Journal of the Egyptian Mathematical Society (2013) 21, 300–304 Egyptian Mathematical Society Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems ORIGINAL ARTICLE Extensions of L-fuzzy closure spaces A.H. Zakari a, S.E. Abbas a b a,* , M.A. Al-Homieyed b Department of Mathematics, Faculty of Science, Jazan University, Saudi Arabia Department of Mathematics, Faculty of Applied Science for Girls, Umm AL-Qura University, Saudi Arabia Received 10 November 2012; revised 17 February 2013; accepted 20 February 2013 Available online 3 May 2013 KEYWORDS L-fuzzy closure space; LC-fuzzy continuous mapping; L-fuzzy c-grill and principle extension of L-fuzzy closure space Abstract In this paper, we study the extension theory to L-fuzzy closure spaces, where L is a strictly two-sided, commutative quantale lattice. We give new notions such as L-fuzzy stack, L-fuzzy c-grill and trace of a point. Also, we construct order relation and equivalence relation between two extensions. Also, We introduce the concept of a principal extension of L-fuzzy closure space and study some of its applications. ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. 1. Introduction In crisp topology, extension theory has been intensively studied for completely regular spaces and is fairly well developed for T0 topological spaces (see for example [1]). Some basic concepts on extensions of closure spaces are introduced and results on embedding of closure spaces in cubes are studied by Gagrat and Thron [2], C˘ech [3] and Thron and Warren [4]. A general theory of extensions of G0 closure spaces is introduced and investigated by Chattopadhyay and Thron [5]. In fuzzy setting, particular type of extension such as compactiﬁcations, completions of fuzzy topological spaces and fuzzy uniform spaces have been studied [6–8]. In [9], Chattopadhyay, Hazra and Samanta introduced for the ﬁrst time, a general concept of extensions of fuzzy topological spaces and provided a method of construction of T0 principal extension of a T0 fuzzy topological space. * Corresponding author. E-mail address: [email protected] (S.E. Abbas). Peer review under responsibility of Egyptian Mathematical Society. In this paper, we need to study the extension of L-fuzzy closure space which deﬁned by Kim [10], as follows: In Section 2, we introduce the notions of L-fuzzy grills, L-fuzzy c-grills and a homeomorphism between two L-fuzzy closure spaces and study some results on it. In addition, we deﬁne a base for the r-L-fuzzy closed sets in L-fuzzy closure space. In Section 3, we study the extension theory to L-fuzzy closure spaces, where L is a strictly two-sided, commutative quantale lattice. We give new notions such as L-fuzzy stack, L-fuzzy c-grill and trace of a point. We present the trace T ðy;EÞ of the point y with respect to the extension E and the trace system XE of the extension E. Also, we study some of its properties. Furthermore, we deﬁne a principal extension of L-fuzzy closure space and provide some results on it. 2. Preliminaries Through this paper, let X be a non-empty set, L = (L, 6, x, ¯, 0, 1) a complete lattice where 0 and 1 denote the least and the greatest elements in L, L0 = L {0}. Production and hosting by Elsevier 1110-256X ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2013.02.013 Extensions of L-fuzzy closure spaces Deﬁnition 2.1 (8,11). A complete lattice (L, 6, x) is called a strictly two-sided, commutative quantal (scq-lattice, for short) iff it satisﬁes the following properties: (1) (L, x) is a commutative semigroup. (2) x = x x 1, for each x 2 L and 1 is the universal upper bound. (3) x is distributive over arbitrary joins, i.e., ! _ _ ri s ¼ ðri sÞ: i2C i2C Deﬁnition 2.2 (8,11). Let (L, 6, x) be an scq-lattice. A map0 ping :L ﬁ L is called a strong negation if it satisﬁes the following conditions: (a) (a0 )0 = a for each a 2 L. (b) If a 6 b, then a0 P b0 , for each a, b 2 L. In this paper, we assume that (L, 6, x, ¯, 0 ) is an scqlattice, where ¯ is deﬁned as follows: 301 An L-fuzzy set k is called an r-L-fuzzy closed set if c(k, r) = k. Let (X, c1) and (Y, c2) be two L-fuzzy closure spaces. A mapping f:(X, c1) ﬁ (Y, c2) is called LC-fuzzy continuous if for each k 2 LX, r 2 L0, fðc1 ðk; rÞÞ 6 c2 ðfðkÞ; rÞ: Deﬁnition 2.5. Let (X, c) be an L-fuzzy closure space, r 2 L0 and b a family of r-L-fuzzy closed sets in (X, c). Then b is said to be a base for the r-L-fuzzy closed sets in (X, c) if each r-Lfuzzy closed set in (X, c) can be expressed as an inﬁmum of a subset of b. Deﬁnition 2.6. Let (X, c) be an L-fuzzy closure space and A X. Deﬁne cA:LA · L0 ﬁ LA by cA(k, r)(x) = c(k, r)(x) for all x 2 A and k 2 LA, r 2 L0. Then (A, cA) is an L-fuzzy closure space and is called a subspace of (X, c). Deﬁnition 2.7. An L-fuzzy closure space (X, c) is called T0 if for each x, y 2 X, x „ y and r 2 L0, there exists an r-L-fuzzy closed set k such that k(x) „ k(y). 0 x y ¼ ðx0 y0 Þ : In particular, the unit interval ([0, 1], 6, x, ¯, 0 ), where x = , ¯ = , is scq-lattice with a strong negation a0 = 1 a for each a 2 [0, 1]. Lemma 2.3 12. Let (L, 6, x, ¯, 0 ) be an seq-lattice with a strong negation. Then for each x, y, z 2 L, {yi:i 2 C} L, we have the following properties: (1) (2) (3) (4) (5) (6) (7) If y 6 z, then (x x y) 6 (x x z). If y 6 z, then (x ¯ y) 6 (x ¯ z). x x y 6 x y. 0V0 = 1, 10= 0 x y 6 x ¯ y. W 0 and 0 Wi2C y i0 ¼ Vi2C y i 0 . i . i ¼ i2CyV i2C yV x i2C y i ¼ i2C ðx y i Þ. Deﬁnition 2.8. Let (X, c) and (Y, c*) be two L-fuzzy closure spaces. A mapping f:(X, c) ﬁ (Y, c*) is called a homeomorphism if f is bijective and f, f1 are LC-fuzzy continuous mappings. Theorem 2.9. Let (X, c) and (Y, c*) be two L-fuzzy closure spaces. A bijective mapping h:(X, c) ﬁ (Y, c*) is a homeomorphism iff hðcðk; rÞÞ ¼ c ðhðkÞ; rÞ for all k 2 LX ; In the following, we deﬁne the notions of L-fuzzy stack, L-fuzzy grill and L-fuzzy c-grill in L-fuzzy closure space. Deﬁnition 2.10. Let X be a nonempty set. A mapping S:LX ﬁ L satisfying Sðk1 Þ P Sðk2 Þ if k1 P k2 All algebraic operations on L can be extended pointwise to the set LX as follows: for all x 2 X and k, l 2 LX, (1) (k x l)(x) = k(x) x l(x). (2) (k ¯ l)(x) = k(x) ¯ l(x). Deﬁnition 2.4 10. An L-fuzzy closure space is an ordered pair (X, c), where c:LX · L0 ﬁ LX is a mapping satisfying the following axioms: (CO1) c(0, r) = 0 for all r 2 L0. (CO2) k 6 c(k, r) for all k 2 LX, r 2 L0. (CO3) If k 6 l and r 6 s, then c(k, r) 6 c(l, s) for all k, l 2 LX. (CO4) c(k ¯ l, r) = c(k, r) ¯ c(l, r) for all k, l 2 LX, r 2 L0.An L-fuzzy closure space (X, c) is called topological if c satisﬁes the following: (CO5) c(c(k, r), r) = c(k, r) for all k 2 LX, r 2 L0. r 2 L0 : for all k1 ; k2 2 LX is called an L-fuzzy stack on X. Deﬁnition 2.11. An L-fuzzy grill G:LX ﬁ L is an L-fuzzy stack on X such that it satisﬁes the following conditions: (i) G(0) = 0. (ii) G(k1 ¯ k2) = G(k1) ¯ G(k2) for all k1, k2 2 LX. (iii) G(1) > 0. An L-fuzzy grill G on X is said to be proper if G(1) = 1. Deﬁnition 2.12. Let G be an L-fuzzy grill in an L-fuzzy closure space (X, c). Then G is called an L-fuzzy c-grill in (X, c) if G(c(k, r)) = G(k) for all k 2 LX, r 2 L0. Deﬁnition 2.13. Let (X, c) be an L-fuzzy closure space. For all x 2 X, deﬁne Gx:LX ﬁ L by Gx ðkÞ ¼ cðk; rÞðxÞ for all k 2 LX ; r 2 L0 : 302 A.H. Zakari et al. Theorem 2.14. Let (X, c) be a T0 L-fuzzy closure space. Then for all x, y 2 X, Gx = Gy implies x = y. Proof. Let x, y 2 X be such that x „ y. Since (X, c) is T0, then there exists r-L-fuzzy closed set k such that k(x) „ k(y). Thus, Gx(k) = c(k, r)(x) = k(x) „ k(y) = c(k, r)(y) = Gy(k). Hence, Gx „ Gy. This completes the proof. h Theorem 2.15. An L-fuzzy closure space (X, c) is topological iff Gx is an L-fuzzy c-grill in (X, c) for each x 2 X. Proof. Let x 2 X and k, l 2 LX, r 2 L0. Then, Gx ðk lÞ ¼ cðk l; rÞðxÞ ¼ ðcðk; rÞ cðl; rÞÞðxÞ ¼ cðk; rÞðxÞ cðl; rÞðxÞ ¼ Gx ðkÞ Gx ðlÞ: Thus, Gx is an L-fuzzy c-grill on X. Conversely, suppose that (X, c) is an L-fuzzy closure space such that Gx is an L-fuzzy c-grill in (X, c) for each x 2 X. Therefore, Gx(c(k, r)) = Gx(k) for all k 2 LX, r 2 L0 and for all x 2 X. Thus, c(c(k, r), r)(x) = c(k, r)(x) for all k 2 LX, r 2 L0 and for all x 2 X i.e., c(c(k, r), r) = c(k, r) for all k 2 LX, r 2 L0. Thus, (X, c) is topological L-fuzzy closure space. h 3. Extension of L-fuzzy closure space In this section, we deﬁne an extension of L-fuzzy closure space, a trace of the point with respect to the extension, a trace system of the extension and a principal extension of L-fuzzy closure space. Also, we provide some results on it. Deﬁnition 3.1. Let (X, c) and (Y, c*) be two L-fuzzy closure * spaces and a:(X, c) ﬁ (Y, c*) be a mapping. Then (a, (Y, c )) is said to be an embedding of (X, c) if a : ðX; cÞ ! aðXÞ; caðXÞ is a homeomorphism. Deﬁnition 3.2. Let (X, c) and (Y, c*) be two L-fuzzy closure spaces and a:(X, c) ﬁ (Y, c*) be a mapping. Then (a, (Y, c*)) is said to be an extension of (X, c) if (i) (a, (Y, c*)) is an embedding of (X, c). (ii) c*(a(1),r) = 1. (iii) a(k ¯ l) = a(k) ¯ a (l) for all k, l 2 LX. An extension E = (a, (Y, c*)) is said to be a principal extension of (X, c) if {c*(a(l), r):l 2 LX, r 2 L0} is a base for the r-L-fuzzy closed sets in (Y, c*). Let E1 ¼ a1 ; Y1 ; c1 and E2 ¼ a2 ; Y2 ; c2 be two extensions of (X, c). Then E1 is said to be greater than or equal to E2 (written as E1 P E2) if there exists an LC-fuzzy continuous mapping f from Y1 ; c1 onto Y2 ; c2 such that f a1 = a2. The E1 ¼ a1 ; Y1 ; c1 is said to be equivalent to extension E2 ¼ a2 ; Y2 ; c2 (written as E1 E2) if there exists a homeomorphism h from Y1 ; c1 onto Y2 ; c2 such that h a1 = a2. Theorem 3.3. Let (X, c) and (Y, c*) be two L-fuzzy closure spaces and a:(X, c) ﬁ (Y, c*) an injective mapping. Then (a, (Y, c*)) is an embedding of (X, c) iff a(c(k, r)) = c* (a(k), r) a(1) for all k 2 LX, r 2 L0. Proof. Let (a, (Y, c*)) be an embedding of (X, c). Then a : ðX; cÞ ! aðXÞ; caðXÞ is a homeomorphism. From Theorem 2.9, a(c(k, r)) = c*(a(k), r) a(1) for all k 2 LX, r 2 L0. Conversely, let a:(X, c) ﬁ (Y, c*) be an injective mapping and a(c(k, r)) = c*(a(k), r) a(1) for all k 2 LX, r 2 L0. Then, a : ðX; cÞ ! aðXÞ; caðXÞ is a homeomorphism. Therefore, (a, (Y, c*)) is an embedding of (X, c). In view of the above theorem, the following result holds. If (X, c) and (Y, c*) are two L-fuzzy closure spaces and a:(X, c) ﬁ (Y, c*) is an injective mapping, then (a, (Y, c*)) is an extension of (X, c) iff (i) a(c(k, r)) = c*(a(k), r) a(1) for all k 2 LX, r 2 L0. (ii) c*(a(1), r) = 1. (iii) a(k ¯ l) = a(k) ¯ a(l) for all k, l 2 LX. h Deﬁnition 3.4. Let E = (a, (Y, c*)) be an extension of (X, c) and y 2 Y. Deﬁne the trace T ðy;EÞ of the point y with respect to the extension E by T ðy;EÞ ðkÞ ¼ c ðaðkÞ; rÞðyÞ for all k 2 LX ; r 2 L0 ; where there is no chance of confusion, we shall simply write T y for T ðy;EÞ . The trace system XE of the extension E is deﬁned by XE ¼ fT y : y 2 Yg: Theorem 3.5. Let E = (a, (Y, c*)) be an extension of (X, c), a is order-preserving and let (Y, c*) be a topological L-fuzzy closure space. Then, (i) T y is an L-fuzzy c-grill in (X, c) for all y 2 Y. (ii) T aðxÞ ¼ Gx for all x 2 X. (iii) If E1 and E2 are two equivalent extensions of X, then X E1 ¼ X E 2 . Proof. (i) Let y 2 Y. Then T y ð0Þ ¼ c ðað0Þ; rÞðyÞ ¼ 0 and T y ð1Þ ¼ c ðað1Þ; rÞðyÞ ¼ 1. Also, for any k, l 2 LX, r 2 L0 T y ðk lÞ ¼ c ðaðk lÞ; rÞðyÞ ¼ c ðaðkÞ aðlÞ; rÞðyÞ ¼ ðc ðaðkÞ; rÞ c ðaðlÞ; rÞÞðyÞ ¼ c ðaðkÞ; rÞðyÞ c ðaðlÞ; rÞðyÞ ¼ T y ðkÞ T y ðlÞ: Thus, T y is an L-fuzzy grill in (X, c). Now, let k 2 LX, r 2 L0. Then Extensions of L-fuzzy closure spaces T y ðkÞ ¼ c ðaðkÞ; rÞðyÞ ¼ c ðc ðaðkÞ; rÞ; rÞðyÞ P c ðc ðaðkÞ; rÞjaðXÞ ; rÞðyÞ ¼ c ðaðcðk; rÞÞ; rÞðyÞ ¼ T y ðcðk; rÞÞ: T aðxÞ ðkÞ ¼ c ðaðkÞ; rÞðaðxÞÞ ¼ c ðaðkÞ; rÞjaðXÞ ðaðxÞÞ ¼ aðcðk; rÞÞðaðxÞÞ ¼ cðk; rÞðxÞ ¼ Gx ðkÞ: Thus, T aðxÞ ¼ Gx for all x 2 X. (iii) Let E1 ¼ a1 ; Y 1 ; c1 and E2 ¼ a2 ; Y 2 ; c2 be two equivalent extensions of (X, c). Then there ; c exists a homeomorphism h from Y 1 1 onto Y 2 ; c2 such that ha1 = a2. Let y 2 Y1 and k 2 LX, r 2 L0. Then, T ðy;E1 Þ ðkÞ ¼ c1 ða1 ðkÞ; rÞðyÞ ¼ h c1 ða1 ðkÞ; rÞ ðhðyÞÞ ¼ c2 ððh a1 ÞðkÞ; rÞðhðyÞÞ ¼ c2 ða2 ðkÞ; rÞðhðyÞÞ ¼ T ðhðyÞ;E2 Þ ðkÞ: 2 Y2 g ¼ XE2 ; since Y2 = {h(y):y 2 Y1}.In what follows we give an example to show that the converse of the result (iii) of Theorem 3.5 need not hold. h Example 3.6. Let X, Y, Z be three inﬁnite sets such that X Y Z and ŒXŒ < ŒYŒ < ŒZŒ, where ŒXŒ denotes the cardinal number of the set X. Let (L, 6, x, ¯, 0 ) = (I, 6, , , 0 ) and let c:IZ · I0 ﬁ IZ be deﬁned by r 2 I0 Clearly, (Z, c) is an L-fuzzy closure space. Let (X, cX) and (Y, cY) be subspaces of (Z, c). Let i1:(X, cX) ﬁ (Z, c) and i2:(X, cX) ﬁ (Y, cY) be the inclusion mappings. Clearly, E1 = (i1, (Z, c)) is an extension of (X, cX) and E2 = (i2, (Y, cY)) is also an extension of (X, cX). Note that for each x 2 X; T ði1 ðxÞ;E1 Þ ¼ Gx ¼ T ði2 ðxÞ;E2 Þ . Deﬁne G:IX ﬁ I by GðkÞ ¼ 1 if kðaÞ ¼ 1 for all a 2 X 0 otherwise: Then it is easy to check that T ðz;E1 Þ ¼ G, for all z 2 Z X and T ðy;E2 Þ ¼ G for all y 2 Y X. Hence XE1 ¼ XE2 . But E1 \ E2, as ŒYŒ < ŒZŒ. Theorem 3.7. Let E = (a, (Y, c*)) be an extension of (X, c). Then Gy1 6 Gy2 implies T y1 6 T y2 for all y1, y2 2 Y. Proof. Suppose that Gy1 6 Gy2 l 2 LX, r 2 L0. Then we have h * Theorem 3.8. Let (Y, c ) be a topological L-fuzzy closure space such that (a, (Y, c*)) is a principal extension of (X, c). Then T y1 6 T y2 iff Gy1 6 Gy2 for all y1, y2 2 Y. Proof. From Theorem 3.7, we have Gy1 6 Gy2 implies T y1 6 T y2 . Conversely, suppose that T y1 6 T y2 . Thus, c*(a(l), r)(y1) 6 c*(a(l), r)(y2) for all l 2 IX. Let k 2 LY. Since (Y, c*) is topological and {c*(a(l), r):l 2 LX, r 2 L0} is a base for the r-L-fuzzy closed sets in (Y, c*), then, we have ^ fc ðaðlÞ; rÞ : l 2 LX ; r ^ 2 L0 ; c ðaðlÞ; rÞ P kgðy1 Þ ¼ fc ðaðlÞ; rÞðy1 Þ : l ^ 2 LX ; r 2 L0 ; c ðaðlÞ; rÞ P kg 6 fc ðaðlÞ; rÞðy2 Þ ^ : l 2 LX ; r 2 L0 ; c ðaðlÞ; rÞ P kg ¼ fc ðaðlÞ; rÞ Gy1 ðkÞ ¼ c ðk; rÞðy1 Þ ¼ r 2 L0 ; c ðaðlÞ; rÞ P kgðy2 Þ ¼ c ðk; rÞðy2 Þ ¼ Gy2 ðkÞ: XE1 ¼ fT ðy;E1 Þ : y 2 Y1 g ¼ fT ðhðyÞ;E2 Þ : y 2 Y1 g ¼ fT ðy;E2 Þ : y 0 if 0 ¼ k 2 IZ ; 1 otherwise: ¼ c ðaðlÞ; rÞðy2 Þ ðsince aðlÞ 2 LY Þ ¼ T y2 ðlÞ: : l 2 LX ; Thus, T ðy;E1 Þ ¼ T ðhðyÞ;E2 Þ . Therefore cðk; rÞ ¼ T y1 ðlÞ ¼ c ðaðlÞ; rÞðy1 Þ ¼ Gy1 ðaðlÞÞ 6 Gy2 ðaðlÞÞ Thus, T y1 6 T y2 . Obviously, T y ðkÞ 6 T y ðcðk; rÞÞ. Thus, T y ðkÞ ¼ T y ðcðk; rÞÞ. Hence, T y is an L-fuzzy c-grill in (X, c) for all y 2 Y. (ii) Let x 2 X and k 2 LX, r 2 L0. Then, ( 303 for all y1, y2 2 Y. Let Thus, Gy1 6 Gy2 . h The following corollary is an easy consequence of the above theorem. Corollary 3.9. Let (Y, c*) be a topological L-fuzzy closure space and (a, (Y, c*)) be a principal extension of (X, c). Then T y1 ¼ T y2 iff Gy1 ¼ Gy2 for all y1, y2 2 Y. Theorem 3.10. Let (Y, c*) be a topological L-fuzzy closure space such that (a, (Y, c*)) is a principal extension of (X, c). Then (Y, c*) is T0 iff T x ¼ T y implies x = y for all x, y 2 Y. Proof. Let (Y, c*) be a T0 and x, y 2 Y be such that T x ¼ T y . Thus, Gx = Gy and hence x = y. Conversely, suppose that T x ¼ T y implies x = y for all x, y 2 Y. Let x, y 2 Y be such that x „ y. Then T x –T y and hence Gx „ Gy. Thus, there exists k 2 IY such that c*(k, r)(x) „ c*(k, r)(y). Since (Y, c*) is topological, then l = c*(k, r) 2 LY is a r-L-fuzzy closed set such that l(x) „ l(y). Therefore, (Y, c*) is T0. h Acknowledgement The authors would like to thank the anonymous refree for his comments, which helped us to improve the ﬁnal version of this paper. 304 References [1] W.J. 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