Connecting Fuzzifying Topologies and Generalized Ideals by Means of Fuzzy Preorders

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 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.

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 let 𝑋,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 subset 𝐴⊆𝑋 also by 𝐴. If 𝜇∈1^𝑋, we define 𝜇^𝛼={𝑥∈𝑋∶𝜇(𝑥)>𝛼}. Let 2^𝑋 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.

A fuzzy relation on a nonempty set is a map 𝑅∶𝑋×𝑋→𝐼. A fuzzy relation 𝑅 is called

(1)reflexive if 𝑅(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 subset 𝐷 of 2^𝑋 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 subset 𝒟 of 𝐼^𝑋 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 preidael 𝒟 is called prime if it satisfies the condition;

∀𝜇,𝜆∈𝐼^𝑋suchthat𝜇𝜈∈𝒟,then𝜇∈𝒟or𝜈∈𝒟. (2.1)

A preideal𝒟 is called a 𝜎-preideal if the following hold.

If {𝜇_𝑛∶𝑛=1,2,3,…} is a countable subcollection of 𝒟, then sup_𝑛𝜇_𝑛∈𝒟.

If 𝒟 is a preideal on 𝑋, we define the characteristic, denoted by 𝑐(𝒟), of 𝒟 by

𝑐(𝒟)=sup_𝜇∈𝒟inf_𝑥∈𝑋𝜇(𝑥). (2.2)

If 𝒟 is a preideal and 𝑐(𝒟)<1, we say that 𝒟 is saturated if for every (𝜇_𝜀)∈𝒟^𝐼1, inf_𝜀∈𝐼1(𝜇_𝜀+𝜀)∈𝒟, equivalently, 𝒟 is saturated if it satisfies

∀𝜀<1,𝜇∈𝐼^𝑋∶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∈𝐼^𝑋. Then 𝜇∈𝒟⟺𝜇^𝛼𝛾∈𝒟,∀𝑐<𝛾<𝛼in𝐼₁. (2.4)

Proof. Let 𝜇∈𝒟. Since 𝜇^𝛼⋀𝛾≤𝜇 for all 𝛾<𝛼∈𝐼₁, then, by (P3), 𝜇^𝛼⋀𝛾∈𝒟 for all 𝛾<𝛼∈𝐼₁.
Conversely, let 𝜇∈𝐼^𝑋 such that 𝜇^𝛼⋀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 𝜇_2𝛼=sup_𝑛(𝜇^𝑛𝛼⋀(𝑛𝛼−𝛼)). Since 𝑛 is finite, then, by (P2), 𝜇_2𝛼∈𝒟. Also, it is easy to see that, for all 𝑥∈𝑋, 0≤𝜇(𝑥)−𝜇_2𝛼(𝑥)≤2𝛼, hence 𝜇+2𝛼≥𝜇_2𝛼+2𝛼≥𝜇. Consequently, 𝜇=∩_0<𝛼<1/2(𝜇_2𝛼+2𝛼)∈𝒟, since 𝒟 is saturated, so we are done.

It should be noticed that the required condition of Lemma 2.3 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 preideal 𝒟, on a subset 𝐴 of 𝑋 may be defined and denoted by 𝑐_𝐴(𝒟)=sup_𝜆∈𝒟inf_𝑥∈𝐴𝜆(𝑥). The following result is useful for our study.

Lemma 2.4. Let 𝒟 on 𝑋. Then, for every 𝐴∈2^𝑋, 𝑐_𝐴(𝒟)=sup{𝑡∈𝐼∶𝐴𝑡∈𝒟}, (2.5) hence, 𝑐=𝑐_𝑋(𝒟)=sup{𝑡∈𝐼∶𝑡∈𝒟}.

Proof. The statement holds when 𝐴=𝜙. Suppose 𝐴≠𝜙, and denote 𝑐=𝑐_𝐴(𝒟) and 𝑏=sup{𝑡∈𝐼∶𝐴⋀𝑡∈𝒟}. Let 𝑡∈𝐼 satisfy 𝐴⋀𝑡∈𝒟. Then 𝑐≥inf_𝑥∈𝐴(𝐴⋀𝑡)(𝑥)=𝑡, that is, 𝑐≥𝑏. Conversely, given 𝜆∈𝒟, let 𝛾=inf_𝑥∈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 function 𝑑∶2^𝑋→𝐼 is called a generalized 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∈2^𝑋;
(G3) if 𝐴⊂𝐵, then 𝑑(𝐴)≥𝑑(𝐵), for each 𝐴,𝐵∈2^𝑋;
The conditions (G2) and (G3) are equivalent to:
(G*) 𝑑(𝐴)⋀𝑑(𝐵)=𝑑(𝐴∪𝐵), for each 𝐴,𝐵∈2^𝑋. Moreover, Since 𝑑 a nonzero function, then by (G3), 𝑑(𝜙)>0.

A generalized ideal 𝑑, is called 𝜎-generalized ideal if 𝑑(∪_𝑖∈𝐽𝐴_𝑖)≥_𝑖∈𝐽𝑑(𝐴_𝑖),foreach(𝐴_𝑖)_𝑖∈𝐽⊆𝐼^𝑋, (2.6) and 𝑑 is called prime if

𝑑(𝐴∩𝐵)≤𝑑(𝐴)∨𝑑(𝐵),foreach𝐴,𝐵∈2^𝑋. (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. Let 𝑅 be a reflexive fuzzy relation on a Boolean lattice 𝑋 with 0. Define 𝑑₁,𝑑₂∶𝐼^𝑋→𝐼, for all 𝐴⊆𝑋 as follows: 𝑑₁(𝐴)=_{𝑥,𝑦∈𝐴}𝑑(1−𝑅(𝑥,1−𝑦));where1−𝑦isthecomplement𝑦,₂(𝐴)=_𝑥∈𝐴(1−𝑅(𝑥,0)). (2.8)(2.9) 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. Let 𝑑 be a generalized ideal on 𝑋. Define 𝒟₁,𝒟₂⊂𝐼^𝑋 by 𝒟₁={𝜇∈𝐼^𝑋∶∀𝐴∈2^𝑋,𝑑(𝐴)≤1−sup_𝑥∉𝐴𝒟𝜇(𝑥)};₂={𝜇∈𝐼^𝑋∶∀𝛼∈𝐼₁,𝑑(𝜇^𝛼)≥𝛼}. (2.10)(2.11) Then both 𝒟₁,𝒟₂ are saturated preideals on 𝑋.

Proof. First, we show that 𝒟₁ a preideal on 𝑋:
(P1) 1∉𝒟₁, otherwise for all 𝐴∈2^𝑋,𝑑(𝐴)≤1−sup_𝑥∉𝐴1(𝑥)=0, that is, 𝑑(𝐴)=0, which is a contradiction, since 𝑑 is a nonzero function;
(P2) let 𝜇,𝜆∈𝒟₁. Then for all 𝐴∈2^𝑋: 1−sup_𝑥∉𝐴(𝜇∨𝜆)(𝑥)=(1−sup_𝑥∉𝐴𝜇(𝑥))(1−sup_𝑥∉𝐴𝜆(𝑥))≥𝑑(𝐴)𝑑(𝐴)=𝑑(𝐴), (2.12) hence 𝜇∨𝜆∈𝒟₁.
(P3) Let 𝜇,𝜆∈𝐼^𝑋 such that 𝜇∈𝒟₁and𝜆≤𝜇. Then for all 𝐴∈2^𝑋, 1−sup_𝑥∉𝐴𝜆(𝑥)≥1−sup_𝑥∉𝐴𝜇(𝑥)≥𝑑(𝐴),hence𝜆∈𝒟₁. (2.13) Let (𝜇_𝜃)_𝜃∈𝐼0 be a family of fuzzy sets in 𝒟₁. Then for all 𝐴∈2^𝑋, 1−sup_𝑥∉𝐴(inf(𝜇_𝜃+𝜃))(𝑥)=1−sup_𝑥∉𝐴(inf_𝜃∈𝐼0(𝜇_𝜃(𝑥)+𝜃))≥1−inf_𝜃∈𝐼0sup_𝑥∉𝐴(𝜇_𝜃(𝑥)+𝜃)=sup_𝜃∈𝐼0(1−sup_𝑥∉𝐴(𝜇_𝜃(𝑥)+𝜃))=sup_𝜃∈𝐼0(1−(sup_𝑥∉𝐴𝜇_𝜃(𝑥)+𝜃))=sup_𝜃∈𝐼0((1−sup_𝑥∉𝐴𝜇_𝜃(𝑥))−𝜃)≥sup_𝜃∈𝐼0(𝑑(𝐴)−𝜃)=𝑑(𝐴), (2.14) therefore, inf_𝜃∈𝐼0(𝜇_𝜃+𝜃)∈𝒟₁, hence 𝒟₁ is saturated.
Second, we prove that 𝒟₂ is saturated preideal on 𝑋:
(P1) 1∉𝒟₂, otherwise 𝑑(1^𝛼)=𝑑(𝑋)=0≥𝛼̸, for all 𝛼∈𝐼₁;
(P2), (P3) direct.
Let (𝜇_𝜃)_𝜃∈𝐼0 be a family of fuzzy sets in 𝒟₂. Then 𝑑((inf_𝜃∈𝐼0(𝜇_𝜃+𝜃))^𝛼)≥𝑑(_𝜃∈𝐼0(𝜇_𝜃+𝜃)^𝛼)≥sup_𝜃∈𝐼0𝑑(𝜇_𝜃^𝛼−𝜃),(by(G3))≥sup_𝜃∈𝐼0(𝛼−𝜃)=𝛼. (2.15) Hence 𝒟₂ is saturated preideal.

Proposition 2.8. Let 𝑑 be a prime generalized ideal on 𝑋. Define 𝒟₃,𝒟₄⊂𝐼^𝑋 by 𝒟₃={𝜇∈𝐼^𝑋∶∀𝛼∈𝐼₁,𝑑((1−𝜇)^𝛼𝒟)≤𝛼};₄={𝜇∈𝐼^𝑋∶∀𝐴∈2^𝑋,𝑑(𝐴)≤sup_𝑥∉𝐴(1−𝜇)(𝑥)}. (2.16) Then 𝒟₃,𝒟₄ are prime saturated preideals on 𝑋. Moreover 𝒟₃=𝒟₄.

Proof. First, we show that 𝒟₃=𝒟₄. Let 𝜇∈𝐼^𝑋 such that 𝑑((1−𝜇)^𝛼)≤𝛼, for all 𝛼∈𝐼₁. If 𝐴∈2^𝑋, we assume that sup_𝑥∉𝐴(1−𝜇)(𝑥)=𝛾∈𝐼₁, hence (1−𝜇)^𝛾⊆𝐴.
Then, by (G3), 𝑑(𝐴)≤𝑑((1−𝜇)^𝛾)≤𝛾=sup_𝑥∉𝐴(1−𝜇)(𝑥), that implies 𝒟₃⊆𝒟₄.
Conversely, let 𝜇∈𝐼^𝑋 such that 𝑑(𝐴)≤sup_𝑥∉𝐴(1−𝜇)(𝑥), for all 𝐴∈2^𝑋. Then 𝑑((1−𝜇)^𝛼)≤sup_{𝑥∉(1−𝜇)}^𝛼(1−𝜇)(𝑥)≤𝛼, for all 𝛼∈𝐼₁ this implies 𝒟₄⊆𝒟₃, thus 𝒟₃=𝒟₄.
Second, we show that 𝒟₃ is a saturated prime preideal.
(P1) Suppose that 1∈𝒟₃. Let 𝑑(𝜙)=𝛾>0. Then, for all 𝛼<𝛾,𝑑(0^𝛼)=𝑑(𝜙)=𝛾>𝛼, which is a contradiction, so 1∉𝒟₃.
(P2) Let 𝜇,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) Let 𝜇,𝜆∈𝐼^𝑋 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 that 𝒟₄ is saturated. Suppose for all 𝜀>0,𝜇−𝜀∈𝐷₃. Then for all 𝐴∈2^𝑋, 𝑑(𝐴)≤sup_𝑥∉𝐴(1−(𝜇−𝜀))(𝑥)=sup_𝑥∉𝐴(1−𝜇)(𝑥)+𝜀, hence 𝑑(𝐴)≤sup_𝑥∉𝐴(1−𝜇)(𝑥), that is, 𝒟₄ is saturated.
Finally, we show that 𝒟₃ is prime. Let 𝜇,𝜆∈𝐼^𝑋 such that 𝜇⋀𝜆∈𝒟₃. Then for all 𝛼∈𝐼₁,𝑑(1−𝜇^𝛼)⋀𝑑(1−𝜆^𝛼)≤𝑑((1−𝜇^𝛼)∪(1−𝜆^𝛼))=𝑑(1−(𝜇^𝛼⋀𝜆^𝛼))=𝑑(1−(𝜇⋀𝜆)^𝛼)≤𝛼, hence, 𝑑(1−𝜇^𝛼)≤𝛼 or 𝑑(1−𝜆^𝛼)≤𝛼, that implies 𝜇∈𝒟₃ or 𝜆∈𝒟₃.

Proposition 2.9. Let 𝒟 be a preideal on 𝑋. Define 𝑑∶2^𝑋→𝐼 by ∀𝐴∈2^𝑋,𝑑(𝐴)=sup{𝑡∈𝐼∶𝑡>𝑐,𝐴𝑡∈𝒟}. (2.17) Then 𝑑 is a generalized ideal on 𝑋.

Proof. (G1) 𝑑(𝑋)=0, since 𝑡∈𝒟 implies 𝑡≤𝑐.
(G2) Let 𝐴,𝐵∈2^𝑋 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 set 𝑋 and the set of generalized ideals on 𝑋.

Proof. Let 𝑑 be a generalized ideal on 𝑋. Denote the preideal 𝒟₂ generated by 𝑑, defined as in Proposition 2.7, by 𝒟_𝑑, and denote the generalized ideal, generated by 𝒟_𝑑, as in Proposition 2.9, by 𝑑_𝒟𝑑, hence for all 𝐴∈2^𝑋𝑑_𝒟𝑑(𝐴)=sup{𝑡∈𝐼₁∶𝑡>𝑐,𝐴𝑡∈𝒟_𝑑},Proposition2.9=sup{𝑡∈𝐼₁∶𝑡>𝑐,𝑑((𝐴𝑡)^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). (2.18) Conversely, let 𝒟 be a saturated preideal on 𝑋. Denote the generalized ideal generated by 𝒟, as in Proposition 2.9, by 𝑑_𝒟, and denote the preideal generated by 𝑑_𝒟, as in (2.11) in Proposition 2.7 by 𝒟_𝑑𝒟. Then, 𝒟_𝑑𝒟={𝜆∈𝐼^𝑋∶𝑑_𝒟(𝜆^𝛼)≥𝛼,∀𝛼∈𝐼₁}={𝜆∈𝐼^𝑋∶sup{𝑡∈𝐼₁,𝑡>𝑐,𝜆^𝛼𝑡∈𝒟}≥𝛼,∀𝛼∈𝐼₁}={𝜆∈𝐼^𝑋∶𝜆^𝛼𝑡∈𝒟,∀c<𝑡<𝛼}={𝜆∈𝐼^𝑋∶𝜆∈𝒟},byLemma2.3=𝒟. (2.19) This completes the proof.

Definition 2.11. A nonzero function 𝔡∶𝐼^𝑋→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∈𝐼^𝑋;
(d3) if 𝜆≥𝜇, then 𝔡(𝜇)≥𝔡(𝜆).
Clearly, a generalized ideal 𝑑 on a set 𝑋 can be regarded as a fuzzy 𝐼-ideal whose domain is 2^𝑋. Precisely, 𝑑(𝐴)=𝔡(𝐴) for each 𝐴⊆𝑋; otherwise 𝑑(𝐴)=0.

Remark 2.12 (Fang and Chen [8]). Defined a generalized fuzzy preorder 𝑅_𝜙, corresponding to a map 𝜙∶𝐼^𝑋→𝐼 as follows: ∀(𝑥,𝑦)∈𝑋×𝑋,𝑅_𝜙(𝑥,𝑦)=_{𝜇∈𝐼𝑋}𝜙(𝜇)→(𝜇(𝑥)→𝜇(𝑦)), (2.20) hence, for a fuzzy 𝐼-ideal 𝔡 on 𝑋, there corresponds a fuzzy preorder 𝑅_𝔡 defined by ∀(𝑥,𝑦)∈𝑋×𝑋,𝑅_𝔡(𝑥,𝑦)=_{𝜇∈𝐼𝑋}𝔡(𝜇)→(𝜇(𝑥)→𝜇(𝑦)). (2.21) 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, 𝑅_𝑑(𝑥,𝑦)=_{𝐴⊆𝑋(𝑥,𝑦)∈𝐴′×𝐴}(𝑑(𝐴)→0). (2.22)

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.

An 𝐼-topology on a set 𝑋 is a crisp subset 𝒯 of 𝐼^𝑋 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 Chang 𝐼-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∶𝐼^𝑋→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∈𝐼^𝑋;(3)ℑ(∨_𝑗∈𝐽𝜇_𝑖)≥⋀_𝑗∈𝐽ℑ(𝜇_𝑗), 𝑗∈𝐽,𝜇_𝑗∈𝐼^𝑋.

By the conditions (1) and (2), we get for 𝜇∈𝐼^𝑋 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∈𝐼^𝑋, and is called homogeneous if the following condition holds: for 𝜇∈𝐼^𝑋 and 𝛼∈𝐼, ℑ(𝜇⋀𝛼)=ℑ(𝜇).

If a map 𝜏∶2^𝑋→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:

(4) for all {𝐴_𝑗}_𝑗∈𝐽⊆2^𝑋,𝜏(∩_𝑗∈𝐽𝐴_𝑗)≥⋀_𝑗∈𝐽𝜏(𝐴_𝑗), then 𝜏 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;(I3)(𝑥→𝑦)∗(𝑦→𝑧)≤𝑥→𝑧;(I4)𝑥→⋀_𝑗∈𝐽𝑦_𝑗=⋀_𝑗∈𝐽(𝑥→𝑦_𝑗), hence, 𝑥→𝑦≤𝑥→𝑧 if 𝑦≤𝑧;(I5)(∨_𝑗∈𝐽𝑦_𝑗)→𝑦=⋀_𝑗∈𝐽(𝑦_𝑗→𝑦), hence 𝑥→𝑧≥𝑦→𝑧 if 𝑥≤𝑦;(I6)𝑥∗∨_𝑗∈𝐽𝑦_𝑗=∨_𝑗∈𝐽(𝑥∗𝑦_𝑗);(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 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 𝒯_𝑅={𝜇∈𝐼^𝑋∶𝑅(𝑥,𝑦)∗𝜇(𝑥)≤𝜇(𝑦),forall𝑥,𝑦∈𝑋}, is an 𝐼-Alexandrov topology induced by 𝑅, that is, 𝒯_𝑅 satisfies the folowing properties: for all F⊆𝒯_𝑅,𝜇∈𝒯_𝑅, and 𝛼∈𝐼,
(a) every constant fuzzy set 𝑋→𝐼 belongs to 𝒯_𝑅;
(b)∨ℱ∈𝒯_𝑅;
(c)⋀ℱ∈𝒯_𝑅;
(d)𝛼∗𝜇∈𝒯_𝑅;
(e)𝛼→𝜇∈𝒯_𝑅.
Moreover, 𝑅 satisfies the equality 𝑅(𝑥,𝑦)=_{𝜇∈𝒯𝑅}𝜇(𝑥)→(𝑦). (3.1) Conversely, for a given 𝐼-Alexandrov topology 𝒯 a set 𝑋, there is a unique ∗-fuzzy preorder 𝑅_𝒯, given by 𝑅_𝒯(𝑥,𝑦)=⋀_𝜇∈𝒯𝜇(𝑥)→𝜇(𝑦), such that 𝒯=𝒯_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 𝜏_𝑅 on 𝑋, given by for all 𝐴⊆𝑋, 𝜏_𝑅(𝐴)=_{(𝑥,𝑦)∈𝐴×𝐴′}(1−𝑅(𝑥,𝑦)), (3.2) conversely, every fuzzifying topology 𝜏 on a set 𝑋 corresponds a fuzzy preorder 𝑅_𝜏, given by: for all 𝑥,𝑦∈𝑋, 𝑅_𝜏(𝑥,𝑦)=_{(𝑥,𝑦)∈𝐴×𝐴′}(1−𝜏(𝐴)). (3.3) 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.

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]). An 𝐼-topological space (𝑋,𝜏) is called a fuzzy neighborhood space if whenever 𝜇∈𝒯, for any 𝛼∈𝐼₁ also 𝛼⋀𝜇^𝛼∈𝒯.
Equivalently, if whenever 𝜇∈𝒯, for any 𝛼,𝑡∈𝐼₁, 𝑡<𝛼 also 𝜇^𝛼⋀𝑡∈𝒯.

Proposition 3.5. Let 𝜏 be a fuzzifying topology on a set 𝑋. Then 𝜏 corresponds an 𝐼-topology 𝒯_𝜏 on 𝑋, defined by 𝒯_𝜏={𝜇∈𝐼^𝑋∶𝜏(𝜇^𝛼)≥𝛼,forall𝛼∈𝐼₁}, and (𝑋,𝒯_𝜏) is a fuzzy neighborhood space.
Conversely, let 𝒯 be an 𝐼-topology on 𝑋. Then 𝜏 corresponds a fuzzifying topology 𝜏_𝒯, on 𝑋, defined by: for all 𝐴⊆𝑋, 𝜏_𝒯(𝐴)=sup{𝛼∈𝐼∶𝐴⋀𝛼∈𝒯}. Moreover,
(1)𝜏_𝒯𝜏=𝜏,
(2)𝒯_𝜏𝒯≤𝒯 and if (𝑋,𝒯) is a fuzzy neighborhood space, then 𝒯_𝜏𝒯=𝒯.

Proof. That is, 𝒯_𝜏 and 𝜏_𝒯 are 𝐼-topology and fuzzifying topology on 𝑋, respectively, and are easy to be verified. To prove that (𝑋,𝒯_𝜏) is a fuzzy neighborhood space, we suppose 𝛼,𝑡∈𝐼 and 𝜇∈𝒯_𝜏, then 𝜏((𝜇^𝑡𝑡)^𝛼)=𝜏(𝜙)=1,if𝛼≥𝑡,𝜏(𝜇^𝑡)≥𝑡,if𝑡>𝛼. (3.4) Therefore, 𝜏((𝜇^𝑡⋀𝑡)^𝛼)≥𝛼,forall𝛼∈𝐼₁, that is, 𝜇^𝑡⋀𝑡∈𝒯_𝜏,forall𝑡∈𝐼₁, so we are done.
To prove (1), let 𝜏 be a fuzzifying topology on 𝑋 then for all ⋀⊆𝑋, 𝜏_𝒯𝜏(𝐴)=sup{𝛼∈𝐼∶𝐴𝛼∈𝒯_𝜏}=sup{𝛼∈𝐼∶𝜏((𝐴𝛼)^𝑡)≥𝑡,∀𝑡∈𝐼₁}=sup{𝛼∈𝐼∶𝜏(𝐴)≥𝑡,∀𝑡<𝛼},since𝜏(𝜙)=1=𝜏(𝐴). (3.5)
To prove (2), suppose that (𝑋,𝒯) is a fuzzy neighborhood space. Then 𝒯_𝜏𝒯={𝜇∈𝐼^𝑋∶𝜏_𝒯(𝜇^𝛼)≥𝛼,∀𝛼∈𝐼₁}={𝜇∈𝐼^𝑋∶sup{𝑡∈𝐼∶𝜇^𝛼𝑡∈𝒯}≥𝛼,∀𝛼∈𝐼₁}={𝜇∈𝐼^𝑋∶𝜇^𝛼𝑡∈𝒯,∀t<𝛼,𝛼∈𝐼₁}≤𝒯,={𝜇∈𝐼^𝑋∶𝜇∈𝒯}=𝒯,if𝒯isafuzzyneighborhoodstructure. (3.6)

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.

Definition 3.7. Given (𝑋,ℑ), an 𝐼-fuzzy topological space, and 𝛼∈𝐼₀, then ℑ_𝛼∶={𝜇∈𝐼^𝑋∶ℑ(𝜇)≥𝛼} is an 𝐼-topology on 𝑋, called the 𝛼-level 𝐼-topology of ℑ.
Consequently, 𝒯_ℑ∶=ℑ₁={𝜇∈𝐼^𝑋∶ℑ(𝜇)=1}, by which we will give an interrelation between 𝐼-topologies and 𝐼-fuzzy topologies.

Proposition 3.8. Let 𝒯 be an 𝐼-topology on 𝑋. Then for all 𝜇∈𝐼^𝑋, ℑ_𝒯(𝜇)∶=sup{𝛼∈𝐼₁∶𝜇⋀𝛼∈𝒯} is a homogeneous 𝐼-fuzzy topology on 𝑋.

Proof. That is, ℑ_𝒯 an 𝐼-fuzzy topology is easy to be verified. Now, let 𝜇∈𝐼^𝑋,𝛼∈𝐼, then we get ℑ_𝒯(𝜇𝛼)∶=sup{𝑡∈𝐼₁∶𝜇𝛼𝑡∈𝒯}≤sup{𝑡∈𝐼₁∶𝑡≤𝛼,𝜇𝛼𝑡∈𝒯}=sup{𝑡∈𝐼₁∶𝜇𝑡∈𝒯}=ℑ_𝒯(𝜇). (3.7) Since ℑ_𝒯 is an 𝐼-fuzzy topology, then ℑ_𝒯(𝜇⋀𝛼)≥ℑ(𝜇), 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 ℑ_𝒯 is a homogeneous 𝐼-fuzzy topology on 𝑋, by Proposition 3.8, and 𝒯_ℑ𝒯(𝜇)={𝜇∈𝐼^𝑋∶ℑ_𝜏(𝜇)=1}={𝜇∈𝐼^𝑋∶sup{𝑡∈𝐼₁∶𝜇𝑡∈𝒯}=1}={𝜇∈𝐼^𝑋∶𝜇𝑡∈𝒯,𝑡<1}=𝒯. (3.8)
Conversely, if ℑ is a homogeneous 𝐼-fuzzy topologies on 𝑋, then 𝒯_ℑ is an 𝐼-topology and ℑ_𝒯ℑ(𝜇)=sup{𝑡∈𝐼₁∶𝜇𝑡∈𝒯_ℑ}=sup{𝑡∈𝐼₁∶ℑ(𝜇𝑡)=1}=1,sinceℑ(𝜇)=ℑ(𝜇𝑡). (3.9) So, we are done.

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.

Lemma 4.1. Let Ω⊆𝐼^𝑋. Define 𝑅_Ω,𝑅^′_Ω∶𝑋×𝑋→𝐼 by for all (𝑥,𝑦)∈𝑋×𝑋,𝑅_Ω(𝑥,𝑦)=inf{𝜇(𝑥)→𝜇(𝑦)∶𝜇∈𝐼^𝑋𝑅,𝜇∈Ω},_′Ω(𝑥,𝑦)=inf{𝜇(𝑦)→𝜇(𝑥)∶𝜇∈𝐼^𝑋,1−𝜇∈Ω}. (4.1) Then 𝑅_Ω,𝑅^′_Ω are fuzzy preorders on 𝑋, especially, if ∗=min=⋀, then for all (𝑥,𝑦)∈𝑋×𝑋, 𝑅_Ω𝑅(𝑥,𝑦)=inf{𝜇(𝑦)∶𝜇∈Ω,𝜇(𝑥)>𝜇(𝑦)},_′Ω(𝑥,𝑦)=inf{𝜇(𝑥)∶1−𝜇∈Ω,𝜇(𝑥)<𝜇(𝑦)}. (4.2)

Proposition 4.2. Let 𝒯 be an 𝐼-topology on a set 𝑋, and 𝒟 be a preideal on 𝑋. Define 𝒯^∗(𝒟)⊆𝐼^𝑋, 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.

However 𝒯^∗(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∈𝐼^𝑋∶𝜇≤sup{𝛼∈𝐼∶𝛼⋀(1−𝜇)∈𝒟}}. Consequently, 𝒯^∗(𝒟) has the following property: for all 𝜇∈𝐼^𝑋 such that 1−𝜇∈𝒟, then 𝜇∈𝒯^∗(𝒟).

Proposition 4.4. Let 𝒟 be a preideal on a set 𝑋. Define ℑ∶𝐼^𝑋→𝐼 by for all 𝜇∈𝐼^𝑋, 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∈𝐼^𝑋. Then ℑ(𝜇⋀𝜆)≥ℑ(𝜇)⋀ℑ(𝜆), by (P2) and (I12). Let {𝜇_𝑖}_𝑖∈𝐽⊆𝐼^𝑋. Then ℑ(∨_𝑖∈𝐽𝜇_𝑖)≥⋀_𝑖∈𝐽ℑ(𝜇_𝑖), by (I5) 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 ℑ_𝒟. It is easy to see that ℑ_𝒟 satisfy the following simple properties:

(1)for all 𝐴⊆𝑋, ℑ_𝒟(𝐴)=1;(2)for all 𝜇∈𝐼^𝑋 and 𝑥∈𝑋 such that (1−𝜇(𝑥))∈𝒟, then ℑ_𝒟(𝜇)=1;(3)restricting the range of ℑ on {0,1}, a Chang 𝐼-topology, associated with a preideal 𝒟, is obtained, namely, ℑ_𝒟={𝜇∈𝐼^𝑋∶𝜇(𝑥)≤sup{𝛼∈𝐼∶𝛼(1−𝜇(𝑥))∈𝒟},∀𝑥∈𝑋}. (4.3)

Proposition 4.5. Let 𝒯 be an 𝐼-topology on a Boolean lattice 𝑋 with 0. Let 𝒟⊆𝐼^𝑋, defined as 𝒟={𝜇∈𝐼^𝑋∶sup_{𝑥∈𝜇𝛼}inf_𝜆∈𝒯(𝜆(𝑥)→𝜆(0))≤1−𝛼,forall𝛼∈𝐼₁}. Then 𝒟 is a saturated preideal on 𝑋 and 𝒟={𝜇∈𝐼^𝑋∶𝑑_𝑅𝒯(𝜇^𝛼)≥𝛼,forall𝛼∈𝐼₁}, where 𝑑_𝑅𝒯 is the generalized ideal corresponding to 𝑅_𝒯.

Proof. Let 𝒯 be an 𝐼-topology on 𝑋. Then 𝑅_𝒯 is a fuzzy preorder, by Theorem 3.1, hence 𝑑_𝑅𝒯 is a generalized ideal, by (2.9) in Proposition 2.6. Thus, 𝒟={𝜇∈𝐼^𝑋∶𝑑_𝑅𝑇(𝜇^𝛼)≥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: 𝒟={𝜇∈𝐼^𝑋∶𝑑_𝑅𝒯(𝜇^𝛼)≥𝛼}={𝜇∈𝐼^𝑋∶_{𝑥∈𝜇𝛼}(1−𝑅_𝒯(𝑥,0))≥𝛼,∀𝛼∈𝐼₁},by(2.9)inProposition2.6={𝜇∈𝐼^𝑋∶∨_{𝑥∈𝜇𝛼}𝑅_𝒯(𝑥,0)≤1−𝛼,∀𝛼∈𝐼₁}={𝜇∈𝐼^𝑋∶∨_{𝑥∉𝜇𝛼𝜆∈𝒯}(𝜆(𝑥)→𝜆(0))≤1−𝛼,∀𝛼∈𝐼₁},byTheorem3.1, (4.4) 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.

Proposition 4.6. Let 𝒟⊆𝐼^𝑋 be a preideal. Define 𝒯_𝒟 by 𝒯_𝒟={𝜆∈𝐼^𝑋∶(_𝜇∈𝒟𝜇(𝑥)→𝜇(𝑦))≤𝜆(𝑥)→𝜆(𝑦)}. (4.5) Then 𝒯_𝒟 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∈𝐼^𝑋∶ℑ(𝜆)=1,forall𝜆≤𝜇}, then 𝒟 is a preideal on 𝑋.

Proof. Straightforward.

Proposition 4.8. Let 𝒟 be a preideal on a set 𝑋. Define 𝒯⊆𝐼^𝑋 by 𝒯={𝜆∈𝐼^𝑋∶𝜆=𝜇∨𝛼,𝜇∈𝒟,𝛼∈𝐼}. Then 𝒯 is an 𝐼-topology if 𝒟 is 𝜎-preideal.

Proof. Since 0∈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 {𝜆_𝑖}_𝑖∈𝐽⊆𝒯, then for all 𝑖∈𝐽, there exists 𝜇_i∈𝒟, 𝛼_𝑖∈𝐼 such that 𝜆_𝑖=𝜇_𝑖∨𝛼_𝑖. Therefore, we get ∨_𝑖∈𝐽𝜆_𝑖=(∨_𝑖∈𝐽𝜇_𝑖)∨(∨_𝑖∈𝐽𝛼_𝑖)∈𝒯, since 𝒟 is a 𝜎-preideal.

Proposition 4.9. Let 𝜏 be a fuzzifying topology on a Boolean lattice. Define 𝑑∶2^𝑋→𝐼 by for all 𝐴⊆𝑋, 𝑑(𝐴)=⋀_{𝑥,𝑦∈𝐴}∨_{𝐵⊆𝑋,𝑥,𝑦′∈𝐵}𝜏(𝐵). Then 𝑑 is a generalized ideal on 𝑋.
Conversely, let 𝑑 be a generalized ideal on 𝑋, then 𝜏_𝑑∶2^𝑋→𝐼, defined by for all 𝐴⊆𝑋, 𝜏(𝐴)=_{(𝑥,𝑦)∈𝐴×𝐴′}∨_{𝐵⊆𝑋′(𝑥,𝑦)∈𝐵×𝐵}𝑑(𝐵) (4.6) 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 𝑅_𝑑∶𝑋×𝑋→𝐼 by for all (𝑥,𝑦)∈𝑋×𝑋, 𝑅_𝑑(𝑥,𝑦)=⋀_{(𝑥,𝑦)∈𝐵×𝐵′}(1−𝑑(𝐵)).
Then, 𝑅_𝑑 is a fuzzy preorder and for all 𝐴⊂𝑋, 𝜏_𝑅𝑑(𝐴)=𝑑(𝐴).

Proof. That 𝑅_𝑑 is a fuzzy preorder follows from Remark 2.12, hence according to Theorem 3.2, for all 𝐴⊆𝑋, 𝜏_𝑅𝑑(𝐴)=_{(𝑥,𝑦)∈𝐴×𝐴′}(1−𝑅_𝑑=(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)). (4.7)
Secondly, we have to show that for any 𝐴⊆𝑋, 𝜏_𝑅𝑑(𝐴)≤𝑑(𝐴).
Let 𝑡∈𝐼₁ such that 𝜏_𝑅𝑑(𝐴)>𝑡. Then for any 𝑥∈𝐴,𝑦∈𝐴^′, there exists 𝑁_𝑦(𝑥)⊆𝑋 such that 𝑥∈𝑁_𝑦(𝑥),𝑦∉𝑁_𝑦(𝑥), and 𝑑(𝑁_𝑦(𝑥))>𝑡. Let 𝑁_𝑦=∪_𝑥∈𝐴𝑁_𝑦(𝑥), then 𝑦∉𝑁_𝑦,𝐴⊆𝑁_𝑦, and 𝑑(𝑁_𝑦)=𝑑(∪_𝑥∈𝐴𝑁_𝑦(𝑥))≥⋀_𝑥∈𝐴𝑑(𝑁_𝑦(𝑥))≥𝑡, since 𝑑 is 𝜎-generelized ideal. Obviously, 𝐴=∩_𝑦∉𝐴𝑁_𝑦. So we would get 𝑑(𝐴)=𝑑(∩_𝑦∉𝐴𝑁_𝑦)≥∨_𝑦∉𝐴𝑑(𝑁_𝑦)≥𝑡, by (G3). Thus 𝑑(𝐴)≥𝜏_𝑅𝑑, which completes the proof.

Example 4.11. (1) Let 𝒟={𝜇∈𝐼^𝑋∶𝜇≤1/2}, and ∗=min, then 𝑥→𝑦=1,𝑥≤𝑦,𝑦,𝑥>𝑦. (4.8) Then 𝒟 is a preideal on 𝑋, and 𝑅_𝒟(𝑥,𝑦)=_𝜇∈𝒟=𝜇(𝑥)→𝜇(𝑦),∀𝑥,𝑦∈𝑋1,𝑥=𝑦,0,𝑥≠𝑦 (4.9) is a fuzzy preorder, hence 𝜏_𝑅𝒟={𝜇∈𝐼^𝑋∶𝑅_𝒟(𝑥,𝑦)𝜇(𝑥)≤𝜇(𝑦),∀𝑥,𝑦∈𝑋}=𝐼^𝐼. (4.10)
Also, another fuzzy preorder ℜ_𝒟 may be dedeuced by 𝒟, Lemma 4.1: ℜ_𝒟(𝑥,𝑦)=inf{𝜇(𝑥)∶1−𝜇∈𝒟,𝜇(𝑥)<𝜇(𝑦),∀𝑥,𝑦∈𝑋}=inf{𝜇(𝑥)∶𝜇∈𝐼^𝐼1,𝜇≥2=1,𝜇(𝑥)<𝜇(𝑦),∀𝑥,𝑦∈𝑋}1,𝑥=𝑦,2,𝑥≠𝑦. (4.11) Then 𝜏_ℜ𝒟={𝜆∈𝐼^𝐼∶ℜ_𝒟(𝑥,𝑦)𝜆(𝑥)≤𝜆(𝑦),∀𝑥,𝑦∈𝐼}={𝜆∈𝐼^𝐼1∶𝜆≥2}{𝛼∈𝐼∶𝛼constant}. (4.12)
(2) Let 𝜏={𝜇∈𝐼^𝑋∶𝜇≥1/2}∪{𝛼∈𝐼constants}, and ∗=min, then 𝒟_𝜏={𝜇∈𝐼^𝐼∶𝜇≤1/2}, by applying Proposition 4.5.

Acknowledgment

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