# New Branch of Intuitionistic Fuzzification in Algebras with Their Applications

Authors: Samaher Adnan Abdul-Ghani; Shuker Mahmood Khalil; Mayadah Abd Ulrazaq; Abu Firas Muhammad Jawad Al-Musawi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

---

## Abstract

The intuitionistic fuzzification inρ-algebras about the concepts of ideals and subalgebras given with several related characterizations is considered. Some new concepts like intuitionistic fuzzy ρ-ideal (IFρi), intuitionistic fuzzy ρ-subalgebra (IFρs), ρ-homomorphism, and intuitionistic fuzzy ρ¯-ideal (IFρ¯i) are introduced and some of their descriptions are given in this work. Further, we show some applications on the family of all intuitionistic fuzzy ρ-subalgebras IFρS(R) in ρ-algebra like the binary relations ≈μ, ≈ν and Γr on IFρS(R). Also, their equivalence classes are given and studied.


---

## Body

## 1. Introduction

The fuzzy set (FS) as suggested by Zadeh [1] in 1965 is a regulation to vagueness and encounter uncertainty. A FS maps each element of the universe of discourse to the interval [0,1]. After the introduction of fuzzy sets theory by him, many mathematicians were conducted on the generalizations of the this concept and studied in the groups, algebras, and soft spaces (see [2–5]). By including a fuzzy set the degree of nonmembership, Atanassov [6] in 1986 suggested the intuitionistic fuzzy set (IFS), which seems more precise for provides opportunities and uncertainty quantification to accurately model a problem based on existing knowledge and monitoring. Also, this notion is discussed in different fields (see [7–11])B C K -algebra, class of algebra of logic, was investigated by Imai and Iseki [12]. After that, the notion of d-algebras was investigated by Neggers and Kim [13]. In 2017, the concepts of ρ-algebra, ρ¯-ideal, ρ-ideal, ρ-subalgebra, and permutation topological ρ-algebra were first proposed by Mahmood and Abud Alradha [14]. Next, they showed the notion of the soft ρ-algebra and soft edge ρ-algebra [15].In the present work, the notions of intuitionistic fuzzyρ-ideal (IFρi), intuitionistic fuzzy ρ-subalgebra (IFρs), ρ-homomorphism, and intuitionistic fuzzy ρ¯-ideal (IFρ¯i) are introduced. Further, we show some applications on the family of all intuitionistic fuzzy ρ-subalgebras IFρS(R) in ρ-algebra like the binary relations ≈μ, ≈ν and Γr on IFρS(R). Also, their equivalence classes are given and studied.


## 2. Preliminaries and Notations

We will recall basic definitions and results to obtain properties developed in this work.Definition 1 (see [16]).
An intuitionistic fuzzy setα (IFS, in short) over the universe R is defined by α=≺a,μαa,vαa≻∣a∈R, where μα(a): R→ [0; 1], vα(a): R→ [0; 1] with 0 ≤μα(a)+vα(a)≤1, ∀a∈R. μα(a) and vα(a) are real numbers and their values represent the degree of membership and nonmembership of a to α, respectively.Definition 2 (see [6]).
The IF whole and empty sets ofR are defined by 1¯=≺a,1,0≻∣a∈R and 0¯=≺a,0,1≻∣a∈R, respectively.


### 2.1. Basic Relations and Operations on Intuitionistic Fuzzy Sets [7]

Assumeα=≺a,μαa,vαa≻∣a∈R and β=≺a,(μβa,vβa≻∣a∈R are two IF sets of R. We deduced the following relations:(1)
[inclusion]α⊆β iff μα(a)≤μβ(a) and vα(a)≥vβ(a), ∀a∈R,(2)
[equality]α=β iff α⊆β and β⊆α,(3)
[intersection]α∩~β={(a,min⁡{μα(a),μβ(a)}, max⁡{vα(a),vβ(a)}):a∈R},(4)
[union]α∪~β={(a,max⁡{μαa,μβa}, min⁡{vαa,vβa}):a∈R},(5)
[complement]αc=a,vαa,μαa,a∈R.Definition 3 (see [14]).
We say(R,•,0) is ρ-algebra if (•) is a binary operation on R with a constant 0∈R and such that(1)
a•a=0,(2)
0•a=0,(3)
a•b=0=b•a imply that a=b,(4)
For alla≠b∈R-{0} imply that a•b=b•a≠0.Definition 4 (see [14]).
Assume(R,•,0) is a ρ-algebra and ϕ≠K⊆R. We say K is a ρ-subalgebra of R if a•b∈K, ∀a,b∈K.Definition 5 (see [14]).
Assume(R,•,0) is ρ-algebra and ϕ≠K⊆R. We say K is ρ-ideal of R if (1)
a,b∈K imply a•b∈K,(2)
a•b∈K and b∈K imply a∈K, ∀a,b∈R.Definition 6 (see [14]).
Assume(R,•,0) is a ρ-algebra and K subset of R. We say K is a ρ¯-ideal of R if (1)
0∈K,(2)
a∈K and b∈R→a•b∈K, ∀a,b∈R.Definition 7 (see [11]).
Assume thatα=≺a,μαa,vαa≻∣a∈R is an IFS in R and r∈[0,1]. The set W(μα,r)=a∈R∣μαa≥r (resp., L(να,r)=a∈R∣ναa≤r) is said to be μ-levelr-cut(resp., ν-levelr-cut) of α.


## 2.1. Basic Relations and Operations on Intuitionistic Fuzzy Sets [7]

Assumeα=≺a,μαa,vαa≻∣a∈R and β=≺a,(μβa,vβa≻∣a∈R are two IF sets of R. We deduced the following relations:(1)
[inclusion]α⊆β iff μα(a)≤μβ(a) and vα(a)≥vβ(a), ∀a∈R,(2)
[equality]α=β iff α⊆β and β⊆α,(3)
[intersection]α∩~β={(a,min⁡{μα(a),μβ(a)}, max⁡{vα(a),vβ(a)}):a∈R},(4)
[union]α∪~β={(a,max⁡{μαa,μβa}, min⁡{vαa,vβa}):a∈R},(5)
[complement]αc=a,vαa,μαa,a∈R.Definition 3 (see [14]).
We say(R,•,0) is ρ-algebra if (•) is a binary operation on R with a constant 0∈R and such that(1)
a•a=0,(2)
0•a=0,(3)
a•b=0=b•a imply that a=b,(4)
For alla≠b∈R-{0} imply that a•b=b•a≠0.Definition 4 (see [14]).
Assume(R,•,0) is a ρ-algebra and ϕ≠K⊆R. We say K is a ρ-subalgebra of R if a•b∈K, ∀a,b∈K.Definition 5 (see [14]).
Assume(R,•,0) is ρ-algebra and ϕ≠K⊆R. We say K is ρ-ideal of R if (1)
a,b∈K imply a•b∈K,(2)
a•b∈K and b∈K imply a∈K, ∀a,b∈R.Definition 6 (see [14]).
Assume(R,•,0) is a ρ-algebra and K subset of R. We say K is a ρ¯-ideal of R if (1)
0∈K,(2)
a∈K and b∈R→a•b∈K, ∀a,b∈R.Definition 7 (see [11]).
Assume thatα=≺a,μαa,vαa≻∣a∈R is an IFS in R and r∈[0,1]. The set W(μα,r)=a∈R∣μαa≥r (resp., L(να,r)=a∈R∣ναa≤r) is said to be μ-levelr-cut(resp., ν-levelr-cut) of α.


## 3. Intuitionistic Fuzzyρ-Subalgebras in ρ-Algebras

In this section, we introduce some new concepts, such as (IFρs), (IFρi), (IFρ¯i), and ρ-homomorphism which are introduced and discussed. Further, some binary relations ≈μ, ≈ν and Γr on IFρS(R) are given, and some basic properties are shown.Definition 8.
Assume(R,•,0) is a ρ-algebra and α=≺a,μαa,ναa≻∣a∈R is IFS of R. We say α is an (IFρs) of R if μα(a•b)≥min⁡{μα(a),μα(b)} and να(a•b)≤max⁡ναa,ναb, ∀a,b∈R.Example 9.
LetR={0,ω,∂,l} be ρ-algebra with Table 1.
Then,α=≺a,μαa,ναa≻∣a∈R={(0,0.9,0.1),(ω,0.4,0.3),(∂,0.7,0.3),(l,0.4,0.2)} is an (IFρs) of R.Table 1 ∗ 0 ω ∂ l 0 0 0 0 0 ω ω 0 ∂ ∂ ∂ ω ∂ 0 ω l l ∂ ω 0Definition 10.
Assume(R,•,0) is ρ-algebra and α=≺a,μαa,ναa≻∣a∈R is IFS of R. We say α is (IFρi) of R if (1)
μα(a•b)≥min⁡{μα(a),μα(b)} and να(a•b)≤max⁡{να(a),να(b)},(2)
μα(a)≥min⁡{μα(a•b),μα(b)} and να(a)≤max⁡{να(a•b),να(b)}, ∀a,b∈R.Example 11.
Let(R,•,0) be ρ-algebra in Example 9 and let α=≺a,μαa,ναa≻∣a∈R={(0,0.8,0.2),(ω,0.3,0.4),(∂,0.2,0.7),(l,0.4,0.3)} be IFS of R. Then, α is (IFρi) of R.Definition 12.
Assume(R,•,0) is ρ-algebra and α=≺a,μαa,ναa≻∣a∈R is IFS of R. We say α is (IFρ¯i) of R if (1)
μα(0)≥μα(a) and να(0)≤να(a),(2)
μα(a•b)≥min⁡{μα(a),μα(b)} and να(a•b)≤max⁡{να(a),να(b)}, ∀a,b∈R.Example 13.
Let(R,•,0) be ρ-algebra in Example 9 and let α=≺a,μαa,ναa≻∣a∈R={(0,0.9,0.1),(ω,0.4,0.3),(∂,0.7,0.3),(l,0.4,0.2)} be IFS of R. Then, α is (IFρ¯i) of R.Remark 14.
(1)
Ifα=≺a,μαa,ναa≻∣a∈R is (IFρi) of R, then α is (IFρs).(2)
Ifα=≺a,μαa,ναa≻∣a∈R is (IFρ¯i) of R, then α is (IFρs).(3)
Ifα=≺a,μαa,ναa≻∣a∈R is (IFρs) of R and satisfies (2) in Definition 10, then α is (IFρi).(4)
Ifα=≺a,μαa,ναa≻∣a∈R is (IFρs) of R and satisfies (1) in Definition 12, then α is (IFρ¯i).Lemma 15.
Ifα=≺a,μαa,ναa≻∣a∈R is (IFρs) of R, then μα(0)≥μα(a) and να(0)≤να(a), ∀a∈R.Proof.
Leta∈R. Then μα(0)=μα(a•a)≥min⁡{μα(a),μα(a)}=μα(a) and να(0)=να(a•a)≤max⁡{να(a),νa(a)}=να(a).Theorem 16.
Ifαi=≺a,μαia,ναia≻∣a∈R,i∈I is any family of (IFρs) of R, then ⋂~i∈Iαi is (IFρ¯i) of R, where ⋂~i∈Iαi=≺a,min⁡μαia,max⁡ναia≻∣a∈R.Proof.
Leta,b∈R. Thus we consider that
min ⁡ { μ α i ( a • b ) } ≥ min ⁡ min ⁡ μ α i a , μ α i b = min ⁡ { min ⁡ { μ α i ( a ) } , min ⁡ { μ α i ( b ) } }. Also max⁡{ναi(a•b)}≤max⁡max⁡ναia,ναib=max⁡{max⁡{ναi(a)},max⁡{ναi(b)}}.
Thus⋂~i∈Iαi=≺a,min⁡μαia,max⁡ναia≻∣a∈R satisfies condition (2) in Definition 12. Also, let a∈R. Hence, we consider that min⁡{μαi(0)}=min⁡{μαi(a•a)}≥min⁡μαia,μαia=min⁡{μαi(a)}. Furthermore, max⁡{ναi(0)}=max⁡{ναi(a•a)}≤max⁡ναia,ναia=max⁡{ναi(a)}. Then (1) in Definition 12 is held and hence ⋂~i∈Iαi is (Iρ¯i) of R.Theorem 17.
Ifα=≺a,μαa,ναa≻∣a∈R is (IFρi) of R, then K=≺a,μα(a),1-μα(a)≻ is (IFρi) of R.Proof.
We need only to show that1-μα(a) satisfies the first and second condition in Definition 10. Assume ∀a,b∈R. Then 1-μα(a•b)≤1-min⁡μαa,μαb=max⁡{1-μα(a),1-μα(b)}. Furthermore, 1-μα(a)≤1-min⁡μαa•b,μαb=max⁡{1-μα(a•b),1-μα(b)}. Hence K is (IFρi) of R.Theorem 18.
Ifα=≺a,μαa,ναa≻∣a∈R is (IFρs) of R, then the sets Tμ=a∈R∣μαa=μα0 and Tν=a∈R∣ναa=να0 are ρ-subalgebras of R.Proof.
Leta,b∈Tμ. Hence μα(a)=μα(0)=μα(b), and μα(a•b)≥min⁡{μα(a),μα(b)}=μα(0). By using Lemma 15, we consider that μα(a•b)=μα(0) or equivalently a•b∈Tμ. Now, let a,b∈Tν. This implies that να(a•b)≤max⁡{να(a),να(b)}=να(0) and, by applying Lemma 15, we conclude that να(a•b)=να(0). Therefore a•b∈Tν.Definition 19.
Assumeα=≺a,μαa,ναa≻∣a∈R is (IFρs) of R. We say α has finite image, if each image of μα and να is with finite cardinality (i.e., Im(μα)=μαa∣a∈R and Im(να)=ναa∣a∈R such that Imμα<∞ and Imνα<∞).Definition 20.
Assume thatα=≺a,μαa,vαa≻∣a∈R is (IFρs) of R and r∈[0,1]. The set W(μα,r)=a∈R∣μαa≥r (resp., L(να,r)=a∈R∣ναa≤r) is said to be μ-levelr-cut(resp., ν-levelr-cut) of α.Theorem 21.
Ifα=≺a,μαa,ναa≻∣a∈R is (IFρs) of R, then W(μα,r)=a∈R∣μαa≥r and L(να,r)=a∈R∣ναa≤r) of α are ρ-subalgebras of R.Proof.
Leta,b∈W(μα,r). Hence μα(b)≥r and μα(b)≥r. This implies that μα(a•b)≥min⁡{μα(a),μα(b)}≥r so that a•b∈W(μα,r). Thus W(μα,r) is ρ-subalgebra of R Now let a,b∈L(να,r). Thus να(a•b)≤max⁡{να(a),να(b)}≤r and a•b∈L(να,r). Therefore Lνα,r is ρ-subalgebra of R.Theorem 22.
Ifα=≺a,μαa,ναa≻∣a∈R is IFS of ρ-algebraR such that the sets W(μα,r) and L(να,r) are ρ-subalgebras of R, then α is an (IFρs) of R.Proof.
Suppose that there are two memberst1 and t2 in R with μαt1•t2<min⁡{μα(t1),μα(t2)}. Let t=[μα(t1•t2)+min⁡{μα(t1),μα(t2)}]/2. Hence μα(t1•t2)<t<min⁡{μα(t1), μα(t2)} and so t1•t2∉W(μα,t), but t1,t2∈W(μα,t). This is a contradiction, and therefore μα(a•b)≥min⁡{μα(a),μα(b)}, ∀a,b∈R. Now assume that να(t1•t2)>min⁡{να(t1),να(t2)} for some t1,t2∈R. Taking k=[να(t1•t2)+min⁡{να(t1),να(t2)}]/2, then we consider that ναt1•t2>k>max⁡{να(t1),να(t2)}. It follows that t1,t2∈L(να,k) and t1•t2∉L(να,k). This is a contradiction. Therefore, we consider that ναa•b≤max⁡{να(a),να(b)}, ∀a,b∈R. Then A is (IFρs) of R.Theorem 23.
IfH is ρ-subalgebra of R, then there exists (IFρs) α of R, where H satisfies both μ-level ρ-subalgebra and ν-level ρ-subalgebra of α in R.Proof.
AssumeH is ρ-subalgebra of R and let μα and να be fuzzy sets in R defined by(1)μαa=k,ifa∈H1,Otherwise,and(2)ναa=m,ifα∈H1,Otherwise,∀a∈R, where k,m∈(0,1) are fixed real numbers with k+m<1. Assume a,b∈R. Then a•b∈H whenever a,b∈H. This implies that μα(a•b)=min⁡{μα(a),μα(b)} and να(a•b)≤max⁡{να(a),να(b)}. If at least one of a or b does not belong to H, then either μα(a)=0 or μα(b)=0 and hence either να(a)=1 or να(b)=1. It follows that μα(a•b)≥0=min⁡{μα(a),μα(b)}, να(a•b)≤1=max⁡{να(a),να(b)}. Hence α=≺a,μαa,ναa≻∣a∈R is (IFρs) of R. Obviously, W(μα,k)=H=L(να,m).Definition 24.
AssumeΘ:R→Y is a mapping of ρ-algebras. We say Θ is ρ-homomorphism if Θ(a•b)=Θ(a)•Θ(b), ∀a,b∈R. And Θ-1(β)=≺a,Θ-1μβa,Θ-1νβa≻∣a∈R is IFS in ρ-algebra R for any IFS β=≺c,μβc,νβc≻∣c∈Y of ρ-algebra Y. Also, if α=≺a,μαa,ναa≻∣a∈R is IFS in ρ-algebra R, then Θ(α) is IFS in Y and defined by
Θ ( α ) = ≺ c , Θ s u p μ α y , Θ i n f ν α c ≻ ∣ c ∈ Y, where(3)Θsupμαc=sup⁡μαa∣a∈Θ-1c,ifΘ-1c≠0,0,Otherwise,and(4)Θinfναc=inf⁡ναa∣a∈Θ-1c,ifΘ-1c≠0,1,Otherwise,,∀c∈Y.Theorem 25.
LetΘ be ρ-homomorphism of ρ-algebra R into ρ-algebra Y and K be (IFρs) of Y. Then Θ-1(K) is (IFρs) of R.Proof.
Assuminga,b∈R, we have μΘ-1K(a•b)=μK(Θ(a•b))=μK(Θ(a)•Θ(b))≥min⁡{μK(Θ(a)),μK(Θ(b))} = min⁡{μΘ-1(K)(a),μΘ-1(K)(b)} and νΘ-1(K)(a•b)=νK(Θ(a•b))=νK(Θ(a)•Θ(b)) ≤ max⁡{νK(Θ(a)),νK(Θ(b))}=max⁡{νΘ-1(K)(a),νΘ-1(K)(b)}. Thus Θ-1K is (IFρs) of R.Theorem 26.
AssumeΘ:R→Y is ρ-homomorphism of ρ-algebra R into ρ-algebra Y and α=≺a,μαa,ναa≻∣a∈R is (IFρs) of R. Then Θ(α)=≺b,(Θsup(μα),Θinf(να))≻ is (IFρs) of Y.Proof.
Letα=≺a,μαa,ναa≻∣a∈R be (IFρs) of R and let t1,t2∈Y. Noticing that {a1•a2∣a1∈Θ-1t1anda2∈Θ-1(t2)}⊆a∈R∣a∈Θ-1t1•t2, we have Θsup(μα)(t1•t2)=sup⁡μαa∣a∈Θ-1t1∗t2≥sup⁡μαa1•a2∣a1∈Θ-1t1anda2∈Θ-1t2 ≥ sup⁡minμαa1,μαa2∣a1∈Θ-1t1anda2∈Θ-1t2 = min⁡sup⁡μαa1∣a1∈Θ-1t1,sup⁡μαa2∣a2∈Θ-1t2 = min⁡{Θsup(μα)(t1),Θsup(μα)(t2)}. Also, we consider that Θinfναt1•t2 = inf⁡ναa∣a∈Θ-1t1•t2 ≤ inf⁡ναa1•a2∣a1∈Θ-1t1anda2∈Θ-1t2 ≤ inf⁡max⁡ναa1,ναa2∣a1∈Θ-1t1anda2∈Θ-1t2 = max⁡inf⁡ναa1∣a1∈Θ-1t1,inf⁡ναa2∣a2∈Θ-1t2 = max⁡{Θsup(να)(t1),Θsup(να)(t2)}. Hence Θ(α)=≺b,(Θsup(μα),Θinf(να))≻ is (IFρs) of Y.Theorem 27.
AssumeΘ:R→Y is ρ-homomorphism of ρ-algebra R into ρ-algebra Y and α=≺a,μαa,ναa≻∣a∈R is (IFρi) of R. Then Θ(α)=≺b,(Θsup(μα),Θinf(να))≻ is (IFρi) of Y.Proof.
Sinceα=≺a,μαa,ναa≻∣a∈R is (IFρi) of R, then by Theorem 26 and Remark 14 we have Θ(α)=≺b,(Θsup(μα),Θinf(να))≻ as (IFρi) of Y. Hence condition (1) in Definition 10 is held. Since Θ is surjective, then for any t1,t2∈Y, ∃a1,a2∈R such that a1∈Θ-1Θ(a1)=Θ-1(t1) and a2∈Θ-1Θ(a2)=Θ-1(t2). Also, a1•a2∈Θ-1(t1)•Θ-1(t1)=Θ-1(t1•t1). Further, noticing that μα(a1)≥min⁡{μα(a1•a2),μα(a2)} and να(a1)≤max⁡{να(a1•a2),να(a2)}, for any t1,t2∈Y, we have Θsup(μα)(t1) = sup⁡μαa∣a∈Θ-1t1 ≥ sup⁡min⁡μαa1•a2,μαa2∣a1•a2∈Θ-1t1•t2anda2∈Θ-1t2 = min⁡sup⁡μαa1•a2∣a1•a2∈Θ-1t1•t2,sup⁡μαa2∣a2∈Θ-1t2 = min⁡{Θsup(μα)(t1•t2),Θsup(μα)(t2)}. Also, Θsup(να)(t1)=sup⁡ναa∣a∈Θ-1t1≤sup⁡max⁡ναa1•a2,ναa2∣a1•a2∈Θ-1t1•t2anda2∈Θ-1t2 = max⁡sup⁡ναa1•a2∣a1•a2∈Θ-1t1•t2,sup⁡ναa2∣a2∈Θ-1t2=max⁡{Θsup(να)(t1•t2),Θsup(να)(t2)}. Thus we consider that Θ(α)=≺b,(Θsup(μα),Θinf(να))≻ is (IFρi) of Y.Theorem 28.
AssumeΘ:R→Y is ρ-homomorphism of ρ-algebra R into ρ-algebra Y and α=≺a,μαa,ναa≻∣a∈R is (IFρ¯i) of R. Then Θ(α)=≺b,(Θsup(μα),Θinf(να))≻ is (IFρ¯i) of Y.Proof.
Sinceα=≺a,μαa,ναa≻∣a∈R is (IFρ¯i) of R. Then by Theorem 26 and Remark 14 we have Θ(α)=≺b,(Θsup(μα),Θinf(να))≻ as (IFρs) of Y. Hence condition (2) in Definition 12 is held. Assume that 0R and 0Y are constants of ρ-algebrasR and Y, respectively. Since α=≺a,μαa,ναa≻∣a∈R is (IFρ¯i) of R, hence μα(0R)≥μα(a) and να(0R)≤να(a), ∀a∈R. Since Θ is ρ-homomorphism of ρ-algebras, then Θ(0R)=0Y, where 0R and 0Y are constants for ρ-algebras R and Y, respectively. Noticing that 0R∈Θ-1(0Y) and a∣a∈Θ-1b⊆a∣a∈R for any b∈Y, then we have Θsup(μα)(0Y)=sup⁡μαa∣a∈Θ-10Y = μα0R≥sup⁡μαa∣a∈R≥sup⁡μαa∣a∈Θ-1b=Θsup(μα)(b). Also, Θsupνα0Y=inf⁡ναa∣a∈Θ-10Y=να0R≤inf⁡ναa∣a∈R≤inf⁡ναa∣a∈Θ-1b=Θsup(να)(b). Hence Θ(α)=≺b,(Θsup(μα),Θinf(να))≻ is (IFρ¯i) of Y.


## 4. Some Applications onIFρS(R)

In this section, some applications onIFρS(R) are shown like the binary relations ≈μ, ≈ν and Γr on IFρS(R). Also, in this section the equivalence classes for theses binary relations are given, and some of their basic properties are studied.


### 4.1. Equivalence Classes Modulo (≈μ/≈ν)

Denote the collection of all (IFρS) of R by IFρS(R) and let r∈[0,1]. Define binary relations ≈μ and ≈ν on IFρS(R) as follows.α ≈ μ β ⇔ W ( μ α , r ) = W ( μ β , r ) and α≈νβ⇔L(να,r)=L(νβ,r), respectively, for α=≺a,μα,να≻ and β=≺a,μβ,νβ≻ in IFρS(R). Moreover, it is clear that ≈μ and ≈ν are equivalence relations on IFρS(R). If α=≺a,μα,να≻∈IFρS(R), then we refer to the equivalence class of α=≺a,μα,να≻ modulo ≈μ (resp., ≈ν) by αμ (resp., αν), and we refer to the family of all equivalence classes of α modulo ≈μ (resp., ≈ν) by IFρS(R)/≈μ (resp., IFρS(X)/≈ν); i.e., IFρS(R)/≈μ=αμ∣α=≺a,μα,να≻∈IFρSR (resp., IFρS(R)/≈ν=αμ∣α=≺a,μα,να≻∈IFρSR). Moreover, denote the collection of all ρ-ideals of R by ρI(R) and let r∈[0,1]. Let σr and ηr be maps from IFρS(R) to ρI(R)∪{ϕ} by σr(α)=W(μα,r) and ηr(α)=L(να,r), respectively, ∀α=≺a,μα,να≻∈IFρS(R). In other words, σr and ηr are well-defined.Theorem 29.
Letσr and ηr be the maps from IFρS(R) to ρI(R)∪{ϕ}. Then σr and ηr are surjective, for each r∈(0,1).Proof.
Letr∈(0,1). Then 0⌢=≺a,0¯,1¯≻ is in IFρS(R), where each one of 0¯ and 1¯ is (FS) in R defined by 0¯(a)=0 and 1¯(a)=1, ∀a∈R. Furthermore, σr(0⌢)=W(0¯,r)=ϕ=L(1¯,r)=ηr(0⌢). Let ϕ≠H∈ρI(R). ∀a∈R, let μH(a)=1,ifa∈H0,ifa∉H, and νH(a)=1-μH(a); thus σr(H⌢)=W(μH,r)=H=L(νH,r)=ηr(H⌢). Now, we want to prove that H⌢=≺x,μH,νH≻∈IFρS(R). Since H∈ρI(R), then by condition (1) in Definition 5 we have H as ρ-subalgebra of R and this implies that W(μH,r) and L(νH,r) are ρ-subalgebras of R. By Theorem 22 we consider H⌢=≺a,μH,νH≻∈IFρS(R). Therefore, ∀H∈ρI(R) we consider σr(H⌢)=H and ηr(H⌢)=H for some H⌢∈IFρS(R). This completes the proof.Theorem 30.
LetIFρS(R)/≈μ and IFρS(R)/≈ν be quotient sets. Then they are equipotent to ρI(R)∪{ϕ}, ∀r∈(0,1).Proof.
Assumer∈(0,1) and let σr′(resp.ηr′) be a map from IFρS(R)/≈μ (resp., IFρS(R)/≈ν) to ρI(R)∪{ϕ} and they are defined by σr′(αμ)=σr(α)(resp.ηr′(αν)=ηr(α)), ∀α=≺a,μα,να≻∈IFρS(R). Hence, α≈μβ and α≈νβ, ∀α=≺a,μα,να≻ and β=≺a,μβ,νβ≻ in IFρS(R), if W(μA,r)=W(μB,r) and L(νA,r)=L(νB,r). Then αμ=βμ and αν=βν. This implies the maps σr′ and ηr′ are injective. Moreover, let ϕ≠H∈ρI(R) and ∀a∈R, let(5)μHa=1,ifa∈H0,ifa∉H,νH(a)=1-μH(a), and thus H⌢=≺a,μH,νH≻∈IFρS(R). We consider that σr′(H⌢μ)=σr(H⌢)=W(μH,r)=H, and ηr′(H⌢ν)=ηr(H⌢)=L(νH,r)=H. Finally, for 0⌢=≺a,0¯,1¯≻∈IFρS(R) we have σr′(0⌢μ)=σr(0⌢)=W(0¯,r)=ϕ and ηr′(0⌢ν)=ηr(0⌢)=L(1¯,r)=ϕ. Therefore σr′ and ηr′ are surjective, and we are done.


### 4.2. Equivalence Class ModuloΓr

Another relationΓr on IFρS(R) is defined by (α,β)∈Γr⇔W(μα,r)∩L(να,r)=W(μβ,r)∩L(νβ,r), ∀r∈[0,1] and, ∀α=≺a,μα,να≻, β=≺a,μβ,νβ≻∈IFρS(R). Moreover, the relation Γr is also an equivalence relation on IFρS(R). Let αΓr denote the equivalence class of α=≺a,μα,να≻ modulo Γr, ∀α=≺a,μα,να≻∈IFρS(R).Theorem 31.
For anyr∈(0,1), the map ψr:IFρS(R)→ρI(R)∪{ϕ} defined by ψr(R)=σr(R)∩ηr(R), ∀α=≺a,μα,να≻∈IFρS(R) is surjective.Proof.
Letr∈(0,1). For 0⌢=≺a,0¯,1¯≻∈IFρS(R), we get ψr(0⌢)=σr(0⌢)∩ηr(0⌢)=W(0¯,r)∩L(1¯,r)=ϕ. For any H∈IFρS(R), there exists H⌢=≺a,μH,νH≻∈IFρS(R), where(6)μHa=1,ifa∈H0,ifa∉Hand νH(a)=1-μH(a) such that ψr(H⌢)=σr(H⌢)∩ηr(H⌢)=W(μH,r)∩L(νH,r)=H. This completes the proof.Theorem 32.
For anyr∈(0,1), the quotient set IFρS(R)/Γr is equipotent to ρI(X)∪{ϕ}.Proof.
Assumer∈(0,1) and ψr′:IFρS(R)/Γr→ρI(R)∪{ϕ} is a map defined by ψr′(αΓr)=ψr(α), ∀αΓr∈IFρS(R)/Γr.
Suppose thatψr′(αΓr)=ψr′(βΓr) for any αΓr,βΓr∈IFρS(R)/Γr. We consider that σr(α)∩ηr(α)=σr(β)∩ηr(β), i.e., W(μα,r)∩L(να,r)=W(μβ,r)∩L(νβ,r). Hence (α,β)∈Γr, and so αΓr=βΓr. Therefore ψr′ is injective. Furthermore, for 0⌢=≺a,0¯,1¯≻∈IFρS(R) we get ψr′(0⌢Γr) = ψr(0⌢) = σr(0⌢)∩ηr(0⌢) = W(0¯,r)∩L(1¯,r) = ϕ. Let H⌢=≺a,μH,νH≻∈IFρS(R), ∀H∈IFρS(R), be the same (IFρS) of X that is defined in the proof of Theorem 22. Then we have ψr′(H⌢Γr)=ψr(H⌢)=σr(H⌢)∩ηr(H⌢)=W(μH,r)∩L(νH,r)=H. Hence ψr′ is surjective. This completes the proof.


## 4.1. Equivalence Classes Modulo (≈μ/≈ν)

Denote the collection of all (IFρS) of R by IFρS(R) and let r∈[0,1]. Define binary relations ≈μ and ≈ν on IFρS(R) as follows.α ≈ μ β ⇔ W ( μ α , r ) = W ( μ β , r ) and α≈νβ⇔L(να,r)=L(νβ,r), respectively, for α=≺a,μα,να≻ and β=≺a,μβ,νβ≻ in IFρS(R). Moreover, it is clear that ≈μ and ≈ν are equivalence relations on IFρS(R). If α=≺a,μα,να≻∈IFρS(R), then we refer to the equivalence class of α=≺a,μα,να≻ modulo ≈μ (resp., ≈ν) by αμ (resp., αν), and we refer to the family of all equivalence classes of α modulo ≈μ (resp., ≈ν) by IFρS(R)/≈μ (resp., IFρS(X)/≈ν); i.e., IFρS(R)/≈μ=αμ∣α=≺a,μα,να≻∈IFρSR (resp., IFρS(R)/≈ν=αμ∣α=≺a,μα,να≻∈IFρSR). Moreover, denote the collection of all ρ-ideals of R by ρI(R) and let r∈[0,1]. Let σr and ηr be maps from IFρS(R) to ρI(R)∪{ϕ} by σr(α)=W(μα,r) and ηr(α)=L(να,r), respectively, ∀α=≺a,μα,να≻∈IFρS(R). In other words, σr and ηr are well-defined.Theorem 29.
Letσr and ηr be the maps from IFρS(R) to ρI(R)∪{ϕ}. Then σr and ηr are surjective, for each r∈(0,1).Proof.
Letr∈(0,1). Then 0⌢=≺a,0¯,1¯≻ is in IFρS(R), where each one of 0¯ and 1¯ is (FS) in R defined by 0¯(a)=0 and 1¯(a)=1, ∀a∈R. Furthermore, σr(0⌢)=W(0¯,r)=ϕ=L(1¯,r)=ηr(0⌢). Let ϕ≠H∈ρI(R). ∀a∈R, let μH(a)=1,ifa∈H0,ifa∉H, and νH(a)=1-μH(a); thus σr(H⌢)=W(μH,r)=H=L(νH,r)=ηr(H⌢). Now, we want to prove that H⌢=≺x,μH,νH≻∈IFρS(R). Since H∈ρI(R), then by condition (1) in Definition 5 we have H as ρ-subalgebra of R and this implies that W(μH,r) and L(νH,r) are ρ-subalgebras of R. By Theorem 22 we consider H⌢=≺a,μH,νH≻∈IFρS(R). Therefore, ∀H∈ρI(R) we consider σr(H⌢)=H and ηr(H⌢)=H for some H⌢∈IFρS(R). This completes the proof.Theorem 30.
LetIFρS(R)/≈μ and IFρS(R)/≈ν be quotient sets. Then they are equipotent to ρI(R)∪{ϕ}, ∀r∈(0,1).Proof.
Assumer∈(0,1) and let σr′(resp.ηr′) be a map from IFρS(R)/≈μ (resp., IFρS(R)/≈ν) to ρI(R)∪{ϕ} and they are defined by σr′(αμ)=σr(α)(resp.ηr′(αν)=ηr(α)), ∀α=≺a,μα,να≻∈IFρS(R). Hence, α≈μβ and α≈νβ, ∀α=≺a,μα,να≻ and β=≺a,μβ,νβ≻ in IFρS(R), if W(μA,r)=W(μB,r) and L(νA,r)=L(νB,r). Then αμ=βμ and αν=βν. This implies the maps σr′ and ηr′ are injective. Moreover, let ϕ≠H∈ρI(R) and ∀a∈R, let(5)μHa=1,ifa∈H0,ifa∉H,νH(a)=1-μH(a), and thus H⌢=≺a,μH,νH≻∈IFρS(R). We consider that σr′(H⌢μ)=σr(H⌢)=W(μH,r)=H, and ηr′(H⌢ν)=ηr(H⌢)=L(νH,r)=H. Finally, for 0⌢=≺a,0¯,1¯≻∈IFρS(R) we have σr′(0⌢μ)=σr(0⌢)=W(0¯,r)=ϕ and ηr′(0⌢ν)=ηr(0⌢)=L(1¯,r)=ϕ. Therefore σr′ and ηr′ are surjective, and we are done.


## 4.2. Equivalence Class ModuloΓr

Another relationΓr on IFρS(R) is defined by (α,β)∈Γr⇔W(μα,r)∩L(να,r)=W(μβ,r)∩L(νβ,r), ∀r∈[0,1] and, ∀α=≺a,μα,να≻, β=≺a,μβ,νβ≻∈IFρS(R). Moreover, the relation Γr is also an equivalence relation on IFρS(R). Let αΓr denote the equivalence class of α=≺a,μα,να≻ modulo Γr, ∀α=≺a,μα,να≻∈IFρS(R).Theorem 31.
For anyr∈(0,1), the map ψr:IFρS(R)→ρI(R)∪{ϕ} defined by ψr(R)=σr(R)∩ηr(R), ∀α=≺a,μα,να≻∈IFρS(R) is surjective.Proof.
Letr∈(0,1). For 0⌢=≺a,0¯,1¯≻∈IFρS(R), we get ψr(0⌢)=σr(0⌢)∩ηr(0⌢)=W(0¯,r)∩L(1¯,r)=ϕ. For any H∈IFρS(R), there exists H⌢=≺a,μH,νH≻∈IFρS(R), where(6)μHa=1,ifa∈H0,ifa∉Hand νH(a)=1-μH(a) such that ψr(H⌢)=σr(H⌢)∩ηr(H⌢)=W(μH,r)∩L(νH,r)=H. This completes the proof.Theorem 32.
For anyr∈(0,1), the quotient set IFρS(R)/Γr is equipotent to ρI(X)∪{ϕ}.Proof.
Assumer∈(0,1) and ψr′:IFρS(R)/Γr→ρI(R)∪{ϕ} is a map defined by ψr′(αΓr)=ψr(α), ∀αΓr∈IFρS(R)/Γr.
Suppose thatψr′(αΓr)=ψr′(βΓr) for any αΓr,βΓr∈IFρS(R)/Γr. We consider that σr(α)∩ηr(α)=σr(β)∩ηr(β), i.e., W(μα,r)∩L(να,r)=W(μβ,r)∩L(νβ,r). Hence (α,β)∈Γr, and so αΓr=βΓr. Therefore ψr′ is injective. Furthermore, for 0⌢=≺a,0¯,1¯≻∈IFρS(R) we get ψr′(0⌢Γr) = ψr(0⌢) = σr(0⌢)∩ηr(0⌢) = W(0¯,r)∩L(1¯,r) = ϕ. Let H⌢=≺a,μH,νH≻∈IFρS(R), ∀H∈IFρS(R), be the same (IFρS) of X that is defined in the proof of Theorem 22. Then we have ψr′(H⌢Γr)=ψr(H⌢)=σr(H⌢)∩ηr(H⌢)=W(μH,r)∩L(νH,r)=H. Hence ψr′ is surjective. This completes the proof.


## 5. Conclusion

In this work, we introduce the notions of (IFρs), (IFρi), (IFρ¯i), and others; then we proved that for any ρ-subalgebra of X can be considered as both μ-level ρ-subalgebra and ν-level ρ-subalgebra of some (IFρs) of R. At the same time, we proved that intersection of any family of (IFρs) of X is (IFρ¯i) of X. Also, we show that if IFS α=≺a,μαa,ναa≻∣a∈R of ρ-algebraX such that the sets W(μα,r) and L(να,r) are ρ-subalgebras of R. Then α is (IFρs) of R. Further, some interesting theorems about ρ-homomorphism are given. Finally, some binary relations ≈μ, ≈ν and Γr on IFρS(R) are obtained, and some of their basic properties are discussed. In future work, we will investigate IF in new types of algebras like BCL+-algebras, BCL+-subalgebras,BCL+-ideals and others. Next, we will study their characteristics.



---

*Source: 5712676-2018-07-10.xml*