# On Interval-Valued Supra-Fuzzy Syntopogenous Structure

Authors: F. M. Sleim; Heba I. Mustafa
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

A new definition of interval-valued supra-fuzzy syntopogenous (resp., supra-fuzzy proximity) space is given. We show that for any interval-valued fuzzy syntopogenous structureS, there is another interval-valued fuzzy syntopogenous structure Sc called the conjugate of S. This leads to introducing the concept of interval-valued bifuzzy syntopogenous space. Finally, we show every interval-valued bifuzzy syntopogenous space induces an interval-valued fuzzy supra-syntopogenous space. Throughout this paper, the family of all fuzzy sets on nonempty set X will be denoted by IX and the family of all interval-valued fuzzy sets on a nonempty set X will be denoted by πX.


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## Body

## 1. Introduction

In [1], Csaszar introduced the concept of a syntopogenous structure to develop a unified approach to the three main structures of set-theoretic-topology: topologies, uniformities, and proximities. This enables him to evolve a theory including the foundations of the three classical theories of topological spaces, uniform spaces, and proximity spaces. In the case of the fuzzy structures, there are at least three notions of fuzzy syntopogenous structures. The first notion worked out in [2–4] presents a unified approach to the theories of Chang fuzzy topological spaces [5], Hutton fuzzy uniform spaces [6], and Liu fuzzy proximity spaces [7]. The second notion worked out in [8] agrees very well with Lowen fuzzy topological spaces [9], Lowen-Hohle fuzzy uniform spaces [10], and Artico-Moresco fuzzy proximity spaces [11]. The third notion worked out in [12] agrees with the framework of a fuzzifying topology [13]. In [14], Kandil et al. introduced the concepts of a supra-fuzzy topological space and supra-fuzzy proximity space. In [15], Ghanim et al. introduced the concepts of supra-fuzzy syntopogenous space as a generalization of the concepts of supra-fuzzy proximity and supra-fuzzy topology.Interval-valued fuzzy sets were introduced independently by Zadeh [16], Grattan-Guiness [17], Jahn [18], and Sambuc [19], in the seventies, in the same year. An interval-valued fuzzy set (IVF) is defined by an interval-valued membership function. As a generalization of fuzzy, the concept of intuitionistic fuzzy sets was introduced by Atanassov [20]. In [21], it is shown that the concepts of interval-valued fuzzy sets and intuitionistic fuzzy sets are equivalent and they are extensions of fuzzy sets.Interval-valued fuzzy sets can be viewed as a generalization of fuzzy sets (Zadeh [22]) that may better model imperfect information which is omnipresent in any conscious decision making [23]. It can be considered as a tool for a more human consistent reasoning under imperfectly defined facts and imprecise knowledge, with an application in supporting medical diagnosis [24, 25]. In [26] new notions of interval-valued supra-fuzzy topology (resp., supra-fuzzy proximity) were introduced by using the notions of interval-valued fuzzy sets.In this paper, we generalize the concept of supra-fuzzy syntopogenous space by using the notion of interval-valued set. Topology and its generalization proximity and syntopogenous are branches of mathematics which have many real life applications. We believe that the generalized topological structure suggested in this paper will be important base for modification of medical diagnosis, decision making, and knowledge discovery. The other parts of this paper are arranged as follows. In Section2, We recall and develop some notions and notations concerning supra-fuzzy topological space, interval-valued fuzzy set, interval-valued fuzzy topology, interval-valued supra-fuzzy topology, and interval-valued supra-fuzzy proximity. In Section 3, we introduce the concept of interval-valued supra-fuzzy syntopogenous (resp., supra-fuzzy proximity) space. We show that there is one-to-one correspondence between family of all interval-valued supra-fuzzy topological spaces and the family of all perfect interval-valued supra-fuzzy topogenous spaces. Also, we prove that there is one-to-one correspondence between the family of all interval-valued supra-fuzzy proximity spaces and the family of all symmetrical interval-valued supra-fuzzy topogenous spaces. We show that for any interval-valued fuzzy topogenous order R on πX there is another interval-valued fuzzy topogenous structure Rc in Section 3. In Section 4, we introduce the concept of interval-valued bifuzzy syntopogenous structure by using two interval-valued fuzzy syntopogenous spaces. In the last section we show that with every interval-valued bifuzzy syntopogenous space there is an interval-valued supra-fuzzy syntopogenous associated with this space.


## 2.  Preliminaries

The concept of a supra-fuzzy topological space has been introduced as follows.Definition 2.1 (see [27]).
A collectionτ*⊂IX is a supra-fuzzy topology on X if 0, 1 ∈τ* and τ* is closed under arbitrary supremum. The pair (X,τ*) is a supra-fuzzy topological space.Definition 2.2 (see [20]).
An interval-valued fuzzy set (IVF set for short) is a setA_=(A1,A2)∈IX×IX such that A1≤A2. The family of all interval-valued fuzzy sets on a given nonempty set X will be denoted by πX.
The IVF set1_ = (1, 1) is called the universal IVF set and the IVF set 0_ = (0, 0) is called the empty IVF set.Proposition 2.3 (see [20]).
The operations onπX are given by the following: Let A_, B_∈πX:(1)
A_=B_⇔A1=B1,A2=B2,(2)
A_≤B_⇔A1≤B1,A2≤B2,(3)
A_∨B_=(A1∨B1,A2∨B2),(4)
A_∧B_=(A1∧B1,A2∧B2),(5)
A_c=(A2c,A1c).Definition 2.4 (see [26]).
The familyη⊆πX is called an interval-valued fuzzy topology (IVF tpology for short) on X if and only if η contains 1_, 0_, and it is closed under finite intersection and arbitrary union. The pair (X,η) is called an interval-valued fuzzy topological space (IVF top. space for short). Any IVF set A_∈η is called an open IVF set and the complement of A_ denoted by A_c is called a closed IVF set. The family of all closed interval-valued fuzzy sets is denoted by ηc.Definition 2.5 (see [26]).
A non empty familyη*⊆πX is called an interval-valued supra-fuzzy topology (IVSF top. for short) on X if it contains 0_, 1_ and it is closed under arbitrary unions.Definition 2.6 (see [26]).
A binary relationδ⊆πX×πX is called an interval-valued supra-fuzzy (IVSF) proximity on X if it satisfies the following conditions: (1)
A_δB_⇒B_δA_,(2)
A_δB_ or A_δC_⇒A_δ(B_∨C_),(3)
0_δ1_,(4)
A_δB_⇒A_≤1-B_,(5)
A_δB_⇒∃C∈πX s.t. A_δC_ and (1-C_)δB_.


## 3.  Interval-Valued Fuzzy Topogenous Order

In this article, we define a new concept of topogenous structure by using the concept of interval-valued fuzzy sets.Definition 3.1.
LetR be a binary relation on πX. Consider the following axioms: (1)
0_R0_,(2)
A_RB_⇒A_⊆B_,(3)
A_1≤A_RB_≤B_1⇒A_1RB_1,(4)
A_1RB_1 and A_2RB_2⇒(A_1∨A_2)R(B_1∨B_2) and (A_1∧A_2)R(B_1∧B_2),(5)
Ai_RBi_ for all i⇒(∨Ai_)R(∨Bi_),(6)
Ai_RBi_ for all i⇒(∧Ai_)R(∧Bi_),(7)
A_RB_⇔(1-B_)R(1-A_).If R satisfies (1)–(4) then it is called an interval-valued fuzzy topogenous order (IVF topogenous order for short) on X.
IfR satisfies (1)–(3) then it is called an interval-valued supra-fuzzy topogenous order (IVSF topogenous order for short) on X.
The interval-valued fuzzy topogenous order is called perfect (resp., biperfect, symmetrical) if it satisfies the condition (5) (resp., (6), (7)).Definition 3.2.
LetX≠ϕ and S be a nonempty family of IVF topogenous orders on X satisfy the following conditions: (s1) R1,R2∈S⇒∃R∈S s.t. R1≤RandR2≤R,(s2) R∈S⇒∃L∈S s.t. R≤L∘L.
ThenS is said to be interval-valued fuzzy (IVF) syntopogenous structure on X. The pair (X,S) is called an interval-valued fuzzy syntopogenous (IVF syntopogenous for short) space. If S satisfy the single element, then S is called an IVF topogenous structure on X and the pair (X,S) is called an IVF topogenous space.Definition 3.3.
LetX≠ϕ and S* be a nonempty family of IVSF topogenous orders on X satisfy the conditions (s1) and (s2). Then S* is called an IVSF syntopogenous space. If S* consists of single element, then it is called an IVSF topogenous structure. If each element of S* is perfect (resp., biperfect symmetrical), then it is called a perfect (resp., biperfect symmetrical) IVSF structure on X.The following proposition investigates the relation between the family of all IVSF topological spaces and the family of all perfect IVSF topogenous spaces.Proposition 3.4.
There is one-to-one correspondence between the family of all IVSF topological spaces and the family of all perfect IVSF topogenous spaces.Proof.
Let(X,τ*) be an IVSF topological space. Define a relation Rτ* by A_Rτ*B_⇔∃C∈τ* s.t. A_≤C_≤B_. Then Rτ* is a perfect IVSF topogenous order on X. In fact, since 0_,1_∈τ*, 0_≤0_≤0_,and1_≤1_≤1_, then 0_Rτ*0_and 1_Rτ*1_. Let A_Rτ*B_⇒∃C∈τ* s.t. A_≤C_≤B_⇒A_≤B_. Assume that A1≤A_Rτ*B_≤B1⇒∃E∈τ* s.t. A1_≤A_≤E_≤B_≤B1_⇒A1_Rτ*B1_. Thus, Rτ* is an IVSF topogenous order on X. We show that Rτ* is perfect. Let Ai_Rτ*Bi_⇒∃Ci_∈τ* s.t. Ai_≤Ci_≤Bi_. Hence ∨Ci_∈τ* and ∨Ai_≤∨Ci_≤∨Bi_. Thus, (∨Ai_)Rτ*(∨Bi_) and consequently Rτ* is perfect. Thus Rτ* is an IVSF topogenous order. Consequently, Sτ*={Rτ*} is a perfect IVSF topogenous structure.
Conversely, let (X,S*) be a perfect IVSF topogenous space. Then S*={R*}, where R* is a perfect IVSF order. Define τR*⊆πX by A_∈τR*⇔A_R*A_. We show that τR* is an IVSF topology on X. Since 0_R*0_ and 1_R*1_, then 0_,1_∈τR*. Also, Ai_∈τR* for all i⇒Ai_R*Ai_ for all i⇒∨Ai_R*∨A_i, because R* is perfect. Hence ∨Ai_∈τR* and, consequently, τR* is an IVSF topology on X.Remark 3.5.
One can easily show thatRτR*=R* and τRτ*=τ*.Example 3.6.
This example is a small form of interval-valued information table of a file containing some patientsX = {Li, Wang, Zhang, Sun}. We consider the set of symptoms S = {Chest-pain, Cough, Stomach-pain, Headache, Temperature}. Each symptom is described by an interval fuzzy sets on X. The symptoms are given in Table 1. Define a binary relation R on πX by A,B∈πXARB if and only if x≤y implies A(x)≤B(y) for all x∈X. Therefore, R is biperfect IVSF topogenous order on X. The set of patients are ordered by Li ≤ Wang ≤ Zhang≤ Sun. So, we have chest-pain R Cough.Table 1
Chest-pain
Cough
Stomach-pain
Headache
Temperature
Li
[0.1, 0.2]
[0.1, 0.3]
[0, 0.2]
[0.3, 0.9]
[0.1, 0.9]
Wang
[0.1, 0.2]
[0.2, 0.3]
[0.3, 0.9]
[0.3, 0.6]
[0.2, 0.2]
Zhang
[0.5, 0.5]
[0.2, 0.3]
[0.4, 0.4]
[0.1, 0.9]
[0.1, 0.9]
Sun
[0.3, 0.6]
[0.7, 0.8]
[0.3, 0.6]
[0.1, 0.6]
[0.3, 0.4]The following proposition shows the relation between the family of all IVSF proximity spaces and the family of all symmetrical IVSF topogenous spaces.Proposition 3.7.
There is one-to-one correspondence between the family of all IVSF proximity spaces and the family of all symmetrical IVSF topogenous spaces.Proof.
Letδ* be an IVSF proximity. Define A_Rδ*B_⇔A_δ*(1-B_). Then Rδ* is a symmetrical IVSF topogenous order. In fact, since 0_δ1_ and 1_δ0_, then 0_Rδ*0_ and 1_Rδ*1_. Also A_Rδ*B_⇒A_δ*1-B_⇒A_≤B_. Finally, A_1≤A_Rδ*B_≤B_1⇒A_1≤A_, A_δ*1-B_ and 1-B_≥1-B_1. Thus A_1δ*1-B_1 and therefore A_1Rδ*B_1. Consequently (X,S*), where S*={Rδ*} is IVSF topogenous space.
Conversely, let(X,S*) a symmetrical IVSF topogenous space. Then S*={R*}, where R* is a symmetrical IVSF topogenous order on X. Define δR* by A_δR*B_⇔A_R*1-B_. We show that δR* is an IVSF topogenous order on X. Assume that A_δR*B_, then A_R*(1-B_). Therefore B_R*(1-A_) because R* is symmetric. Hence B_δR*A_. Since 0R*0 and 1R*1, then 0δR*0. Assume that A_δR*B_ or A_δR*C_. Then A_R*(1-B_) or A_R*(1-C_). Suppose that A_δR*B_∨C_, then A_R*(1-(B_∨C_)). Thus A_R*(1-B_)∧(1-C_). Consequently A_R*(1-B_) and A_R*(1-C_). So, A_δR*B_ and A_δR*C_ a contradiction. Now A_δR*B_ implies A_R*(1-B_) and hence A_≤(1-B_).


## 4. Interval-Valued Bifuzzy Syntopogenous Space

Proposition 4.1.
LetR be an IVF topogenous order on X. Then the relation Rc on πX defined by A_RcB_⇔(1-B_)R(1-A_) is an IVF topogenous order on X.Proof.
Since0_R0_ and 1_R1_, then 1_Rc1_ and 0_Rc0_. Also, A_RcB_⇒(1-B_)R(1-A_)⇒1-B_≤1-A_⇒A_≤B_. Furthermore, A1≤A_RcB_≤B_1⇒(1-B_)R(1-A_), 1-B_1≤1-B_,1-A_≤1-A_1⇒(1-B_1)R(1-A_1)⇒A_1RcB_1. Finally, A_1RcB_1andA_2RcB2⇒(1-B_1)R(1-A_1) and (1-B_2)R(1-A_2). Since R is a topogenous order, then (1-B_1)∧(1-B_2)R(1-A_1)∧(1-A_2) and hence (1-(B_1∨B_2))R(1-(A_1∨A_2)). Consequently, (A_1∨A_2)Rc(B_1∨B_2). Similarly, we can show that (A_1∧A_2)Rc(B_1∧B_2). Thus Rc is an IVF topogenous order on X.Remark 4.2.
(1) The relationRc defined in the previous proposition is called the conjugate of R.
(2) IfR is perfect or biperfect, then Rc is also.Proposition 4.3.
LetR1 and R2 be two IVF topogenous orders on X. If R1⊆R2, then R1c⊆R2c and (R1∘R2)c=R2c∘R1c.Proof.
Assume that,R1⊆R2. Then A_R1cB_⇒(1-B_)R1(1-A_)⇒(1-B_)R2(1-A_)⇒A_R2cB_. Hence, R1c⊆R2c. Also, A_(R1∘R2)cB⇒(1-B_)(R1∘R2)(1-A_)⇒∃C_∈πX s.t. (1-B_)R1C_∧C_R2(1-A_)⇒∃C_∈πX s.t. (1-C_)R1cB_∧A_R2c(1-C_)⇒∃1-C_∈πX s.t. A_R2c1-C_∧(1-C_)R1cB_⇒A_(R2c∘R1c)B_⇒(R1∘R2)c≤R2c∘R1c. Similarly, we can show that R2c∘R1c≤(R1∘R2)c. Consequently, (R1∘R2)c=R2c∘R1c.Proposition 4.4.
LetS be an IVF syntopogenous structure on X. Then the family Sc={Rc:R∈S} is also an IVF syntopogenous structure.Proof.
SinceS≠ϕ, then Sc≠ϕ. Let R1c,R2c∈Sc, then R1,R2∈S. Hence ∃R∈S s.t. R1≤R and  R2≤R. Therefore ∃Rc∈Sc s.t. R1c≤Rc and  R2c≤Rc. Also, Rc∈Sc⇒R∈S⇒∃R1∈S s.t. R≤(R1∘R1)⇒Rc≤(R1∘R1)c=R1c∘R1c. Thus ∃R1c∈Sc s.t.  Rc≤R1c∘R1c. Consequently, Sc is also IVF syntopogenous structure on X.
Hence, we have the result that on a nonempty setX we have two interval-valued fuzzy syntopogenous structures.Definition 4.5.
LetX a nonempty set. Let S1,S2 be two IVF syntopogenous structures on X. The triple (X,S1,S2) is called an interval-valued bifuzzy syntopogenous space. If each S1,S2 is perfect (resp., biperfect symmetrical), then the space (X,S1,S2) is perfect (resp. biperfect symmetrical).
The following proposition shows that two IVF topogenous orders onX can induce an IVSF topogenous order on X.Proposition 4.6.
LetR1,R2 be two interval-valued fuzzy topogenous orders on X. Then the order R=R1∨R2 defined by
(4.1)ARB⟺AR1BorAR2B
is IVSF topogenous order on πX.Proof.
Since,0_Ri0_ and 1_Ri1_ for all  i=1,2, then 0_R0_ and 1_R1_. Also, A_1≤A_RB_≤B_1⇒A_1≤A_RiB_≤B_1 for some i=1,2⇒A_1RiB_1 for some i=1,2⇒A_1RB_1. Finally, A_RB_⇒A_RiB_ for some i=1,2⇒A_≤B_. Thus, R is an IVSF topogenous order on πX.


## 5. Interval-Valued Supra-Fuzzy Syntopogenous Space Associated with Interval-Valued Bifuzzy Syntopogenous Spaces

Proposition 5.1.
Let(X,S1,S2) be interval-valued bifuzzy syntopogenous space. Then the family S12*={R=R1∨R2:R1∈S1,R2∈S2} is an IVSF syntopogenous structure on X.Proof.
(s1) Let R,L∈S12*⇒R=R1∨R2 and  L=L1∨L2RiandLi∈Si for some i=1,2⇒∃K1∈S1 s.t. R1∨L1=K1 and ∃K2∈S1 s.t. R2∨L2=K2. So, ∃K=K1∨K2∈S12* s.t. R∨L≤K1∨K2=K.
(s2) R∈S12*⇒∃R1∈S1, R2∈S2 s.t. R=R1∨R2⇒∃L1∈S1 s.t. R1≤L1∘L1 and ∃L2∈S2 s.t. R2≤L2∘L2. Since L1,L2≤L1∨L2, then L1∘L1≤L1∨L2∘L1∨L2 and L2∘L2≤L1∨L2∘L1∨L2. Therefore R1≤L1∘L1≤L1∨L2∘L1∨L2 and R2≤L2∘L2≤L1∨L2∘L1∨L2. So R1∨R2≤(L1∨L2)∘(L1∨L2) and consequently S12* is an IVSF syntopogenous structure on X.Remark 5.2.
The structureS12* is called the interval-valued supra-fuzzy syntopogenous associated with the space (X,S1,S2).Proposition 5.3.
Let(X,Ti)i=1,2 be an IVF topological spaces. Then the family T12*={A_=A_1∨A_2:A_i∈Tii=1,2} is an IVSF topology on X.Proof.
Since0_,1_∈Ti for all i=1,2, then 0_,1_∈T12* Also, A_i∈T12*⇒A_i=B_1i∨B_2i, B_1i∈T1,B_2i∈T2⇒∨iA_i=∨i(B_1i∨B_2i)=(∨iB_1i)∨(∨iB_2i). Since, Tii=1,2 is an IVSF topology, then (∨iB_1i)∈T1 and (∨iB_2i)∈T2. Therefore, ∨iA_i∈T12* and consequently, T12* is an IVSF topology on X.Proposition 5.4.
Let(X,S1,S2) be an interval-valued bifuzzy syntopogenous space. Assume that Ti is the interval-valued fuzzy topology associated with Si for all i=1,2 and T12* is interval-valued supra-fuzzy topology associated with the space (X,T1,T2). Then T12* is the IVSF topology associated with the space (X,S12*), that is, T12*=TS12*.Proof.
The proof is obvious.Remark 5.5.
In crisp case, letR be a fuzzy topogenous order on IX. Define a binary relation Rp as follows:
(5.1)μRpρ⟺∃family{μi:i∈I}s.t.μ=∨iμi,μiRρ∀i.
Then, Rp is a perfect fuzzy topogenous order on X finer than R and coarser than any perfect fuzzy semitopogenous order on IX which is finer than R.
Let S be a fuzzy syntopogenous structure onX. Define a relation RS on IX by
(5.2)μRSρ⟺μRρforsomeR∈S.
The binary relation RS as defined above is a perfect topogenous order on IX. Let Rp be the coarsest perfect topogenous order on IX finer than each member of S. (X,RSp) is the topogenous space. Let Ro=RSp.


## 6. Conclusion

Computer scientists used the relation concepts in many areas of life such as artificial intelligence, knowledge discovery, decision making and medical diagnosis [28]. One of the important theories depend on relations is the rough set theory [29]. Many authors studied the topological properties of rough set [30, 31]. In this paper we generalized the topological concepts by introducing the notions of IVSF syntopogenous structures. We can use the relations obtained from an interval-valued information systems to generate an IVSF syntopogenous structure and use the mathematical properties of these structures to discover the knowledge in the information modelings.



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