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On the Level Decompositions by the Range of an L-fuzzy Topological Space

On the Level Decompositions by the Range of an L-fuzzy Topological Space Fuzzy Inf. Eng. (2011) 3: 225-234 DOI 10.1007/s12543-011-0079-4 ORIGINAL ARTICLE On the Level Decompositions by the Range of an L-fuzzy Topological Space Ginu Varghese· Sunil C. Mathew Received: 14 November 2010/ Revised: 15 July 2011/ Accepted: 15 August 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract This paper is on the level decompositions corresponding to the elements in the range of an L-fuzzy topology on a given set and an investigation into the lat- tice structure of the same. In general, this lattice is not complete and distributive. However, certain necessary and sufficient conditions for it to be modular, distribu- tive, complete and complemented are derived. Atoms and dual atoms of it along with the conditions for their existence are obtained. Certain related properties of it are also discussed here. Keywords Complete L-fuzzy topology · Lattice · Atom· F-upper set 1. Introduction Ever since Ho ¨ hle and S ostak [4] introduced the concept of L-fuzzy topological space in 1995, many authors [7, 11] have studied the level decompositions of it and obtained certain representation theorems for L-fuzzy topologies on a given set X. It has been proved that these level decompositions are actually L-topologies on X and the given L-fuzzy topological space can be characterized by them [9, 12]. Later in [8], the authors have shed more light on this topic by establishing the lattice structure of these L-topologies. It is proved that this characterizing lattice L ofagiven L-fuzzy topological space is complete but not distributive. The purpose of this paper is to study the level decompositions corresponding to the range elements of a given L-fuzzy topological space. Contrary to the case of Ginu Varghese() Centre for Mathematical Sciences, Pala Campus, Arunapuram P.O.-686 574, Kottayam, Kerala, India email: ginuvettuparampil@gmail.com Sunil C. Mathew() Department of Mathematics, St.Thomas College Palai, Arunapuram P.O.-686 574, Kottayam, Kerala, India email: sunilcmathew@rediffmail.com, sunil@stcp.ac.in 226 Ginu Varghese· Sunil C. Mathew (2011) L, the set of all L-topologies given by these level decompositions need not always form a lattice. However, we could characterize the situation under which they form a lattice L . Further, we have inquired also the circumstances in which the lattice L is completely determined by a proper subset of L leading to the concept of a complete L- fuzzy topological space. Even thoughL is always a subset of L, there are situations in which it is not a proper sublattice of L. It is also noted that L may be a lattice without being a sublattice of L. It is proved that the equality of the two lattices L and L is a necessary condition for an L-fuzzy topological space to be complete. In addition, we have also obtained a characterization for their equality. and obtained certain Finally, we have investigated the structural properties of L necessary and sufficient conditions for L to be modular, distributive, complete and complemented. We have also found the atoms and dual atoms of L along with the conditions for their existence. 2. Preliminaries Throughout this paper, X stands for a non-empty ordinary set and L for a completely distributive lattice with an order reversing involution. We denote the constant function in L taking the valueα ∈ L byα. The characteristic function of A ⊆ X is denoted by μ . The fundamental definitions of L-fuzzy set theory and L-topology are assumed to be familiar to the reader. The following are some important definitions and results from [2, 5]. Definition 2.1 Let P be a poset with order relation≤ . Then x < yif x ≤ y and x  y. Definition 2.2 An element of a lattice L is called an atom if it is the minimal element of L\{0}. Definition 2.3 An element of a lattice L is called a dual atom if it is the maximal element of L\{1}. Definition 2.4 A lattice is said to be bounded if it possesses 0 and 1. Definition 2.5 A bounded lattice L is said to be join complemented if for all x in L, there exists y in L such that x∨ y = 1. Definition 2.6 A bounded lattice L is said to be meet complemented if for all x in L, there exists y in L such that x∧ y = 0. Definition 2.7 A bounded lattice is said to be complemented if it is both join comple- mented and meet complemented. Definition 2.8 Let L and L be two lattices. A map f : L −→ L is said to be a 1 2 1 2 lattice homomorphism if f (a ∨ b) = f (a) ∨ f (b) and f (a ∧ b) = f (a) ∧ f (b) for all a, b ∈ L. Definition 2.9 A bijective lattice homomorphism is called a lattice isomorphism. Lemma 2.1 Let L be a lattice with an order reversing involution. Then de-Morgan’s laws hold for all a, b ∈ L, (a∨ b) = a ∧ b and (a∧ b) = a ∨ b . The following pentagon and diamond lattices are usually denoted by N and M , 5 3 respectively. Fuzzy Inf. Eng. (2011) 3: 225-234 227 Fig. 1 The pentagon lattice N Fig. 2 The diamond lattice M 5 3 Theorem 2.1 Let L be a lattice. (i) L is non-modular if and only if N  L. (ii) L is non-distributive if and only if N  Lor M  L, where N  L means that 5 3 L contain a sublattice isomorphic to N. Whenever we write P ⊂ Q, we mean P as a proper subset of Q. Also, if A ⊆ X by μ , we mean the characteristic function associated with A. 3. Complete L-fuzzy Topological Spaces Definition 3.1 [1, 3, 6, 10] An L-fuzzy topology on X is a map F : L → L satisfying (i) F(0)=F(1)= 1, (ii) F( f ∧ g) ≥ F( f )∧ F(g), for all f, g ∈ L , (iii) F( f ) ≥ F( f ), for all f ∈ L . i i i i∈Δ i∈Δ The pair (X, F) is called an L-fuzzy topological space (in short L-fts) and L = {a ∈ L| a = F(g) for some g ∈ L } is called the range of F. It should be noted that L is a subset of L, containing the greatest element of L. However, the smallest element of L need not be in L . The smallest element of L is denoted by 0 . X th Definition 3.2 [8] Let (X,F) be an L-fts. Then δ = { f ∈ L : F( f ) ≥ a}, the a level of F is an L-topology on X. Definition 3.3 [8] Let (X, F) be an L-fuzzy topological space and a ∈ L. Then an element b in L is said to be an F-upper element of a if a ≤ b and there exist no c ∈ L such that a < c < b. The set of all F-upper elements of a is denoted by ⇑ a. That is, ⇑ a={b ∈ L | a ≤ b and there exist no c ∈ L such that a < c < b}. Notation 1 [8] ⇑ a = {y ∈ L | y ≥ x}. Theorem 3.1 [8] Let (X, F) be an L-fuzzy topological space such that a, b ∈ L with a < b,a ∈ L . Thenδ ⊂ δ . b a 228 Ginu Varghese· Sunil C. Mathew (2011) Theorem 3.2 [8] Let (X, F) be an L-fuzzy topological space. Then L = {δ : a ∈ L} forms a complete lattice under the natural order of set inclusion. Theorem 3.3 [8] If a , a ∈ L such that a ≤ a . Then δ ⊇ δ . 1 2 1 2 a a 1 2 Converse of the above theorem is not always true as shown in [8], but in the fol- lowing theorem, we provide a situation where the converse is also true. Theorem 3.4 Let (X, F) be an L-fts. Then for a, b ∈ L ,δ ⊆ δ if and only if a ≥ b. a b Proof Necessity: Suppose thatδ ⊆ δ . If possible, let a, b ∈ L are such that a  b. a b Let A = { f ∈ L : F( f ) = a}. Since a ∈ L , we have A  ∅. Then any f ∈ A is such that f  δ . Since A ⊂ δ , this contradicts the fact thatδ ⊆ δ . b a a b Sufficiency: Follows from Theorem 3.3. In general, for a given L-fts, δ = δ need not imply a = b. However, if a, b ∈ L , a b we have the following: Theorem 3.5 Let (X, F) be an L-fts. Then for a, b ∈ L ,δ = δ if and only if a = b. a b Proof Straightforward. Remark 1 For any L-fts, by Theorem 3.2, the level decompositions corresponding to a ∈ L form a lattice. However, this is not true in the case of level decompo- sitions corresponding to a ∈ L . For example, consider the diamond type lattice L = {0, a, b, 1} given by the following Hasse diagram and X = {x, y}. Fig. 3 The diamond type lattice Let (X, F) be the L-fts determined by the following level decompositions. δ = L ,δ = {0, 1, b,μ , (b, a), (b, 1), (0, b), (b, 0), (0, a)}, 0 b {y} δ = {0, 1, a,μ , (a, b), (1, a), (a, 0), (a, 1), (1, b)},δ = {0, 1}, a {x} 1 where (a, b) denotes the function which carries x to a and y to b. Clearly, here {δ : t ∈ L } = {δ ,δ ,δ } is not a lattice. Further, in the light of the following theorem, it 1 a b is worth to note that L = {a, b, 1} is also not a lattice. Theorem 3.6 Let (X, F) be an L-fts. Then L = {δ : a ∈ L } forms a lattice under set inclusion if and only if L is a lattice. Proof Suppose that L is a lattice. Letδ ,δ ∈L . We claim thatδ ∨δ = δ and a b a b ab δ ∧δ = δ , where a  b, a  b denote the join and meet of a and b in L . Clearly, a b ab Fuzzy Inf. Eng. (2011) 3: 225-234 229 a  b ≤ a∧ b ≤ a, b. Then by Theorem 3.3, δ ,δ ⊆ δ . If possible, suppose there a b ab exist δ ∈L such that δ ,δ ⊂ δ ⊂ δ . Then c ∈ L is such that a, b > c > a  b, c a b c ab which is not possible. Consequently,δ ∨δ = δ . a b ab Similarly, it can be shown that δ ∧ δ = δ so that L is a lattice under set a b ab inclusion. Conversely, suppose that L is a lattice under set inclusion. Let a, b ∈ L be such that a b  L . Then clearly, a, b are incomparable and there exist no c ∈ L such that X X c ≤ a, b. Let A = { f ∈ L |F( f ) = a} and B = {g ∈ L |F(g) = b}. Since a, b ∈ L , we have A, B  ∅. Then any f ∈ A and g ∈ B are such that f  δ and g  δ . Since b a A ⊆ δ and B ⊆ δ , this shows that δ and δ are incomparable. Since L is a lattice, a b a b δ ∨ δ = δ for some c ∈ L , c ≤ a, b which is a contradiction. Similar is the case a b c when a  b  L . Note 1 1) It should be noted that if L is a sublattice of L, then a∨b = ab, a∧b = ab so that for δ ,δ ∈L , we have δ ∨δ = δ and δ ∧δ = δ . For the rest a b a b a∧b a b a∨b of the paper, when only the lattice L is under consideration, we use ∨ and ∧ instead of  and  even if L is not a sublattice of L. 2) Even if a and b are incomparable elements of L, δ and δ may be compa- a b rable in L. However, from the proof of the above theorem, it follows that the incomparability of two elements in L is carried over to the corresponding level decompositions in L . Remark 2 Even though L is a lattice, it need not be a sublattice of L. For instance, let X = {x, y, z} and L be the lattice given in the following Hasse diagram. Fig. 4 Hasse diagram of L Let (X, F) be the L-fts determined by the level decompositions δ = L ,δ = 0 a {0, 1,μ ,μ },δ = δ = {0, 1,μ },δ = δ = {0, 1,μ },δ = δ = {0, 1}. Then {x,y} {y} b d {x,y} c e {y} f 1 L = {0, d, e, 1} is a lattice, but not a sublattice of L. In case L is a sublattice of L, it need not be complete. For example, consider L = [0, 1] and X be any singleton. 1 1 Define F : L −→ L by F(0) = 1, F(x) = for all x ∈ (0, ), F(y) = y for all 2 2 230 Ginu Varghese· Sunil C. Mathew (2011) 1 3 3 3 3 3 y ∈ ( , ), F(z) = for all z ∈ ( , ), F(w) = w for all w ∈ ( , 1). Then (X, F)isan 2 5 4 5 4 4 L-fts. Here, it should be noted that L  L.If L is a complete sublattice of L, then we define F to be complete, the justification of which will follow later. Definition 3.4 An L-fts (X, F) is said to be a complete L-fts if L is a complete sub- lattice of L. Theorem 3.7 Let (X, F) be a complete L-fts. Then L = L . Proof Clearly L ⊆L. Now let δ ∈L.If a ∈ L , then δ ∈L . So we may take a a a ∈ L \ L . Then we claim that there exist exactly one F-upper element of a say, d. Firstly, we note that ⇑ a ∅, for if ⇑ a= ∅, then for d ∈ L , with d > a (since 1 ∈ L F F such a d always exists) there exist infinitely many t ∈ L such that a < t < d.Now since L is complete, t ∈ L ,If t > a, then clearly t ∈⇑ a , which is a<t<d a<t<d a<t<d not possible. Consequently, t = a ∈ L which is again a contradiction. Thus a<t<d ⇑ a contains at least one element. If possible, suppose there exist d  d ,∈⇑ a. F 1 2 F Then d ≥ a, and d ∈ L for i = 1, 2. Also d ∧ d ≥ a and a  L . But by i i 1 2 assumption d ∧ d ∈ L and so a < d ∧ d . Since d  d ,∈⇑ a, we must have 1 2 1 2 1 2 F d  d ∧ d ; i = 1, 2. Thus we get a < d ∧ d < d, for i = 1, 2. Since d ∧ d ∈ L , i 1 2 1 2 i 1 2 this implies that d ⇑ a for i = 1, 2 which is a contradiction. Hence ⇑ a= {d}, for i F F some d > a. Further since L is complete and a  L , we must have δ = δ , so that a d δ ∈L . Hence the name ‘complete’ is suitable because even if L ⊂ L, we get L = L. That is, the lattice L is completely determined by a proper sublattice of L. Remark 3 The converse of the above theorem is not true. Consider the lattice L and the set X as given in Remark 2. Let F be the L-fts determined by δ = L ,δ = 0 a {0, 1,μ },δ = {0, 1,μ ,μ },δ = δ = {0, 1,μ ,δ = δ = δ = {0, 1}. Then {y} b {x} {x,y} f c {x} d e 1 L = {0, a, b, f, 1} and L = L . But (X, F) is not a complete L-fts. However the following theorem provides a necessary condition for the equality ofL andL . Theorem 3.8 Let (X, F) be an L-fts such thatL = L . Then⇑ a is a singleton for all a ∈ L. Proof Suppose that L = L . Then for every a ∈ L,δ = δ , for some d ∈ L . a d Since d ∈ L and δ = δ , we must have d ∈⇑ a. If possible, let t  d,∈⇑ a then δ a d F F t and δ are incomparable which is a contradiction since δ = δ ⊃ δ . Hence ⇑ a is a d d a t F singleton. Converse of the above theorem is not always true. For instance, consider L = 1 3 [0, 1] ∪ 0 , where < 0 < and 0 is incomparable to all t ∈ L such that t ∈ 4 4 1 3 1 ( , ) and X be any singleton. Define F : L −→ L by F(0) = 1, F(x) = ; x ∈ 4 4 5 1 1 1 1 (0, ], F(y) = 0 ; y ∈ ( , ]∪{0 }, F(z) = z; z ∈ ( , 1]. Then (X, F)isan L-fts such 5 5 4 4 that ⇑ a is a singleton for all a in L andL  L. Corollary 1 Let (X, F) be a complete L-fts. Then for every a ∈ L,⇑ a is a singleton. F Fuzzy Inf. Eng. (2011) 3: 225-234 231 Since the converse of Theorem 3.8 is not true, it follows by Theorem 3.7 that the converse of the above corollary is also not true. Theorem 3.9 Let (X, F) be an L-fts such that for every a ∈ L, ⇑ a is a singleton. ThenL = L if and only if ⇑a has no element incomparable to the element in⇑ a. Proof If possible, let ⇑a has an element b incomparable to the unique element say, d in ⇑ a. Clearly, then d  a and b  d < a.Now in L(= L ), we have δ ∨δ = δ ⊃ δ . But δ ∨δ ⊆ δ , since a < b, d, which is a contradiction. b d bd a b d a Conversely, let a ∈ L \ L and ⇑ a= {d}. Then d > a. Take t > a,∈ L .If t < d, then a < t < d, which is not possible and if t is incomparable to d then t, d ∈⇑ a which is again not possible. Hence t ≥ d and soδ = δ . a d Corollary 2 Let (X, F) be an L-fts such that L is a chain. Then L = L if and only if for every a ∈ L, ⇑ a is a singleton. Even thoughL is a lattice, there are situations in which it is not a proper sublattice ofL. The following theorem accounts for such a situation. Theorem 3.10 Let (X, F) be an L- f ts such that ⇑ a ∅ for all a ∈ L and L  L . ThenL is not a sublattice of L. Proof If possible, let L be a proper sublattice of L. Then by Theorem 3.9, there exist an a ∈ L such that either (i) ⇑ a is not a singleton or (ii) ⇑ a has at least one element incomparable to the element in ⇑ a. In Case (i) we have|⇑ a|≥ 2 so that F F a  L . Then there exist d  d ∈⇑ a such that δ ,δ ∈L . But in L d  d < a 1 2 F d d 1 2 1 2 so that in L, by Theorem 3.1 δ ⊃ δ and in L,δ ∨δ = δ . Since L is a d d a d d d d 1 2 1 2 1 2 lattice and δ ,δ ⊂ δ ,wehave δ ∨δ ⊆ δ ⊂ δ , which is a contradiction. In d d a d d a d d 1 2 1 2 1 2 Case (ii) there is a b ∈⇑a, which is incomparable to the unique element d, different from a in ⇑ a. Since b ⇑ a there exist infinitely many t’s in L such that a < t < b. F F Clearly, all these t’s are incomparable to d, Otherwise b will become comparable to d. For any such t, d  t < a. Also δ = δ ∨δ . Now in L, δ ∨δ ⊆ δ ⊂ δ ,by dt d t d t a dt Theorem 3.1 which is a contradiction. Corollary 3 Let (X, F) be an L-fts such that L is a sublattice of L and ⇑ a ∅ for all a ∈ L. Then L = L . Corollary 4 Let (X, F) be an L-fts such that L is a proper sublattice of L. Then ⇑ a= ∅ for some a ∈ L. 4. Structural Properties of the LatticeL We have already seen that L is always a complete lattice. However, this is not true in the case of L . For instance, consider the lattice L = [0, 1] and X be any single- 1 1 ton. Define F : L −→ L by F(0) = 1, F(x) = x for all x ∈ [ , 1], F(y) = for 2 n 1 1 all x ∈ ( , ], n ≥ 2. Then (X, F)isan L-fts. Here L is a lattice which is not n+ 1 n complete. 232 Ginu Varghese· Sunil C. Mathew (2011) Theorem 4.1 Let (X, F) be an L-fts. ThenL is complete if and only if L is complete. Proof Suppose L is complete. Let G be any arbitrary subset of L and A = {a ∈ L : δ ∈G}. We claim that G = δ , where the meet is taken over all a ∈ A and a ∧a G = δ , where the join is taken over all a ∈ A. Since t ≤ a, for all a ∈ A, ∨a t∈A by Theorem 3.3, we have for every a ∈ A, δ ⊆ δ , where the meet is taken over all a ∧t t ∈ A. Consequently,G⊆ δ . Now to prove that δ is the smallest element in L ∧t ∧t containing G. If possible, suppose there exist δ ∈L such thatG⊂ δ ⊂ δ . Then b b ∧t by Theorem 3.4 and Theorem 3.5, we have a ≥ b for all a ∈ A. Then t ≥ b. Also, t∈A δ ⊂ δ =⇒ t ≤ b. Thus t = b. Hence by Theorem 3.5, δ = δ . In a similar b ∧ b ∧t t∈A t∈A way, we get G = δ , where the join is taken over all a ∈ A. Since L is complete, ∨a G, G∈L . HenceL is complete. Conversely, suppose that L is complete. Let H be an arbitrary subset of L . Con- sider H = {δ : a ∈ H}. By assumption H has a join and a meet in L . Also H = δ , where the meet is taken over all a ∈ H and H = δ , where the join ∧a ∨a is taken over all a ∈ H. Hence a and a belong to L . a∈H a∈H Theorem 4.2 Let (X, F) be an L-fts. Then L has atoms if and only if L has dual atoms. Proof Let δ be an atom of L . Then δ ⊂ δ =⇒ δ = δ , for all δ ∈L . Hence a b a b 1 b by Theorem 3.4 and Theorem 3.5, we get a < b and b = 1. Thus a is a dual atom of L . Conversely, suppose thatL has no atom. Then for anyδ ∈L , there existδ  δ a b 1 such that δ ⊂ δ . Hence by Theorem 3.4 and Theorem 3.5, we have for any a ∈ L , b a there exist b  1 such that b > a, i.e. L has no dual atom. In a similar way, we can prove the following. Theorem 4.3 Let (X, F) be an L-fts. Then L has dual atoms if and only if L has atoms. An atom ofL is of the formδ where a is a dual atom of L and a dual atom ofL is of the form δ where b is an atom of L . Remark 4 In general, L need not be complemented. For example, take L and X as in Remark 2 and F as the L-fuzzy topology determined by δ = L ,δ = 0 a {0, 1,μ ,μ },δ = δ = δ = δ = {0, 1},δ = {0, 1,μ ,μ },δ = {0, 1,μ }. {x} {x,y} b d f 1 c {y} {x,y} e {x,y} HereL is not complemented. The following theorem provides a necessary and sufficient condition for L to be complemented. Theorem 4.4 Let (X, F) be an L-fts. Then L is complemented if and only if L is complemented. Proof L is complemented ⇐⇒ for all δ ∈L , there existδ ∈L , such that δ ∨δ = δ andδ ∧δ = δ . a b a b 0 a b 1 Fuzzy Inf. Eng. (2011) 3: 225-234 233 ⇐⇒ for all δ ∈L , there existδ ∈L , such thatδ = δ andδ = δ . a b a∧b 0 a∨b 1 ⇐⇒ for all a ∈ L , there exist b ∈ L , such that a∧ b = 0 and a∨ b = 1 by Theorem 3.5. Remark 5 L need not always be modular. For example, with respect to L and X given in Remark 2, consider the L-fts determined by δ = L ,δ = {0, 1,μ },δ = 0 a {z} b δ = {0, 1,μ },δ = {0, 1,μ ,μ } δ = δ = δ = {0, 1}. Here it can be easily f {z,y} c {y} {z,y} e d 1 verified that L is not modular. Theorem 4.5 Let (X, F) be an L-Fts. ThenL is modular if and only if L is modular. Proof L is non modular ⇐⇒ L contains a sublattice isomorphic to N ⇐⇒ there exist two incomparable elements b, c and a d in L such that b < d with b∧ c = d ∧ c and b∨ c = d∨ c. ⇐⇒ there exist two incomparable elements δ ,δ and a δ in L such that δ ⊃ δ b c d b d with δ = δ andδ = δ . b∧c d∧c b∨c d∨c ⇐⇒ there exist two incomparable elements δ ,δ and a δ in L such that δ ⊃ δ b c d b d with δ ∨δ = δ ∨δ andδ ∧δ = δ ∧δ . b c d c b c d c ⇐⇒ L is non modular. Theorem 4.6 Let (X, F) be an L-Fts. Then L is distributive if and only if L is distributive. Proof L has a sublattice isomorphic to M ⇐⇒ L contains three incomparable elements a, b, c such that a∨ b = a∨ c = b∨ c and a∧ b = b∧ c = a∧ c. ⇐⇒ there exist three incomparable elements δ ,δ ,δ ∈L such that a b c δ = δ = δ andδ = δ = δ . a∧b b∧c a∧c a∨b b∨c a∨c ⇐⇒ there exist three incomparable elementsδ ,δ ,δ ∈L such that a b c δ ∨δ = δ ∨δ = δ ∨δ andδ ∧δ = δ ∧δ = δ ∧δ . a b b c a c a b b c a c ⇐⇒ L has a sublattice isomorphic to M . Also from the proof of the above theorem, it follows that L contains a sublattice isomorphic to N if and only if L contains a sublattice isomorphic to N . Now the 5 5 result follows from Theorem 2.1. Remark 6 In the light of the above theorems, one may think that L is isomorphic to L . However, this is not true in general. For example, let X and L be as in Re- mark 2, and F be the L-fts determined by δ = L ,δ = {0, 1,μ ,μ },δ = 0 a {x} {x,y} c {0, 1,μ ,μ },δ = {0, 1,μ } δ = δ = δ = δ = {0, 1}. Here L is not isomor- {x,y} {y} e {x,y} b d f 1 phic to L . However, if there is an order reversing involution in L , then L becomes isomorphic to L . Theorem 4.7 Let (X, F) be an L-fts with an order reversing involution in L . ThenL is isomorphic toL . Proof Define f : L −→ L by f (δ ) = i(a), i denote the order reversing involution in L. Then f (δ ∨ δ ) = f (δ ) = i(a ∧ b) = i(a) ∨ i(b) = f (δ ) ∨ f (δ ). Also a b a∧b a b f (δ ∧δ ) = f (δ ) = i(a∨ b) = f (δ )∧ f (δ ). Clearly, f is one-one and onto. Hence a b a∨b a b the proof. 234 Ginu Varghese· Sunil C. Mathew (2011) 5. Conclusion We have analyzed the lattice structure of the level decompositions corresponding to the range of an L-fuzzy topology on a given set and have obtained characterizations for certain properties of it. Further, the notion of complete L-fuzzy topological space introduced, sheds more light on the structure of this lattice. Thus, the study reveals more about the interplay between fuzzy topology and lattice theory. Acknowledgements The first author places on record her gratitude to the Department of Science and Technology, Government of India, New Delhi, for the support towards this study under the project No: SR/S4/MS: 287/05 and the Centre for Mathematical Sciences, Pala Campus for providing all the facilities. The authors would like to thank the referees for their valuable comments. References 1. Arzu Ari A, Halis Augu ¨ n (2008) Strong compactness and P-closedness in smooth L-Fuzzy topologi- cal spaces. Int. J. Contemp. Math. Sciences 3(5): 199-212 2. Davey B A, Priestley H A (2002) Introduction to lattices and order. Cambridge University Press 3. Gregori V, Vidal A (1998) Gradation of openness and Chang’s fuzzy topologies. Fuzzy Sets and Systems 109: 233-244 4. Ulrich Ho ¨ hle, Alexander S ostak (1995) A general theory of fuzzy topological spaces. Fuzzy Sets and Systems 73: 131-149 5. Liu Y M, Luo M K (1997) Fuzzy topology. World Scientific 6. Ramadan A A (1992) On smooth topological spaces. Fuzzy Sets and Systems 48: 371-375 7. Rodabaugh S E (1991) Point-set lattice-theoretic topology. Fuzzy Sets and Systems 40(2): 297-345 8. Ginu Varghese, Sunil C. Mathew (2010) On the characterizing lattice of an L-fuzzy topological space. Far East Journal of Mathematical Sciences 39(1): 15-27 9. Wuyts P (1984) On the determination of fuzzy topological spaces and fuzzy neighborhood spaces by their level topologies. Fuzzy Sets and Systems 12: 71-85 10. Yue Y L (2007) Lattice valued induced fuzzy topological spaces. Fuzzy Sets and Systems 158: 1461- 11. Zahran A M, Azab Abd-Allah M, Abd El-Nasser G. Abd El-Rahman (2008) On L-fuzzy closure spaces. Fuzzy Math. 16(2): 361-376 12. Zhang J, Shi F G, Zheng C Y (2005) On L-fuzzy topological spaces. Fuzzy Sets and Systems 149: 473-484 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

On the Level Decompositions by the Range of an L-fuzzy Topological Space

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Publisher
Taylor & Francis
Copyright
© 2011 Taylor and Francis Group, LLC
ISSN
1616-8666
eISSN
1616-8658
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10.1007/s12543-011-0079-4
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Fuzzy Inf. Eng. (2011) 3: 225-234 DOI 10.1007/s12543-011-0079-4 ORIGINAL ARTICLE On the Level Decompositions by the Range of an L-fuzzy Topological Space Ginu Varghese· Sunil C. Mathew Received: 14 November 2010/ Revised: 15 July 2011/ Accepted: 15 August 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract This paper is on the level decompositions corresponding to the elements in the range of an L-fuzzy topology on a given set and an investigation into the lat- tice structure of the same. In general, this lattice is not complete and distributive. However, certain necessary and sufficient conditions for it to be modular, distribu- tive, complete and complemented are derived. Atoms and dual atoms of it along with the conditions for their existence are obtained. Certain related properties of it are also discussed here. Keywords Complete L-fuzzy topology · Lattice · Atom· F-upper set 1. Introduction Ever since Ho ¨ hle and S ostak [4] introduced the concept of L-fuzzy topological space in 1995, many authors [7, 11] have studied the level decompositions of it and obtained certain representation theorems for L-fuzzy topologies on a given set X. It has been proved that these level decompositions are actually L-topologies on X and the given L-fuzzy topological space can be characterized by them [9, 12]. Later in [8], the authors have shed more light on this topic by establishing the lattice structure of these L-topologies. It is proved that this characterizing lattice L ofagiven L-fuzzy topological space is complete but not distributive. The purpose of this paper is to study the level decompositions corresponding to the range elements of a given L-fuzzy topological space. Contrary to the case of Ginu Varghese() Centre for Mathematical Sciences, Pala Campus, Arunapuram P.O.-686 574, Kottayam, Kerala, India email: ginuvettuparampil@gmail.com Sunil C. Mathew() Department of Mathematics, St.Thomas College Palai, Arunapuram P.O.-686 574, Kottayam, Kerala, India email: sunilcmathew@rediffmail.com, sunil@stcp.ac.in 226 Ginu Varghese· Sunil C. Mathew (2011) L, the set of all L-topologies given by these level decompositions need not always form a lattice. However, we could characterize the situation under which they form a lattice L . Further, we have inquired also the circumstances in which the lattice L is completely determined by a proper subset of L leading to the concept of a complete L- fuzzy topological space. Even thoughL is always a subset of L, there are situations in which it is not a proper sublattice of L. It is also noted that L may be a lattice without being a sublattice of L. It is proved that the equality of the two lattices L and L is a necessary condition for an L-fuzzy topological space to be complete. In addition, we have also obtained a characterization for their equality. and obtained certain Finally, we have investigated the structural properties of L necessary and sufficient conditions for L to be modular, distributive, complete and complemented. We have also found the atoms and dual atoms of L along with the conditions for their existence. 2. Preliminaries Throughout this paper, X stands for a non-empty ordinary set and L for a completely distributive lattice with an order reversing involution. We denote the constant function in L taking the valueα ∈ L byα. The characteristic function of A ⊆ X is denoted by μ . The fundamental definitions of L-fuzzy set theory and L-topology are assumed to be familiar to the reader. The following are some important definitions and results from [2, 5]. Definition 2.1 Let P be a poset with order relation≤ . Then x < yif x ≤ y and x  y. Definition 2.2 An element of a lattice L is called an atom if it is the minimal element of L\{0}. Definition 2.3 An element of a lattice L is called a dual atom if it is the maximal element of L\{1}. Definition 2.4 A lattice is said to be bounded if it possesses 0 and 1. Definition 2.5 A bounded lattice L is said to be join complemented if for all x in L, there exists y in L such that x∨ y = 1. Definition 2.6 A bounded lattice L is said to be meet complemented if for all x in L, there exists y in L such that x∧ y = 0. Definition 2.7 A bounded lattice is said to be complemented if it is both join comple- mented and meet complemented. Definition 2.8 Let L and L be two lattices. A map f : L −→ L is said to be a 1 2 1 2 lattice homomorphism if f (a ∨ b) = f (a) ∨ f (b) and f (a ∧ b) = f (a) ∧ f (b) for all a, b ∈ L. Definition 2.9 A bijective lattice homomorphism is called a lattice isomorphism. Lemma 2.1 Let L be a lattice with an order reversing involution. Then de-Morgan’s laws hold for all a, b ∈ L, (a∨ b) = a ∧ b and (a∧ b) = a ∨ b . The following pentagon and diamond lattices are usually denoted by N and M , 5 3 respectively. Fuzzy Inf. Eng. (2011) 3: 225-234 227 Fig. 1 The pentagon lattice N Fig. 2 The diamond lattice M 5 3 Theorem 2.1 Let L be a lattice. (i) L is non-modular if and only if N  L. (ii) L is non-distributive if and only if N  Lor M  L, where N  L means that 5 3 L contain a sublattice isomorphic to N. Whenever we write P ⊂ Q, we mean P as a proper subset of Q. Also, if A ⊆ X by μ , we mean the characteristic function associated with A. 3. Complete L-fuzzy Topological Spaces Definition 3.1 [1, 3, 6, 10] An L-fuzzy topology on X is a map F : L → L satisfying (i) F(0)=F(1)= 1, (ii) F( f ∧ g) ≥ F( f )∧ F(g), for all f, g ∈ L , (iii) F( f ) ≥ F( f ), for all f ∈ L . i i i i∈Δ i∈Δ The pair (X, F) is called an L-fuzzy topological space (in short L-fts) and L = {a ∈ L| a = F(g) for some g ∈ L } is called the range of F. It should be noted that L is a subset of L, containing the greatest element of L. However, the smallest element of L need not be in L . The smallest element of L is denoted by 0 . X th Definition 3.2 [8] Let (X,F) be an L-fts. Then δ = { f ∈ L : F( f ) ≥ a}, the a level of F is an L-topology on X. Definition 3.3 [8] Let (X, F) be an L-fuzzy topological space and a ∈ L. Then an element b in L is said to be an F-upper element of a if a ≤ b and there exist no c ∈ L such that a < c < b. The set of all F-upper elements of a is denoted by ⇑ a. That is, ⇑ a={b ∈ L | a ≤ b and there exist no c ∈ L such that a < c < b}. Notation 1 [8] ⇑ a = {y ∈ L | y ≥ x}. Theorem 3.1 [8] Let (X, F) be an L-fuzzy topological space such that a, b ∈ L with a < b,a ∈ L . Thenδ ⊂ δ . b a 228 Ginu Varghese· Sunil C. Mathew (2011) Theorem 3.2 [8] Let (X, F) be an L-fuzzy topological space. Then L = {δ : a ∈ L} forms a complete lattice under the natural order of set inclusion. Theorem 3.3 [8] If a , a ∈ L such that a ≤ a . Then δ ⊇ δ . 1 2 1 2 a a 1 2 Converse of the above theorem is not always true as shown in [8], but in the fol- lowing theorem, we provide a situation where the converse is also true. Theorem 3.4 Let (X, F) be an L-fts. Then for a, b ∈ L ,δ ⊆ δ if and only if a ≥ b. a b Proof Necessity: Suppose thatδ ⊆ δ . If possible, let a, b ∈ L are such that a  b. a b Let A = { f ∈ L : F( f ) = a}. Since a ∈ L , we have A  ∅. Then any f ∈ A is such that f  δ . Since A ⊂ δ , this contradicts the fact thatδ ⊆ δ . b a a b Sufficiency: Follows from Theorem 3.3. In general, for a given L-fts, δ = δ need not imply a = b. However, if a, b ∈ L , a b we have the following: Theorem 3.5 Let (X, F) be an L-fts. Then for a, b ∈ L ,δ = δ if and only if a = b. a b Proof Straightforward. Remark 1 For any L-fts, by Theorem 3.2, the level decompositions corresponding to a ∈ L form a lattice. However, this is not true in the case of level decompo- sitions corresponding to a ∈ L . For example, consider the diamond type lattice L = {0, a, b, 1} given by the following Hasse diagram and X = {x, y}. Fig. 3 The diamond type lattice Let (X, F) be the L-fts determined by the following level decompositions. δ = L ,δ = {0, 1, b,μ , (b, a), (b, 1), (0, b), (b, 0), (0, a)}, 0 b {y} δ = {0, 1, a,μ , (a, b), (1, a), (a, 0), (a, 1), (1, b)},δ = {0, 1}, a {x} 1 where (a, b) denotes the function which carries x to a and y to b. Clearly, here {δ : t ∈ L } = {δ ,δ ,δ } is not a lattice. Further, in the light of the following theorem, it 1 a b is worth to note that L = {a, b, 1} is also not a lattice. Theorem 3.6 Let (X, F) be an L-fts. Then L = {δ : a ∈ L } forms a lattice under set inclusion if and only if L is a lattice. Proof Suppose that L is a lattice. Letδ ,δ ∈L . We claim thatδ ∨δ = δ and a b a b ab δ ∧δ = δ , where a  b, a  b denote the join and meet of a and b in L . Clearly, a b ab Fuzzy Inf. Eng. (2011) 3: 225-234 229 a  b ≤ a∧ b ≤ a, b. Then by Theorem 3.3, δ ,δ ⊆ δ . If possible, suppose there a b ab exist δ ∈L such that δ ,δ ⊂ δ ⊂ δ . Then c ∈ L is such that a, b > c > a  b, c a b c ab which is not possible. Consequently,δ ∨δ = δ . a b ab Similarly, it can be shown that δ ∧ δ = δ so that L is a lattice under set a b ab inclusion. Conversely, suppose that L is a lattice under set inclusion. Let a, b ∈ L be such that a b  L . Then clearly, a, b are incomparable and there exist no c ∈ L such that X X c ≤ a, b. Let A = { f ∈ L |F( f ) = a} and B = {g ∈ L |F(g) = b}. Since a, b ∈ L , we have A, B  ∅. Then any f ∈ A and g ∈ B are such that f  δ and g  δ . Since b a A ⊆ δ and B ⊆ δ , this shows that δ and δ are incomparable. Since L is a lattice, a b a b δ ∨ δ = δ for some c ∈ L , c ≤ a, b which is a contradiction. Similar is the case a b c when a  b  L . Note 1 1) It should be noted that if L is a sublattice of L, then a∨b = ab, a∧b = ab so that for δ ,δ ∈L , we have δ ∨δ = δ and δ ∧δ = δ . For the rest a b a b a∧b a b a∨b of the paper, when only the lattice L is under consideration, we use ∨ and ∧ instead of  and  even if L is not a sublattice of L. 2) Even if a and b are incomparable elements of L, δ and δ may be compa- a b rable in L. However, from the proof of the above theorem, it follows that the incomparability of two elements in L is carried over to the corresponding level decompositions in L . Remark 2 Even though L is a lattice, it need not be a sublattice of L. For instance, let X = {x, y, z} and L be the lattice given in the following Hasse diagram. Fig. 4 Hasse diagram of L Let (X, F) be the L-fts determined by the level decompositions δ = L ,δ = 0 a {0, 1,μ ,μ },δ = δ = {0, 1,μ },δ = δ = {0, 1,μ },δ = δ = {0, 1}. Then {x,y} {y} b d {x,y} c e {y} f 1 L = {0, d, e, 1} is a lattice, but not a sublattice of L. In case L is a sublattice of L, it need not be complete. For example, consider L = [0, 1] and X be any singleton. 1 1 Define F : L −→ L by F(0) = 1, F(x) = for all x ∈ (0, ), F(y) = y for all 2 2 230 Ginu Varghese· Sunil C. Mathew (2011) 1 3 3 3 3 3 y ∈ ( , ), F(z) = for all z ∈ ( , ), F(w) = w for all w ∈ ( , 1). Then (X, F)isan 2 5 4 5 4 4 L-fts. Here, it should be noted that L  L.If L is a complete sublattice of L, then we define F to be complete, the justification of which will follow later. Definition 3.4 An L-fts (X, F) is said to be a complete L-fts if L is a complete sub- lattice of L. Theorem 3.7 Let (X, F) be a complete L-fts. Then L = L . Proof Clearly L ⊆L. Now let δ ∈L.If a ∈ L , then δ ∈L . So we may take a a a ∈ L \ L . Then we claim that there exist exactly one F-upper element of a say, d. Firstly, we note that ⇑ a ∅, for if ⇑ a= ∅, then for d ∈ L , with d > a (since 1 ∈ L F F such a d always exists) there exist infinitely many t ∈ L such that a < t < d.Now since L is complete, t ∈ L ,If t > a, then clearly t ∈⇑ a , which is a<t<d a<t<d a<t<d not possible. Consequently, t = a ∈ L which is again a contradiction. Thus a<t<d ⇑ a contains at least one element. If possible, suppose there exist d  d ,∈⇑ a. F 1 2 F Then d ≥ a, and d ∈ L for i = 1, 2. Also d ∧ d ≥ a and a  L . But by i i 1 2 assumption d ∧ d ∈ L and so a < d ∧ d . Since d  d ,∈⇑ a, we must have 1 2 1 2 1 2 F d  d ∧ d ; i = 1, 2. Thus we get a < d ∧ d < d, for i = 1, 2. Since d ∧ d ∈ L , i 1 2 1 2 i 1 2 this implies that d ⇑ a for i = 1, 2 which is a contradiction. Hence ⇑ a= {d}, for i F F some d > a. Further since L is complete and a  L , we must have δ = δ , so that a d δ ∈L . Hence the name ‘complete’ is suitable because even if L ⊂ L, we get L = L. That is, the lattice L is completely determined by a proper sublattice of L. Remark 3 The converse of the above theorem is not true. Consider the lattice L and the set X as given in Remark 2. Let F be the L-fts determined by δ = L ,δ = 0 a {0, 1,μ },δ = {0, 1,μ ,μ },δ = δ = {0, 1,μ ,δ = δ = δ = {0, 1}. Then {y} b {x} {x,y} f c {x} d e 1 L = {0, a, b, f, 1} and L = L . But (X, F) is not a complete L-fts. However the following theorem provides a necessary condition for the equality ofL andL . Theorem 3.8 Let (X, F) be an L-fts such thatL = L . Then⇑ a is a singleton for all a ∈ L. Proof Suppose that L = L . Then for every a ∈ L,δ = δ , for some d ∈ L . a d Since d ∈ L and δ = δ , we must have d ∈⇑ a. If possible, let t  d,∈⇑ a then δ a d F F t and δ are incomparable which is a contradiction since δ = δ ⊃ δ . Hence ⇑ a is a d d a t F singleton. Converse of the above theorem is not always true. For instance, consider L = 1 3 [0, 1] ∪ 0 , where < 0 < and 0 is incomparable to all t ∈ L such that t ∈ 4 4 1 3 1 ( , ) and X be any singleton. Define F : L −→ L by F(0) = 1, F(x) = ; x ∈ 4 4 5 1 1 1 1 (0, ], F(y) = 0 ; y ∈ ( , ]∪{0 }, F(z) = z; z ∈ ( , 1]. Then (X, F)isan L-fts such 5 5 4 4 that ⇑ a is a singleton for all a in L andL  L. Corollary 1 Let (X, F) be a complete L-fts. Then for every a ∈ L,⇑ a is a singleton. F Fuzzy Inf. Eng. (2011) 3: 225-234 231 Since the converse of Theorem 3.8 is not true, it follows by Theorem 3.7 that the converse of the above corollary is also not true. Theorem 3.9 Let (X, F) be an L-fts such that for every a ∈ L, ⇑ a is a singleton. ThenL = L if and only if ⇑a has no element incomparable to the element in⇑ a. Proof If possible, let ⇑a has an element b incomparable to the unique element say, d in ⇑ a. Clearly, then d  a and b  d < a.Now in L(= L ), we have δ ∨δ = δ ⊃ δ . But δ ∨δ ⊆ δ , since a < b, d, which is a contradiction. b d bd a b d a Conversely, let a ∈ L \ L and ⇑ a= {d}. Then d > a. Take t > a,∈ L .If t < d, then a < t < d, which is not possible and if t is incomparable to d then t, d ∈⇑ a which is again not possible. Hence t ≥ d and soδ = δ . a d Corollary 2 Let (X, F) be an L-fts such that L is a chain. Then L = L if and only if for every a ∈ L, ⇑ a is a singleton. Even thoughL is a lattice, there are situations in which it is not a proper sublattice ofL. The following theorem accounts for such a situation. Theorem 3.10 Let (X, F) be an L- f ts such that ⇑ a ∅ for all a ∈ L and L  L . ThenL is not a sublattice of L. Proof If possible, let L be a proper sublattice of L. Then by Theorem 3.9, there exist an a ∈ L such that either (i) ⇑ a is not a singleton or (ii) ⇑ a has at least one element incomparable to the element in ⇑ a. In Case (i) we have|⇑ a|≥ 2 so that F F a  L . Then there exist d  d ∈⇑ a such that δ ,δ ∈L . But in L d  d < a 1 2 F d d 1 2 1 2 so that in L, by Theorem 3.1 δ ⊃ δ and in L,δ ∨δ = δ . Since L is a d d a d d d d 1 2 1 2 1 2 lattice and δ ,δ ⊂ δ ,wehave δ ∨δ ⊆ δ ⊂ δ , which is a contradiction. In d d a d d a d d 1 2 1 2 1 2 Case (ii) there is a b ∈⇑a, which is incomparable to the unique element d, different from a in ⇑ a. Since b ⇑ a there exist infinitely many t’s in L such that a < t < b. F F Clearly, all these t’s are incomparable to d, Otherwise b will become comparable to d. For any such t, d  t < a. Also δ = δ ∨δ . Now in L, δ ∨δ ⊆ δ ⊂ δ ,by dt d t d t a dt Theorem 3.1 which is a contradiction. Corollary 3 Let (X, F) be an L-fts such that L is a sublattice of L and ⇑ a ∅ for all a ∈ L. Then L = L . Corollary 4 Let (X, F) be an L-fts such that L is a proper sublattice of L. Then ⇑ a= ∅ for some a ∈ L. 4. Structural Properties of the LatticeL We have already seen that L is always a complete lattice. However, this is not true in the case of L . For instance, consider the lattice L = [0, 1] and X be any single- 1 1 ton. Define F : L −→ L by F(0) = 1, F(x) = x for all x ∈ [ , 1], F(y) = for 2 n 1 1 all x ∈ ( , ], n ≥ 2. Then (X, F)isan L-fts. Here L is a lattice which is not n+ 1 n complete. 232 Ginu Varghese· Sunil C. Mathew (2011) Theorem 4.1 Let (X, F) be an L-fts. ThenL is complete if and only if L is complete. Proof Suppose L is complete. Let G be any arbitrary subset of L and A = {a ∈ L : δ ∈G}. We claim that G = δ , where the meet is taken over all a ∈ A and a ∧a G = δ , where the join is taken over all a ∈ A. Since t ≤ a, for all a ∈ A, ∨a t∈A by Theorem 3.3, we have for every a ∈ A, δ ⊆ δ , where the meet is taken over all a ∧t t ∈ A. Consequently,G⊆ δ . Now to prove that δ is the smallest element in L ∧t ∧t containing G. If possible, suppose there exist δ ∈L such thatG⊂ δ ⊂ δ . Then b b ∧t by Theorem 3.4 and Theorem 3.5, we have a ≥ b for all a ∈ A. Then t ≥ b. Also, t∈A δ ⊂ δ =⇒ t ≤ b. Thus t = b. Hence by Theorem 3.5, δ = δ . In a similar b ∧ b ∧t t∈A t∈A way, we get G = δ , where the join is taken over all a ∈ A. Since L is complete, ∨a G, G∈L . HenceL is complete. Conversely, suppose that L is complete. Let H be an arbitrary subset of L . Con- sider H = {δ : a ∈ H}. By assumption H has a join and a meet in L . Also H = δ , where the meet is taken over all a ∈ H and H = δ , where the join ∧a ∨a is taken over all a ∈ H. Hence a and a belong to L . a∈H a∈H Theorem 4.2 Let (X, F) be an L-fts. Then L has atoms if and only if L has dual atoms. Proof Let δ be an atom of L . Then δ ⊂ δ =⇒ δ = δ , for all δ ∈L . Hence a b a b 1 b by Theorem 3.4 and Theorem 3.5, we get a < b and b = 1. Thus a is a dual atom of L . Conversely, suppose thatL has no atom. Then for anyδ ∈L , there existδ  δ a b 1 such that δ ⊂ δ . Hence by Theorem 3.4 and Theorem 3.5, we have for any a ∈ L , b a there exist b  1 such that b > a, i.e. L has no dual atom. In a similar way, we can prove the following. Theorem 4.3 Let (X, F) be an L-fts. Then L has dual atoms if and only if L has atoms. An atom ofL is of the formδ where a is a dual atom of L and a dual atom ofL is of the form δ where b is an atom of L . Remark 4 In general, L need not be complemented. For example, take L and X as in Remark 2 and F as the L-fuzzy topology determined by δ = L ,δ = 0 a {0, 1,μ ,μ },δ = δ = δ = δ = {0, 1},δ = {0, 1,μ ,μ },δ = {0, 1,μ }. {x} {x,y} b d f 1 c {y} {x,y} e {x,y} HereL is not complemented. The following theorem provides a necessary and sufficient condition for L to be complemented. Theorem 4.4 Let (X, F) be an L-fts. Then L is complemented if and only if L is complemented. Proof L is complemented ⇐⇒ for all δ ∈L , there existδ ∈L , such that δ ∨δ = δ andδ ∧δ = δ . a b a b 0 a b 1 Fuzzy Inf. Eng. (2011) 3: 225-234 233 ⇐⇒ for all δ ∈L , there existδ ∈L , such thatδ = δ andδ = δ . a b a∧b 0 a∨b 1 ⇐⇒ for all a ∈ L , there exist b ∈ L , such that a∧ b = 0 and a∨ b = 1 by Theorem 3.5. Remark 5 L need not always be modular. For example, with respect to L and X given in Remark 2, consider the L-fts determined by δ = L ,δ = {0, 1,μ },δ = 0 a {z} b δ = {0, 1,μ },δ = {0, 1,μ ,μ } δ = δ = δ = {0, 1}. Here it can be easily f {z,y} c {y} {z,y} e d 1 verified that L is not modular. Theorem 4.5 Let (X, F) be an L-Fts. ThenL is modular if and only if L is modular. Proof L is non modular ⇐⇒ L contains a sublattice isomorphic to N ⇐⇒ there exist two incomparable elements b, c and a d in L such that b < d with b∧ c = d ∧ c and b∨ c = d∨ c. ⇐⇒ there exist two incomparable elements δ ,δ and a δ in L such that δ ⊃ δ b c d b d with δ = δ andδ = δ . b∧c d∧c b∨c d∨c ⇐⇒ there exist two incomparable elements δ ,δ and a δ in L such that δ ⊃ δ b c d b d with δ ∨δ = δ ∨δ andδ ∧δ = δ ∧δ . b c d c b c d c ⇐⇒ L is non modular. Theorem 4.6 Let (X, F) be an L-Fts. Then L is distributive if and only if L is distributive. Proof L has a sublattice isomorphic to M ⇐⇒ L contains three incomparable elements a, b, c such that a∨ b = a∨ c = b∨ c and a∧ b = b∧ c = a∧ c. ⇐⇒ there exist three incomparable elements δ ,δ ,δ ∈L such that a b c δ = δ = δ andδ = δ = δ . a∧b b∧c a∧c a∨b b∨c a∨c ⇐⇒ there exist three incomparable elementsδ ,δ ,δ ∈L such that a b c δ ∨δ = δ ∨δ = δ ∨δ andδ ∧δ = δ ∧δ = δ ∧δ . a b b c a c a b b c a c ⇐⇒ L has a sublattice isomorphic to M . Also from the proof of the above theorem, it follows that L contains a sublattice isomorphic to N if and only if L contains a sublattice isomorphic to N . Now the 5 5 result follows from Theorem 2.1. Remark 6 In the light of the above theorems, one may think that L is isomorphic to L . However, this is not true in general. For example, let X and L be as in Re- mark 2, and F be the L-fts determined by δ = L ,δ = {0, 1,μ ,μ },δ = 0 a {x} {x,y} c {0, 1,μ ,μ },δ = {0, 1,μ } δ = δ = δ = δ = {0, 1}. Here L is not isomor- {x,y} {y} e {x,y} b d f 1 phic to L . However, if there is an order reversing involution in L , then L becomes isomorphic to L . Theorem 4.7 Let (X, F) be an L-fts with an order reversing involution in L . ThenL is isomorphic toL . Proof Define f : L −→ L by f (δ ) = i(a), i denote the order reversing involution in L. Then f (δ ∨ δ ) = f (δ ) = i(a ∧ b) = i(a) ∨ i(b) = f (δ ) ∨ f (δ ). Also a b a∧b a b f (δ ∧δ ) = f (δ ) = i(a∨ b) = f (δ )∧ f (δ ). Clearly, f is one-one and onto. Hence a b a∨b a b the proof. 234 Ginu Varghese· Sunil C. Mathew (2011) 5. Conclusion We have analyzed the lattice structure of the level decompositions corresponding to the range of an L-fuzzy topology on a given set and have obtained characterizations for certain properties of it. Further, the notion of complete L-fuzzy topological space introduced, sheds more light on the structure of this lattice. Thus, the study reveals more about the interplay between fuzzy topology and lattice theory. Acknowledgements The first author places on record her gratitude to the Department of Science and Technology, Government of India, New Delhi, for the support towards this study under the project No: SR/S4/MS: 287/05 and the Centre for Mathematical Sciences, Pala Campus for providing all the facilities. The authors would like to thank the referees for their valuable comments. References 1. Arzu Ari A, Halis Augu ¨ n (2008) Strong compactness and P-closedness in smooth L-Fuzzy topologi- cal spaces. Int. J. Contemp. Math. Sciences 3(5): 199-212 2. Davey B A, Priestley H A (2002) Introduction to lattices and order. Cambridge University Press 3. Gregori V, Vidal A (1998) Gradation of openness and Chang’s fuzzy topologies. Fuzzy Sets and Systems 109: 233-244 4. Ulrich Ho ¨ hle, Alexander S ostak (1995) A general theory of fuzzy topological spaces. Fuzzy Sets and Systems 73: 131-149 5. Liu Y M, Luo M K (1997) Fuzzy topology. World Scientific 6. Ramadan A A (1992) On smooth topological spaces. Fuzzy Sets and Systems 48: 371-375 7. Rodabaugh S E (1991) Point-set lattice-theoretic topology. Fuzzy Sets and Systems 40(2): 297-345 8. Ginu Varghese, Sunil C. Mathew (2010) On the characterizing lattice of an L-fuzzy topological space. Far East Journal of Mathematical Sciences 39(1): 15-27 9. Wuyts P (1984) On the determination of fuzzy topological spaces and fuzzy neighborhood spaces by their level topologies. Fuzzy Sets and Systems 12: 71-85 10. Yue Y L (2007) Lattice valued induced fuzzy topological spaces. Fuzzy Sets and Systems 158: 1461- 11. Zahran A M, Azab Abd-Allah M, Abd El-Nasser G. Abd El-Rahman (2008) On L-fuzzy closure spaces. Fuzzy Math. 16(2): 361-376 12. Zhang J, Shi F G, Zheng C Y (2005) On L-fuzzy topological spaces. Fuzzy Sets and Systems 149: 473-484

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Sep 1, 2011

Keywords: Complete L -fuzzy topology; Lattice; Atom; F -upper set

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