On completely homogeneous L-topological spaces

In this paper we investigate completely homogeneous Ltopological spaces. The smallest completely homogeneous Ltopology on a set X containing an Lset f is called the principal completely homogeneous Ltopology generated by f . Here we also study the principal completely homogeneous Ltopological spaces generated by an Lset and characterize completely homogeneous Alexandroff discrete Ltopological spaces. 2010 AMS Classification: 54A40, 03E10


Introduction
In 1965, L. A. Zadeh introduced fuzzy set theory describing fuzziness mathematically for the first time [16].Based on this notion C. L. Chang introduced fuzzy topology and studied its properties [3].Later an extensive study of fuzzy topological space has been carried out by several mathematicians and they developed a theory of fuzzy topological spaces.Larson studied the concept of complete homogeneity in topological spaces and characterized all spaces which are minimum and maximum with respect to a topological property [12].He also determined a characterization of completely homogeneous topological spaces.In [5] T. P. Johnson defined the concept of a completely homogeneous fuzzy topological space in an analogous way and generalized the result of Larson R. E. for fuzzy topological spaces.He characterized all fuzzy topological spaces which are minimum and maximum with respect to a fuzzy topological property.He considered the lattice of completely homogeneous fuzzy topologies also [7].In [10] the authors extended complete homogeneity to L-topological spaces.
In this paper we continue the study of completely homogeneous L-topological spaces.We characterize Alexandroff discrete completely homogeneous L-topological spaces when L is a complete chain.Also we introduce the principal completely homogeneous L-topological spaces generated by an L-set and obtain certain properties when L is an F -lattice.
If we take L = {0, 1}, then it is clear that the lattice L X is isomorphic to the lattice P (X), the power set of X and the L-topologies become crisp.The simplest F -lattice other than L = {0, 1} is L = {0, a, 1} where a is different from 0 and 1.Also we study the completely homogeneous L-topological spaces and characterize principal completely homogeneous L-topological spaces when L = {0, a, 1}.

Preliminaries
Here we include certain definitions and known results needed for the subsequent development.

Definition 2.1 ([13]
).A completely distributive lattice L is called an F -lattice, if L has an order reversing involution : L → L.
Throughout this paper X stands for a nonempty set, L for an F -lattice with the smallest element 0 and largest element 1 and L X for the lattice of L-fuzzy sets or L-sets of X.An L-set with constant membership α ∈ L is denoted by α.The Lsubset x l with x ∈ X and l ∈ L, l = 0 defined by is called an L-point in X with support x and value l.An L-set x l , with x ∈ X and l ∈ L, l = 1 is defined by 13]).Let X and Y be two sets and h : X → Y be a function.

Then for any
For an L-set g in Y , we define h −1 (g)(x) = g(h(x)), for all x ∈ X.

Definition 2.3 ([13]
).Let X be a nonempty set, L an F -lattice, δ ⊂ L X .Then δ is called an L-fuzzy topology or L-topology on X, and (X, δ) is called an L-fuzzy topological space, or L-topological space for short, if δ satisfies the following three conditions: Every element in δ is called an L-open subset in X.
We say h is an L-continuous function from X to Y , if h −1 (f ) ∈ δ, for every f in δ and h is said to be open, if h(f ) ∈ δ , for every f ∈ δ.
A bijection h from (X, δ) onto (Y, δ ) is called an L-homeomorphism if both h and h −1 are L-continuous.So a necessary and sufficient condition for a permutation h of a set X to be an L-homeomorphism of (X, δ) on to itself is that f ∈ δ if and only if f • h ∈ δ.The set of all L-homeomorphisms of an L-topological space (X, δ) onto itself is a group under composition, which is a subgroup of the group of all permutations on the set X.It is called the group of L-homeomorphisms of (X, δ) and is denoted by LF H(X, δ).
An L-topological space (X, δ) is called homogeneous if for any two points x and y in X, there exists an L-fuzzy homeomorphism h in (X, δ) such that h(x) = y [4].Several authors studied homogeneity in L-topological spaces [4,8,9,10].The order of the LF H(X, δ) depends on the structure of the L-topological space.For example, if the L-topological space is homogeneous, the order of the group of Lhomeomorphisms on (X, δ) is greater than or equal to the cardinality of X.Thus if the number of L-fuzzy homeomorphisms increase, the homogeneity of the L-topological space also increases.The extremity of this happens when the group of Lhomeomorphisms equals S(X), the set of all permutations of the set X. Then we say that (X, δ) is a completely homogeneous L-topological space.Definition 2.5 ([6]).An L-topological space (X, δ) is called a completely homogeneous space if every bijection of X onto itself is an L-homeomorphism.
From the definition of completely homogeneous L-topological space we immediately see that indiscrete space {0, 1}, discrete space 2 X , L-discrete space L X and L-topological space generated by L-points having same membership value are completely homogeneous L-topological spaces.Also we have every completely homogeneous L-topological space is homogeneous [10].

Notations
• If A is a given set, we will use |A| to denote the cardinality of A.

Completely homogeneous L-topological spaces
In this section we study some properties of completely homogeneous L-topological spaces.Let X be a set and f be an L-set of X.We define for c ∈ L \ {1}, the set The relation between completely homogeneous L-topological space and its level topologies is given in the following theorem.Theorem 3.1.Let (X, δ) be an L-topological space which is completely homogeneous, then all the level topologies of δ are completely homogeneous.705 Proof.First we claim that the group of all L-homeomorphisms LF H(X, δ) of an L-topological space (X, δ) is a subgroup of the group of homeomorphisms of the level topologies.We have h ∈ LF H(X, δ) if and only if h is a bijection and both h(f ) and h −1 (f ) are in δ for all f ∈ δ.Let (X, T [c] ) be a level topology of (X, δ) and for some f in δ and So h is a homeomorphism on (X, T [c] ).Hence the group of L-homeomorphisms of an L-topological space is a subgroup of the group of homeomorphisms of the level topologies.Here the group of L-homeomorphisms of (X, δ) is the group of all permutations on X.So the group of homeomorphisms of the level topologies are also the group of all permutations on X. Therefore the level topologies of a completely homogeneous L-topological space (X, δ) are completely homogeneous.
Remark 3.2.The converse of the theorem 3.1 is not true.
Example 3.3.Let X = {a, b, c}, L = {1, .5, 0} with usual order and δ be the L-topology having base where It is easy to verify that LF H(X, δ) = {(a, b, c), (a, c, b), I} where I is the identity permutation on X and the level topologies T [ 1 2 ] and T [0] are the discrete topology on X.Here all the level topologies are completely homogeneous but the L-topology is not completely homogeneous.
Corollary 3.4.Let (X, δ) be an L-topological space and Λ f ⊆ {0, c} for every f ∈ δ\{1}, where c is a nonzero element in L. Then the group L-homeomorphisms of an L-topological space is equal to the group of homeomorphisms of the level topology Proof.We have the group of L-homeomorphisms LF H(X, δ) of an L-topological space (X, δ) is a subgroup of the group of homeomorphisms H(X, T . Since f takes only one non zero value, h(f ) ∈ δ.Similarly we can prove that h −1 (f ) ∈ δ.Thus h is an Lhomeomorphism.So H(X, T [0] ) ⊆ LF H(X, δ).Hence the result holds.
An immediate consequence of the Theorem 3.1 is the following.Remark 3.5.Observe that if (X, δ) is an L-topological space and Λ f ⊆ {0, c} for every f ∈ δ \ {1}, where c is a nonzero element in L, then the L-topological space (X, δ) is completely homogeneous if and only if the level topology T [0] is completely homogeneous.
The next definition appears in [1].Definition 3.6.A nonempty subset R, not containing 0, of L is said to be tirreducible, if no element of R can be written as the finite meet or arbitrary join of members of L \ R. The complement of a t-irreducible set is called a t-complete set.
From the definition, it is clear that a subset A of L is said to be t complete, if 0 ∈ A, A is closed under finite meet and arbitrary join operations.
Let δ be a completely homogeneous L-topology on X.Then clearly {f (x) : x ∈ X, f ∈ δ} is a t-complete subset of L.
Conversely, using t-complete subset of L, we can define more than one completely homogeneous L-topology on X. Theorem 3.7.let A be a t-complete subset of L such that 1 ∈ A. Define δ = {f ∈ L X : f (x) ∈ A for all x ∈ X}.Then δ is a completely homogeneous L-topology on X.
Proof.This can be easily verified.Theorem 3.8.Let A be a t-complete subset of L. Define δ = {l : l ∈ A}.Then δ is a completely homogeneous L-topology on X.
Proof.Proof is trivial.
For any t-complete subset A of L, there exist atleast two completely homogeneous L-topologies on X. Remark 3.9.If L is a finite chain, then we can define another completely homogeneous L-topology on X, for each t-complete subset A of L defined by δ = {f ∈ L X : f (x) ∈ A \ {0}, for all x ∈ X} ∪ {0}.This is not true in the case of general lattice L. The following example illustrates this.Then δ = {f ∈ L X : f (x) ∈ A \ {0} for all x ∈ X} ∪ {0} is not even an Ltopological space, where A is L itself.
Let(X, δ) be an L-topology on X.Then the L-topology δ on X generated by δ ∪ {α : α ∈ L} is called the stratification of δ and (X, δ ) is called the stratification of (X, δ) [13].
Proposition 3.11.Let (X, δ) be an L-topological space.Then δ is a completely homogeneous L-topology on X if and only if the stratification of δ is completely homogeneous L-topology on X.
Proof.Let (X, δ ) be the stratification of the space (X, δ).Then every f ∈ δ is of the form g∨α or g∧β where g ∈ δ and α, β ∈ L. So for any h ∈ S(X), h(f ) = h(g)∨α or h(g) ∧ β.First assume that (X, δ) is a completely homogeneous L-topological space.Then h(g) and h −1 (g) are in δ for all g ∈ δ.So h(f ) ∈ δ .
Similarly, we can prove that h −1 (f ) ∈ δ .Thus δ is a completely homogeneous L-topology on X.Now assume that δ is completely homogeneous L-topology on X.Then clearly δ is completely homogeneous L-topology on X.
In a completely homogeneous topological space, supersets of nonempty open sets are open [14].But this is not true in the case of an L-topology.See the following example.
where f and g are L-sets of X defined by Here δ is a completely homogeneous L-topology.Let h be an L-set defined by Here h is a super set of a 1 4 but h / ∈ δ.
Now we prove that super sets of L-open sets having the same range are open in completely homogeneous L-topological space as we see in the next lemma.Lemma 3.13.Let (X, δ) be a completely homogeneous L-topological space and f ∈ δ such that f = 0. Let g be an L-set such that g ≥ f and Λ g ⊂ Λ f .Then g ∈ δ.
Proof.If g = f , there is nothing to prove.So assume that f < g.Let Y = {x ∈ X : f (x) < g(x)}.Now choose two points y and z such that y ∈ Y and z ∈ X, where f (z) = g(y).We can choose such a point z since f = 0 and range of g is a subset of range of f .Now define f y = f • h y , where h y is a function from X to X such that Then f y ∈ δ and hence f ∨ f y ∈ δ.Observe that This completes the proof.
An L-topological space (X, δ) is said to be Alexandroff discrete L-topological space if ∧A ∈ δ for all A ⊂ δ.
Next we give a characterization for a completely homogeneous Alexandroff discrete L-topological space when L is a complete chain.Theorem 3.14.Let (X, δ) be an Alexandroff discrete L-topological space where L is a complete chain.Then (X, δ) is a completely homogeneous L-topological space on X if and only if Λ X f ⊆ δ for all f ∈ δ.Proof.Let (X, δ) be a completely homogeneous Alexandroff discrete L-topological space where L be a complete chain and f ∈ δ.Define l 1 = ∧ l∈Λ f l and We claim that x li ∈ δ for all l i ∈ Λ f .Here l 1 ∈ Λ δ .Here we consider two cases.Case(1): where h x is a function from X onto itself which maps x to x 0 , x 0 to x and keeping all other elements fixed.Then Case(2): l 1 / ∈ Λ f .In this case we construct an L-set using f which takes the value l 1 .Fix some x 0 in X and define f x = f • h x for all x ∈ X, where h x is a function as defined above.Then Thus we get an L-subset of X which takes the value l 1 .Now proceeding as in the case (1), we can easily prove that x l ∈ δ.So in both cases any f ∈ Λ X f can be expressed as a join of x l .So Λ X f ⊆ δ for all f ∈ δ.Conversely, assume that Λ X f ⊆ δ for all f ∈ δ.Then f • h ∈ δ for all f ∈ δ.Thus δ is a completely homogeneous L-topology on X.So the result holds.
An L-topological space (X, δ) is said to be finite if the underlying set X is finite.With the characterization theorem of completely homogeneous Alexandroff discrete L-topological space, we list finite completely homogeneous L-topological spaces when the membership lattice L = {0, 1  2 , 1} with the usual order.Corollary 3.15.Let X be a finite set and L = {0, 1  2 , 1}.Then the only completely homogeneous L-topologies on X are the following.
Remark 3.16.It is not possible to drop the chain condition on the lattice from the hypothesis of the Theorem 3.14.The following example illustrates this.
Example 3.17.Let X = {a, b} and L be the diamond type lattice.Define where Here δ is a completely homogeneous L-topology on X, but Λ X f1 δ.

Principal completely homogeneous Ltopological space generated by an Lset
In this section, we define principal completely homogeneous L-topological space generated by single L-set and study some of its properties.Also we characterize the principal completely homogeneous L-topological space generated by an L-subset when the membership lattice L = {0, 1  2 , 1} with the usual order.In this section we use some set theoretic results.Let A and B be two subsets of set X and |A| = |B|, it does not necessarily follow that there exists a bijection of X which maps A onto B. In order to exist such a function, we must also know that |X \ A| = |X \ B|.If X is an infinite set, it is possible to choose A and B such that A ∪ B = X, A ∩ B = φ and |A| = |X| = |B| since for any infinite cardinal number α, we have α + α = α [15].Definition 4.1.Let f ∈ L X .Then the smallest completely homogeneous L-topology containing f is called the principal completely homogeneous L-topology generated by f and is denoted by CHLF T (f ).Here the L-set f is called the generator of the principal completely homogeneous L-topology.
A completely homogeneous L-topological space (X, δ) is called principal completely homogeneous L-topological space if δ = CHLF T (f ) for some L-set f ∈ L X .
Observe that the intersection of all completely homogeneous L-topological spaces containing an L-set is completely homogeneous, which asserts the existence of CHLF T (f ) for all f ∈ L X .
An immediate consequence of the Definition 4.1 is the following.2 )| < α}, where ℵ 0 ≤ α ≤ |X| and α is not a limit cardinal.Thus we determined the principal completely homogeneous L-topological space generated by an L-set when L = {0, 1  2 , 1}.We conclude this section by listing all principal completely homogeneous Ltopologies on X when L = {0, 1 2 , 1}.Corollary 4.17.Let X be an infinite set.Then the only principal completely homogeneous L-topologies on X when L = {0, 1  2 , 1} are the following.

Conclusion
We have studied completely homogeneous L-topological spaces and identified some of its properties.We obtained a characterization for Alexandroff discrete completely homogeneous L-topological spaces when L is a complete chain.Also the concept of principal completely homogeneous L-topological spaces is introduced.