Afr. Mat.DOI 10.1007/s13370-012-0065-y

α∗-Uniformities and their order structure

Surabhi Tiwari

Received: 22 July 2011 / Accepted: 29 February 2012© African Mathematical Union and Springer-Verlag 2012

Abstract The aim of this paper is to introduce α∗-uniformity on a non-empty set X usingα∗-covers of X in L-fuzzy set theory and investigate its lattice structure. Equivalent con-ditions for an α∗-uniformity and basis of an α∗-uniformity, respectively, are obtained. Thetopology generated by α∗-uniformity, called α∗-topology, is also given. Various examples ofα∗-uniformities on X are given and the α∗-topologies induced by them are also considered.

Keywords α∗-Uniformity · Basis · α∗-Topology · Lattice structure

Mathematics Subject Classification 54E15 · 54A40 · 54D80

1 Introduction

Various nearness-like structures on a non-empty set, such as topology, uniformity, proximity,merotopy, contiguity, near sets and generalizations and variations of these concepts have beencreated to handle problems of a “topological” nature (see [7,15–18,24,34]). Generalizationsof proximity, merotopy and contiguity in the L-fuzzy theory can be seen in [12–14,41,42].Uniform spaces are the carriers of notions such as uniform convergence, uniform continuity,precompactness, etc. In the case of metric spaces, these notions were easily defined. How-ever, for general topological spaces, structures such as distance- or size- related conceptscannot be defined unless we have somewhat more structures than topology itself provides.So uniform spaces lie between pseudo-metric spaces and topological spaces, in the sense thata pseudo-metric induces a uniformity and a uniformity induces a topology.

The notion of uniform spaces was introduced by Weil [44] as a generalization of metricspaces. The approach of Weil is called the “entourage” or “surrounding” approach (see [45]).Given uniform spaces X and Y and a set UC(X, Y ) of uniformly continuous functions on

S. Tiwari (B)Department of Mathematics, Motilal Nehru National Institute of Technology,Allahabad, 211004 UP, Indiae-mail: au.surabhi@gmail.com

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X to Y, Naimpally proved that graph topology on UC(X, Y ) contains the uniform topology[27, Sect. 4, p. 279]. In 1940, Tukey [43] introduced another approach to uniformity throughuniform coverings. This approach is particularly favored by Isbell [10]. See also [5], whereGinsburg and Isbell give results concerning a class of uniform spaces defined by the con-dition that every uniformly locally uniform covering is uniform. Weil’s entourage approachand Tukey’s covering approach to defining uniform structures have recently been shown tobe duals of each other [38]. The entourage approach and pseudo-metric approach have beengeneralized to the fuzzy context by Lowen [25], Höhle [8], Hutton [9], Katsaras [11], Liang[21–23], García et al. [3,4], etc. The covering to uniformity in fuzzy theory has been givenby Chandrika and Meenakshi [1,2] using Chang’s definition of fuzzy covers. The coveringapproach to uniformity on L , where L is a frame (a complete lattice satisfying the first infi-nitely distributive law [26]) has also been done (see [39]). In 1988, Smyth [40] suggested thatuniform spaces provide a suitable vehicle for the study of problems in Theoretical ComputerScience. For a comprehensive account with references, we refer to Section 11 (Applicationsto Theoretical Science) in [19].

In the present paper, we characterize uniformity in L-fuzzy theory via α∗-covers of anon-empty set X , where L is a completely distributive complete lattice with order reversinginvolution. The main objective of this paper is to study the lattice structure of a family ofα∗-uniformities on X. In Sect. 2, some basic definitions that are used throughout this paperare collected. In Sect. 3, the notions of α∗-uniformity, separated α∗-uniformity and basisfor α∗-uniformity on X are axiomatized. An equivalent condition for a family of α∗-coversof X to be a basis for some α∗-uniformity on X is obtained. In Sect. 4, α∗-topology isobtained from α∗-uniformity when α is a

∨-prime element of L . It has been shown that a

separated α∗-uniformity on X induces an α-T2 L-fuzzy topology on X. Various examplesof α∗-uniformity on X and the α∗-topology induced by them are given. Indeed, when α isa

∨-prime element of L , α∗-uniformities on X lie between L-fuzzy pseudo-quasi metrices

(p.q. metrices) and α∗-topologies. For α = 1, it is shown that an α∗-topology induced by anα∗-uniformity is different from that of [2]. The lattice structure of a family of α∗-uniformitieshas been discussed; the supremum and infimum of a family of α∗-uniformities have beenprovided precisely.

2 Preliminaries

Let X be a non-empty ordinary set and L be a completely distributive complete lattice withorder reversing involution (′ : L → L), largest element 1 and smallest element 0. Denote byℵ0 the first infinite cardinal number. For the definitions of an L-fuzzy point, L-fuzzy interioroperator and L-fuzzy mappings, we refer to [26]. For any t ∈ L , the mapping which sendseach x ∈ X to t is denoted by t. That is, 0 is the mapping which sends each x ∈ X to 0 ∈ L ,

and 1 is the mapping which sends each x ∈ X to 1 ∈ L . For α ∈ L , A ⊂ L X is called anα∗-cover of X, if and only if, for all x ∈ X , there exists f ∈ A such that α ≤ f (x). For A, Bsubsets of L X , we say A ∧ B = { f ∧ g : f ∈ A, g ∈ B};A refines B(A ≺ B), if and onlyif, for all f ∈ A there exists g ∈ B such that f ≤ g. A mapping ρ : L X × L X → [0,∞]is an L-fuzzy psuedo-quasi (p.q.) metric on X, if and only if, it satisfies the followingconditions:

(1) f �= 0 ⇒ ρ(0, f ) = ∞,

ρ( f, 0) = ρ( f, f ) = 0;(2) ρ( f, g) ≤ ρ( f, h)+ ρ(h, g);

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(3) (i) f ≤ g ⇒ ρ( f, h) ≥ ρ(g, h);(ii) ρ( f,

∨i∈ I

gi ) =∨i∈ I

ρ( f, gi ), where I is an arbitrary index set;(4) if ρ( fi , g) < r ⇒ g ≤ h, for all g ∈ L X and for all i ∈ I, then the following holds

for every k ∈ L X : ρ(∨

i∈ Ifi , k) < r ⇒ k ≤ h.

Let ρ be an L-fuzzy p.q. metric on X. Define mapping Nr : L X → L X , for all r ∈ R (the setof real numbers), r > 0 as follows: for every f ∈ L X , Nr ( f ) = ∨{g ∈ L X : ρ( f, g) < r}.Call Dρ∗ = {Nr : r > 0} the associated neighborhood mappings of ρ. Then ρ is calledan L-fuzzy pseudo-metric (p. metric), if and only if, for every Nr ∈ Dρ∗ , Nr = N�

r , whereN�

r =∧{g ∈ L X : Nr (g ′) ≤ f ′}. An L-filter on X is a non-empty subset F of L X

satisfying: 0 �∈ F ; if f ∈ F and f ≤ g, then g ∈ F ; if f ∈ F and g ∈ F , then f ∧ g ∈ F(see [26]).

A non-zero element α ∈ L is called∨

-prime (∨-prime) if, for (finite) M ⊂ L , α ≤∨ Mimplies there exists m ∈ M such that α ≤ m. Note that each singleton in (P(X),⊂) is∨

-prime and each atom of a lattice is∨

-prime.

3 α∗-Uniformities

Throughout this paper, we take X to be a non-empty ordinary set and L a completely dis-tributive complete lattice with order reversing involution (′ : L → L), largest element 1 andsmallest element 0; also I denotes an arbitrary index set and α is a non-zero element of L .

Definition 3.1 Define a relation ≤α on L X as follows:f ≤α g, if and only if, for every x ∈ X , if xα ∈ f , then xα ∈ g.

We obtain:

(i) If f ≤ g, then f ≤α g.(ii) The relation ≤α is reflexive, transitive but not antisymmetric. For example, let L =

X = I ≡ [0, 1], A ≡ [0, 12 ]. Define f, g ∈ I I as follows:

for x ∈ X,

f (x) ={

12 , if x ∈ A,

112 , otherwise,

and

g(x) ={

12 , if x ∈ A,

215 , otherwise.

Let α = 13 . Then f ≤α g and g ≤α f , but f �= g.

(iii) If fi ≤α gi , for every i ∈ I, then∧

i∈ Ifi ≤α

∧i ∈ I

gi . Also if α is∨

-prime, then∨i ∈ I

fi ≤α

∨i ∈ I

gi . Note that if I is finite and α is∨-prime, then∨i ∈ I fi ≤α ∨i ∈ I gi .

Definition 3.2 Let A, B be subsets of L X . We say that A α∗-refines B, denoted by A ≺α B,if and only if, for each f ∈ A, there exists g ∈ B such that f ≤α g.

The following observations are obvious:

(i) If A ≺ B, then A ≺α B.

(ii) The relation≺α is reflexive, transitive but not antisymmetric. For instance, A∧A ≺α

A ≺α A ∧A but A �= A ∧A, in general, if |A| ≥ 2.(iii) If C is an α∗-cover of X and C ≺α E , then E is also an α∗-cover of X .

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Definition 3.3 If C is an α∗-cover of X , we definest (xα, C) =∨{ f ∈ C : xα ∈ f },st ( f, C) = f ∨ (

∨{st (xα, C) : xα ∈ f }),C∗ = {st (xα, C) : x ∈ X}, andSt (C) = {st ( f, C) : f ∈ C}.Clearly, for any α∗-covers C and E of X, C ≺α C∗; C ≺α St (C); if C ≺α E , then C∗ ≺α

E∗, St (C) ≺α St (C) and (C ∧ E)∗ ≺α C∗ ∧ E∗. Also, st ( f, C ∧ E) = st ( f, C)∧ st ( f, E), forany f ∈ L X .

Definition 3.4 An α∗-uniformity (or an α∗-uniform structure) on X is a non-empty collec-tion U of α∗-covers of X satisfying the following conditions:

(FU1) if C ∈ U and E ∈ U , then C ∧ E ∈ U;(FU2) if C ∈ U and C ≺α E , then E ∈ U;(FU3) for every C ∈ U , there exists E ∈ U such that E∗ ≺α C.

We call (X, U) an α∗-uniform space. Further, an α∗-uniformity U on X is called separated,if and only if, it satisfies the following additional condition:(FU4) for every x, y ∈ X , there is an α∗-covering C ∈ U such that every element of C doesnot simultaneously contain xα and yα.

We call (X, U) a separated α∗-uniform space. If α is a∨

-prime element of L , then E∗ in(FU3) can be interchanged with St(E).

Definition 3.5 A family B of α∗-covers of X is a basis for an α∗-uniformity U on X, if andonly if, B ⊂ U and each C ∈ U is α∗-refined by some α∗-cover E ∈ B.

Let B be a family of α∗-covers of X satisfying (FU1) and (FU3). Then U = {C : C1 ≺α

C, for some C1 ∈ B} is an α∗-uniformity on X .

Definition 3.6 Define a relation ≺∗α as follows:C ≺∗α E, if and only if, C∗ ≺α E , for any α∗-covers C and E .

Proposition 3.1 Let B be a family of α∗-covers of X. Then B is basis for some α∗-uniformityon X, if and only if, the following condition is satisfied:

(FB) if C1 ∈ B and C2 ∈ B, then there exists C3 ∈ B such that C3 ≺∗α (C1 ∧ C2).

Proof Let B be basis for some α∗-uniformity U on X . The proof follows by noting that forα∗-covers C1, C2, C3 and C4, if C1 ≺α C2 and C3 ≺α C4, then (C1 ∧ C3) ≺α (C2 ∧ C4) andC∗1 ≺α C∗2 ; also (C3 ∧ C4)

∗ ≺α (C∗3 ∧ C∗4 ). Conversely, if B satisfies (FB), then it is a basis forthe α∗-uniformity U = {C : C1 ≺α C, for some C1 ∈ B} on X .

Lemma 3.1 Let B be a basis for some α∗-uniformity U on X. Then U = {C : C1 ≺α

C, for some C1 ∈ B} = {C : C1 ≺∗α C, for some C1 ∈ B}.Proof It is straightforward.

Lemma 3.2 A family U of α∗-covers of X is an α∗-uniformity on X, if and only if, (U, ≺∗α)

is a filter on X.

Proof It follows from Lemma 3.1.

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Theorem 3.1 Every separated α∗-uniform structure U on X induces a separated uniformstructure U∗α(U) on X, where, U∗α(U) = {U∗α(A) : A ∈ U} and U∗α(A) = {{x ∈ X : f (x) ≥α} : f ∈ A}. Conversely, given a separated uniform space (X, μ), there is a separatedα∗-uniform space (X, ω∗α(μ)).

Proof Let U be an α∗-uniformity on X . Then U∗α(A) is a cover of X , for every A ∈ U . Toshow that U∗α(U) is an uniformity on X , it is sufficient to show that (U∗α(U),≺∗) is a filter onP(X). Firstly, ∅ /∈ U∗α(U), since ∅ /∈ U . Next, let U∗α(A), U∗α(B) ∈ U∗α(U), where A, B ∈ U .

Now, U∗α(A)∧ U∗α(B) = U∗α(A∧ B). Since A∧ B ∈ U , therefore U∗α(A)∧ U∗α(B) ∈ U∗α(U).

Finally, let U∗α(A) ≺∗ E and U∗α(A) ∈ U∗α(U). That is, (U∗α(A))∗ ≺ E . Evidently, E is acover of X . Let E ∈ E . Define fE : X → L as follows:for x ∈ X ,

fE (x) ={

α, if x ∈ E,

0, otherwise.

(Notice that the map fE is not uniquely defined.) Then E = {{x ∈ X : fE (x) ≥ α} : fE ∈ B},where B = { fE : E ∈ E}. Since U∗α(A) ≺ E , therefore for all C ∈ U∗α(A), there exists E ∈ Esuch that C ⊂ E . Thus, if f ∈ A and α ≤ f (x), then α ≤ fE (x), for some fE ∈ B andfor all x ∈ X. Consequently, for all x ∈ X , if xα ∈ f , then xα ∈ fE . Therefore, for allf ∈ A, there exists fE ∈ B such that f ≤α fE . Thus, A ≺α B. Since A ∈ U , thereforeB ∈ U , which gives E = U∗α(B) ∈ U∗α(U). Further, if U is a separated α∗-uniformity on Xand x, y ∈ X , then there exists U∗α(A) ∈ U∗α(U) such that each element of U∗α(A) does notsimultaneously contain x and y.

Conversely, let (X, μ) be a uniform space. Define

ω∗α(μ) = {ω∗α(A) : A ∈ μ},ω∗α(A) = { f A : A ∈ A},

where, for xi ∈ X (i ∈ I),

f A(xi ) ={

βi ≥ α, if xi ∈ A,

γi � α, otherwise,

for βi , γi ∈ L . Since A is a cover of X , therefore ω∗α(A) is an α∗-cover of X , for allA ∈ μ. Then by Lemma 3.2, it is sufficient to show that (ω∗α(U),≺∗α) is a filter on X. Firstly,∅ /∈ ω∗α(μ). Next, let ω∗α(A) ∈ ω∗α(μ) and ω∗α(B) ∈ ω∗α(μ). Then ω∗α(A) ∧ ω∗α(B) ={ f A ∧ fB : f A ∈ ω∗α(A) and fB ∈ ω∗α(B)} = { f A

⋂B : f A

⋂B ∈ ω∗α(A ∧ B)}. Since

A ∧ B ∈ μ, therefore ω∗α(A ∧ B) ∈ ω∗α(μ).

Finally, let ω∗α(A) ∈ ω∗α(μ) and ω∗α(A) ≺∗α B. Then ω∗α(A) ≺α B. Thus, B is an α∗-coverof X . Let C = {{x ∈ X : g(x) ≥ α} : g ∈ B}. Since ω∗α(A) ≺α B, therefore A ≺ C. Forg ∈ B, denote Cg = {x ∈ X : g(x) ≥ α}. Define fCg = g, for all g ∈ B. Thus, B = ω∗α(C).

Since C ∈ μ, we get B ∈ ω∗α(μ). Further, if μ is separated, then ω∗α(μ) is separated.

4 α∗-Uniform topologies and order structure of α∗-uniformities

In this section, we obtain the L-fuzzy topology induced by an α∗-uniformity on X wheneither α is a

∨-prime element of L or α = 1. When α = 1, it is shown that the L-fuzzy

topology given by an α∗-uniformity on X in our sense is different from that given in [2].

Definition 4.1 Let α be a∨

-prime element of L and U an α∗-uniformity on X . Then f ∈ L X

is a neighborhood of xα ∈ f, if and only if, for some α∗-covering C ∈ U, st (xα, C) ≤α f.

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Theorem 4.1 Let α be a∨

-prime element of L and U an α∗-uniformity on X. Then τ ={ f ∈ L X : f is neighborhood of all xα ∈ f } is an L-fuzzy topology on X.

Proof Clearly, 0 ∈ τ and 1 ∈ τ . If xα /∈ f ∧ g for every x ∈ X , then f ∧ g ∈ τ , vacuously.If xα ∈ f ∧ g for some x ∈ X , then xα ∈ f and xα ∈ g. Therefore, there exist C ∈ U andE ∈ U such that st (xα, C) ≤α f and st (xα, E) ≤α g. Thus, st (xα, C)∧ st (xα, E) ≤α f ∧ g.Consequently, st (xα, C ∧ E) ≤ f ∧ g. Since C ∧ E ∈ U , therefore f ∧ g ∈ τ.

Finally, let { fi ∈ τ : i ∈ I} and xα ∈∨i∈ I

fi . Then there exists i1 ∈ I such that xα ∈ fi1 .

Since fi1 ∈ τ , therefore there exists C ∈ U such that st (xα, C) ≤ fi1 ≤∨

i∈ Ifi . Thus,∨

i∈ Ifi ∈ τ .

Definition 4.2 The above L-fuzzy topology induced by an α∗-uniformity is called anα∗- uniform topology on X , or α∗-topology on X simply. Further, an L-fuzzy topologi-cal space is called α-T2, if and only if, for all x, y ∈ X there exists a neighborhood f of xα

and g of yα such that f ∧ g = 0.

Observe that, if U is a separated α∗-uniformity, then τ is an α-T2 L-fuzzy topologicalspace.

Proposition 4.1 Let α be a∨

-prime element of L. If τ is an α∗-topology induced by anα∗-uniformity U on X, then τ ∗α (τ ) = {{x ∈ X : f (x) ≥ α} : f ∈ τ } is the topology inducedby the uniformity U∗α(U) on X. Conversely, if τ is a induced by an uniformity U on X, thenω∗α(τ ) = {χA : A ∈ τ } is the α∗-topology induced by the α∗-uniformity ω∗α(U) on X.

Proof It follows by noting that for any α∗-cover C of X st (x, U∗α(C)) = {y ∈ X :st (xα, C)(y) ≥ α} and for any cover C of X , st (xα, ω∗α(C)) ≤α fst (x, C) where the mapfst (x, C) is defined in the similar manner as the map f A in the proof of Theorem 3.1.

Proposition 4.2 Let α = 1. Define i : L X → L X such that i( f ) =∨{x1 ∈ L X : there existsC ∈ U satisfying st (x1, C) ≤1 f }, for every f ∈ L X . Then i induces an L-fuzzy topologyon X.

Proof It follows by noting that i(1) = 1; i( f ) ≤ f and i( f ∧ g) = i( f ) ∧ i(g). ��Further, note that for any constant function t ∈ L X such that t �= 1, i(t) = 0 in our sense

but i(t) = t according to [2]. Therefore, our L-fuzzy topology is different from that of [2].

Definition 4.3 Let UX and UY be α∗-uniformities on X and Y , respectively. An L-fuzzymapping T→ : (X, UX ) → (Y, UY ) is called α∗-continuous if, T←( f ) is neighborhood ofT←(xα) ∈ L X , whenever f is neighborhood of xα ∈ LY . Further, T→ is called α∗-uniformlycontinuous if, T←(C) ∈ UX , for every C ∈ UY .

Proposition 4.3 Every α∗-uniformly continuous one-one map T→ : (X, UX )→ (Y, UY ) isα∗-continuous.

Proof Let g be a neighborhood of T→(xα) = [ T (x)]α ∈ LY . Then there exists C ∈ UY

such that st (T→(xα), C) ≤ g. As T→ is injective, T←(st (T→(xα), C)) = st (xα, T←(C)),and therefore st (xα, T←(C)) ≤ T←(g). Since T is α∗-uniformly continuous and C ∈ UY ,therefore T←(C) ∈ UX . Thus, T←(g) is a neighborhood of xα ∈ L X . Hence, T→ isα∗-continuous.

Definition 4.4 Let S be a family of α∗-coverings of X. Then S is called a subbasis for someα∗-uniformity U on X if {∧ C : C ⊆ S and |C| < ℵ0} is a basis for U, where

∧ C = {∧ Ei :Ei ∈ C} and

∧ Ei =∧f j∈Ei

f j .

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Definition 4.5 A family N of α∗-coverings of X is called a normal family, if for all C ∈ N ,

there exists E ∈ N such that E∗ ≺α C.

Proposition 4.4 Every normal family of α∗-coverings is a subbasis for some α∗-uniformityon X.

Proof Let N be a normal family. Then to show that N is a subbasis for some α∗-uniformityon X, it is sufficient to show that U = {C : Cis anα∗-cover ofX and

∧ E ≺α C, where E ⊆ Nand |E| < ℵ0} is an α∗-uniformity on X . We will only show (FU3). Let C ∈ U . Then there isE ⊆ N with |E| < ℵ0 and

∧ E ≺α C. Since N is normal, therefore for each α∗-cover Ei ∈ E ,there exists Ai ∈ N satisfying Ai ≺∗α Ei . Consequently,

∧ Ai ≺∗α∧ Ei ≺α C, which gives

the desired result.

Theorem 4.2 Let (T→i : X → (Yi , UYi ))i∈I (where I is an arbitrary index set) be a familyof L-fuzzy mappings. Then there is a coarest α∗-uniformity UX on X, including all inverseimages of α∗-coverings in UYi , for all i ∈ I, under these L-fuzzy maps.

Proof Let BX = {T←i (Ci ) : Ci ∈ UYi , i ∈ I}. Then T←i is an α∗-uniformly continuous map,for all i ∈ I. Suppose that E∗ ≺α C, for some α∗-covers E and C of Yi , i ∈ I. Then for i ∈ I,

st (xα, T←i (E)) =∨{

T←i ( f ) ∈ T←i (E) : xα ∈ T←i ( f )}

≤α

∨ {T←i ( f ) : f ∈ Eandyα ∈ f

}

= T←i ({ f ∈ E : yα ∈ f })= T←i (st (yα, E)).

Hence (T←i (E))∗ ≺α T←i (E∗) ≺α T←i (C). As a result, {T←i (C) : C ∈ UYi , i ∈ I} is a normalfamily and therefore a subbasis for an α∗-uniformity UX , which is the coarest possible. ��

We now present some examples of α∗-uniformities to guarentee the existence of suchstructures.

Example (1) The family D = {C ⊂ L X : C is an α∗-cover of X} is a separated α∗-unifor-mity called the uniformly discrete α∗-uniformity on X . It follows by noting that for theα∗-cover E = {xα : x ∈ X} of X , E = E∗ and E ≺∗α C, for every α∗-cover C of X .Further, if α is a

∨-prime element of L , then the α∗-uniform topology induced by D is

the discrete L-fuzzy topology on X.

(2) The family I = {C ⊂ L X : {1} ≺α C} is an α∗-uniformity on X , called the indiscreteα∗-uniformity on X. It follows by noting that st (xα, {1}) = 1, for all x ∈ X and, conse-quently, {1} is the base for the α∗-uniformity I. Further, if α is a

∨-prime element of L ,

then the α∗-uniform topology induced by I, called the indiscrete α∗-uniform topologyon X , is τ(I) = {1} ∪ { f ∈ L X : xα /∈ f, for all x ∈ X}.

(3) Let L be a boolean algebra, α a ∨-prime element of L and F an L-filter on X. Then theL-filter α∗-uniformity UF on X has all α∗-covers { f } ∨ {xα : xα ∈ f c} (where f ∈ Fand f c is the complement of f in L X , that is, f ∧ f c = 0 and f ∨ f c = 1 (see [26])as a basis.

(4) Let α be a∨

-prime element of L . Then the family B = {{Nr (xα) : x ∈ X} : r > 0} isa base for an L-fuzzy p.q. metric α∗-uniformity Uρ on X , where the α∗-topology τρ onX induced by L-fuzzy p.q. metric is defined as follows: f ∈ τρ, if and only if, for everyxα ∈ f , there exists r > 0 such that Nr (xα) ≤ f. Thus, when α is a

∨-prime element

of L , an α∗-uniformity lies between an L-fuzzy p.q. metric and an α∗-topology on X .

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(5) Let (X, i) be an L-fuzzy topological space generated by an L-fuzzy interior operatori on X. Then U = {C ⊂ L X : C is an α∗-cover and

∨f ∈C i( f ) ≥ xα , for every

x ∈ X} is an α∗-uniformity on X. Further, if α is a∨

-prime element of L , thenU = {C ⊂ L X :∨ f ∈C i( f ) ≥ xα , for every x ∈ X} is an α∗-uniformity on X.

Having obtained an adequate number of α∗-uniformities, it is now relevant to introduceordering between α∗-uniformities on X.

Definition 4.6 Let U1 and U 2 be two α∗-uniformities on X. We say that U1 is coarser thanU 2, and write U1 < U 2, if and only if, U1 ⊂ U 2.

Theorem 4.3 The family of all α∗-uniformities on X forms a complete lattice with respectto the order < . The zero of this lattice (i.e. the least element) is the indiscrete α∗ -uniformityI while the unit (i.e. the largest element) is the uniformly discrete α∗-uniformity D on X.

Further, if α is∨

-prime element of L, then τ(∨

i∈ IU i ) =∨

i∈ Iτ(U i ).

Proof The uniformly discrete α∗-uniformity D on X and the indiscrete α∗-uniformity I onX are the unit and the zero of the poset {Ui : Ui is an α∗-uniformity on X, i ∈ I}, respectively.Suppose that {Ui : i ∈ I} is an arbitrary family of α∗-uniformities on X. Then

⋂{Ui : i ∈ I}is the greatest lower bound of this family. Let B consist of all finite meets of elements of⋃{Ui : i ∈ I} and C ∈ B. Without loss of generality, we can assume that C = (Ci1 ∧ Ci2)

such that Ci1 ∈ Ui1 and Ci2 ∈ Ui2 , i1, i2 ∈ I. Therefore, there exists Ei1 ∈ Ui1 and Ei2 ∈ Ui2

such that E∗i1≺α Ci1 and E∗i2

≺α Ci2 . Consequently, (E∗i1∧ E∗i2

) ≺α (Ci1∧ Ci2) = C. Hence,(Ei1 ∧ Ei2)

∗ ≺α C. Clearly, (Ei1 ∧ Ei2) ∈ B is the finite meet of elements of⋃

i∈ IUi . Thus, B

is a basis for an α∗-uniformity on X. Let U be the α∗-uniformity generated by the basis B.

Then, by construction, U is an upper bound of the family {Ui : i ∈ I}. But any upper bound ofthis family would have to contain all finite meets of elements of

⋃i∈ I

U i . So, U =∨i∈ I

U i .

Concluding remark

The present theory is a unified study of the classical theory and the generalized L-fuzzytheory. In this paper, we aim to study the order structure of the family of the new structure(α∗-uniformity) defined. It is known that an L-topological space can be induced by a Huttonuniformity if and only if, it is completely regular [26]. So, our next project is to study theseparation axiom (i.e., complete regularity for an α∗-uniform space) required to induce anL-topology from an α∗-uniformity on X .

The importance of nearness-like structures, and therefore uniformity, on a perceptual basiswas pointed out informally in 2002 [29] and formally in [31,36], motivated by image analysisand inspired by a study of the perception of the nearness of familiar physical objects (seee.g., [31,32,35,36]). More precisely, near sets grew out generalization of the approach to theclassification of objects proposed by Pawlak during the early 1980s (see [28,30]). In addition,near sets cater to the needs of science and engineering, where the focus in on the discovery ofaffinities between observed, sample specimens found either in the laboratory (e.g., featuresof sets of microscope images) or in field work (e.g., physical features of fossils found inBaltic amber or physical features of members of animal populations). For more applicationsof the tools from nearness-like structures in image analysis, we refer to [6,20,37], and infeature selection, we refer to [33]. A natural consequence of the near set approach is therealization that affinities between specimens are discovered by comparing the description ofpairs of objects in perceptual granules such as images. In other words, the focus in near sets is

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α∗-Uniformities and their order structure

on discovering relations between two or more samples relative to the degree of closeness oftheir specific descriptions. Since an approach nearness space measures the degree of nearnessbetween a collection of sets, therefore it may serve as a better tool for such studies in thefield of computer science, engineering and science.

Acknowledgments The author is grateful to the Referees for their valuable suggestions towards the improve-ment of this paper. The author also extends her profound thanks to Prof. Mona Khare for her importantsuggestions concerning topics in this paper.

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