# Connecting Fuzzifying Topologies and Generalized Ideals by Means of Fuzzy Preorders

Authors: G. Elsalamony
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

The present paper investigates the relations between fuzzifying topologies and generalized ideals of fuzzy subsets, as well as constructing generalized ideals and fuzzifying topologies by means of fuzzy preorders. Furthermore, a construction of generalized ideals from preideals, and vice versa, is obtained. As a particular consequence of the results in this paper, a construction of  fuzzifying topology  generated by generalized ideals of fuzzy subsets  via a givenI-topology is given. The notion of σ-generalized ideal is introduced and hence every σ-generalized ideal is shown to be a fuzzifying topology induced by some fuzzy preorder.


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## Body

## 1. Introduction

The concepts of preideals and generalized ideals were introduced by Ramadan et al. [1], as a consistent approach to the ideas of fuzzy mathematics. The authors investigated the interrelations between these two concepts. This paper is devoted to finding fuzzifying topologies derived from preideals, and vise versa, by means of fuzzy preorders and residual implications. The contents of this paper are arranged as follows. In Section 2, we recall some basic notions of ideals, preideals, and generalized ideals. The notion of saturation of preideals is introduced, hence, a one-to-one correspondence between the set of saturated preideals and the set of generalized ideals on the same set is obtained. In fact, some studies were done on the correspondence of such two sets, see [2, 3], for example, based on the duality between fuzzy filters and fuzzy ideals. In our paper, we prove such correspondence independent of the duality principle. Also, given X a Boolean lattice, we construct  a generalized ideal, in terms of fuzzy preorders. Section 3 is concerned with the relationship between the fuzzy preorders and each of I-topologies [4], fuzzifying topologies [5], and I-fuzzy topologies [6], respectively. Thus, an interesting relation between a special type of I-topological spaces, called fuzzy neighborhood spaces, introduced by Lowen [7], and fuzzifying topological spaces is established. In Section 4, we introduce the notion of  σ-generalized ideal, therefore, a characterization of this special kind of ideals is shown to be a fuzzifying topology generated by some fuzzy preorder. Also, we study the relations between preideals and generalized topological structures, for example, I-topologies, fuzzifying topologies, and I-fuzzy topologies. Also, I-topologies are constructed  from preideals via given I-topologies.


## 2. Generalized Ideal Structures

We recall some basic definitions. We should letX,I denote a nonempty set and the closed unit interval [0,1], respectively, and we let I0=(0,1],I1=[0,1).We denote the characteristic function of a subsetA⊆X also by A. If μ∈1X, we define μα={x∈X:μ(x)>α}. Let 2X be the collection of all subsets of X. The fuzzy set which assigns to each element in X the value α,0≤α≤1, is also denoted by α. In this paper, the concepts of triangular norm and residual implication are applied.A triangular norm, a t-norm in short, on the unit interval I is a binary operator *:I×I→I which is symmetric, associative, order-preserving on each place and has 1 as the unit element.Afuzzy relation on a nonempty set is a map R:X×X→I. A fuzzy relation R is called(1)
reflexive ifR(x,x)=1 for each x∈X;(2)
*-transitive if R(x,y)*R(y,z)≤R(x,z)forallx,y,z∈X.A reflexive and*-transitive fuzzy relation is called *-fuzzy preorder. If * is assumed to be the  t-norm min=⋀, we will drop the *-.This section presents a review of some fundamental notions of ideals, preideals, and generalized ideals. We refer to Ramadan et al. [1] for details.Definition 2.1.
A nonempty subsetD of 2X is an ideal of subsets of X, simply, an ideal on X, if it satisfies the following conditions:(D1)
X∉D;(D2)
A,B∈D⇒A∪B∈D;(D3)
A∈D,B⊂A⇒B∈D.Definition 2.2.
A nonempty subset𝒟 of IX is a preideal of subsets of X, simply, a preideal on X, if it satisfies the following conditions:(P1)
1∉𝒟;(P2)
μ,ν∈𝒟⇒μ∨ν∈𝒟;(P3)
ν∈𝒟,μ≤ν⇒μ∈𝒟.A preidael𝒟 is called prime if it satisfies the condition;(2.1)∀μ,λ∈IXsuchthatμ⋀ν∈𝒟,thenμ∈𝒟orν∈𝒟.A preideal𝒟 is called a σ-preideal  if the following hold.If{μn:n=1,2,3,…} is a countable subcollection of 𝒟, then supnμn∈𝒟.If𝒟 is a preideal on X, we define thecharacteristic, denoted by c(𝒟), of 𝒟 by(2.2)c(𝒟)=supμ∈𝒟infx∈Xμ(x).If𝒟 is a preideal and c(𝒟)<1, we say that 𝒟 is saturated  if for every (με)∈𝒟I1, infε∈I1(με+ε)∈𝒟, equivalently, 𝒟 is saturated if it satisfies(2.3)∀ε<1,μ∈IX:μ-ε∈𝒟⇒μ∈𝒟.We provide a useful characterization of a saturated preideal𝒟 in the following lemma.Lemma 2.3.
Let𝒟 be a saturated preideal on X, μ∈IX. Then
(2.4)μ∈𝒟⟺μα⋀γ∈𝒟,∀c<γ<αinI1.Proof.
Letμ∈𝒟. Since μα⋀γ≤μ for all γ<α∈I1, then, by (P3), μα⋀γ∈𝒟 for all γ<α∈I1.
Conversely, letμ∈IX such that μα⋀γ∈𝒟 for all c<γ<α in I1. For every α∈(0,1/2) and 0<n(integer)<1/α, we consider the fuzzy subsets μ2α=supn(μnα⋀(nα-α)). Since n is finite, then, by (P2), μ2α∈𝒟. Also, it is easy to see that, for all x∈X, 0≤μ(x)-μ2α(x)≤2α, hence μ+2α≥μ2α+2α≥μ. Consequently, μ=∩0<α<1/2(μ2α+2α)∈𝒟, since 𝒟 is saturated, so we are done.It should be noticed that the required condition of Lemma2.3 holds, trivially, for all μ∈𝒟 and γ≤c, therefore, we provide γ>c, so that the above condition is significant.Also, the characteristic of a preideal𝒟, on a subset A of X may be defined and denoted by cA(𝒟)=supλ∈𝒟infx∈Aλ(x). The following result is useful for our study.Lemma 2.4.
Let𝒟 on X. Then, for every A∈2X,
(2.5)cA(𝒟)=sup{t∈I:A⋀t∈𝒟},
hence, c=cX(𝒟)=sup{t∈I:t∈𝒟}.Proof.
The statement holds whenA=ϕ. Suppose A≠ϕ, and denote c=cA(𝒟) and b=sup{t∈I:A⋀t∈𝒟}. Let t∈I satisfy A⋀t∈𝒟. Then c≥infx∈A(A⋀t)(x)=t, that is, c≥b. Conversely, given λ∈𝒟, let γ=infx∈Aλ(x), hence A⋀γ≤λ. This implies, by (P3), A⋀γ∈𝒟, so b≥γ. Thus, by definition of c, we get b≥c, this shows that the equality holds.Definition 2.5.
A nonzero functiond:2X→I is called ageneralized ideal  of fuzzy subsets of X, simply, a generalized ideal on X, if it satisfies the following conditions:
(G1)d(X)=0;
(G2)d(A)⋀d(B)≤d(A∪B), for each A,B∈2X;
(G3) ifA⊂B, then d(A)≥d(B), for each A,B∈2X;
The conditions (G2) and (G3) are equivalent to:
(G*)d(A)⋀d(B)=d(A∪B), for each A,B∈2X. Moreover, Since d a nonzero function, then by (G3), d(ϕ)>0.A generalized ideald, is called σ-generalized ideal  if
(2.6)d(∪i∈JAi)≥⋀i∈Jd(Ai),foreach(Ai)i∈J⊆IX,
and d is called prime  if(2.7)d(A∩B)≤d(A)∨d(B),foreachA,B∈2X.A generalized ideal on a setX can be derived by fuzzy relations in different ways, as we see in the following.Proposition 2.6.
LetR be a reflexive fuzzy relation on a Boolean lattice X with 0. Define d1,d2:IX→I, for all A⊆X as follows:
(2.8)d1(A)=⋀x,y∈A(1-R(x,1-y));where1-yisthecomplementy,(2.9)d2(A)=⋀x∈A(1-R(x,0)).
Then d1,d2 are generalized ideal on X.Proof.
Straightforward.Now, we will construct different types of preideals derived from a generalized ideals.Proposition 2.7.
Letd be a generalized ideal on X. Define 𝒟1,𝒟2⊂IX by
(2.10)𝒟1={μ∈IX:∀A∈2X,d(A)≤1-supx∉Aμ(x)};(2.11)𝒟2={μ∈IX:∀α∈I1,d(μα)≥α}.
Then both 𝒟1,𝒟2 are saturated preideals on X.Proof.
First, we show that𝒟1 a preideal on X:
(P1)1∉𝒟1, otherwise for all A∈2X,d(A)≤1-supx∉A1(x)=0, that is, d(A)=0, which is a contradiction, since d is a nonzero function;
(P2) letμ,λ∈𝒟1. Then for all A∈2X:
(2.12)1-supx∉A(μ∨λ)(x)=(1-supx∉Aμ(x))⋀(1-supx∉Aλ(x))≥d(A)⋀d(A)=d(A),
hence μ∨λ∈𝒟1.
(P3) Letμ,λ∈IX such that μ∈𝒟1andλ≤μ. Then for all A∈2X,
(2.13)1-supx∉Aλ(x)≥1-supx∉Aμ(x)≥d(A),henceλ∈𝒟1.
Let (μθ)θ∈I0 be a family of fuzzy sets in 𝒟1. Then for all A∈2X,
(2.14)1-supx∉A(inf(μθ+θ))(x)=1-supx∉A(infθ∈I0(μθ(x)+θ))≥1-infθ∈I0supx∉A(μθ(x)+θ)=supθ∈I0(1-supx∉A(μθ(x)+θ))=supθ∈I0(1-(supx∉Aμθ(x)+θ))=supθ∈I0((1-supx∉Aμθ(x))-θ)≥supθ∈I0(d(A)-θ)=d(A),
therefore, infθ∈I0(μθ+θ)∈𝒟1, hence 𝒟1 is saturated.
Second, we prove that𝒟2 is saturated preideal on X:
(P1)1∉𝒟2, otherwise d(1α)=d(X)=0≱α, for all α∈I1;
(P2), (P3) direct.
Let(μθ)θ∈I0 be a family of fuzzy sets in 𝒟2. Then
(2.15)d((infθ∈I0(μθ+θ))α)≥d(⋂θ∈I0(μθ+θ)α)≥supθ∈I0d(μθα-θ),(by(G3))≥supθ∈I0(α-θ)=α.
Hence 𝒟2 is saturated preideal.Proposition 2.8.
Letd be a prime generalized ideal on X. Define 𝒟3,𝒟4⊂IX by
(2.16)𝒟3={μ∈IX:∀α∈I1,d((1-μ)α)≤α};𝒟4={μ∈IX:∀A∈2X,d(A)≤supx∉A(1-μ)(x)}.
Then 𝒟3,𝒟4 are prime saturated preideals on X. Moreover 𝒟3=𝒟4.Proof.
First, we show that𝒟3=𝒟4. Let μ∈IX such that d((1-μ)α)≤α, for all α∈I1. If A∈2X, we assume that supx∉A(1-μ)(x)=γ∈I1, hence (1-μ)γ⊆A.
Then, by (G3),d(A)≤d((1-μ)γ)≤γ=supx∉A(1-μ)(x), that implies 𝒟3⊆𝒟4.
Conversely, letμ∈IX such that d(A)≤supx∉A(1-μ)(x), for all A∈2X. Then d((1-μ)α)≤supx∉(1-μ)α(1-μ)(x)≤α, for all α∈I1 this implies 𝒟4⊆𝒟3, thus 𝒟3=𝒟4.
Second, we show that𝒟3 is a saturated prime preideal.
(P1) Suppose that1∈𝒟3. Let d(ϕ)=γ>0. Then, for all α<γ,d(0α)=d(ϕ)=γ>α, which is a contradiction, so 1∉𝒟3.
(P2) Letμ,λ∈𝒟3. Then for all α∈I1, d(1-(μ∨λ)α)=d((1-μ)α∩(1-λ)α)≤d((1-μ)α)∨d((1-λ)α)≤α, since d is prime. This implies μ∨λ∈𝒟3.
(P3) Letμ,λ∈IX such that μ∈𝒟3 and λ≤μ. Then d((1-λ)α)≤d((1-μ)α)≤α, by (G3), hence λ∈𝒟3.
Now, we will show that𝒟4 is saturated. Suppose for all ε>0,μ-ε∈D3. Then for all A∈2X, d(A)≤supx∉A(1-(μ-ε))(x)=supx∉A(1-μ)(x)+ε, hence d(A)≤supx∉A(1-μ)(x), that is, 𝒟4 is saturated.
Finally, we show that𝒟3 is prime. Let μ,λ∈IX such that μ⋀λ∈𝒟3. Then for all α∈I1,d(1-μα)⋀d(1-λα)≤d((1-μα)∪(1-λα))=d(1-(μα⋀λα))=d(1-(μ⋀λ)α)≤α, hence, d(1-μα)≤α or d(1-λα)≤α, that implies μ∈𝒟3 or λ∈𝒟3.Proposition 2.9.
Let𝒟 be a preideal on X. Define d:2X→I by
(2.17)∀A∈2X,d(A)=sup{t∈I:t>c,A⋀t∈𝒟}.
Then d is a generalized ideal on X.Proof.
(G1)d(X)=0, since t∈𝒟 implies t≤c.
(G2) LetA,B∈2X and α,β∈I,α>c,β>c such that A⋀α∈𝒟,B⋀β∈𝒟.
Then, by (P2),(A⋀α)∨(B⋀β)∈𝒟. Let γ=min(α,β). Then, by (P3), we get (A∪B)⋀γ=(A⋀γ)∨(B⋀γ)∈𝒟, and γ=α⋀β>c. Thus sup{t:t>c,(A∪B)⋀t∈𝒟}≥γ=α⋀β, hence sup{t:t>c,(A∪B)⋀t∈𝒟}≥sup{t:t>c,A⋀t∈𝒟}⋀sup{t:t>c,B⋀t∈𝒟}, so d(A∪B)≥d(A)⋀d(B).
Now we come to the main result of this section.Theorem 2.10.
There is a one-to-one correspondence between the set saturated preideals on a nonempty setX and the set of generalized ideals on X.Proof.
Letd be a generalized ideal on X. Denote the preideal 𝒟2 generated by d, defined as in Proposition 2.7, by 𝒟d, and denote the generalized ideal, generated by 𝒟d, as in Proposition 2.9, by d𝒟d, hence for all A∈2X(2.18)d𝒟d(A)=sup{t∈I1:t>c,A⋀t∈𝒟d},Proposition2.9=sup{t∈I1:t>c,d((A⋀t)α)≥α,∀α∈I1},(2.11)inProposition2.7=sup{t∈I1:t>c,d(A)≥α,∀α<t},sinced(ϕ)≥d(A),∀A⊆X=sup{t∈I1:t>c,d(A)≥t}=d(A).
Conversely, let 𝒟 be a saturated preideal on X. Denote the generalized ideal generated by 𝒟, as in Proposition 2.9, by d𝒟, and denote the preideal generated by d𝒟, as in (2.11) in Proposition 2.7 by 𝒟d𝒟. Then,
(2.19)𝒟d𝒟={λ∈IX:d𝒟(λα)≥α,∀α∈I1}={λ∈IX:sup{t∈I1,t>c,λα⋀t∈𝒟}≥α,∀α∈I1}={λ∈IX:λα⋀t∈𝒟,∀c<t<α}={λ∈IX:λ∈𝒟},byLemma2.3=𝒟.
This completes the proof.Definition 2.11.
A nonzero function𝔡:IX→I is called a fuzzy I-ideal on X if it satisfies the following conditions:
(d1)𝔡(1)=0;
(d2)𝔡(λ∨μ)≥𝔡(λ)⋀𝔡(μ), for λ,μ∈IX;
(d3) ifλ≥μ, then 𝔡(μ)≥𝔡(λ).
Clearly, a generalized ideald on a set X can be regarded as a fuzzy I-ideal whose domain is 2X. Precisely, d(A)=𝔡(A) for each A⊆X; otherwise d(A)=0.Remark 2.12 (Fang and Chen [8]).
Defined a generalized fuzzy preorderRϕ, corresponding to a map ϕ:IX→I as follows:
(2.20)∀(x,y)∈X×X,Rϕ(x,y)=⋀μ∈IXϕ(μ)→(μ(x)→μ(y)),
hence, for a fuzzy I-ideal 𝔡 on X, there corresponds a fuzzy preorder R𝔡 defined by
(2.21)∀(x,y)∈X×X,R𝔡(x,y)=⋀μ∈IX𝔡(μ)→(μ(x)→μ(y)).
Especially, for a generalized ideal d on X, the corresponding fuzzy preorder of d is given by the following: For every (x,y)∈X×X,
(2.22)Rd(x,y)=⋀A⊆X(x,y)∈A′×A(d(A)→0).


## 3. Connecting Fuzzy Preorders andI-fuzzy Topologies

In this section, we recall some basic results about the connection between fuzzy preorders and each ofI-topologies, fuzzifying topologies, and I-fuzzy topologies. Also, the relations among these different types of topologies are investigated.AnI-topology on a set X is a crisp subset 𝒯 of IX which is closed with respect to finite meets and arbitrary joins and which contains all the constant functions from X to I. If the constants are 0,1 only, we will call it a ChangI-topology.Another more consistent approach to the fuzziness has been developed. According to Shostak [6], an I-fuzzy topology on a set X is a map ℑ:IX→I, satisfying the following conditions:(1)
ℑ(α)=1 for all constant functions α:X→I;(2)
ℑ(μ⋀λ)≥ℑ(μ)⋀ℑ(λ), for all μ,λ∈IX;(3)
ℑ(∨j∈Jμi)≥⋀j∈Jℑ(μj), j∈J,μj∈IX.By the conditions (1) and (2), we get for μ∈IX and α∈I0, ℑ(μ⋀α)≥ℑ(μ). An I-fuzzy topology on a set X is called trivial, if ℑ(λ)=1, for all λ∈IX, and is called homogeneous  if the following condition holds: for μ∈IX and α∈I, ℑ(μ⋀α)=ℑ(μ).If a mapτ:2X→I satisfies the similar conditions of I-fuzzy topology on a set X, then τ is called a fuzzifyingtopology  Ying [9]. Furthermore, if a fuzzifying topology τ satisfies the following condition:(4) for all  {Aj}j∈J⊆2X,τ(∩j∈JAj)≥⋀j∈Jτ(Aj), then τ is called a saturated fuzzifying topology.For a given left-continuoust-norm *:I×I→I, the corresponding residual implication →:I×I→I is given by x→y=sup{z∈I:x*z≤y}.Many properties of residual implications are found in litrature, but we will recall some of them, which will be used in this paper:(I1)
x→y=1 if and only if x→y;(I2)
1→x=x;(I3)
(x→y)*(y→z)≤x→z;(I4)
x→⋀j∈Jyj=⋀j∈J(x→yj), hence, x→y≤x→z if y≤z;(I5)
(∨j∈Jyj)→y=⋀j∈J(yj→y), hence x→z≥y→z if x≤y;(I6)
x*∨j∈Jyj=∨j∈J(x*yj);(I7)
x→(y→z)=y→(x→z);(I8)
(x→y)*(u→v)≤x*u→y*v;(I9)
x≤(x→y)→y;(I10)
x→y=⋀λ∈I((λ→x)→(λ→y));(I11)
x→y=⋀λ∈I((y→λ)→(x→λ));(I12)
(x⋀y)→(z⋀w)≥(x→z)⋀(y→w).Clearly, by (I1) and (I3), the residual implication→  with respect to a t-norm * is a fuzzy preorder on I and called thecanonical fuzzy preorder on I.The relationships between fuzzy preorders andI-topologies, fuzzifying topologies, and I-fuzzy topologies, respectively, have been investigated by many authors. For example, we recall the main results of Lai and Zhang [10], Fang and Chen [8], and Fang [11].Theorem 3.1 (Lai and Zhang [10]).
For a fuzzy preorderR on a set X, the family 𝒯R={μ∈IX:R(x,y)*μ(x)≤μ(y),forallx,y∈X}, is an I-Alexandrov topology   induced by R, that is, 𝒯R satisfies the folowing properties: for all F⊆𝒯R,μ∈𝒯R, and α∈I,
(a) every constant fuzzy setX→I belongs to 𝒯R;
(b)∨ℱ∈𝒯R;
(c)⋀ℱ∈𝒯R;
(d)α*μ∈𝒯R;
(e)α→μ∈𝒯R.
Moreover,R satisfies the equality
(3.1)R(x,y)=⋀μ∈𝒯Rμ(x)→(y).
Conversely, for a given I-Alexandrov topology 𝒯 a set X, there is a unique *-fuzzy preorder R𝒯, given by R𝒯(x,y)=⋀μ∈𝒯μ(x)→μ(y), such that 𝒯=𝒯R𝒯.
Thus, there is a one-to-one correspondence between the set ofI-Alexandrov topologies on a set X and *-fuzzy preorders.Theorem 3.2 (Fang and Chen [8]).
Every fuzzy preorderR on a set X corresponds a saturated fuzzifying topology τR on X, given by for all A⊆X,
(3.2)τR(A)=⋀(x,y)∈A×A′(1-R(x,y)),
conversely, every fuzzifying topology τ on a set X corresponds a fuzzy preorder Rτ, given by: for all x,y∈X,
(3.3)Rτ(x,y)=⋀(x,y)∈A×A′(1-τ(A)).
Thus, there is a one-to-one correspondence between the set of saturated fuzzifying topologies on a set X and the set of fuzzy preorders on X.By the above two theorems, one can dedeuce, at once, the following corollary.Corollary 3.3.
The set of saturated fuzzifying toplogies on a setX and the set of I-Alexandrov topologies on X are in one-to-one correspondence.A special type ofI-topological spaces, called fuzzy neighborhood spaces, was introduced by Lowen [7]. The interrelations between these spaces and the fuzzifying toplogical spaces are interesting to study. Although a study of fuzzifying topological spaces and their relation with fuzzy neighborhood spaces has been investigated, sestematically, see, for example, [12], we give, in this section, alternative methods. We recall the folowing definition.Definition 3.4 (Wuyts et al. [13]).
AnI-topological space (X,τ) is called a fuzzy neighborhood space if whenever μ∈𝒯, for any α∈I1 also α⋀μα∈𝒯.
Equivalently, if wheneverμ∈𝒯, for any α,t∈I1, t<α also μα⋀t∈𝒯.Proposition 3.5.
Letτ be a fuzzifying topology on a set X. Then τ corresponds an I-topology 𝒯τ on X, defined by  𝒯τ={μ∈IX:τ(μα)≥α,forallα∈I1}, and (X,𝒯τ) is a fuzzy neighborhood space.
Conversely, let𝒯 be an I-topology on X. Then τ corresponds a fuzzifying topology τ𝒯, on X, defined by: for all A⊆X, τ𝒯(A)=sup{α∈I:A⋀α∈𝒯}. Moreover,
(1)τ𝒯τ=τ,
(2)𝒯τ𝒯≤𝒯 and if (X,𝒯) is a fuzzy neighborhood space, then 𝒯τ𝒯=𝒯.Proof.
That is,𝒯τ and τ𝒯 are I-topology and fuzzifying topology on X, respectively, and are easy to be verified. To prove that (X,𝒯τ) is a fuzzy neighborhood space, we suppose α,t∈I and μ∈𝒯τ, then
(3.4)τ((μt⋀t)α)={τ(ϕ)=1,ifα≥t,τ(μt)≥t,ift>α.
Therefore, τ((μt⋀t)α)≥α,forallα∈I1, that is, μt⋀t∈𝒯τ,forallt∈I1, so we are done.
To prove (1), let τ be a fuzzifying topology on X then for all ⋀⊆X,
(3.5)τ𝒯τ(A)=sup{α∈I:A⋀α∈𝒯τ}=sup{α∈I:τ((A⋀α)t)≥t,∀t∈I1}=sup{α∈I:τ(A)≥t,∀t<α},sinceτ(ϕ)=1=τ(A).
To prove (2), suppose that (X,𝒯) is a fuzzy neighborhood space. Then
(3.6)𝒯τ𝒯={μ∈IX:τ𝒯(μα)≥α,∀α∈I1}={μ∈IX:sup{t∈I:μα⋀t∈𝒯}≥α,∀α∈I1}={μ∈IX:μα⋀t∈𝒯,∀t<α,α∈I1}≤𝒯,={μ∈IX:μ∈𝒯}=𝒯,if𝒯isafuzzyneighborhoodstructure.Therefore, as a direct result of Proposition3.5, we have the following.Theorem 3.6.
There is a one-to-one correspondence between the set of fuzzifying topologies and the set of fuzzy neighborhood structures on the same set.Definition 3.7.
Given(X,ℑ), an I-fuzzy topological space, and α∈I0, then ℑα:={μ∈IX:ℑ(μ)≥α} is an I-topology on X, called the α-level I-topology of ℑ.
Consequently,𝒯ℑ:=ℑ1={μ∈IX:ℑ(μ)=1}, by which we will give an interrelation between I-topologies and I-fuzzy topologies.Proposition 3.8.
Let𝒯 be an I-topology on X. Then for all μ∈IX, ℑ𝒯(μ):=sup{α∈I1:μ⋀α∈𝒯} is a homogeneous I-fuzzy topology on X.Proof.
That is,ℑ𝒯 an I-fuzzy topology is easy to be verified. Now, let μ∈IX,α∈I, then we get
(3.7)ℑ𝒯(μ⋀α):=sup{t∈I1:μ⋀α⋀t∈𝒯}≤sup{t∈I1:t≤α,μ⋀α⋀t∈𝒯}=sup{t∈I1:μ⋀t∈𝒯}=ℑ𝒯(μ).
Since ℑ𝒯 is an I-fuzzy topology, then ℑ𝒯(μ⋀α)≥ℑ(μ), so the equality holds.Theorem 3.9.
There is a one-to-one correspondence between the set ofI-topologies on a set X and the set of homogeneous I-fuzzy topologies on X.Proof.
Let𝒯 be an I-topology on X, then ℑ𝒯 is a homogeneous I-fuzzy topology on X, by Proposition 3.8, and
(3.8)𝒯ℑ𝒯(μ)={μ∈IX:ℑτ(μ)=1}={μ∈IX:sup{t∈I1:μ⋀t∈𝒯}=1}={μ∈IX:μ⋀t∈𝒯,t<1}=𝒯.
Conversely, ifℑ is a homogeneous I-fuzzy topologies on X, then 𝒯ℑ is an I-topology and
(3.9)ℑ𝒯ℑ(μ)=sup{t∈I1:μ⋀t∈𝒯ℑ}=sup{t∈I1:ℑ(μ⋀t)=1}=1,sinceℑ(μ)=ℑ(μ⋀t).
So, we are done.


## 4. ConnectingI-fuzzy Topologies and Preideals

In this section, we associate for each a fuzzifying topology, a generalized ideal. Also, we construct a preideal corresponding to anI-topology, by means of the residuated implication. Conversely, an I-topology, a fuzzifying topology, and I-fuzzy topology are derived by old ones via preideals. We made use of the results of Section 2, beside the useful following lemma.Lemma 4.1.
LetΩ⊆IX. Define RΩ,RΩ′:X×X→I by for all (x,y)∈X×X,(4.1)RΩ(x,y)=inf{μ(x)→μ(y):μ∈IX,μ∈Ω},RΩ′(x,y)=inf{μ(y)→μ(x):μ∈IX,1-μ∈Ω}.
Then RΩ,RΩ′ are fuzzy preorders on X, especially, if *=min=⋀, then for all (x,y)∈X×X,
(4.2)RΩ(x,y)=inf{μ(y):μ∈Ω,μ(x)>μ(y)},RΩ′(x,y)=inf{μ(x):1-μ∈Ω,μ(x)<μ(y)}.Proposition 4.2.
Let𝒯 be an I-topology on a set X, and 𝒟 be a preideal on X. Define 𝒯*(𝒟)⊆IX, by μ∈𝒯*(𝒟) if and only if μ≤sup{λ∈𝒯:λ⋀(1-μ)∈𝒟}. Then 𝒯*(𝒟) is Chang I-topology on X.Proof.
Direct.However𝒯*(𝒟) will be called the I-fuzzy preideal topology. As a special case.Example 4.3.
If*=min=⋀, and  𝒯={α∈I:αconstans}, then 𝒯*(𝒟)={μ∈IX:μ≤sup{α∈I:α⋀(1-μ)∈𝒟}}. Consequently, 𝒯*(𝒟) has the following property: for all μ∈IX such that 1-μ∈𝒟, then μ∈𝒯*(𝒟).Proposition 4.4.
Let𝒟 be a preideal on a set X. Define ℑ:IX→I by for all μ∈IX, ℑ(μ)=⋀x∈X(μ(x)→sup{α∈I:α⋀(1-μ(x))∈𝒟}). Then ℑ is a Chang I-fuzzy topology on X.Proof.
Clearly,ℑ(1)=ℑ(0)=1, by (I1). Let μ,λ∈IX. Then ℑ(μ⋀λ)≥ℑ(μ)⋀ℑ(λ), by (P2) and (I12). Let {μi}i∈J⊆IX. Then ℑ(∨i∈Jμi)≥⋀i∈Jℑ(μi), by (I5) and (P3), so we are done.TheI-fuzzy topology ℑ, given in the above proposition, is called the I-fuzzy topology associated with a preideal and denoted by ℑ𝒟. It is easy to see that ℑ𝒟 satisfy the following simple properties:(1)
for allA⊆X, ℑ𝒟(A)=1;(2)
for allμ∈IX and x∈X such that (1-μ(x))∈𝒟, then ℑ𝒟(μ)=1;(3)
restricting the range ofℑ on {0,1}, a Chang I-topology, associated with a preideal 𝒟, is obtained, namely,
(4.3)ℑ𝒟={μ∈IX:μ(x)≤sup{α∈I:α⋀(1-μ(x))∈𝒟},∀x∈X}.Proposition 4.5.
Let𝒯 be an I-topology on a Boolean lattice X with 0. Let 𝒟⊆IX, defined as 𝒟={μ∈IX:supx∈μαinfλ∈𝒯(λ(x)→λ(0))≤1-α,forallα∈I1}. Then 𝒟 is a saturated preideal on X and 𝒟={μ∈IX:dR𝒯(μα)≥α,forallα∈I1}, where dR𝒯 is the generalized ideal corresponding to R𝒯.Proof.
Let𝒯 be an I-topology on X. Then R𝒯 is a fuzzy preorder, by Theorem 3.1, hence dR𝒯 is a generalized ideal, by (2.9) in Proposition 2.6. Thus, 𝒟={μ∈IX:dRT(μα)≥α} is a saturated preideal by (2.11) in Proposition 2.7. Secondly, we prove the equality of the two forms of 𝒟 as follows:
(4.4)𝒟={μ∈IX:dR𝒯(μα)≥α}={μ∈IX:⋀x∈μα(1-R𝒯(x,0))≥α,∀α∈I1},by(2.9)inProposition2.6={μ∈IX:∨x∈μαR𝒯(x,0)≤1-α,∀α∈I1}={μ∈IX:∨x∉μα⋀λ∈𝒯(λ(x)→λ(0))≤1-α,∀α∈I1},byTheorem3.1,
which completes the proof.As a direct consequence of Lemma4.1 and Theorem 3.1, we may construct an I-topology from a preideal, as follows.Proposition 4.6.
Let𝒟⊆IX be a preideal. Define 𝒯𝒟 by
(4.5)𝒯𝒟={λ∈IX:(⋀μ∈𝒟μ(x)→μ(y))≤λ(x)→λ(y)}.
Then 𝒯𝒟 is an I-Alexandrov topology on X, containing 𝒟.Proposition 4.7.
Letℑ be a nontrivial I-fuzzy topology on a set X. Define 𝒟:={μ∈IX:ℑ(λ)=1,forallλ≤μ}, then 𝒟 is a preideal on X.Proof.
Straightforward.Proposition 4.8.
Let𝒟 be a preideal on a set X. Define 𝒯⊆IX by 𝒯={λ∈IX:λ=μ∨α,μ∈𝒟,α∈I}. Then 𝒯 is an I-topology if 𝒟 is σ-preideal.Proof.
Since0∈𝒟, then 𝒯 contains all the constants fuzzy subsets on X. Secondly, if λ1,λ2∈𝒯, then there exist μ1,μ2∈𝒟 and α1,α2∈I such that λ1=μ1∨α1 and λ2=μ2∨α2, then λ1⋀λ2=(μ1⋀μ2)∨(μ1⋀α2)∨(μ2⋀α1)∨(α1⋀α2). By (P3) and (P2), we get (μ1⋀μ2)∨(μ1⋀α2)∨(μ2⋀α1)∈𝒟, hence λ1⋀λ2∈𝒯.
Let{λi}i∈J⊆𝒯, then for all i∈J, there exists μi∈𝒟, αi∈I such that λi=μi∨αi. Therefore, we get ∨i∈Jλi=(∨i∈Jμi)∨(∨i∈Jαi)∈𝒯, since 𝒟 is a σ-preideal.Proposition 4.9.
Letτ be a fuzzifying topology on a Boolean lattice. Define d:2X→I by for all A⊆X, d(A)=⋀x,y∈A∨B⊆X,x,y′∈Bτ(B). Then d is a generalized ideal on X.
Conversely, letd be a generalized ideal on X, then τd:2X→I, defined by for all A⊆X,
(4.6)τ(A)=⋀(x,y)∈A×A′∨B⊆X(x,y)∈B×B′d(B)
is a fuzzifying topology on X.Proof.
It is a direct composition of the two maps defined by Proposition2.6 and Theorem 3.2. Conversely, follows from Remark 2.12 and Theorem 3.2.Proposition 4.10.
Letd be a σ-generalized ideal on a set X. Define Rd:X×X→I by for all (x,y)∈X×X, Rd(x,y)=⋀(x,y)∈B×B′(1-d(B)).
Then,Rd is a fuzzy preorder and for all A⊂X, τRd(A)=d(A).Proof.
ThatRd is a fuzzy preorder follows from Remark 2.12, hence according to Theorem 3.2, for all A⊆X,
(4.7)τRd(A)=⋀(x,y)∈A×A′(1-Rd(x,y))=⋀(x,y)∈A×A′(1-⋀(x,y)∈B×B′(d(B)→0)),byRemark2.12=⋀(x,y)∈A×A′(∨(x,y)∈B×B′(1-(d(B)→0))≥1-(d(A)→0)≥1-(1-d(A))=d(A)).
Secondly, we have to show that for anyA⊆X, τRd(A)≤d(A).
Lett∈I1 such that τRd(A)>t. Then for any x∈A,y∈A′, there exists Ny(x)⊆X such that x∈Ny(x),y∉Ny(x), and d(Ny(x))>t. Let Ny=∪x∈ANy(x), then y∉Ny,A⊆Ny, and d(Ny)=d(∪x∈ANy(x))≥⋀x∈Ad(Ny(x))≥t, since d is σ-generelized ideal. Obviously, A=∩y∉ANy. So we would get d(A)=d(∩y∉ANy)≥∨y∉Ad(Ny)≥t, by (G3). Thus d(A)≥τRd, which completes the proof.Example 4.11.
(1) Let 𝒟={μ∈IX:μ≤1/2}, and *=min, then
(4.8)x→y={1,x≤y,y,x>y.
Then 𝒟 is a preideal on X, and
(4.9)R𝒟(x,y)=⋀μ∈𝒟μ(x)→μ(y),∀x,y∈X={1,x=y,0,x≠y
is a fuzzy preorder, hence
(4.10)τR𝒟={μ∈IX:R𝒟(x,y)⋀μ(x)≤μ(y),∀x,y∈X}=II.
Also, another fuzzy preorderℜ𝒟 may be dedeuced by 𝒟, Lemma 4.1:
(4.11)ℜ𝒟(x,y)=inf{μ(x):1-μ∈𝒟,μ(x)<μ(y),∀x,y∈X}=inf{μ(x):μ∈II,μ≥12,μ(x)<μ(y),∀x,y∈X}={1,x=y,12,x≠y.
Then
(4.12)τℜ𝒟={λ∈II:ℜ𝒟(x,y)⋀λ(x)≤λ(y),∀x,y∈I}={λ∈II:λ≥12}⋃{α∈I:αconstant}.
(2) Let τ={μ∈IX:μ≥1/2}∪{α∈Iconstants}, and *=min, then 𝒟τ={μ∈II:μ≤1/2}, by applying Proposition 4.5.



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*Source: 567482-2009-07-28.xml*