Connecting Fuzzifying Topologies and Generalized Ideals by Means of Fuzzy Preorders

Home , Alexandrov topology

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 567482, 16 pages doi:10.1155/2009/567482

Research Article Connecting Fuzzifying Topologies and Generalized Ideals by Means of Fuzzy Preorders

G. Elsalamony

Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

Correspondence should be addressed to G. Elsalamony, [email protected]

Received 17 February 2009; Accepted 25 May 2009

Recommended by Etienne Kerre

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 given I-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.

Copyright q 2009 G. Elsalamony. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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,andI-fuzzy topologies 6, respectively. Thus, an interesting relation between a special type of I-topological spaces, 2 International Journal of Mathematics and Mathematical Sciences 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 let X, I denote a nonempty set and the closed , I , ,I , unit interval 0 1 , respectively, and we let 0 0 1 1 0 1 . X We denote the characteristic function of a subset A ⊆ X also by A.Ifμ ∈ 1 , we deﬁne α X μ {x ∈ X : μx >α}.Let2 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,at-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. A fuzzy relation on a nonempty set is a map R : X × X → I. A fuzzy relation R is called

1 reﬂexive if Rx, x1 for each x ∈ X; 2 ∗-transitive if Rx, y ∗ Ry, z ≤ Rx, z for all x, 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 subset D 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.

X Deﬁnition 2.2. A nonempty subset D of I 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.

ApreidaelD is called prime if it satisﬁes the condition;

X ∀μ, λ ∈ I such that μ ∧ ν ∈D, then μ ∈Dor ν ∈D. 2.1

ApreidealD is called a σ-preideal if the following hold. International Journal of Mathematics and Mathematical Sciences 3

If {μn : n 1, 2, 3,...} is a countable subcollection of D, then supn μn ∈D. If D is a preideal on X, we deﬁne the characteristic, denoted by cD,ofD by

cD sup inf μx. 2.2 μ∈D x∈X

I If D is a preideal and cD < 1, we say that D is saturated if for every με ∈D1 , μ ε ∈D D infε∈I1 ε , equivalently, is saturated if it satisﬁes

X ∀ε<1,μ∈ I : μ − ε ∈D⇒ μ ∈D. 2.3

We provide a useful characterization of a saturated preideal D in the following lemma.

Lemma 2.3. Let D be a saturated preideal on X, μ ∈ IX.Then

μ ∈D⇐⇒μα ∧ γ ∈D, ∀c<γ<α I . in 1 2.4

μ ∈D μα ∧ γ ≤ μ γ<α∈ I μα ∧ γ ∈D γ<α∈ I Proof. Let . Since for all 1, then, by P3 , for all 1. μ ∈ IX μα ∧ γ ∈D c<γ<α I α ∈ , / Conversely, let such that for all in 1. For every 0 1 2

It should be noticed that the required condition of Lemma 2.3 holds, trivially, for all μ ∈Dand γ ≤ c, therefore, we provide γ>c, so that the above condition is signiﬁcant. Also, the characteristic of a preideal D,onasubsetA of X may be deﬁned and denoted by cADsupλ∈D infx∈A λx. The following result is useful for our study.

X Lemma 2.4. Let D on X. Then, for every A ∈ 2 ,

cAD sup{t ∈ I : A ∧ t ∈D}, 2.5

hence, c cXDsup{t ∈ I : t ∈D}.

Proof. The statement holds when A φ. Suppose A / φ, and denote c cAD and b sup{t ∈ I : A ∧ t ∈D}.Lett ∈ I satisfy A ∧ t ∈D. Then c ≥ infx∈AA ∧ txt,thatis,c ≥ b. Conversely, given λ ∈D,letγ infx∈A λx, hence A ∧ γ ≤ λ. This implies, by P3, A ∧ γ ∈D, so b ≥ γ. Thus, by deﬁnition of c,wegetb ≥ c, this shows that the equality holds. 4 International Journal of Mathematics and Mathematical Sciences

X Deﬁnition 2.5. A nonzero function d :2 → I is called a generalized ideal of fuzzy subsets of X, simply, a generalized ideal on X, if it satisﬁes the following conditions: G1 dX0; X G2 dA ∧ dB ≤ dA ∪ B, for each A, B ∈ 2 ; G3 if A ⊂ B, then dA ≥ dB, for each A, B ∈ 2X; The conditions G2 and G3 are equivalent to: ∗ X G dA ∧ dBdA ∪ B, for each A, B ∈ 2 . Moreover, Since d a nonzero function, then by G3, dφ > 0.

A generalized ideal d, is called σ-generalized ideal if

X d ∪ Ai ≥∧dAi, for each Aii∈J ⊆ I , 2.6 i∈J i∈J and d is called prime if

X dA ∩ B ≤ dA ∨ dB, for each A, B ∈ 2 . 2.7

A generalized ideal on a set X can be derived by fuzzy relations in different ways, as we see in the following. Proposition 2.6. R X d ,d Let be a reflexive fuzzy relation on a Boolean lattice with 0. Define 1 2 : IX → I, for all A ⊆ X as follows:

d A ∧ 1 − R x, 1 − y ; where 1 − y is the complement y, 1 x,y∈A 2.8

d A ∧ 1 − Rx, 0. 2 x∈A 2.9

d ,d X Then 1 2 are generalized ideal on .

Proof. Straightforward.

Now, we will construct diﬀerent types of preideals derived from a generalized ideals. Proposition 2.7. d X D , D ⊂ IX Let be a generalized ideal on . Deﬁne 1 2 by

D μ ∈ IX ∀A ∈ X,dA ≤ − μ x 1 : 2 1 sup ; 2.10 x /∈ A D μ ∈ IX ∀α ∈ I ,dμα ≥ α . 2 : 1 2.11

D , D X Then both 1 2 are saturated preideals on . International Journal of Mathematics and Mathematical Sciences 5

D X Proof. First, we show that 1 apreidealon : ∈D A ∈ X,dA ≤ − x dA P1 1 / 1, otherwise for all 2 1 supx /∈ A1 0, that is, 0, which is a contradiction, since d is a nonzero function; μ, λ ∈D A ∈ X P2 let 1. Then for all 2 : 1 − sup μ ∨ λ x 1 − sup μx ∧ 1 − sup λx ≥ dA ∧ dA dA, 2.12 x /∈ A x /∈ A x /∈ A

μ ∨ λ ∈D hence 1. μ, λ ∈ IX μ ∈D λ ≤ μ A ∈ X P3 Let such that 1 and . Then for all 2 ,

− λ x ≥ − μ x ≥ d A , λ ∈D. 1 sup 1 sup hence 1 2.13 x /∈ A x /∈ A

X Let μθ be a family of fuzzy sets in D . Then for all A ∈ 2 , θ∈I0 1 1 − sup inf μθ θ x 1 − sup inf μθx θ θ∈I x /∈ A x /∈ A 0 ≥ 1 − inf sup μθx θ θ∈I 0 x /∈ A sup 1 − sup μθx θ x ∈ A θ∈I0 / 2.14

sup 1 − sup μθx θ x ∈ A θ∈I0 /

sup 1 − supμθx − θ x ∈ A θ∈I0 /

≥ supdA − θ dA, θ∈I0

μ θ ∈D D therefore, infθ∈I0 θ 1, hence 1 is saturated. D X Second, we prove that 2 is saturated preideal on : ∈D d αdX ≥α α ∈ I P1 1 / 2, otherwise 1 0/ , for all 1; P2, P3 direct. Let μθ be a family of fuzzy sets in D . Then θ∈I0 2 α α d inf μθ θ ≥ d μθ θ θ∈I 0 θ∈I 0 2.15 α−θ ≥ sup d μθ , by G3 ≥ sup α − θ α. θ∈I0 θ∈I0

D Hence 2 is saturated preideal. 6 International Journal of Mathematics and Mathematical Sciences

Proposition 2.8. d X D , D ⊂ IX Let be a prime generalized ideal on . Deﬁne 3 4 by

D μ ∈ IX ∀α ∈ I ,d − μ α ≤ α 3 : 1 1 ; 2.16 D μ ∈ IX ∀A ∈ X,dA ≤ − μ x . 4 : 2 sup 1 x /∈ A

D , D X D D Then 3 4 are prime saturated preideals on . Moreover 3 4. D D μ ∈ IX d − μα ≤ α, α ∈ I Proof. First, we show that 3 4.Let such that 1 for all 1.If A ∈ X − μxγ ∈ I − μγ ⊆ A 2 , we assume that supx /∈ A 1 1, hence 1 . dA ≤ d − μγ ≤ γ − μx D ⊆D Then, by G3 , 1 supx /∈ A 1 , that implies 3 4. X X Conversely, let μ ∈ I such that dA ≤ supx /∈ A1 − μx, for all A ∈ 2 . Then α d − μ ≤ α − μx ≤ α α ∈ I D ⊆D D D 1 supx /∈ 1−μ 1 , for all 1 this implies 4 3,thus 3 4. D Second, we show that 3 is a saturated prime preideal. ∈D dφγ> α<γ,d αdφγ>α P1 Suppose that 1 3.Let 0. Then, for all 0 , ∈D which is a contradiction, so 1 / 3. μ, λ ∈D α ∈ I d − μ ∨ λαd − μα ∩ − λα ≤ P2 Let 3. Then for all 1, 1 1 1 d − μα ∨ d − λα ≤ α d μ ∨ λ ∈D 1 1 ,since is prime. This implies 3. μ, λ ∈ IX μ ∈D λ ≤ μ d − λα ≤ d − μα ≤ α P3 Let such that 3 and . Then 1 1 ,by λ ∈D G3 , hence 3. D ε> ,μ− ε ∈ D Now, we will show that 4 is saturated. Suppose for all 0 3. Then for all X A ∈ 2 , dA ≤ supx /∈ A1−μ−εxsupx /∈ A1−μxε, hence dA ≤ supx /∈ A1−μx, D that is, 4 is saturated. D μ, λ ∈ IX μ ∧ λ ∈D Finally, we show that 3 is prime. Let such that 3. Then for all α ∈ I ,d − μα ∧ d − λα ≤ d − μα ∪ − λα d − μα ∧ λα d − μ ∧ λα ≤ α 1 1 1 1 1 1 1 , d − μα ≤ α d − λα ≤ α μ ∈D λ ∈D hence, 1 or 1 , that implies 3 or 3.

Proposition 2.9. Let D be a preideal on X. Deﬁne d :2X → I by

X ∀A ∈ 2 ,dA sup{t ∈ I : t>c,A∧ t ∈D}. 2.17

Then d is a generalized ideal on X.

Proof. G1 dX0, since t ∈Dimplies t ≤ c. X G2 Let A, B ∈ 2 and α, β ∈ I, α > c, β > c such that A ∧ α ∈D,B∧ β ∈D. Then, by P2, A∧α∨B∧β ∈D.Letγ minα, β. Then, by P3,wegetA∪B∧γ A ∧ γ ∨ B ∧ γ ∈D,andγ α ∧ β>c.Thussup{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 dA ∪ B ≥ dA ∧ dB. 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 set X and the set of generalized ideals on X. International Journal of Mathematics and Mathematical Sciences 7

d X D d Proof. Let be a generalized ideal on . Denote the preideal 2 generated by , deﬁned as in Proposition 2.7,byDd, and denote the generalized ideal, generated by Dd,asin d A ∈ X Proposition 2.9,by Dd , hence for all 2

d A {t ∈ I t>c,A∧ t ∈D }, . Dd sup 1 : d Proposition 2 9 t ∈ I t>c,d A ∧ t α ≥ α, ∀α ∈ I , . . sup 1 : 1 2 11 in Proposition 2 7 t ∈ I t>c,d A ≥ α, ∀αc,d A ≥ t sup{ 1 : } dA.

Conversely, let D be a saturated preideal on X. Denote the generalized ideal generated by D,asinProposition 2.9,bydD, and denote the preideal generated by dD,asin2.11 in D Proposition 2.7 by dD . Then, D λ ∈ IX d λα ≥ α, ∀α ∈ I dD : D 1 λ ∈ IX t ∈ I ,t>c,λα ∧ t ∈D ≥ α, ∀α ∈ I :sup{ 1 } 1 X α λ ∈ I : λ ∧ t ∈D, ∀c

This completes the proof.

X Deﬁnition 2.11. A nonzero function d : I → I is called a fuzzy I-ideal on X if it satisﬁes the following conditions: d1 d10; X d2 dλ ∨ μ ≥ dλ ∧ dμ,forλ, μ ∈ I ; d3 if λ ≥ μ, then dμ ≥ dλ. Clearly, a generalized ideal d on a set X can be regarded as a fuzzy I-ideal whose X domain is 2 .Precisely,dAdA for each A ⊆ X; otherwise dA0.

Remark 2.12 Fang and Chen 8. Deﬁned a generalized fuzzy preorder Rφ, corresponding X to a map φ : I → I as follows: ∀ x, y ∈ X × X, Rφ x, y ∧ φ μ −→ μx → μ y , 2.20 μ∈IX hence, for a fuzzy I-ideal d on X, there corresponds a fuzzy preorder Rd deﬁned by ∀ x, y ∈ X × X, Rd x, y ∧ d μ −→ μx → μ y . 2.21 μ∈IX 8 International Journal of Mathematics and Mathematical Sciences

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, Rd x, y ∧ dA → 0. 2.22 A⊆X x,y∈A×A

3. Connecting Fuzzy Preorders and I-fuzzy Topologies

In this section, we recall some basic results about the connection between fuzzy preorders and each of I-topologies, fuzzifying topologies, and I-fuzzy topologies. Also, the relations among these diﬀerent types of topologies are investigated. X An I-topology on a set X is a crisp subset T of I which is closed with respect to ﬁnite meets and arbitrary joins and which contains all the constant functions from X to I.Ifthe constants are 0, 1 only, we will call it a Chang I-topology. Another more consistent approach to the fuzziness has been developed. According to X Shostak 6,anI-fuzzy topology on a set X is a map I : I → I, satisfying the following conditions:

1 Iα1 for all constant functions α : X → I; X 2 Iμ ∧ λ ≥ Iμ ∧ Iλ, for all μ, λ ∈ I ; X 3 I∨j∈J μi ≥∧j∈J Iμj , j ∈ J, μj ∈ I . μ ∈ IX α ∈ I Iμ ∧ α ≥ Iμ I By the conditions 1 and 2 ,wegetfor and 0, .An -fuzzy X topology on a set X is called trivial,ifIλ1, for all λ ∈ I , and is called homogeneous if the X following condition holds: for μ ∈ I and α ∈ I, Iμ ∧ αIμ. X If a map τ :2 → I satisﬁes the similar conditions of I-fuzzy topology on a set X, then τ is called a fuzzifying topology Ying 9. Furthermore, if a fuzzifying topology τ satisﬁes the following condition: {A } ⊆ X,τ∩ A ≥∧ τA τ 4 for all j j∈J 2 j∈J j j∈J j , then is called a saturated fuzzifying topology. For a given left-continuous t-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; I3x → y ∗ y → z ≤ x → z; I4 x →∧j∈J yj ∧j∈J x → yj , hence, x → y ≤ x → z if y ≤ z; I5∨j∈J yj → y ∧j∈J yj → y, hence x → z ≥ y → z if x ≤ y; I6 x ∗∨j∈J yj ∨j∈J x ∗ yj ; I7 x → y → zy → x → z; I8x → y ∗ u → v ≤ x ∗ u → y ∗ v; I9 x ≤ x → y → y; I10 x → y ∧λ∈I λ → x → λ → y; International Journal of Mathematics and Mathematical Sciences 9

I11 x → y ∧λ∈I y → λ → x → λ;

I12x ∧ 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 the canonical fuzzy preorder on I. The relationships between fuzzy preorders and I-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 preorder R on a set X, the family TR {μ ∈ X I : Rx, y ∗ μx ≤ μy, for all x, y ∈ X},isanI-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 set X → I belongs to TR; b ∨F ∈ TR; c ∧F ∈ TR; d α ∗ μ ∈TR; e α → μ ∈TR. Moreover, R satisﬁes the equality

R x, y ∧ μx −→ y . 3.1 μ∈TR

Conversely, for a given I-Alexandrov topology T a set X, there is a unique ∗-fuzzy preorder RT,given R x, y∧ μx → μy T T by T μ∈T , such that RT . Thus, there is a one-to-one correspondence between the set of I-Alexandrov topologies on a set X and ∗-fuzzy preorders.

Theorem 3.2 Fang and Chen 8. Every fuzzy preorder R on a set X corresponds a saturated fuzzifying topology τR on X, given by for all A ⊆ X,

τRA ∧ 1 − R x, y , 3.2 x,y∈A×A

conversely, every fuzzifying topology τ on a set X corresponds a fuzzy preorder Rτ , given by: for all x, y ∈ X,

Rτ x, y ∧ 1 − τA. 3.3 x,y∈A×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 set X and the set of I-Alexandrov topologies on X are in one-to-one correspondence. 10 International Journal of Mathematics and Mathematical Sciences

A special type of I-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 μ ∈T α ∈ I α ∧ μα ∈T space if whenever , for any 1 also . μ ∈T α, t ∈ I t<α μα ∧ t ∈T Equivalently, if whenever , for any 1, also .

Proposition 3.5. Let τ be a fuzzifying topology on a set X.Thenτ corresponds an I-topology Tτ on X T {μ ∈ IX τμα ≥ α, α ∈ I } X, T , deﬁned by τ : for all 1 , and τ is a fuzzy neighborhood space. Conversely, let T be an I-topology on X.Thenτ corresponds a fuzzifying topology τT,onX, deﬁned by: for all A ⊆ X, τTAsup{α ∈ I : A ∧ α ∈T}. Moreover, τ τ 1 Tτ , T ≤T X, T T T 2 τT and if is a fuzzy neighborhood space, then τT .

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

⎧ ⎨τ φ , α ≥ t, t α 1 if τ μ ∧ t 3.4 ⎩ t τ μ ≥ t, if t>α.

τμt ∧ tα ≥ α, α ∈ I μt ∧ t ∈T , t ∈ I Therefore, for all 1,thatis, τ for all 1, so we are done. To prove 1,letτ be a fuzzifying topology on X then for all ∧⊆X,

τ A {α ∈ I A ∧ α ∈T } Tτ sup : τ α ∈ I τ A ∧ α t ≥ t, ∀t ∈ I sup : 1 3.5 sup{α ∈ I : τA ≥ t, ∀t<α}, since τ φ 1 τA.

To prove 2, suppose that X, T is a fuzzy neighborhood space. Then

T μ ∈ IX τ μα ≥ α, ∀α ∈ I τT : T 1 μ ∈ IX t ∈ I μα ∧ t ∈T ≥ α, ∀α ∈ I :sup : 1 3.6 μ ∈ IX μα ∧ t ∈T, ∀ <α,α∈ I ≤T, : t 1 X μ ∈ I : μ ∈T T, if T is a fuzzy neighborhood structure. International Journal of Mathematics and Mathematical Sciences 11

Therefore, as a direct result of Proposition 3.5, 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. X, I I α ∈ I I {μ ∈ IX Deﬁnition 3.7. Given ,an -fuzzy topological space, and 0, then α : : Iμ ≥ α} is an I-topology on X, called the α-level I-topology of I. T I {μ ∈ IX Iμ } Consequently, I : 1 : 1 , by which we will give an interrelation between I-topologies and I-fuzzy topologies. Proposition 3.8. T I X μ ∈ IX I μ {α ∈ I μ∧α ∈T} Let be an -topology on . Then for all , T : sup 1 : is a homogeneous I-fuzzy topology on X.

X Proof. That is, IT an I-fuzzy topology is easy to be veriﬁed. Now, let μ ∈ I ,α∈ I, then we get I μ ∧ α t ∈ I μ ∧ α ∧ t ∈T T : sup 1 : ≤ t ∈ I t ≤ α, μ ∧ α ∧ t ∈T sup 1 : 3.7 t ∈ I μ ∧ t ∈T sup 1 : IT μ .

Since IT is an I-fuzzy topology, then ITμ ∧ α ≥ Iμ, so the equality holds.

Theorem 3.9. There is a one-to-one correspondence between the set of I-topologies on a set X and the set of homogeneous I-fuzzy topologies on X.

Proof. Let T be an I-topology on X, then IT is a homogeneous I-fuzzy topology on X,by Proposition 3.8,and T μ μ ∈ IX I μ IT : τ 1 μ ∈ IX t ∈ I μ ∧ t ∈T :sup 1 : 1 3.8 X μ ∈ I : μ ∧ t ∈T,t<1

Conversely, if I is a homogeneous I-fuzzy topologies on X, then TI is an I-topology and I μ t ∈ I μ ∧ t ∈T TI sup 1 : I t ∈ I I μ ∧ t sup 1 : 1 3.9 1, since I μ I μ ∧ t .

So, we are done. 12 International Journal of Mathematics and Mathematical Sciences

4. Connecting I-fuzzy Topologies and Preideals

In this section, we associate for each a fuzzifying topology, a generalized ideal. Also, we construct a preideal corresponding to an I-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.

X Lemma 4.1. Let Ω ⊆ I . Deﬁne RΩ,RΩ : X × X → I by for all x, y ∈ X × X, X RΩ x, y inf μx → μ y : μ ∈ I ,μ∈ Ω , 4.1 X RΩ x, y inf μ y → μx : μ ∈ I , 1 − μ ∈ Ω .

Then RΩ,RΩ are fuzzy preorders on X, especially, if ∗ min ∧, then for all x, y ∈ X × X, RΩ x, y inf μ y : μ ∈ Ω,μx >μ y , 4.2 RΩ x, y inf μx :1− μ ∈ Ω,μx <μ y .

Proposition 4.2. Let T be an I-topology on a set X, and D be a preideal on X. Deﬁne T∗D ⊆ IX, ∗ ∗ by μ ∈TD if and only if μ ≤ sup{λ ∈T: λ ∧ 1 − μ ∈D}.ThenT D is Chang I-topology on X.

Proof. Direct.

∗ However T D will be called the I-fuzzy preideal topology. As a special case.

∗ X Example 4.3. If ∗ min ∧,andT {α ∈ I : α constans}, then T D{μ ∈ I : 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 ∈TD.

X X Proposition 4.4. Let D be a preideal on a set X. Deﬁne I : I → I by for all μ ∈ I , Iμ ∧x∈Xμx → sup{α ∈ I : α ∧ 1 − μx ∈D}.ThenI is a Chang I-fuzzy topology on X.

X Proof. Clearly, I1I01, by I1.Letμ, λ ∈ I . Then Iμ ∧ λ ≥ Iμ ∧ Iλ,byP2 and X I12.Let{μi}i∈J ⊆ I . Then I∨i∈J μi ≥∧i∈J Iμi,byI5 and P3, so we are done.

The I-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 all A ⊆ X, IDA1; X 2 for all μ ∈ I and x ∈ X such that 1 − μx ∈D, then IDμ1; 3 restricting the range of I on {0, 1},aChangI-topology, associated with a preideal D, is obtained, namely, X ID μ ∈ I : μx ≤ sup α ∈ I : α ∧ 1 − μx ∈D , ∀x ∈ X . 4.3 International Journal of Mathematics and Mathematical Sciences 13

Proposition 4.5. Let T be an I-topology on a Boolean lattice X with 0. Let D⊆IX, deﬁned as D {μ ∈ IX λx → λ ≤ − α, α ∈ I } D :supx∈μα infλ∈T 0 1 for all 1 .Then is a saturated X D {μ ∈ IX d μα ≥ α, α ∈ I } d preideal on and : RT for all 1 ,where RT is the generalized ideal corresponding to RT. T I X R d Proof. Let be an -topology on . Then T is a fuzzy preorder, by Theorem 3.1, hence RT is D {μ ∈ IX d μα ≥ α} a generalized ideal, by 2.9 in Proposition 2.6.Thus, : RT is a saturated preideal by 2.11 in Proposition 2.7. Secondly, we prove the equality of the two forms of D as follows:

D μ ∈ IX d μα ≥ α : RT X μ ∈ I ∧ − RTx, ≥ α, ∀α ∈ I , . . : x∈μα 1 0 1 by 2 9 in Proposition 2 6 4.4 X μ ∈ I ∨ RTx, ≤ − α, ∀α ∈ I : x∈μα 0 1 1 X μ ∈ I : ∨ ∧ λx → λ0 ≤ 1 − α, ∀α ∈ I , by Theorem 3.1, x /∈ μα λ∈T 1 which completes the proof.

As a direct consequence of Lemma 4.1 and Theorem 3.1, we may construct an I- topology from a preideal, as follows.

X Proposition 4.6. Let D⊆I be a preideal. Deﬁne TD by

X TD λ ∈ I : ∧ μx → μ y ≤ λx → λ y . 4.5 μ∈D

Then TD is an I-Alexandrov topology on X, containing D.

Proposition 4.7. Let I be a nontrivial I-fuzzy topology on a set X. Deﬁne D : {μ ∈ IX : Iλ 1, for all λ ≤ μ},thenD is a preideal on X.

Proof. Straightforward.

Proposition 4.8. Let D be a preideal on a set X. Deﬁne T⊆IX by T {λ ∈ IX : λ μ ∨ α, μ ∈ D,α∈ I}.ThenT is an I-topology if D is σ-preideal. ∈D T X λ ,λ ∈T, Proof. Since 0 , then contains all the constants fuzzy subsets on . Secondly, if 1 2 μ ,μ ∈D α ,α ∈ I λ μ ∨ α λ μ ∨ α λ ∧ λ then there exist 1 2 and 1 2 such that 1 1 1 and 2 2 2, then 1 2 μ ∧μ ∨μ ∧α ∨μ ∧α ∨α ∧α μ ∧μ ∨μ ∧α ∨μ ∧α ∈D 1 2 1 2 2 1 1 2 .By P3 and P2 ,weget 1 2 1 2 2 1 , λ ∧ λ ∈T hence 1 2 . {λ } ⊆T i ∈ J, μ ∈D α ∈ I λ μ ∨ α Let i i∈J , then for all there exists i , i such that i i i. Therefore, we get ∨i∈J λi ∨i∈J μi ∨ ∨i∈J αi ∈T,sinceD is a σ-preideal. 14 International Journal of Mathematics and Mathematical Sciences

X Proposition 4.9. Let τ be a fuzzifying topology on a Boolean lattice. Deﬁne d :2 → I by for all A ⊆ X, dA∧x,y∈A ∨B⊆X, x,y∈B τB.Thend is a generalized ideal on X. X Conversely, let d be a generalized ideal on X,thenτd :2 → I, deﬁned by for all A ⊆ X,

τA ∧ ∨ dB 4.6 x,y∈A×A B⊆X x,y∈B×B

is a fuzzifying topology on X.

Proof. It is a direct composition of the two maps deﬁned by Proposition 2.6 and Theorem 3.2. Conversely, follows from Remark 2.12 and Theorem 3.2.

Proposition 4.10. Let d be a σ-generalized ideal on a set X. Deﬁne Rd : X × X → I by for all x, y ∈ X × X, Rdx, y∧x,y∈B×B 1 − dB. R A ⊂ X τ AdA Then, d is a fuzzy preorder and for all , Rd .

Proof. That Rd is a fuzzy preorder follows from Remark 2.12, hence according to Theorem 3.2, for all A ⊆ X,

τ A ∧ − R x, y Rd 1 d x,y∈A×A ∧ 1 −∧dB → 0 , by Remark 2.12 x,y∈A×A x,y∈B×B 4.7 ∧ ∨ 1 − dB → 0 x,y∈A×A x,y∈B×B

≥ 1 − dA → 0

≥ 1 − 1 − dA dA.

A ⊆ X τ A ≤ dA Secondly, we have to show that for any , Rd . t ∈ I τ A >t x ∈ A, y ∈ A N x ⊆ X Let 1 such that Rd . Then for any , there exists y such that x ∈ Nyx,y/∈ Nyx,anddNyx >t.LetNy ∪x∈A Nyx, then y /∈ Ny,A⊆ Ny, and dNyd∪x∈A Nyx ≥∧x∈A dNyx ≥ t,sinced is σ-generelized ideal. Obviously, A ∩y /∈ A Ny. So we would get dAd∩y /∈ A Ny ≥∨y /∈ A dNy ≥ t,byG3.ThusdA ≥ τ Rd , which completes the proof.

X Example 4.11. 1 Let D {μ ∈ I : μ ≤ 1/2}, and ∗ min, then

⎧ ⎨1,x≤ y, x −→ y 4.8 ⎩y, x > y. International Journal of Mathematics and Mathematical Sciences 15

Then D is a preideal on X,and RD x, y ∧ μx −→ μ y , ∀x, y ∈ X μ∈D ⎧ ⎨1,x y, 4.9 ⎩ 0,x/ y is a fuzzy preorder, hence τ μ ∈ IX R x, y ∧ μx ≤ μ y , ∀x, y ∈ X II . RD : D 4.10

Also, another fuzzy preorder RD may be dedeuced by D, Lemma 4.1: RD x, y inf μx :1− μ ∈D,μx <μ y , ∀x, y ∈ X I 1 inf μx : μ ∈ I ,μ≥ ,μx <μ y , ∀x, y ∈ X 2 ⎧ 4.11 ⎨⎪1,x y, ⎪1 ⎩ ,x/ y. 2

Then τ λ ∈ II R x, y ∧ λx ≤ λ y , ∀x, y ∈ I RD : D 4.12 I 1 λ ∈ I : λ ≥ {α ∈ I : α constant}. 2

X I 2 Let τ {μ ∈ I : μ ≥ 1/2}∪{α ∈ I constants},and∗ min, then Dτ {μ ∈ I : μ ≤ 1/2}, by applying Proposition 4.5.

Acknowledgment

The author is grateful to Professor Etienne Kerre for his warm encouragement.

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