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The Upgrade of Topological Group Based on a New Hyper-topology

The Upgrade of Topological Group Based on a New Hyper-topology Fuzzy Inf. Eng. (2011) 2: 137-146 DOI 10.1007/s12543-011-0072-y ORIGINAL ARTICLE The Upgrade of Topological Group Based on a New Hyper-topology Yu-bin Zhong · Hong-xing Li Received: 12 November 2010/ Revised: 19 April 2011/ Accepted: 20 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract Wang et al [1] have put forward the problem of upgrade of topology and obtained a series of systematic theories and applicable results. Li et al [2] have put forwarded the problem of upgrade of group and also obtained a series of results. Con- sequently, the problem of upgrade of mathematic structure has drawn more attention of researchers. However, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. Based on a kind of intuitive convergent way, the paper has proposed a new hyper-topology and upgraded two mathematic structures of topological group to power set and fuzzy power set respectively. It has proved that multiplication and contradiction operations continues in the new hyper-topology after upgrading and a series of results are ob- tained creating hyper-topological group and fuzzy hyper-topological group. Accord- ingly, hyper-topological group and fuzzy hyper-topological group can be created, ob- taining a breakthrough to upgrade topological group to its power set and fuzzy power set. Keywords Upgrade· Power set· Fuzzy power set· Power group· Hyper-topological group· Fuzzy hyper-topological group 1. Introduction Algebra structure, sequence structure and topology are the three major structures of modern mathematics. Hyperalgebra is an important algebraic one. Hypergroup plays Yu-bin Zhong () School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, P.R.China email: Zhong yb@163.com Hong-xing Li () School of Mathematics Sciences, Beijing Normal University, Beijing 100875, P.R.China College of Electronic and Information Engineering, Dalian University of Technology, Dalian , P.R.China email: lhxqx@bnu.edu.cn 138 Yu-bin Zhong · Hong-xing Li (2011) an important role in modern analysis and its application, and especially has an irre- placeable role in the quantum mechanics and string theory of theoretical physics [3]. In the theory study of mathematical structure upgrading, Wang et al [1] put forward the problem of upgrading of topology and obtained a series of systematic theories and applied results [2, 4, 5, 6]. Li et al [2] have put forward the problem of upgrad- ing of group and also obtained a series of results [7-16]. However, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. Based on a kind of intuitive convergent way, this paper has proposed a new hyper-topology and upgraded two mathematic structures of topological group to power set and fuzzy power set respectively. Well then, do multi- plication and contradiction operations continue in hyper-topology? Do multiplication and contradiction operations of fuzzy power group in fuzzy power set F (G) about fuzzy hyper-topology continue? The following is to discuss these problems, and the answers are true. Finally, we obtain a series of results creating hyper-topological group and fuzzy hyper-topological group, which can create hyper-topological and fuzzy hyper-topological groups. For the convenience of depiction and research, we introduce the following def- inition and illumination. For topological group (G,•, T ), fuzzy topology (G, T ) can be obtained by topology structure (G, T ) and fuzzy power group can be ob- tained by upgrading group (G,•) to fuzzy power setF (G). In addition, the following diagram (Figure 1) can be used to show the forming of hyper-topology and fuzzy hyper-topology in detail (Figure 1): Educe to hyper-topological (P(G), T ) (G, T ) Creat Educe to fuzzy hyper-topological (F (G), T ) (G, T ) Fig.1 Forming of hyper-topology and fuzzy hyper-topology 2. A New Hyper-topology Based on Intuitive Convergent Way Firstly, we start on the problem of convergence in two-dimensional real plane R .For 2 2 2 example, for the setα = {(x, y)|x + y < 1} in two-dimensional real plane R ,weget 2 2 γ = {(x, y)|x + y < 1}, according to hyper-topology T defined in reference [1], for any open set U  α, sinceγ = {β|β ⊆ γ} is the minimal radix open set containingα, 2 2 we obtain α ∈ γ ⊆ U . On the other hand, we get W (n) = {(x, y)|x + y ≤ 1− 1/n}, then for any natural number n, we get W (n) ∈ β . Consequently, according to the 2 2 hyper-topology presented in reference [1], W (n) is converged in {(x, y)|x + y < 1} 2 2 instead of {(x, y)|x + y ≤ 1}. However, based on intuitive convergent way, W (n) 2 2 should be converged in{(x, y)|x + y ≤ 1}. To get over this situation, we study on this problem. Based on intuitive convergent way, we put forward a new hyper-topology and show it as follows: Fuzzy Inf. Eng. (2011) 2: 137-146 139 Definition 2.1 For a given topological space (G, T ), with T = {U |U ∈ T } as ∗ ∗ radix, where U = {α|close f old α ⊆ U ∈ T }, it can educe a kind of topol- ogy (P (G), T ), and (P (G), T ) is called hyper-topology of (G, T ), which is a new hyper-topology differing from reference [1]. − − − For ∀U , V ∈ T , (A ∈ U V ) ⇒ (A ⊆ U and A ⊆ V ) ⇒(A ⊆ V ∈ T ) ⇒ ∗ ∗ ∗ ∗ ∃(U V ) ∈ T ,we have A ∈ (U V ) ⊆ U V , so it is reasonable to treat T ∗ ∗ ∗ ∗ ∗ ∗ defined above as topological radix. If we study the convergence situation of W (n) with the new hyper-topology pre- 2 2 sented in this paper, W (n) is converged in {(x, y)|x + y ≤ 1}, which meets our tra- ditional intuitive idea. So it is a very useful hyper-topology. The study result of this new hyper-topology will be discussed in the other paper. For the convenience of study, we introduce the following marks and definitions. In the discussion of this paper, we assume that (G, T ) is topological space, (G,•, T ) topological group, the multiplication operation f is G × G → G, f (x, y) = xy and the −1 contradiction operation g is G → G, g(x) = x . Definition 2.2 (G,•, T ) is called topological group ⇐⇒ (G,•) is a group, (G, T ) is a topological space, and the multiplication operation f : G × G → G, f (x, y) = xy −1 and contradictory operation g : G → G, g(x) = x are continuous. Until now, research about hyper-topology and power group has already obtained a series of systematic theories, however, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. The paper presents the way of upgrading two mathematic structures of topological group to power group and fuzzy power group respectively, taking multiplication and contradiction operations into consideration. We have studied the problem of continu- ity of these two operations about the new hyper-topology above and obtained a series of basic theorems. Additionally, we will give the corresponding results. 3. Topological Group Upgrade to Power Set In this section, we upgrade two mathematic structures of topological group to power set respectively, and then take multiplication and contradiction operations into con- sideration. We study the problem of continuity of these two operations about the new hyper-topology above and obtained a series of basic results. −1 Lemma 3.1 Let g : G → G, g(x) = x . Then g is the homeomorphism mapping from (G, T ) to (G, T ). −1 −1 −1 Proof Because (g ◦ g) = x( ) , then g = g is a correspondence one by one. (x) on the other hand, g is continuous mapping from (G, T )to(G, T ), thus g is homeo- morphism mapping. Lemma 3.2 For ∀A ∈ P (G), close fold and contradictory set operations can be −  -1  -1 − exchanged, namely we get (A ) = (A ) . -1 − −  − Proof Because g is homeomorphism mapping and A is closed⇒ (A ) = g(A )is −  -1 −  -1 − a closed set. And g(A ) ⊇ g(A) = A ,so (A ) = g(A ) ⊇ {F|F is closed set, and F -1 -1 -1 -1 -1 -1 -1 -1 −  −    −  − − ⊇ A } = (A ) . On the other hand, ((A ) ) ⊇ ((A ) ) ,so(A ) = (A ) . 140 Yu-bin Zhong · Hong-xing Li (2011) −1  -1 Lemma 3.3 Suppose any element A of power group G satisfies A = A , then for − − − − − ∀A, B ∈ G , we gain f (A , B ) = A • B = (A• B) . −  − Proof (i) Since A = A A , B = B B , we obtain − −     − A • B = A• B A • B A• B A • B , (A• B) = A• B (A• B) . For ∀xy ∈ A • B ⇒ x ∈ A , y ∈ B ⇒∃ net {x }⊆ A,{y }⊆ B, we make n n x → x, y → y ⇒{x y }⊆ A • B. To prove x y → xy. Since f is continuous, n n n n n n for ∀U ∈ T and x, y ∈ U,we have ∃V, W ∈ T making f (V, W ) ∈ V • W ⊆ U , where x ∈ V, y ∈ W . In addition, x → x ∈ V, y → y ∈ W , therefore ∃N,at n > N, n n we have x ∈ V, y ∈ W , and then x y ∈ V • W ⊆ U , namely, x y → xy. Since n n n n n n −   −  − xy ∈ (A• B) , namely, A • B ⊆ (A• B) , according to the same way, A • B ⊆ (A• B) − − − and A• B ⊆ (A• B) , then we get A = B . -1 −1 (ii) For∀A ∈ G , we gain A = A , and then − −1 − − − −1 − − −1 − A ⊇ A = A• B• B ⇒ A = (A ) ⊇ (A• B• B ) ⊇ (A• B) • (B ) −  -1 − = (A•) • (B ) − −  -1 − − − −  -1 − − = (A• B) • (B ) ⇒ A • B ⊇ (A• B) • (B ) • B ⊇ (A• B) •{e} = (A• B) ˊ − − − Sum up (i) (ii), we have (A• B) = A • B . Theorem 3.1 Suppose (G,•, T ) is a topological group, (G, T ) is compact T space −1 and (G,•) is the power group of (G,•). If the contradictory element A is equal to -1 −1 the contradictory set A = {x |x ∈ A} for any element A of (G,•), then (G,•, T ) composes topological group, and we call (G,•, T ) the hyper-topological group of (G,•, T ), where the radix of sub-hyper-topology T is G T = {G U |U ∈ G ∗ ∗ T }ˊ Proof (1) g is a continuous mapping from (G, T )to(G, T ). For ∀A ∈ G U G G ∗ -1  -1 − −  -1  -1 − g(A) = A ˈwhere A ∈ G , then we obtain U ⊇ (A ) = (A ) , namely, U ⊇ A. -1 − Set V = U = g(U ) ∈ T , then A ⊆ V ⇒ A ∈ G V ∈ G T . -1 -1 -1 − −1 −  − − In addition, as ∀B ∈ G V ⇒ B ⊆ V ⇒ (B ) = (B ) = (B ) ⊆ V = −1 U ⇒ g(B) = B ∈ G U , we can get the conclusion that g is a continuous mapping from (G, T )to(G, T ). G G (2) f is a continuous mapping from (G × G, T )to(G, T ). For ∀G U G×G G ∗ − − − f (A, B) = A • B, where A, B ∈ G ⇒ (A • B) ⊆ U ⇒∀ab ∈ A • B ⊆ U and − − (a, b) ∈ A × B , because of the continuity of f (a, b) = ab, we get∃V (a, b), W (a, b) ∈ T , where V (a, b)  a, W (a, b) ∈ b, making f (V (a, b)), W (a, b)) = V (a, b)• W (a, b) ⊆ U ⇒ ( f (V (a, b), W (a, b)) ⊆ U . Within the compact T space (G, T ), com- − − (a,b)∈A ×B − − pact set is equal to closed set, so A × B is compact set. And because of the existence of finite coverage, we gain∃V, W ∈ T , i = 1, 2,··· , nˈmaking i i − − (V × W ) ⊇ A × B i i i=1 and (V • W ) ⊆ f (V (a, b), W (a, b)) ⊆ U for∀a ∈ A . i i − − i=1 (a,b)∈A ×B Fuzzy Inf. Eng. (2011) 2: 137-146 141 If we set V = {V|V  a, i = 1, 2,··· , n}. Since V is a finite intersection set, (a) i i (a) we get V  T . Suppose V = V ∈ T ˈwe have V ⊇ A . (a) (a) a∈A In the same way, for ∀b ∈ B , suppose W (b) = {W|W  b, i = 1, 2,··· , n}∈ T i i and W = W (b) ∈ T then we have W ⊇ B . Therefore A× B ∈ (G × G ) (V × b∈B W ) ˊ We affirm that (V × W ) obtained in this way meets (V × W ) ⊆ (V, W ). To i i i=1 prove ∀(v, w) ∈ V × W ⇒ v ∈ V , then ∃a ∈ A , we obtain v ∈ V and w  w, (a) − − − and then ∃b ∈ B , w ∈ W (b). On the other hand, since (a, b) ∈ A × B , we get ∃V × W  (a, b), where 1 ≤ k ≤ n, namely: k k V  a ⇒ V ⊇ V = {V|V  a, i = 1, 2,··· , n}, k k a i i W  b ⇒ W ⊇ W = {W|W  b, i = 1, 2,··· , n}. k k a i i Therefore, (v, w) ∈ V × W (b) ⊆ V × W ⊆ (V × W ), namely, a k k i i i=1 V × W ⊆ (V × W ). i i i=1 So f (V × W ) = V • W ⊆ (V • W ) ⊆ U . i i i=1 ∀C × D ∈ (G × G ) (V × W ) ⇒ C ⊆ V, − − − − D ⊆ W ⇒ (C • D) = C • D ⊆ V • W ⊆ U ⇒ C • D ∈ G U . Namely, f is a continuous mapping from (G × G, T )to(G, T ). G×G G Theorem 3.2 Suppose that (G,•, T ) is topological group, (G, T ) is compact T space and (G,•) is the power group of (G,•). If the unit E of (G,•) is the subgroup of G, then (G,•, T ) forms the hyper-topological group and we call (G,•, T ) is the G G hyper-topological group of (G,•, T ). −  -1 Proof In the power group (G,•), according to reference [6], for∀A ∈ G, A = A ,if only if, the unit E of f is the sub-group of G. Therefore, Theorem 3.2 can be regarded as the direct conclusion of Theorem 3.1. Comparing the conditions of Theorem 3.1 with Theorem 3.2’s, it is clear that con- ditions of Theorem 3.2 is more intuitive and can be used for further research of up- grade of topological group. Until now, we have completed the breakthrough to up- grade topological group to power group and obtained the hyper-topological group. 4. Topological Group Upgrade to Fuzzy Power Set The purpose in this section is to upgrade the two mathematic structures of topologi- cal group to fuzzy power set respectively. It shows that the multiplication operation and the contradictory operation about fuzzy hyper-topology are continuous when the topological group (G,•, T ) upgraded to fuzzy power set, then obtain the fuzzy hyper- topological group. For convenience to description and proof, we introduce the fol- lowing definitions firstly. 142 Yu-bin Zhong · Hong-xing Li (2011) Definition 4.1 Suppose (G, T ) is fuzzy topological space, for fuzzy dot x (λ) ∈ (0, 1), if A(x)>λ, then we say that x strongly belongs to fuzzy set A, marked as x ∈A. λ λ Definition 4.2 Suppose that (X, T ), (Y, T ) relates to fuzzy topological space, the X Y −1 mapping f : X → Y is fuzzy continuous, if only the original image f (c) of every open set C in T is the open set of T . If f is a correspondence one by one, and f , Y X −1 f are fuzzy continuous, then we say that f denotes a fuzzy homeomorphism map- ping. Definition 4.3 Suppose that (G, T ) stands for a fuzzy topological space, if every open bestrow of (G, T ) has limited subbestrow, then we say that (G, T ) is fuzzy compact. For the fuzzy compactness, the following conclusion is available: (G, T ) is fuzzy space, with the result A is fuzzy compact set, i.e., A is fuzzy close set. compact T Definition 4.4 Suppose that (G,•, T ) is topological group, and (G, T ) is fuzzy −1 topology created by (G, T ),f : G× G → G, f (x, y) = xy, g : G → G, g(x) = x .If f is fuzzy continuous mapping: (G, T )× (G, T ) → (G, T ), and g is fuzzy continuous mapping from: (G, T ) → (G, T ) too. Then the (G,•, T ) is called fuzzy topological group. Definition 4.5 With the fuzzy topology (G, T ) created by the given topological space (G, T ), and the set T = {U |U ∈ T } as radix, where U = {α| close f old α ⊆ ∗ ∗ ∗ ∗ U ⊆ T } , a new kind of hyper-topology (F (G), T ) can be educed, usually we call it the fuzzy hyper-topology created by (G, T ). About the problem of continuity of the above-mentioned fuzzy hyper-topology, the linchpin to prove the theorems is to prove the following. -1 −1 For f : G × G → G, f (A, B) = AB, g : G → G, g(A) = A = A , f and g are continuous about subhyper-topology T , where the radix of T has the form as G G −1 G U = G {A| closed fold A ⊆ U ∈ T }. In order to prove these two points, we prove the several lemmas as follows firstly: −1 Lemma 4.1 Let f : G × G → G, f (x, y) = xy, g : G → G, g(x) = x . Then f, g is fuzzy continuous mapping of (G, T ) × (G, T ) → (G, T ) and (G, T ) → (G, T ) respectively. −1 Proof For any U ∈ T , then U = λH (λ) , where H (λ) ∈ T ⇒ f (U ) = U U λ∈[0,1] −1 −1 −1 λ f (H (λ)), g (U ) = g (H (λ)). For the continuity of f , g ⇒ U U λ∈[0,1] λ∈[0,1] −1 −1 f (H (λ)), g (U ) ∈ T . −1 −1 Hence, f (U ), g (U ) ∈ T . Namely f and g are fuzzy continuous mappings. Lemma 4.2 g of Lemma 4.1 is the fuzzy homeomorphism mapping from (G, T ) to (G, T ). −1 −1 −1 Proof Since g ◦ g(x) = (x ) = x, then g = g is a correspondence one by one. In another hand, g is fuzzy continuous mapping from (G, T )to(G, T ), so g is Fuzzy Inf. Eng. (2011) 2: 137-146 143 homeomorphism mapping. -1 −  -1 Lemma 4.3 For any A ∈ G , then (A ) = (A ) . − −  -1 Proof Because g is fuzzy homeomorphism mapping and A is the closed, (A ) = -1 -1 − −  −  − g(A ) is closed. And g(A ) ⊇ g(A) = A , then (A ) = g(A ) ⊇ {F|F is closed, -1  -1 − and F ⊇ A } = (A ) . -1 −  -1  -1  -1 − −  -1 − −  -1 On the other hand, ((A ) ) ⊇ ((A ) ) = A ⇒ (A ) ⊇ (A ) . -1 − −  -1 Therefore, (A ) = (A ) . − − Lemma 4.4 If A = λH (λ),then A = λH (λ). λ∈[0,1] λ∈[0,1] − − Proof Suppose B = λH (λ). Because B = H (α) is closed, B is closed A A λ∈[0,1] α<λ and B ⊇ A ⇒ B ⊇ A . As far as for any closed set C ⊇ A ⇒ C ⊇ A ⊇ H (λ) and C is closed, we have λ λ A λ C ⊇ H (λ), (∀λ ∈ [0, 1]) ⇒ C ⊇ B. − − − Hence B = A , namely, A = λH (λ). λ∈[0,1] − − − Lemma 4.5 A • B = (A• B) . Proof Suppose A = λH (λ)|(λ), B = λH (λ)|(λ). Due to expansion prin- A B λ∈[0,1] λ∈[0,1] − − − − ciple III, we get A • B = A = λ(H (λ) • H (λ)). And due to Lemma 3 in A B λ∈[0,1] − − − − Reference [1], we obtain A • B = λ(H (λ)• H (λ)) = (A• B) . A B λ∈[0,1] Theorem 4.1 Suppose that (G,•, T ) is topological group, (G, T ) educes fuzzy com- pact T space (G, T ), and (G,•) fuzzy power group of (G,•). If the contradictory -1 −1 element A is equal to the contradictory set A for any element A of (G,•) , then (G,•, T ) is topological group. And we call (G,•, T ) the fuzzy hyper-topological G G group of (G,•, T ), where G T = {G U |U ∈ T },U is the topological base of ∗ ∗ subhypertopology T . Proof (1) g is continuous mapping from (G, T )to(G, T ). G G -1 -1 -1 -1 −1   −1 For ∀G U  g(A) = A ,we have U ⊇ (A ) = (A ) . Namely, U ⊇ A . -1 − Let V = U = g(U ) ∈ T . Since g is fuzzy homeomorphism, then A ⊆ V ⇒ A ∈ G V ∈ T . ∗ G Because -1 -1 -1 −    − ∀B ∈ G V ⇒ B ⊆ V ⇒ U = V ⊇ (B) = (B ) , -1 then g(B) = B ∈ G U . Namely, g is continuous mapping from (G,•, T )to ∗ G (G,•, T ). (2) f is continuous mapping from (G,•, T )× (G,•, T )to(G,•, T ). G G G − −  − For ∀G U  f (A, B) = A • B, where A, B ∈ G ⇒ A • B ⊆ (A • B) ⊆ U . Since f is fuzzy continuous mapping of (G, T ), we get 144 Yu-bin Zhong · Hong-xing Li (2011) −1 −1 f (U ) ∈ T × T ⇒ f (U ) = (V × W ), t t t∈T where V , W ∈ T . t t Therefore − − −1 − − −1 − − −1 A × B ⊆ f . f (A × B ) = f (A × B ) ⊆ f (U ) = (V × W ). t t t∈T Sine (G, T ) is fuzzy compact, with the finite bestrew, we have (V , W ) ⊇ A × i i i=1 −1 −1 B , where V × W ⊆ f (U ) and because V • W = f (V , W ) ⊆ f ◦ f (U ) ⊆ U,we i i i i i i have V • W ⊆ U . i i i=1 For any fuzzy dot a ∈ ˙ A , suppose V (a ) = {V |V  ˙ a }∈ T which is finite λ λ i i λ intersection, then we get V (a )a . Ifweset V = V (a ) ∈ T , then V ⊇ A . λ λ λ a ∈ ˙ A ˙ ˙ For any fuzzy dot b ∈B , suppose W (b ) = {W |V b }∈ T , then we obtain λ λ i i λ W (b )b . Ifweset W = W (b ) ∈ T , thenW ⊇ B . λ λ λ b ∈B To prove: V × W ⊆ (V × W ). i i i=1 For any fuzzy dot ⎪ ˙ ˙ ˙ V ∈V ⇒∃a ∈A , V ∈V (a ), ⎪ λ α λ α (V , W )∈V × W ⇒ λ γ ⎪ ⎩ W ∈ ˙ V ⇒∃b ∈ ˙ B , W ∈ ˙ W (b ). γ β γ β − − Because (a , b )∈ ˙ A × B ⊆ (V × W ). We have ∃V × W  ˙ (a , b ), where α β i i k k α β i=1 1 ≤ k ≤ n, namely V  ˙ a ⇒ V ⊇ V (a )• W  ˙ b ⇒ W ⊇ W (b ) k α k α k β k β n   n ⇒ (V , W )∈V × W ⊆ (V × W ) ⇒ V × W ⊆ (V × W ). λ γ k k i i i i i=1 i=1 So f (V × W ) = V • W ⊆ (V • W ) ⊆ U. i i i=1 − − And because V ⊇ A ⇒ A ∈ G V ∈ T , and W ⊇ B ⇒ B ∈ G W ∈ T , ∗ G ∗ G we have that ∀C ∈ G V , D ∈ G W ⇒ C ⊆ V ∗ ∗ and − − − − D ∈ W ⇒ C • D = (C • D) ⊆ V · W ⊆ U ⇒ C • D ∈ G U . Namely f is continuous mapping from (G, T )× (G, T )to(G, T ). Up until now, G G G we have completed the proof of Theorem 4.1. Theorem 4.2 Suppose that (G,•, T ) is a topological group, (G, T ) educes fuzzy compact T space (G, T ), (G,•) denotes a fuzzy power group of (G,•) and the unit E stands for the fuzzy subgroup of G, then (G,•, T ) is the fuzzy hyper-topological group of (G,•, T ). Fuzzy Inf. Eng. (2011) 2: 137-146 145 −1  -1 Proof According to Reference [5], in power group (G,•), for∀A ∈ G, A = A ⇔ the unit element E of G is the subgroup of G. From the process of prove, we get that -1 in fuzzy power group (G,•), for ∀A ∈ G , the contradictory set A is equal to the −1 contradictory element A , if only the unit element E of G is the fuzzy subgroup of G. Thus Theorem 4.2 can be regarded as the direct deduction of Theorem 4.1. Comparing Theorem 4.1’s conditions with Theorem 4.2’s, it is clear that the con- ditions of Theorem 4.2 are more intuitive and easier to prove. The conditions can be regarded as conditions to upgrade topological group to fuzzy topological group to convenient for the future research. Now, we have completed the breakthrough to upgrade the topological group to fuzzy power group and obtained the fuzzy hyper- topological group. 5. Conclusions This paper has proposed a new hyper-topology based on a kind of intuitive convergent way, and then upgraded two mathematic structures of topological group to power set and fuzzy power set respectively. It has proved that multiplication and contradiction operations are continuous in the new hyper-topology after upgrading and a series of result creating hyper-topological group and fuzzy hyper-topological group are ob- tained. Accordingly, hyper-topological group and fuzzy hyper-topological group can be created, which has finished the breakthrough work to upgrade topological group to its power set and fuzzy power set. In fact, the upgrade of topology has already obtained broad application. It lay the theoretic foundation for Wang et al to set up the triple inverted pendulum successfully and internationally for the first time in 1990 , and for Li et al to set up four level inverted pendulum in 2002. We believe that the breakthrough work in this paper will also provide foundation for operation research and control. 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Fuzzy System and Mathematics 8(1): 1-9 9. Zhong Y B (1994) The structure of hypergroup on the cyclical group. Chinese Quarterly Journal of Mathematics 9(1)˖26-31 146 Yu-bin Zhong · Hong-xing Li (2011) 10. Zhong Y B (1994) Isomorphism on hypergroup. Chinese Quarterly Journal of Mathematics 9(2)˖60- 11. Zhong Y B (1997) The homomorphism and isomorphism on hypergroup. Chinese Quarterly Journal of Mathematics 12(3)˖7-14 12. Zhong Y B (1998) The cardinal number theorem of fuzzy hypergroup. Fuzzy System and Mathemat- ics 12(2)˖61-69 13. Zhong Y B (2000) The existence of HX-RING. Applied Mathematics Journal of Chinese University 15(2): 134-138 14. Zhong Y B (2006) Fuzzy hyper-ring. Journal of Guangzhou University 5(4): 1-5 15. Yao B X, Zhong Y B (2007) The construction of power ring. Advance in Soft Computing 40(1):181- 16. Zhong Y B, Zheng J, Cao B Y (2010) The structures and expressions of power group. Fuzzy Infor- mation and Engineering 2(2): 213-224 17. Wang P Z, Sanchez E (1982) Treating a fuzzy subset as a projectable random subset. Fuzzy Informa- tion and Decision 2(2): 213-219 18. Wang P Z, Sugeno M (1982) The factor fields and background structure for fuzzy sets. Fuzzy Math 2(2): 45-54 19. Wang P Z (1983) From the fuzzy statistics to the falling random subsets. Advance in Fuzzy Set Theory and Appl. 1(1): 81-96 20. Wang P Z (1982) Random sets and fuzzy sets. International Encyclopedia on Systems and Control, Pergamon Press: 63-70 21. Wang P Z (1982) Fuzzy contactability and fuzzy varlables. Fuzzy Sets and Systems 8(1): 81-92 22. Wang P Z, Sanchez E (1983) Hyperfields and random sets. Proc. of IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis: 325-330 23. Wang P Z, Liu X H, Chen Y Y (1984) Fuzzy force and fuzzy dynamic analysis. Proc. of IEEE Symposium on Man, Machine and Cyber: 132-141 24. Wang P Z, Liu X H, Chuan K (1984) Set-valued statistics. Proc. of FIP-84 Symposium on Fuzzy Information Processing: 121-129 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

The Upgrade of Topological Group Based on a New Hyper-topology

Fuzzy Information and Engineering , Volume 3 (2): 10 – Jun 1, 2011

The Upgrade of Topological Group Based on a New Hyper-topology

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AbstractWang et al [1] have put forward the problem of upgrade of topology and obtained a series of systematic theories and applicable results. Li et al [2] have put forwarded the problem of upgrade of group and also obtained a series of results. Consequently, the problem of upgrade of mathematic structure has drawn more attention of researchers. However, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. Based...
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Fuzzy Inf. Eng. (2011) 2: 137-146 DOI 10.1007/s12543-011-0072-y ORIGINAL ARTICLE The Upgrade of Topological Group Based on a New Hyper-topology Yu-bin Zhong · Hong-xing Li Received: 12 November 2010/ Revised: 19 April 2011/ Accepted: 20 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract Wang et al [1] have put forward the problem of upgrade of topology and obtained a series of systematic theories and applicable results. Li et al [2] have put forwarded the problem of upgrade of group and also obtained a series of results. Con- sequently, the problem of upgrade of mathematic structure has drawn more attention of researchers. However, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. Based on a kind of intuitive convergent way, the paper has proposed a new hyper-topology and upgraded two mathematic structures of topological group to power set and fuzzy power set respectively. It has proved that multiplication and contradiction operations continues in the new hyper-topology after upgrading and a series of results are ob- tained creating hyper-topological group and fuzzy hyper-topological group. Accord- ingly, hyper-topological group and fuzzy hyper-topological group can be created, ob- taining a breakthrough to upgrade topological group to its power set and fuzzy power set. Keywords Upgrade· Power set· Fuzzy power set· Power group· Hyper-topological group· Fuzzy hyper-topological group 1. Introduction Algebra structure, sequence structure and topology are the three major structures of modern mathematics. Hyperalgebra is an important algebraic one. Hypergroup plays Yu-bin Zhong () School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, P.R.China email: Zhong yb@163.com Hong-xing Li () School of Mathematics Sciences, Beijing Normal University, Beijing 100875, P.R.China College of Electronic and Information Engineering, Dalian University of Technology, Dalian , P.R.China email: lhxqx@bnu.edu.cn 138 Yu-bin Zhong · Hong-xing Li (2011) an important role in modern analysis and its application, and especially has an irre- placeable role in the quantum mechanics and string theory of theoretical physics [3]. In the theory study of mathematical structure upgrading, Wang et al [1] put forward the problem of upgrading of topology and obtained a series of systematic theories and applied results [2, 4, 5, 6]. Li et al [2] have put forward the problem of upgrad- ing of group and also obtained a series of results [7-16]. However, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. Based on a kind of intuitive convergent way, this paper has proposed a new hyper-topology and upgraded two mathematic structures of topological group to power set and fuzzy power set respectively. Well then, do multi- plication and contradiction operations continue in hyper-topology? Do multiplication and contradiction operations of fuzzy power group in fuzzy power set F (G) about fuzzy hyper-topology continue? The following is to discuss these problems, and the answers are true. Finally, we obtain a series of results creating hyper-topological group and fuzzy hyper-topological group, which can create hyper-topological and fuzzy hyper-topological groups. For the convenience of depiction and research, we introduce the following def- inition and illumination. For topological group (G,•, T ), fuzzy topology (G, T ) can be obtained by topology structure (G, T ) and fuzzy power group can be ob- tained by upgrading group (G,•) to fuzzy power setF (G). In addition, the following diagram (Figure 1) can be used to show the forming of hyper-topology and fuzzy hyper-topology in detail (Figure 1): Educe to hyper-topological (P(G), T ) (G, T ) Creat Educe to fuzzy hyper-topological (F (G), T ) (G, T ) Fig.1 Forming of hyper-topology and fuzzy hyper-topology 2. A New Hyper-topology Based on Intuitive Convergent Way Firstly, we start on the problem of convergence in two-dimensional real plane R .For 2 2 2 example, for the setα = {(x, y)|x + y < 1} in two-dimensional real plane R ,weget 2 2 γ = {(x, y)|x + y < 1}, according to hyper-topology T defined in reference [1], for any open set U  α, sinceγ = {β|β ⊆ γ} is the minimal radix open set containingα, 2 2 we obtain α ∈ γ ⊆ U . On the other hand, we get W (n) = {(x, y)|x + y ≤ 1− 1/n}, then for any natural number n, we get W (n) ∈ β . Consequently, according to the 2 2 hyper-topology presented in reference [1], W (n) is converged in {(x, y)|x + y < 1} 2 2 instead of {(x, y)|x + y ≤ 1}. However, based on intuitive convergent way, W (n) 2 2 should be converged in{(x, y)|x + y ≤ 1}. To get over this situation, we study on this problem. Based on intuitive convergent way, we put forward a new hyper-topology and show it as follows: Fuzzy Inf. Eng. (2011) 2: 137-146 139 Definition 2.1 For a given topological space (G, T ), with T = {U |U ∈ T } as ∗ ∗ radix, where U = {α|close f old α ⊆ U ∈ T }, it can educe a kind of topol- ogy (P (G), T ), and (P (G), T ) is called hyper-topology of (G, T ), which is a new hyper-topology differing from reference [1]. − − − For ∀U , V ∈ T , (A ∈ U V ) ⇒ (A ⊆ U and A ⊆ V ) ⇒(A ⊆ V ∈ T ) ⇒ ∗ ∗ ∗ ∗ ∃(U V ) ∈ T ,we have A ∈ (U V ) ⊆ U V , so it is reasonable to treat T ∗ ∗ ∗ ∗ ∗ ∗ defined above as topological radix. If we study the convergence situation of W (n) with the new hyper-topology pre- 2 2 sented in this paper, W (n) is converged in {(x, y)|x + y ≤ 1}, which meets our tra- ditional intuitive idea. So it is a very useful hyper-topology. The study result of this new hyper-topology will be discussed in the other paper. For the convenience of study, we introduce the following marks and definitions. In the discussion of this paper, we assume that (G, T ) is topological space, (G,•, T ) topological group, the multiplication operation f is G × G → G, f (x, y) = xy and the −1 contradiction operation g is G → G, g(x) = x . Definition 2.2 (G,•, T ) is called topological group ⇐⇒ (G,•) is a group, (G, T ) is a topological space, and the multiplication operation f : G × G → G, f (x, y) = xy −1 and contradictory operation g : G → G, g(x) = x are continuous. Until now, research about hyper-topology and power group has already obtained a series of systematic theories, however, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. The paper presents the way of upgrading two mathematic structures of topological group to power group and fuzzy power group respectively, taking multiplication and contradiction operations into consideration. We have studied the problem of continu- ity of these two operations about the new hyper-topology above and obtained a series of basic theorems. Additionally, we will give the corresponding results. 3. Topological Group Upgrade to Power Set In this section, we upgrade two mathematic structures of topological group to power set respectively, and then take multiplication and contradiction operations into con- sideration. We study the problem of continuity of these two operations about the new hyper-topology above and obtained a series of basic results. −1 Lemma 3.1 Let g : G → G, g(x) = x . Then g is the homeomorphism mapping from (G, T ) to (G, T ). −1 −1 −1 Proof Because (g ◦ g) = x( ) , then g = g is a correspondence one by one. (x) on the other hand, g is continuous mapping from (G, T )to(G, T ), thus g is homeo- morphism mapping. Lemma 3.2 For ∀A ∈ P (G), close fold and contradictory set operations can be −  -1  -1 − exchanged, namely we get (A ) = (A ) . -1 − −  − Proof Because g is homeomorphism mapping and A is closed⇒ (A ) = g(A )is −  -1 −  -1 − a closed set. And g(A ) ⊇ g(A) = A ,so (A ) = g(A ) ⊇ {F|F is closed set, and F -1 -1 -1 -1 -1 -1 -1 -1 −  −    −  − − ⊇ A } = (A ) . On the other hand, ((A ) ) ⊇ ((A ) ) ,so(A ) = (A ) . 140 Yu-bin Zhong · Hong-xing Li (2011) −1  -1 Lemma 3.3 Suppose any element A of power group G satisfies A = A , then for − − − − − ∀A, B ∈ G , we gain f (A , B ) = A • B = (A• B) . −  − Proof (i) Since A = A A , B = B B , we obtain − −     − A • B = A• B A • B A• B A • B , (A• B) = A• B (A• B) . For ∀xy ∈ A • B ⇒ x ∈ A , y ∈ B ⇒∃ net {x }⊆ A,{y }⊆ B, we make n n x → x, y → y ⇒{x y }⊆ A • B. To prove x y → xy. Since f is continuous, n n n n n n for ∀U ∈ T and x, y ∈ U,we have ∃V, W ∈ T making f (V, W ) ∈ V • W ⊆ U , where x ∈ V, y ∈ W . In addition, x → x ∈ V, y → y ∈ W , therefore ∃N,at n > N, n n we have x ∈ V, y ∈ W , and then x y ∈ V • W ⊆ U , namely, x y → xy. Since n n n n n n −   −  − xy ∈ (A• B) , namely, A • B ⊆ (A• B) , according to the same way, A • B ⊆ (A• B) − − − and A• B ⊆ (A• B) , then we get A = B . -1 −1 (ii) For∀A ∈ G , we gain A = A , and then − −1 − − − −1 − − −1 − A ⊇ A = A• B• B ⇒ A = (A ) ⊇ (A• B• B ) ⊇ (A• B) • (B ) −  -1 − = (A•) • (B ) − −  -1 − − − −  -1 − − = (A• B) • (B ) ⇒ A • B ⊇ (A• B) • (B ) • B ⊇ (A• B) •{e} = (A• B) ˊ − − − Sum up (i) (ii), we have (A• B) = A • B . Theorem 3.1 Suppose (G,•, T ) is a topological group, (G, T ) is compact T space −1 and (G,•) is the power group of (G,•). If the contradictory element A is equal to -1 −1 the contradictory set A = {x |x ∈ A} for any element A of (G,•), then (G,•, T ) composes topological group, and we call (G,•, T ) the hyper-topological group of (G,•, T ), where the radix of sub-hyper-topology T is G T = {G U |U ∈ G ∗ ∗ T }ˊ Proof (1) g is a continuous mapping from (G, T )to(G, T ). For ∀A ∈ G U G G ∗ -1  -1 − −  -1  -1 − g(A) = A ˈwhere A ∈ G , then we obtain U ⊇ (A ) = (A ) , namely, U ⊇ A. -1 − Set V = U = g(U ) ∈ T , then A ⊆ V ⇒ A ∈ G V ∈ G T . -1 -1 -1 − −1 −  − − In addition, as ∀B ∈ G V ⇒ B ⊆ V ⇒ (B ) = (B ) = (B ) ⊆ V = −1 U ⇒ g(B) = B ∈ G U , we can get the conclusion that g is a continuous mapping from (G, T )to(G, T ). G G (2) f is a continuous mapping from (G × G, T )to(G, T ). For ∀G U G×G G ∗ − − − f (A, B) = A • B, where A, B ∈ G ⇒ (A • B) ⊆ U ⇒∀ab ∈ A • B ⊆ U and − − (a, b) ∈ A × B , because of the continuity of f (a, b) = ab, we get∃V (a, b), W (a, b) ∈ T , where V (a, b)  a, W (a, b) ∈ b, making f (V (a, b)), W (a, b)) = V (a, b)• W (a, b) ⊆ U ⇒ ( f (V (a, b), W (a, b)) ⊆ U . Within the compact T space (G, T ), com- − − (a,b)∈A ×B − − pact set is equal to closed set, so A × B is compact set. And because of the existence of finite coverage, we gain∃V, W ∈ T , i = 1, 2,··· , nˈmaking i i − − (V × W ) ⊇ A × B i i i=1 and (V • W ) ⊆ f (V (a, b), W (a, b)) ⊆ U for∀a ∈ A . i i − − i=1 (a,b)∈A ×B Fuzzy Inf. Eng. (2011) 2: 137-146 141 If we set V = {V|V  a, i = 1, 2,··· , n}. Since V is a finite intersection set, (a) i i (a) we get V  T . Suppose V = V ∈ T ˈwe have V ⊇ A . (a) (a) a∈A In the same way, for ∀b ∈ B , suppose W (b) = {W|W  b, i = 1, 2,··· , n}∈ T i i and W = W (b) ∈ T then we have W ⊇ B . Therefore A× B ∈ (G × G ) (V × b∈B W ) ˊ We affirm that (V × W ) obtained in this way meets (V × W ) ⊆ (V, W ). To i i i=1 prove ∀(v, w) ∈ V × W ⇒ v ∈ V , then ∃a ∈ A , we obtain v ∈ V and w  w, (a) − − − and then ∃b ∈ B , w ∈ W (b). On the other hand, since (a, b) ∈ A × B , we get ∃V × W  (a, b), where 1 ≤ k ≤ n, namely: k k V  a ⇒ V ⊇ V = {V|V  a, i = 1, 2,··· , n}, k k a i i W  b ⇒ W ⊇ W = {W|W  b, i = 1, 2,··· , n}. k k a i i Therefore, (v, w) ∈ V × W (b) ⊆ V × W ⊆ (V × W ), namely, a k k i i i=1 V × W ⊆ (V × W ). i i i=1 So f (V × W ) = V • W ⊆ (V • W ) ⊆ U . i i i=1 ∀C × D ∈ (G × G ) (V × W ) ⇒ C ⊆ V, − − − − D ⊆ W ⇒ (C • D) = C • D ⊆ V • W ⊆ U ⇒ C • D ∈ G U . Namely, f is a continuous mapping from (G × G, T )to(G, T ). G×G G Theorem 3.2 Suppose that (G,•, T ) is topological group, (G, T ) is compact T space and (G,•) is the power group of (G,•). If the unit E of (G,•) is the subgroup of G, then (G,•, T ) forms the hyper-topological group and we call (G,•, T ) is the G G hyper-topological group of (G,•, T ). −  -1 Proof In the power group (G,•), according to reference [6], for∀A ∈ G, A = A ,if only if, the unit E of f is the sub-group of G. Therefore, Theorem 3.2 can be regarded as the direct conclusion of Theorem 3.1. Comparing the conditions of Theorem 3.1 with Theorem 3.2’s, it is clear that con- ditions of Theorem 3.2 is more intuitive and can be used for further research of up- grade of topological group. Until now, we have completed the breakthrough to up- grade topological group to power group and obtained the hyper-topological group. 4. Topological Group Upgrade to Fuzzy Power Set The purpose in this section is to upgrade the two mathematic structures of topologi- cal group to fuzzy power set respectively. It shows that the multiplication operation and the contradictory operation about fuzzy hyper-topology are continuous when the topological group (G,•, T ) upgraded to fuzzy power set, then obtain the fuzzy hyper- topological group. For convenience to description and proof, we introduce the fol- lowing definitions firstly. 142 Yu-bin Zhong · Hong-xing Li (2011) Definition 4.1 Suppose (G, T ) is fuzzy topological space, for fuzzy dot x (λ) ∈ (0, 1), if A(x)>λ, then we say that x strongly belongs to fuzzy set A, marked as x ∈A. λ λ Definition 4.2 Suppose that (X, T ), (Y, T ) relates to fuzzy topological space, the X Y −1 mapping f : X → Y is fuzzy continuous, if only the original image f (c) of every open set C in T is the open set of T . If f is a correspondence one by one, and f , Y X −1 f are fuzzy continuous, then we say that f denotes a fuzzy homeomorphism map- ping. Definition 4.3 Suppose that (G, T ) stands for a fuzzy topological space, if every open bestrow of (G, T ) has limited subbestrow, then we say that (G, T ) is fuzzy compact. For the fuzzy compactness, the following conclusion is available: (G, T ) is fuzzy space, with the result A is fuzzy compact set, i.e., A is fuzzy close set. compact T Definition 4.4 Suppose that (G,•, T ) is topological group, and (G, T ) is fuzzy −1 topology created by (G, T ),f : G× G → G, f (x, y) = xy, g : G → G, g(x) = x .If f is fuzzy continuous mapping: (G, T )× (G, T ) → (G, T ), and g is fuzzy continuous mapping from: (G, T ) → (G, T ) too. Then the (G,•, T ) is called fuzzy topological group. Definition 4.5 With the fuzzy topology (G, T ) created by the given topological space (G, T ), and the set T = {U |U ∈ T } as radix, where U = {α| close f old α ⊆ ∗ ∗ ∗ ∗ U ⊆ T } , a new kind of hyper-topology (F (G), T ) can be educed, usually we call it the fuzzy hyper-topology created by (G, T ). About the problem of continuity of the above-mentioned fuzzy hyper-topology, the linchpin to prove the theorems is to prove the following. -1 −1 For f : G × G → G, f (A, B) = AB, g : G → G, g(A) = A = A , f and g are continuous about subhyper-topology T , where the radix of T has the form as G G −1 G U = G {A| closed fold A ⊆ U ∈ T }. In order to prove these two points, we prove the several lemmas as follows firstly: −1 Lemma 4.1 Let f : G × G → G, f (x, y) = xy, g : G → G, g(x) = x . Then f, g is fuzzy continuous mapping of (G, T ) × (G, T ) → (G, T ) and (G, T ) → (G, T ) respectively. −1 Proof For any U ∈ T , then U = λH (λ) , where H (λ) ∈ T ⇒ f (U ) = U U λ∈[0,1] −1 −1 −1 λ f (H (λ)), g (U ) = g (H (λ)). For the continuity of f , g ⇒ U U λ∈[0,1] λ∈[0,1] −1 −1 f (H (λ)), g (U ) ∈ T . −1 −1 Hence, f (U ), g (U ) ∈ T . Namely f and g are fuzzy continuous mappings. Lemma 4.2 g of Lemma 4.1 is the fuzzy homeomorphism mapping from (G, T ) to (G, T ). −1 −1 −1 Proof Since g ◦ g(x) = (x ) = x, then g = g is a correspondence one by one. In another hand, g is fuzzy continuous mapping from (G, T )to(G, T ), so g is Fuzzy Inf. Eng. (2011) 2: 137-146 143 homeomorphism mapping. -1 −  -1 Lemma 4.3 For any A ∈ G , then (A ) = (A ) . − −  -1 Proof Because g is fuzzy homeomorphism mapping and A is the closed, (A ) = -1 -1 − −  −  − g(A ) is closed. And g(A ) ⊇ g(A) = A , then (A ) = g(A ) ⊇ {F|F is closed, -1  -1 − and F ⊇ A } = (A ) . -1 −  -1  -1  -1 − −  -1 − −  -1 On the other hand, ((A ) ) ⊇ ((A ) ) = A ⇒ (A ) ⊇ (A ) . -1 − −  -1 Therefore, (A ) = (A ) . − − Lemma 4.4 If A = λH (λ),then A = λH (λ). λ∈[0,1] λ∈[0,1] − − Proof Suppose B = λH (λ). Because B = H (α) is closed, B is closed A A λ∈[0,1] α<λ and B ⊇ A ⇒ B ⊇ A . As far as for any closed set C ⊇ A ⇒ C ⊇ A ⊇ H (λ) and C is closed, we have λ λ A λ C ⊇ H (λ), (∀λ ∈ [0, 1]) ⇒ C ⊇ B. − − − Hence B = A , namely, A = λH (λ). λ∈[0,1] − − − Lemma 4.5 A • B = (A• B) . Proof Suppose A = λH (λ)|(λ), B = λH (λ)|(λ). Due to expansion prin- A B λ∈[0,1] λ∈[0,1] − − − − ciple III, we get A • B = A = λ(H (λ) • H (λ)). And due to Lemma 3 in A B λ∈[0,1] − − − − Reference [1], we obtain A • B = λ(H (λ)• H (λ)) = (A• B) . A B λ∈[0,1] Theorem 4.1 Suppose that (G,•, T ) is topological group, (G, T ) educes fuzzy com- pact T space (G, T ), and (G,•) fuzzy power group of (G,•). If the contradictory -1 −1 element A is equal to the contradictory set A for any element A of (G,•) , then (G,•, T ) is topological group. And we call (G,•, T ) the fuzzy hyper-topological G G group of (G,•, T ), where G T = {G U |U ∈ T },U is the topological base of ∗ ∗ subhypertopology T . Proof (1) g is continuous mapping from (G, T )to(G, T ). G G -1 -1 -1 -1 −1   −1 For ∀G U  g(A) = A ,we have U ⊇ (A ) = (A ) . Namely, U ⊇ A . -1 − Let V = U = g(U ) ∈ T . Since g is fuzzy homeomorphism, then A ⊆ V ⇒ A ∈ G V ∈ T . ∗ G Because -1 -1 -1 −    − ∀B ∈ G V ⇒ B ⊆ V ⇒ U = V ⊇ (B) = (B ) , -1 then g(B) = B ∈ G U . Namely, g is continuous mapping from (G,•, T )to ∗ G (G,•, T ). (2) f is continuous mapping from (G,•, T )× (G,•, T )to(G,•, T ). G G G − −  − For ∀G U  f (A, B) = A • B, where A, B ∈ G ⇒ A • B ⊆ (A • B) ⊆ U . Since f is fuzzy continuous mapping of (G, T ), we get 144 Yu-bin Zhong · Hong-xing Li (2011) −1 −1 f (U ) ∈ T × T ⇒ f (U ) = (V × W ), t t t∈T where V , W ∈ T . t t Therefore − − −1 − − −1 − − −1 A × B ⊆ f . f (A × B ) = f (A × B ) ⊆ f (U ) = (V × W ). t t t∈T Sine (G, T ) is fuzzy compact, with the finite bestrew, we have (V , W ) ⊇ A × i i i=1 −1 −1 B , where V × W ⊆ f (U ) and because V • W = f (V , W ) ⊆ f ◦ f (U ) ⊆ U,we i i i i i i have V • W ⊆ U . i i i=1 For any fuzzy dot a ∈ ˙ A , suppose V (a ) = {V |V  ˙ a }∈ T which is finite λ λ i i λ intersection, then we get V (a )a . Ifweset V = V (a ) ∈ T , then V ⊇ A . λ λ λ a ∈ ˙ A ˙ ˙ For any fuzzy dot b ∈B , suppose W (b ) = {W |V b }∈ T , then we obtain λ λ i i λ W (b )b . Ifweset W = W (b ) ∈ T , thenW ⊇ B . λ λ λ b ∈B To prove: V × W ⊆ (V × W ). i i i=1 For any fuzzy dot ⎪ ˙ ˙ ˙ V ∈V ⇒∃a ∈A , V ∈V (a ), ⎪ λ α λ α (V , W )∈V × W ⇒ λ γ ⎪ ⎩ W ∈ ˙ V ⇒∃b ∈ ˙ B , W ∈ ˙ W (b ). γ β γ β − − Because (a , b )∈ ˙ A × B ⊆ (V × W ). We have ∃V × W  ˙ (a , b ), where α β i i k k α β i=1 1 ≤ k ≤ n, namely V  ˙ a ⇒ V ⊇ V (a )• W  ˙ b ⇒ W ⊇ W (b ) k α k α k β k β n   n ⇒ (V , W )∈V × W ⊆ (V × W ) ⇒ V × W ⊆ (V × W ). λ γ k k i i i i i=1 i=1 So f (V × W ) = V • W ⊆ (V • W ) ⊆ U. i i i=1 − − And because V ⊇ A ⇒ A ∈ G V ∈ T , and W ⊇ B ⇒ B ∈ G W ∈ T , ∗ G ∗ G we have that ∀C ∈ G V , D ∈ G W ⇒ C ⊆ V ∗ ∗ and − − − − D ∈ W ⇒ C • D = (C • D) ⊆ V · W ⊆ U ⇒ C • D ∈ G U . Namely f is continuous mapping from (G, T )× (G, T )to(G, T ). Up until now, G G G we have completed the proof of Theorem 4.1. Theorem 4.2 Suppose that (G,•, T ) is a topological group, (G, T ) educes fuzzy compact T space (G, T ), (G,•) denotes a fuzzy power group of (G,•) and the unit E stands for the fuzzy subgroup of G, then (G,•, T ) is the fuzzy hyper-topological group of (G,•, T ). Fuzzy Inf. Eng. (2011) 2: 137-146 145 −1  -1 Proof According to Reference [5], in power group (G,•), for∀A ∈ G, A = A ⇔ the unit element E of G is the subgroup of G. From the process of prove, we get that -1 in fuzzy power group (G,•), for ∀A ∈ G , the contradictory set A is equal to the −1 contradictory element A , if only the unit element E of G is the fuzzy subgroup of G. Thus Theorem 4.2 can be regarded as the direct deduction of Theorem 4.1. Comparing Theorem 4.1’s conditions with Theorem 4.2’s, it is clear that the con- ditions of Theorem 4.2 are more intuitive and easier to prove. The conditions can be regarded as conditions to upgrade topological group to fuzzy topological group to convenient for the future research. Now, we have completed the breakthrough to upgrade the topological group to fuzzy power group and obtained the fuzzy hyper- topological group. 5. Conclusions This paper has proposed a new hyper-topology based on a kind of intuitive convergent way, and then upgraded two mathematic structures of topological group to power set and fuzzy power set respectively. It has proved that multiplication and contradiction operations are continuous in the new hyper-topology after upgrading and a series of result creating hyper-topological group and fuzzy hyper-topological group are ob- tained. Accordingly, hyper-topological group and fuzzy hyper-topological group can be created, which has finished the breakthrough work to upgrade topological group to its power set and fuzzy power set. In fact, the upgrade of topology has already obtained broad application. It lay the theoretic foundation for Wang et al to set up the triple inverted pendulum successfully and internationally for the first time in 1990 , and for Li et al to set up four level inverted pendulum in 2002. We believe that the breakthrough work in this paper will also provide foundation for operation research and control. 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Proc. of FIP-84 Symposium on Fuzzy Information Processing: 121-129

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Jun 1, 2011

Keywords: Upgrade; Power set; Fuzzy power set; Power group; Hyper-topological group; Fuzzy hyper-topological group

References