# Fuzzy Annihilator Ideals ofC-Algebra

Authors: Wondwosen Zemene Norahun; Teferi Getachew Alemayehu; Gezahagne Mulat Addis
Journal: Advances in Fuzzy Systems
 (2021)

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## Abstract

In this paper, we introduce the concept of relative fuzzy annihilator ideals inC-algebras and investigate some its properties. We characterize relative fuzzy annihilators in terms of fuzzy points. It is proved that the class of fuzzy ideals of C-algebras forms Heything algebra. We observe that the class of all fuzzy annihilator ideals can be made as a complete Boolean algebra. Moreover, we study the concept of fuzzy annihilator preserving homomorphism. We provide a sufficient condition for a homomorphism to be a fuzzy annihilator preserving.


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## Body

## 1. Introduction

Fuzzy set theory was guided by the assumption that classical sets were not natural, appropriate, or useful notions in describing the real-life problems because every object encountered in this real physical world carries some degree of fuzziness. A lot of work on fuzzy sets has come into being with many applications to various fields such as computer science, artificial intelligence, expert systems, control systems, decision making, medical diagnosis, management science, operations research, pattern recognition, neural network, and others (see [1–4]). Many papers on fuzzy algebras have been published since Rosenfeld [5] introduced the concept of fuzzy group in 1971. In particular, fuzzy subgroups of a group (see [6–8]), fuzzy ideals of lattices and MS-algebra (see [9–16]), fuzzy ideals of C-algebras (see [17, 18]), and intuitionistic fuzzy ideals of BCK-algebra, BG-algebra, and BCI-algebra (see [19–21]).On the contrary, Guzman and Squier, in [22], introduced the variety of C-algebras as the variety generated by the three-element algebra C=T,F,U with the operations ∧,∨,′ of type 2,2,1, which is the algebraic form of the three-valued conditional logic. They proved that the two-element Boolean algebras B and C are the only subdirectly irreducible C-algebras and that the variety of C-algebras is a minimal cover of the variety of Boolean algebras. In [23], U. M. Swamy et al. studied the center BA of a C-algebra A and proved that the center of a C-algebra is a Boolean algebra. In [24], Rao and Sundarayya studied the concept of C-algebra as a poset. In a series of papers (see [25–28]), Vali et al. studied the concept of ideals, principal ideals, and prime ideals of C-algebras as well as the concept of prime spectrum, ideal congruences, and annihilators of C-algebras. Later, Rao carried out a study on annihilator ideals of C-algebras [29].In this paper, we study the concept of relative fuzzy annihilator ideals inC-algebras. We characterize relative fuzzy annihilators in terms of fuzzy points. Using the concept of the relative fuzzy annihilator, we prove that the class of fuzzy ideals of C-algebras forms the Heything algebra. We also study fuzzy annihilator ideals. Basic properties of fuzzy annihilator ideals are also studied. It is shown that the class of all fuzzy annihilator ideals forms a complete Boolean algebra. Moreover, we study the concept of fuzzy annihilator preserving homomorphism and derived a sufficient condition for a homomorphism to be a fuzzy annihilator preserving. Finally, we prove that the image and preimage of fuzzy annihilator ideals are again fuzzy annihilator ideals.


## 2. Preliminaries

In this section, we recall some definitions and basic results onc−algebras.Definition 1 (see [22]).
An algebraA,∨,∧,′ of type 2,2,1 is called a c-algebra if it satisfies the following axioms:(1)
a″=b(2)
a∧b′=a′∨b′(3)
a∧b∧c=a∧b∧c(4)
a∧b∨c=a∧b∨a∧c(5)
a∨b∧c=a∧c∨a′∧b∧c(6)
a∨a∧b=a(7)
a∧b∨b∧a=b∧a∨a∧b, for all a,b,c∈AExample 1.
The three-element algebraC=T,F,U with the operations given by by the following tables is a C-algebra.
Note: the identities 2.1 (a) and 2.1 (b) imply that the variety of C-algebras satisfies all the dual statements of 2.1 (2) to 2.1 (7) in this view.Lemma 1 (see [22]).
EveryC−algebra satisfies the following identities:(1)
x∧x=x(2)
x∧x′=x′∧x(3)
x∧y∧x=x∧y(4)
x∧x′∧y=x∧x′(5)
x∧y=x′∨y∧x(6)
x∧y=x∧y∨x′(7)
x∧y=x∧x′∨y(8)
x∧y∧x′=x∧y∧y′(9)
x∨y∧x=x∨y∧x(10)
x∧x′∨x=x′∨x∧x=x∨x′∧xThe dual statements of the above identities are also valid in aC-algebra.Definition 2 (see [22]).
An elementz of a C-algebra A is called a left zero for ∧ if z∧x=z, for all x∈A.Definition 3 (see [26]).
A nonempty subsetI of a C-algebra A is called an ideal of A if(1)
a,b∈I⇒a∨b,a∧x∈I(2)
a∈I⇒x∧a∈I, for each x∈A.
It can also be observed thata∧x∈I, for all a∈I and all x∈A. For any subset S⊆A, the smallest ideal of A containing S is called the ideal of A generated by S and is denoted by S. Note that 〈S=∨yi∧xi:yi∈A,xi∈S,i=1,…,nfor somen∈Z+
IfS=a, then we write a for S. In this case, a=x∧a:x∈A. Moreover, it is observed in [26] that the set I0=x∧x′:x∈A is the smallest ideal in A.Definition 4.
LetA,∨,∧,′,I0 and A′,∨,∧,′,I0′ be two C-algebras. Then, a mapping f:A⟶A′ is called a homomorphism if it satisfies the following conditions:(1)
fa∨b=fa∨fb(2)
fa∧b=fa∧fb(3)
fa′=fa′, for all a,b∈A
Here,I0=z∈A:zis a left zero for∧ and I0′=y∈A′:yis a left zero for∧.
I0 and I0′ are the smallest ideals of the C-algebras A and A′, respectively. The kernel of the homomorphism is defined as Kerf=x∈A:fx∈I0′.
Remember that, for any setA, a function μ:A⟶0,1 is called a fuzzy subset of A. For each t∈0,1, the set,(1)μt=x∈A:μx≥t,is called the level subset of μ at t [30]. For numbers α and β in 0,1, we write α∧β for minα,β and α∨β for maxα,β.Definition 5 (see [17]).
A fuzzy subsetλ of A is called a fuzzy ideal of A if(1)
λz=1, for all z∈I0(2)
λa∨b≥λa∧λb(3)
λa∨b≥λb, for all a,b∈A
We denote the class of all fuzzy ideals ofA by FIA.Lemma 2 (see [17]).
Letλ be a fuzzy ideal of A. Then, the following hold, for all a,b∈A:(1)
λa∧b≥λa(2)
λa∧b≥λb∧a(3)
λa∧x∧b≥λa∧b, for each x∈A(4)
λa≥λa∨b; hence, λa∧λb∧λb∨a(5)
Ifx∈a, then λx≥λaLetμ be a fuzzy subset of A. Then, the fuzzy ideal generated by μ is denoted by μ.Theorem 1 (see [17]).
Ifλ and ν are fuzzy ideals of a C-algebra, then their supremum is given by(2)λ∨νx=Sup∧i=1nλbi∨νbi:x=∨i=1nbi,bi∈A.We define the binary operations “+” and “.” on the set of all fuzzy subsets of A as(3)λ+νx=Supλy∧νz:y,z∈A,y∨z=x,λ⋅νx=Supλy∧νz:y,z∈A,y∧z=x.Ifλ and ν are fuzzy ideals of A, then λ⋅ν is a fuzzy ideal and λ⋅ν=λ∩ν. However, in a general case, λ+ν is not a fuzzy ideal.Definition 6 (see [5]).
Letf be a function from X to Y, μ be a fuzzy subset of X, and θ be a fuzzy subset of Y.(1)
The image ofμ under f, denoted by fμ, is a fuzzy subset of Y defined, for each y∈Y, by(4)fμy=Supμx:x∈f−1y,iff−1y≠ϕ,0,otherwise.(2)
The preimage ofθ under f, denoted by f−1θ, is a fuzzy subset of X defined, for each x∈X, by(5)f−1θx=θfx.Theorem 2 (see [31]).
Letf be a function from X to Y. Then, the following assertions hold:(1)
For all fuzzy subsetμi of X,i∈I, f∪i∈Iμi=∪i∈Ifμi, so μ1⊆μ2⇒fμ1⊆fμ2.(2)
For all fuzzy subsetθj of Y,j∈J, f−1∪j∈Jθj=∪j∈Jf−1θj,f−1∩j∈Jθj=∩j∈Jf−1θj, and therefore, θ1⊆θ2⇒f−1θ1⊆f−1θ2.(3)
μ⊆f−1fμ. In particular, if f is an injection, then μ=f−1fμ, for all fuzzy subset μ of X.(4)
ff−1θ⊆θ. In particular, if f is a surjection, then ff−1θ=θ, for all fuzzy subset θ of Y.(5)
fμ⊆θ⇔μ⊆f−1θ, for all fuzzy subsets μ and θ of X and Y, respectively.The class of fuzzy ideals of aC-algebra is denoted by FIA.Note: throughout the rest of this paper, A stands for a C-algebra.


## 3. Relative Fuzzy Annihilator

In this section, we study the concept of relative fuzzy annihilator ideals in aC-algebra. Basic properties of relative fuzzy annihilator ideals are also studied. We characterize relative fuzzy annihilator in terms of fuzzy points. Finally, we prove that the class of fuzzy ideals of a C-algebra forms the Heyting algebra.Definition 7.
For any fuzzy subsetλ of A and a fuzzy ideal ν, we define(6)λ:ν=∪η:η∈0,1A,η⋅λ⊆ν.
A fuzzy subsetλ:ν is called fuzzy annihilator of λ relative to ν.
For anyx∈A,(7)λ:νx=Supηx:η∈0,1A,η⋅λ⊆ν.
For simplicity, we write(8)λ:ν=Supη:η∈0,1A,η⋅λ⊆ν.Lemma 3.
For any two fuzzy subsetsλ and ν of a C-algebra A, we have(9)λ⋅ν=λ∧ν.Now, we prove the following lemma.Lemma 4.
For any fuzzy subsetλ of A and a fuzzy ideal ν, we have(10)λ:ν=∪η:η∈FIA,η⋅λ⊆ν.Proof.
Clearly,∪η:η∈FIA,η⋅λ⊆ν⊆∪δ:δ∈0,1A,δ⋅λ⊆ν. Since η⋅λ=η∧λ, we can easily show that the other inclusion holds. Thus,(11)λ:ν=Supη:η∈FIA,η⋅λ⊆ν.Theorem 3.
For any fuzzy subsetλ of A and a fuzzy ideal ν, λ:ν is a fuzzy ideal of A.Proof.
Sinceλ⋅ν⊆ν and ν is a fuzzy ideal, we get νz=1, for all left zero element z for ∧.
Letx,y∈A. Then,(12)λ:νx∧λ:νy=Supηx:η∈FIA,η⋅λ⊆ν∧Supσy:σ∈FIA,σ⋅λ⊆ν=Supηx∧σy:η,σ∈FIA,η⋅λ⊆ν,σ⋅λ⊆ν≤Supη∨σx∧η∨σy:η,σ∈FIA,η⋅λ⊆ν,σ⋅λ⊆ν.
Sinceη,σ∈FIA, η⋅λ⊆ν, and σ⋅λ⊆ν, we get that η∨σ∈FIA and η∨σ⋅λ⊆ν. Then,(13)λ:νx∧λ:νy≤Supγx∧γy:γ∈FIA,λ⋅λ⊆ν=Supγx∨y:γ∈FIA,γ⋅λ⊆ν=λ:νx∨y.
Thus,λ:νx∨y≥λ:νx∧λ:νy.
On the contrary, letx∈A. Then,(14)λ:νx=Supηx:η∈FIA,η⋅λ⊆ν≤Supηx∧y:η∈FIA,η⋅λ⊆ν=λ:νx∧y.
Similarly,λ:νy≤λ:νx∧y. So, λ:νx∧y≥λ:νx∨λ:νy. Hence, λ:ν is a fuzzy ideal of A.
In the following theorem, we characterize relative fuzzy annihilators in terms of fuzzy points.Theorem 4.
Letλ be a fuzzy subset of A and ν be a fuzzy ideal. Then, for each x∈A,(15)λ:νx=Supα∈0,1:xα⋅λ⊆ν.Proof.
For eachx∈A, let us define two sets Cx and Bx as follows:(16)Cx=ηx:η∈0,1A,η⋅λ⊆ν,Bx=α∈0,1:xα⋅λ⊆ν.
Sinceλ⋅ν⊆ν, then both Cx and Bx are nonempty subsets of 0,1. Now, we proceed to show that ∨Cx=∨Bx. Let α∈Cx. Then, α=ηx for some fuzzy subset η of A satisfying η⋅λ⊆ν. If α=0, then we can find β∈Bx such that α≤β. On the contrary, suppose that α≠0. Then, xα is a fuzzy point of A such that xα⊆η, which implies xα⋅λ⊆η⋅ν and α∈Bx. Thus, Cx⊆Bx. So, ∨Cx≤∨Bx.
To show∨Bx≤∨Cx, let β∈Bx. Then, xβ is a fuzzy point of A such that xβ⋅λ⊆ν. This shows that β∈Cx. Thus, Bx⊆Cx. So, ∨Bx≤∨Cx. So, ∨Bx=∨Cx. Hence, the result is obtained.Example 2.
LetA=T,F,U, and define ∨, ∧, and ′ on A as follows.
Now, consider theC-algebra C=A×A=a1,a2,a3,a4,a5,a6,a7,a8,a9, where a1=T,U,a2=F,U,a3=U,T,a4=U,F,a5=U,U,a6=T,T,a7=F,F,a8=T,F,anda9=F,T. Then, the set of left zero for ∧ is I0=a2,a4,a5,a7.
If we define two fuzzy subsetsθ and μ of C as(17)θa2=θa4=θa5=θa7=1,θa1=θa8=0.6,θa3=θa6=θa9=0.5,μa1=μa2=μa3=0.8,μa4=μa5=μa6=0.7,μa7=μa8=μa9=0.4,then θ is a fuzzy ideal of C and μ:θa2=μ:θa4=μ:θa5=μ:θa7=1, μ:θa1=μ:θa8=0.6 and μ:θa3=μ:θa6=μ:θa9=0.5. Thus, μ:θ is a fuzzy ideal of C.
In the following lemma, some basic properties of relative fuzzy annihilators can be observed.Lemma 5.
Letη and ξ be fuzzy subsets and λ,ν, and γ be fuzzy ideals of A. Then,(1)
η:λ=χA⇔η⊆λ(2)
ν⊆η:ν(3)
η⊆ξ⇒ξ:λ⊆η:λ(4)
λ⊆ν⇒ξ:λ⊆ξ:ν(5)
η:λ∩ν=η:λ∩η:ν(6)
η:λ=η:λ(7)
η∪ξ:λ=η:λ∩ξ:λ(8)
λ∨ν:γ=λ:γ∩ν:γ(9)
λ:ν=λ∨ν:ν=λ:λ∧ν(10)
η∩ν⊆λ⇔ν⊆η:λProof.
The proof of (3) and (4) is straightforward. Now, we proceed to prove the following.(1)
Letη:λ=χA. To show η⊆λ, assume that η⊈λ. Then, there is x∈A such that ηx>μx. This implies that δx≤λx, for each δ such that δ⋅η⊆λ. Thus, λx is an upper bound of δx:δ⋅η⊆λ. This shows that λx≥1. Thus, λx≥ηx; it is a contradiction. So, η⊆λ. The converse part is trivial.(2)
Sinceν∧η⊆ν, we get that ν⋅η⊆ν. Thus, ν⊆η:ν.(3)
(18)η:λ∩η:ν=Supγ1∈FIA,γ1⋅η⊆λ∧Supγ2∈FIA,γ2⋅η⊆ν=Supγ1∧γ2:γ1⋅η⊆λ,γ2⋅η⊆ν=Supγ:γ1⋅η⊆λ,γ2⋅η⊆ν,Whereγ=γ1∧γ2Supγ:γ∈FIA,γ⋅η⊆λ∩ν=η:λ∩ν.(4)
Sinceγ⋅η=γ∧η, for every γ∈FIA, we can easily verified that η:λ=η:λ.(5)
By property (3), we have thatη∪ξ,λ⊆η,λ∩ξ,λ. On the contrary,(19)η:λ∩ξ:λ=η:λ∩ξ:λ=Supγ1∈FIA:γ1∧η⊆λ∧Supγ2∈FIA,γ2∧ξ⊆λ=Supγ1∧γ2:γ1,γ2∈FIL,γ1∧η⊆λ,γ2∧ξ⊆λ.Sinceγ1∧η⊆λ and γ2∧ξ⊆λ, there exists a fuzzy ideal γ of A contained in γ1 and γ2 such that γ∧η⊆λ and γ∧ξ⊆λ. This implies that γ∧η∨ξ⊆λ. This shows that(20)η:λ∩ξ:λ⊆⊆Supγ:γ∧η∨ξ⊆λ⊆Supγ:γ⋅η∪ξ⊆λ=η∪ξ:λ.(6)
Sinceλ∨ν=λ∪ν, by (6), we get λ∨ν:γ=λ∪ν:γ. Thus,(21)λ∨ν:γ=λ:γ∩ν:γ.(7)
Sinceθ:θ=χA, by (8), we get λ∨ν:ν=λ:ν. On the contrary, let γ∧λ⊆ν for some fuzzy ideal γ of A. Since γ∧λ⊆λ, we get that γ∧λ⊆λ∧ν. Thus, λ:ν⊆λ:λ∧ν. Since λ∧ν⊆ν, by (4), we have λ:λ∧ν⊆λ:ν. Hence, λ:ν=λ:λ∧ν. So, λ:ν=λ∨ν:ν=λ:λ∧ν.(8)
Ifη∩ν⊆λ, then, by (6) and by the definition of relative fuzzy annihilator, ν⊆η:λ. Conversely, suppose ν⊆η:λ. Since ν is a fuzzy ideal, we can express ν as follows:(22)ν=∨xα⊆νxα.
Letxα be a fuzzy point of A such that xα⊆ν. Since ν⊆η:λ, we get xα⊆η:λ. Thus, xα⋅η⊆λ. Now,(23)η∩ν=∨xα⊆νxα∩η=∨xα⊆νxα⋅ηby Lemma3⊆ν.
Thus,η∩ν⊆λ.Theorem 5.
Letν be a fuzzy ideal of A. If λαα∈Δ is a class of fuzzy ideals of A, then(24)∪α∈Δλα:ν=∩α∈Δλα:ν.Proof.
We know thatλα⊆∪α∈Δλα for each α∈Δ. Thus, by Lemma 5 (3), we get ∪α∈Δλα:ν⊆λα:ν for each α∈Δ. Thus,(25)∪α∈Δλα:ν⊆∩α∈Δλα:ν.
On the contrary, putη=∩α∈Δλα:ν. Then, η⊆λα:ν, for each α∈Δ. By Lemma 5 (10), we have λα∩η⊆ν, for each α∈Δ. This implies(26)∨α∈Δλα∩η=∨α∈Δλα∩η⊆ν.
So, by Lemma5 (10), we have η⊆∨α∈Δλα:ν. Thus,(27)∩α∈Δλα:ν⊆∪α∈Δλα:ν.
So,(28)∪α∈Δλα:ν=∩α∈Δλα:ν.
In the following theorem, we prove thatλ:ν is a relative pseudocomplement of λ and ν in the class of FIA.Theorem 6.
Letη be a fuzzy subset and λ and ν fuzzy ideals of A. Then,(1)
η:λ is the largest fuzzy ideal such that η∩η:λ⊆λ(2)
λ:ν is the largest fuzzy ideal such that λ∩λ:ν⊆νProof.
First, we have to show thatη∩η:λ⊆λ. For any x∈A,(29)η∩η:λx=ηx∧Supγx:γ∈FIA,γ∧η⊆λ=Supηx∧γx:γ∧η⊆λ=Supη∩γx:γ∩η⊆λ≤λx.
Thus,η∩η:λ⊆λ.
Now, we show thatη:λ is the largest fuzzy ideal satisfying η∩η:λ⊆λ. Suppose not. Then, there exists a fuzzy ideal γ properly containing η:λ such that η∩γ⊆λ. Then, by Lemma 5 (10), we get that γ⊆η:λ, which is a contradiction. Therefore, η:λ is the largest fuzzy ideal, satisfying η∩η:λ⊆λ.
In [18], Alaba and Addis introduced the concept of fuzzy ideals of C-algebra, and they proved that the class of all fuzzy ideals of a C-algebra is a complete distributive lattice. In the following theorem, using the concept of relative fuzzy annihilator ideals of a C-algebra, we prove that the class of fuzzy ideals of a C-algebra forms the Heyting algebra.Theorem 7.
The setFIA of all fuzzy ideals of A is the Heyting algebra.Proof.
We know that the setFIA,∨,∩,χI0,χA of all fuzzy ideals of A is a complete distributive lattice. For any fuzzy ideals λ and ν of A, by Theorem 6, λ:ν is the largest fuzzy ideal of γ∈FIA:γ∩λ⊆ν. Thus,(30)λ⟶ν=λ:ν.
So,FIA,∨,∩,⟶,χI0,χA is the Heyting algebra.


## 4. Fuzzy Annihilator Ideals

In this section, we study fuzzy annihilator ideals inC-algebras. Some basic properties of fuzzy annihilator ideals are also studied. It is proved that the set of all fuzzy annihilator ideals forms a complete Boolean algebra.Definition 8.
For any fuzzy subsetλ of A, the fuzzy subset λ:χI0 is a fuzzy ideal denoted by λ∗ and λ∗ is called a fuzzy annihilator of λ.Lemma 6.
Letλ be a fuzzy subset of A. Then,(1)
χI0⊆λ∗(2)
λ⋅λ∗⊆χI0(3)
λ⋅λ∗=χI0, whenever λz=1, where z∈I0(4)
λ∗∧λ∗∗=χI0Proof.
Here, it is enough to prove property (3). Let λ be any fuzzy subset of A and x∈A. Then,(31)λ⋅λ∗x=Supλa∧λ∗b:x=a∧b=Supλa∧Supηb:η∈FIA,η⋅λ⊆χI0:x=a∧b=SupSupλa∧ηb:η⋅λ⊆χI0:x=a∧b=SupSupλa∧ηb:x=a∧b:η⋅λ⊆χI0=Supλ⋅ηx:η⋅λ⊆χI0≤χI0x.
This shows thatλ⋅λ∗⊆χI0. If λz=1, for z∈I0, then λ⋅λ∗z=1 and χI0=λ⋅λ∗.Lemma 7.
Letλ and ν be fuzzy subsets of A. Then,(1)
λ⊆ν⇒ν∗⊆λ∗(2)
ν⋅λ⊆χI0⇔ν⊆λ∗(3)
ν⋅λ=χI0⇔ν⊆λ∗, whenever λz=1=νz for z∈I0(4)
λ⊆λ∗∗(5)
λ∗=λ∗∗∗(6)
λ∗=λ∗(7)
λ∪ν∗=λ∗∩ν∗Lemma 8.
For any fuzzy idealsλ and ν of A, we have(1)
λ∨ν∗=λ∗∩ν∗(2)
λ∨λ∗∗=χI0(3)
χI0∗=χA(4)
χA∗=χI0Theorem 8.
The setFIA of all fuzzy ideals of A is a pseudocomplemented lattice.Proof.
Letλ be a fuzzy ideal of A. Then, it is clear that λ∗ is a fuzzy ideal of A and that λ∩λ∗=χI0. Suppose now θ∈FIA such that λ∩ν=χI0. Then, by Lemma 7 (2), ν⊆λ∗, and consequently, λ∗ is the pseudocomplement of λ.Lemma 9.
Ifλi∈0,1A, for every i∈I, then(32)∪i∈Iλi∗=∩i∈Iλi∗.Proof.
Letλi:i∈I be family of fuzzy subsets of L. Since λi⊆∪i∈Iλi for each i∈I, by Lemma 7 (1), we have ∪i∈Iλi∗⊆λi∗. Thus,(33)∪i∈Iλi∗⊆∩i∈Iλi∗.
To prove∩i∈Iλi∗⊆∪i∈Iλi∗, it is enough to show that ∩i∈Iλi∗⋅∪j∈Iλj⊆χI0. For any x∈A,(34)∩i∈Iλi∗⋅∪j∈Iλjx=Sup∩i∈Iλi∗a∧∪j∈Iλjb:a∧b=x=Sup∧i∈Iλi∗a∧∨j∈Iλjb:a∧b=x=Sup∨j∈I∧i∈Iλi∗a∧λjb:a∧b=x≤Sup∨j∈Iλj∗a∧λjb:a∧b=x≤Sup∨j∈Iλj∗⋅λjx:a∧b=x≤Sup∨j∈IχI0x=χI0x.
Thus, by Lemma7 (2), we get that ∩i∈Iλi∗⊆∪i∈Iλi∗. So,(35)∩i∈Iλi∗=∪i∈Iλi∗.
Now, we define the fuzzy annihilator ideal.Definition 9.
A fuzzy idealλ of A is called a fuzzy annihilator ideal if λ=ν∗, for some fuzzy subset ν of A, or equivalently, if λ=λ∗∗.
We denote the class of all fuzzy annihilator ideals ofA by FI∗A.Example 3.
Consider the three-elementC-algebra A=T,F,U and C=A×A=a1,a2,a3,a4,a5,a6,a7,a8,a9 given in Example 2. If we define a fuzzy subset λ of C as(36)λa1=λa2=λa4=λa5=λa7=λa8=1,λa3=λa6=λa9=0.4,then λ is a fuzzy ideal of C and λ=λ∗∗. Thus, λ is a fuzzy annihilator ideal of C.Lemma 10.
Letλ,ν∈FI∗A. Then,(1)
λ∩ν=λ∗∨ν∗∗(2)
λ∩ν=λ∩ν∗∗The result (2) of the above lemma can be generalized as given in the following.Corollary 1.
Ifλi:i∈Δ is a family of fuzzy annihilator ideals of A, then(37)∩i∈Δλi∗∗=∩i∈Δλi.Theorem 9.
A mapα:FIA⟶FIA defined by αλ=λ∗∗,∀λ∈FIA is a closure operator on FIA. That is,(1)
λ⊆αλ(2)
ααλ=αλ(3)
λ⊆ν⇒αλ⊆αν, for any two fuzzy ideals λ,ν of AFuzzy annihilator ideals are simply the closed elements with respect to the closure operator.Lemma 11.
Ifλ,ν∈FI∗L, the supremum of λ and ν is given by(38)λ⊔ν=λ∗∩ν∗∗.Proof.
First, we need to showλ⊔ν is a fuzzy annihilator ideal. Clearly λ⊔ν is a fuzzy ideal of A. Since λ∗∩ν∗⊆λ∗, we get λ=λ∗∗⊆ν∗⊆λ∗∗=λ⊔ν. Similarly, ν⊆λ⊔ν. This implies λ⊔ν is an upper bound of λ and ν. Suppose that γ is a fuzzy annihilator ideals of A such that λ⊆γ and ν⊆γ. Then, we get γ∗⊆λ∗ and γ∗⊆ν∗ and γ⊆λ∗∩ν∗. This implies λ⊔ν=λ∗∩ν∗∗⊆γ∗∗=γ. Hence, λ⊔ν is the smallest fuzzy annihilator ideal containing λ and ν.Corollary 2.
Letλii∈I be a family of fuzzy annihilator ideals of A. Then, ⊔i∈Iλi is the smallest fuzzy annihilator ideal containing each λi.In the following theorem, we prove that the class of all fuzzy annihilator ideals forms a complete Boolean algebra.Theorem 10.
The setFI∗A of all fuzzy annihilator ideals of A forms a complete Boolean algebra.Proof.
Clearly,FI∗A,∩,⊔,χI0,χA is a complete bounded lattice. To show the distributivity, let λ,ν,η∈FI∗A. Then,(39)λ⊔ν∩η=λ∗∩ν∩η∗∗=λ∗∗∗∩ν∗∗∩η∗∗∗∗=λ∗∩ν∗∨η∗∗∗∗=λ∗∩ν∗∨λ∗∩η∗∗=λ∗∩ν∗∗∩λ∗∩η∗∗=λ⊔ν∩λ⊔η.
Hence,FI∗A,∩,⊔,χI0,χA is a complete distributive lattice. For any λ∈FI∗A, we have λ∩λ∗=χI0 and λ∧λ∗=λ⊔λ∗∗=χA. Hence, λ∗ is the complement of λ in FI∗A. Therefore, FI∗A,∩,⊔,χI0,χA is a complete Boolean algebra.Definition 10.
A fuzzy idealμ of A is called dense fuzzy ideal if μ∗=χI0.


## 5. Fuzzy Annihilator Preserving Homomorphism

In this section, we study some basic properties of fuzzy annihilator preserving homomorphisms. We give a sufficient condition for a homomorphism to be fuzzy annihilator preserving. Finally, we show that the images and inverse images of fuzzy annihilator ideals are again fuzzy annihilator ideals.Throughout this section,A and A′ denote C-algebras with the smallest ideals I0 and I0′, respectively, and f:A⟶A′ denotes a C-algebra homomorphism.Lemma 12.
InA, the following conditions hold:(1)
χI0⊆χKerf(2)
fχI0⊆χI0′(3)
χKerf is a fuzzy ideal of ALemma 13.
Ifλ is any fuzzy subset of A and ν is a fuzzy ideal of A, then(40)fλ:ν⊆fλ:fν.In particular, ifν=χI0, then fλ∗⊆fλ∗.Proof.
Letλ be any fuzzy subset of A and ν be a fuzzy ideal of A. For any y∈A′,(41)fλ:νy=Supλ:νx:x∈f−1y=SupSupηx:η∈FIA,η⋅λ⊆ν:x∈f−1y=Supfηy:η⋅λ⊆ν≤Supγy:γ∈FIA′,λ⋅fλ⊆fν=fλ:fνy.
Then,fλ:ν⊆fλ:fν.Definition 11.
For any fuzzy subsetλ of L, f is said to be a fuzzy annihilator preserving if fλ∗=fλ∗.
In the following theorem, we give a sufficient condition for a homomorphism to be fuzzy annihilator preserving.Theorem 11.
IfKerf=I0 and f is onto, then f is a fuzzy annihilator preserving.Proof.
Letλ be any fuzzy subset of A. Then, fλ∗⊆fλ∗. Since Kerf=I0 and f is onto, f−1χI0′=χI0 and ν=ff−1ν, for all ν∈0,1A′. Let y∈A′. Then,(42)fλ∗y=Supνy:ν∈FIA′,ν⋅fλ⊆χI0′=Supνy:ff−1ν⋅fλ⊆χI0′=Supνy:ff−1ν⋅λ⊆χI0′=Supνy:f−1ν⋅λ⊆f−1χI0′=Supνfx:f−1ν⋅λ⊆χI0,fx=y≤SupSupf−1νx:f−1ν⋅λ⊆χI0:x∈f−1y≤SupSupηx:η∈FIA,η⋅λ⊆χI0:x∈f−1y=fλ∗y.
Thus,fλ∗⊆fλ∗. So, f preserves fuzzy annihilator.Theorem 12.
IfKerf=I0 and f is onto, then f−1 preserves the fuzzy annihilator.Proof.
Letλ be any fuzzy subset of A′ and x∈A. Then,(43)f−1λ∗x=λ∗fx=Supνfx:ν∈FIA′,ν⋅λ⊆χI0′=Supνfx:ff−1ν⋅ff−1λ⊆χI0′=Supf−1νx:f−1ν⋅f−1λ⊆χI0≤Supηx:η∈FIA,η⋅f−1λ⊆χI0=f−1λ∗x.
Thus,f−1λ∗⊆f−1λ∗. Similarly, f−1λ∗⊆f−1λ∗. Therefore, f−1λ∗=f−1λ∗.Theorem 13.
For anyC-algebra A, the following conditions hold:(1)
Iff is fuzzy annihilator preserving and onto, then fλ is a fuzzy annihilator ideal of A′, for every fuzzy annihilator ideal λ of A(2)
Iff−1 preserves the annihilator, then f−1λ is a fuzzy annihilator ideal of A, for every fuzzy annihilator ideal λ of A′(3)
χKerf is a fuzzy annihilator ideal of A


## 6. Conclusion

In this work, we studied the concept of relative fuzzy annihilator ideals ofC-algebras. We characterized relative fuzzy annihilators in terms of fuzzy points. We proved that the class of fuzzy ideals of C-algebras forms the Heything algebra. We also studied fuzzy annihilator ideals and investigate some its properties. It is shown that the class of all fuzzy annihilator ideals forms a complete Boolean algebra. Moreover, we study the concept of fuzzy annihilator preserving homomorphism. Our future work will focus on fuzzy congruence relation on C-algebras.



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