in (X; @L (c)) is @L (c ). So we have a map @L : CL (X ) → C(X ) de2ned by c → @L (c) and a functor @L : L-CLOSURE → CLOSURE: The functors !L and @L possess many interesting properties which is shown in the following proposition. Proposition 7.2. The mappings !L ; @L possess the following properties: (1) For every c ∈ C(X ); (@L ◦ !L )(c) = c. (2) For every c ∈ CL (X ); (!L ◦ @L )(c) ¡ c. Particularly, if c = !L (c∗ ), then (!L ◦ @L )(c) = c, where c∗ ∈ C(X ). Proof. (1) Because {>a (B) : a ∈ L; B ∈ !L (=)} is the subbase of @L (!L (c)), (@L ◦ !L )(F) = F for every F ∈ =. So (@L ◦ !L )(=) ⊆ = and hence (@L ◦ !L )(c) 4 c. Conversely, for every F ∈ =, let A = AF , the characteristic function of F. Then A ∈ !L (=). Let a = 1 ∈ L. Then F = >a (A) = {x ∈ X : AF (x)¿1} ∈ {>a (B) : a ∈ L; B ∈ !L (=)}. So (@L ◦ !L )(=) ⊇ = and hence (@L ◦ !L )(c) ¡ c. Therefore ∀c ∈ C(X ); (@L ◦ !L )(c) = c. (2) Let c ∈ CL (X ), A ∈ c . Then ∀a ∈ L; >a (A) ∈ @L (c ). This shows that A is an upper semicontinuous function on (X; @L (c)). So A ∈ (!L ◦ @L )(c ) and hence (!L ◦ @L )(c) ¡ c. Especially, if c = !L (c∗ ), then (!L ◦ @L )(c) = (!L ◦ @L )(!L (c∗ )) = !L ((@L ◦ !L )(c∗ )) = !L (c∗ ) = c. From the concept of full subcategory, one can easily obtain the following result whose proof is omitted. Proposition 7.3. Category L-INDCLOSURE is a full subcategory of L-CLOSURE where each object of L-INDCLOSURE is (LX ; !L (c)) on LX induced by some crisp closure space (X; c) on X and each morphism of L-INDCLOSURE is the L-continuous mapping. If we restrict @L on the category L-INDCLOSURE, then we can obtain another important result in this paper. For this, we introduce the concept of adjunction of categories, :rstly. De nition 7.2 (Adjunction of categories HRohle and Rodabaugh [2], Mac Lane [5]). Let F : C → D and G : D → C be functors. We say F is left-adjoint to G if the following two criteria are satis2ed: (1) Lifting/continuity criterion: ∀A ∈ |C|; ∃ : A → GF(A); ∀B ∈ |D|; ∀f : A → G(B); Q ◦ : ∃!fQ : F(A) → B; f = G(f)
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(2) Naturality criterion: ∀f : A1 → A2
in C; F(f) = (or ≡)A2 ◦ f;
where the “≡” option allows F to only be de2ned initially on objects, in which (1) and “≡” will stipulate an action of F on morphisms such that F G. We also say that G is right-adjoint to F or that (F; G) is an adjunction, or we may write F G. The map is the unit of the adjunction; and the dual D morphism C of in the duals of the above statements is the counit of the adjunction. Finally, we state and proof another important result in this paper which shows us the relationship between two functors !L and @L . Theorem 7.4. The functor !L : CLOSURE → L-INDCLOSURE is the left-adjoint of the functor @L : L-INDCLOSURE → CLOSURE. Proof. (1) For every (X; c1 ) ∈ |CLOSURE|, let = @L ◦ !L . Then is the identity mapping by Proposition 7.2. ∀(LX ; !L (c2 )) ∈ |L-INDCLOSURE| and every continuous mapping f : (X; c1 ) → (X; @L (!L Q = f by Proposition 7.2. Therefore the Lifting=Continuity cri(c2 ))), taking fQ = !L (f), then ◦ @L (f) terion is satis2ed. (2) For every (X; c1 ); (X; c2 ) ∈ |CLOSURE|, (@L ◦ !L )((X; c1 )) = (X; c1 ) and (@L ◦ !L )((X; c2 )) = (X; c2 ) by Proposition 7.2. Taking A1 = A2 = id|CLOSURE| , then ∀f : (X; c1 ) → (X; c2 ), A2 ◦ f = f. So taking A2 ◦ f = !L (f), then A1 ◦ (@L (A2 ◦ f)) = A2 ◦ f. Therefore the Naturality criterion is satis2ed. By (1) and (2), the proof of this theorem is completed.
8. Conclusion In this paper, we have introduced the concepts of L-closure space and continuous generation order homomorphism and studied many important properties of L-closure space, such as category properties, especially. On the one hand, the concept of L-closure space in this paper is a generalization of fuzzy closure space introduced in [6,8,9]. On the other hand, the results about the category properties of L-closure space in this paper are deeper than in [6,8,9]. However, separation and compactness properties in L-closure spaces have not been studied in this paper and remain for further research.
Acknowledgements The author wishes to thank the Main Editor, Prof. S.E. Rodabaugh and the referees for their valuable comments and helpful suggestions.
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