# (Fuzzy) Ideals of BN-Algebras

Authors: Grzegorz Dymek; Andrzej Walendziak
Journal: The Scientific World Journal
 (2015)

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## Abstract

The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy
set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained.


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## Body

## 1. Introduction

In 1966, Imai and Iséki [1] introduced the notion of a BCK-algebra. There exist several generalizations of BCK-algebras, such as BCI-algebras [2], BCH-algebras [3], BCC-algebras [4], BH-algebras [5], and d-algebras [6]. Neggers et al. defined B/BM/BG-algebras [7–9] and showed that the class of all B-algebras is a proper subclass of the class of all BG-algebras. They also proved that an algebra is a BM-algebra if and only if it is a 0-commutative B-algebra (therefore, every BM-algebra is a B-algebra). In [10], it is shown that the class of 0-commutative B-algebras is the class of p-semisimple BCI-algebras (and hence any BM-algebra is a BCI-algebra). The class of BM-algebras contains Coxeter algebras (see [8]). Some other connections between BM-algebras and its related topics are studied in [11]. Walendziak introduced in [12] the concept of BF-algebras, which is a generalization of B-algebras and BN-algebras defined by C. B. Kim and H. S. Kim [13]. An interesting result of [13] states that an algebra is a BN-algebra if and only if it is a 0-commutative BF-algebra.We will denote byBCK (resp., BCI/BCH/BH/B/BM/BG/BF/BN) the class of all BCK-algebras (resp., BCI/BCH/BH/B/BM/BG/BF/BN-algebras). The interrelationships between some classes of algebras mentioned before are visualized in Figure 1 (an arrow indicates proper inclusion; that is, if X and Y are classes of algebras, then X→Y means X⊂Y).Figure 1In this paper we consider ideals and fuzzy ideals in BN-algebras. In Section2, similarly to BCK/BCI/BCH/BH/BF-algebras (see [5, 12, 14–16]), we define the concept of an ideal. We also introduce the notions of normal ideals and normal congruences. We investigate the properties of them and prove that there is a one-to-one correspondence between normal ideals and normal congruences of a BN-algebra. Moreover we obtain the isomorphism theorem for BN-algebras. In this section we also give all what is necessary, to make the paper self-contained. In Section 3 we define fuzzy ideals in BN-algebras (fuzzy ideals of BCK/BCI/BCC/BF-algebras are considered in [17–20]) and provide conditions for a fuzzy set to be a fuzzy ideal. Given a fuzzy set μ, we make the least fuzzy ideal containing μ. This leads us to show that the set of fuzzy ideals of a BN-algebra is a complete lattice. Moreover, the homomorphic properties of fuzzy ideals are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras in terms of fuzzy ideals are given in Section 4. Noetherian BCK/BCI/BM/BF-algebras are studied in [21–26].


## 2. On BN-Algebras: Ideals

An algebra(A;*,0) of type (2,0) is called aCoxeter algebra [27] if for all x,y,z∈A the following identities hold:(B1)
x*x=0,(B2)
x*0=x,(C)
(x*y)*z=x*(y*z).Remark 1.
Kim et al. [27] showed that a Coxeter algebra is equivalent to an abelian group all of whose elements have order 2. Therefore, if (A;*,0) is a Coxeter algebra, then the operation * is commutative and associative.We say that(A;*,0) is aBM-algebra [8] if it satisfies (B2) and(BM)
(z*x)*(z*y)=y*x.Proposition 2 (see [8]).
If(A;*,0) is a BM-algebra, then, for any x,y,z∈A,(i)
0*(x*y)=y*x,(ii)
(x*y)*z=(x*z)*y.Proposition 3.
If(A;*,0) is a BM-algebra, then, for all x,y∈A,(i)
x*(x*y)=y,(ii)
ifx*y=0, then x=y.Proof.
(i) Substitutingz=x and x=0 in (BM), we have (x*0)*(x*y)=y*0. Applying (B2) we get (i).
(ii) follows from (i) and (B2).An algebra(A;*,0) of type (2,0) is called aBF-algebra [12] if it satisfies (B1), (B2), and(BF)
0*(x*y)=y*x.Definition 4.
An algebra(A;*,0) of type (2,0) is called aBN-algebra [13] if (B1), (B2), and(BN)
(x*y)*z=(0*z)*(y*x)hold for all x,y,z∈A.Let(A;*,0) be a BN-algebra. We define a binary relation ≤ on A by x≤y if and only if x*y=0. It is easy to see that, for any x∈A, if x≤0, then x=0.Example 5.
LetR be the set of real numbers and let R=(R;*,0) be the algebra with the operation * defined by
(1)x*y=x,ify=0,y,ifx=0,0,otherwise.
Then R is a BN-algebra.Example 6 (see [13]).
LetA={0,1,2,3} and * be defined by the following:
(2)*012300123110112210133110
Then A;*,0 is a BN-algebra.Example 7.
LetG;+,0 be an abelian group. If we define x*y=x-y, for all x,y∈G, then, by Theorem 2.15 of [13], G;*,0 is a BN-algebra.Proposition 8 (see [13]).
If(A;*,0) is a BN-algebra, then, for all x,y∈A, (i)
0*(0*x)=x,(ii)
0*(x*y)=y*x,(iii)
y*x=(0*x)*(0*y),(iv)
x*y=0⇒y*x=0,(v)
0*x=0*y⇒x=y.From [13] we obtain the following relation: BM⊂BN⊂BF.From now on,A always denotes a BN-algebra (A;*,0). We introduce the notion of an ideal in BN-algebras.Definition 9.
A subsetI of A is called anideal of A if it satisfies (I1)
0∈I,(I2)
x*y∈I and y∈I imply x∈I for any x,y∈A.We will denote byId(A) the set of all ideals of a BN-algebra A.Example 10.
LetR=(R;*,0) be the BN-algebra given in Example 5. Observe that Id(R)={{0},R}. Indeed, let I be an ideal of R and suppose that I≠{0}. Then a∈I for some a∈R-{0}. Let b∈R. Obviously, b*a∈I and hence b∈I. Therefore, I=R.Proposition 11.
LetI∈Id(A) and x,y∈A. If x≤y and y∈I, then x∈I.Proof.
Letx≤y and y∈I. Then x*y=0∈I and y∈I. Hence x∈I.A nonempty subsetS of A is called asubalgebra of A if x*y∈S for any x,y∈S. It is easy to see that if S is a subalgebra of A, then 0∈S.Example 12.
LetZ=(Z;-,0), where Z is the set of all integers. We know that Z is a BN-algebra (see Example 7). Obviously, a nonempty subset S⊆Z is a subalgebra of Z if and only if a-b∈S for all a,b∈S; that is, S is a subgroup of the group of integers. Therefore,
(3)∅≠S⊆Zis a subalgebra ofZ⇔S=mk:k∈Zfor somem∈N∪0.Remark 13.
Consider the BN-algebraA=A;*,0 given in Example 6. We have Id(A)={{0},{0,2},{0,3},{0,2,3},A}. Observe that the ideal I={0,2,3} is not a subalgebra of A and S={0,1} is a subalgebra of A but it is not an ideal.Remark 14.
It is easy to prove that the intersection of an arbitrary number of ideals of a BN-algebraA is an ideal of A. It is also not hard to show that the union of an ascending sequence of ideals of A is an ideal of A.A nonempty subsetN of A is said to benormal in A if
(4)x*a*y*b∈N
for any x*y,a*b∈N. We say that an ideal I of A (resp., a subalgebra S of A) isnormal if the set I (resp., the set S) is normal. We will denote by IdN(A) the set of all normal ideals of A.Remark 15.
It is easy to see that{0}, A∈Id(A). The ideal A is normal, but {0} is not normal in general (see the example below).Example 16.
Consider the setA={0,1,2,3} with the operation * defined by the following:(5)*012300123110302230233020
It is easily seen that A;*,0 is a BN-algebra. The ideal I={0} is not normal because 1*3=0∈I, but (2*1)*(2*3)=3*2=2∉I.Proposition 17.
IfI is a normal ideal of A, then I is a subalgebra of A.Proof.
Letx,y∈I. Since I is normal, we conclude that (x*y)*(x*0)∈I; that is, (x*y)*x∈I. From the definition of an ideal we have x*y∈I. Thus I is a subalgebra of A.Remark 18.
The BN-algebraA;*,0 given in Example 16 shows that the converse of Proposition 17 does not hold. Indeed, {0} is a subalgebra of A but it is not a normal ideal.Proposition 19.
Every ideal of a Coxeter algebra is normal.Proof.
LetA=A;*,0 be a Coxeter algebra and let I be an ideal of A. Let x,y,a,b∈A and suppose that x*y,a*b∈I. Since I is a subalgebra, we have (x*y)*(a*b)∈I. From Remark 1 it follows that (x*a)*(y*b)∈I. Thus I is normal.Lemma 20.
LetI be a normal ideal of a BN-algebra A and x,y∈A. Then (a)
x∈I⇒0*x∈I,(b)
x*y∈I⇒y*x∈I.Proof.
(a) Letx∈I. Then x=x*0∈I. Since I is normal, (0*x)*(0*0)∈I. Hence, 0*x∈I.
(b) Letx*y∈I. By (a), 0*(x*y)∈I. Applying Proposition 8(ii) we have y*x∈I.Proposition 21.
LetA be a BN-algebra and let S⊆A. Then S is a normal subalgebra of A if and only if S is a normal ideal.Proof.
LetS be a normal subalgebra of A. Clearly, 0∈S. Suppose that x*y∈S and y∈S. Then 0*y∈S. Since S is normal, we have (x*0)*(y*y)∈S. By (B1) and (B2), (x*0)*(y*y)=x. Therefore x∈S, and thus S is an ideal. The converse follows from Proposition 17.Remark 22.
LetI be a normal ideal of a BN-algebra A. For any x,y∈A, we define
(6)x~Iy⟺x*y∈I.
Then ~I is a congruence of A by the proof of Theorem 3.5 of [13].Definition 23.
Letθ be a congruence of a BN-algebra A. One says that θ isnormal if
(7)x*yθ0,a*bθ0implyx*a*y*bθ0
for arbitrary x,y,a,b∈A.Example 24.
LetA={0,1,2,3} and * be defined by the following:(8)*012300123110322230133210
It is easy to check that A=A;*,0 is a BN-algebra and θ={(0,0),(1,1),(2,2),(3,3),(1,0),(0,1),(2,3),(3,2)} is a normal congruence of A.ByCN(A) we denote the set of all normal congruences of A.Remark 25.
LetA be a BN-algebra. It is easily seen that ~I∈CN(A), where I is a normal ideal of A. In particular, A2=~A∈CN(A).Proposition 26.
In BM-algebras all congruences are normal.Proof.
LetA be a BM-algebra and let θ be a congruence of A. Let x,y,a,b∈A and suppose that (x*y)θ0 and (a*b)θ0. Hence, ((x*y)*(a*b))θ0. Observe that
(9)x*y*a*b=x*a*y*b.
Indeed, we have
(10)x*y*a*b=0*y*x*a*bby Proposition2i=0*a*b*y*xby Proposition2ii=b*a*y*xby Proposition2i=y*a*y*b*y*xbyBM=y*a*y*x*y*bby Proposition2ii=x*a*y*bbyBM.
Therefore, ((x*a)*(y*b))θ0 and thus θ is normal.Remark 27.
LetA=(A;*,0) be the BN-algebra from Example 16. Observe that the least congruence on A, namely, the identity relation ω={(a,a):a∈A}, is not normal. Indeed, we get (1*3,0)∈ω and (2*2,0)∈ω but ((1*2)*(3*2),0)=(2,0)∉ω.LetA be a BN-algebra and let θ be a congruence on A. For x∈A, by [x]θ we denote the congruence class containing x; that is, [x]θ={y∈A:xθy}.Proposition 28.
Letθ be a congruence on A. Then θ∈CN(A) if and only if [0]θ is a normal ideal of A.Proof.
SetI=[0]θ and let θ∈CN(A). It follows easily that 0∈I. Let x and y be elements of A such that x*y,y∈I. Then, (x*y)θ0 and yθ0. Since yθ0, we obtain (x*y)θ(x*0) and hence x=x*0∈I. Consequently, I is an ideal. Now, suppose that x*y,a*b∈I, where x,y,a,b∈A. Then, (x*y)θ0 and (a*b)θ0. By the definition of a normal congruence, ((x*a)*(y*b))θ0; that is, (x*a)*(y*b)∈I. Thus, I=[0]θ is normal.
The converse is obvious.Theorem 29.
There is a bijection between the set of normal ideals and the set of normal congruences of a BN-algebra.Proof.
LetA be a BN-algebra. We consider functions f:IdN(A)→CN(A) and g:CN(A)→IdN(A) given as follows:
(11)fI=~Ifor anyI∈IdNA,gθ=0θfor anyθ∈CNA.
Since ~I∈CNA and [0]θ∈IdNA, we conclude that f and g are well-defined. Next, note that I=[0]~I. Indeed,
(12)x∈I⟺x*0∈I⟺x~I0⟺x∈0~I.
Hence,
(13)g∘fI=g~I=0~I=I∀I∈IdNA.
Further, observe that
(14)xθy⟺x*yθ0⟺x*y∈0θ
and from this we obtain
(15)f∘gθ=f0θ=θ∀θ∈CNA.
We deduce from 13 and 15 that g∘f=idIdN(A) and f∘g=idCN(A). Thus, f and g are inverse bijections between IdN(A) and CN(A).LetA=(A,*,0A) and B=(B,*,0B) be BN-algebras. A mapping f:A→B is called ahomomorphism from A into B if f(x*y)=f(x)*f(y) for any x,y∈A.Observe thatf(0A)=0B. Indeed, f(0A)=f(0A*0A)=f(0A)*f(0A)=0B. We denote by ker⁡f the subset {x∈A:f(x)=0B} of A (it is thekernel of the homomorphism f).Lemma 30.
Letf:A→B be a homomorphism from A into B. Then ker⁡f is an ideal of A.Proof.
Obviously,0A∈ker⁡f; that is, (I1) holds. Let x*y∈ker⁡f and y∈ker⁡f. Then 0B=f(x*y)=f(x)*f(y)=f(x)*0B=f(x). Consequently, x∈ker⁡f. Therefore, (I2) is satisfied. Thus I is an ideal of A.Remark 31.
The kernel of a homomorphism is not always a normal ideal. LetA;*,0 be the algebra given in Example 16. Clearly, idA:A→A is a homomorphism and the ideal ker⁡(idA)={0} is not normal.Proposition 32.
LetA=(A,*,0A) be a BN-algebra and let B=(B,*,0B) be a BM-algebra. Let f:A→B be a homomorphism from A into B. Then ker⁡f is a normal ideal.Proof.
By Lemma30, ker⁡f is an ideal of A. Let x,y,a,b∈A and x*y,a*b∈ker⁡f. Then 0B=f(x*y)=f(x)*f(y). From Proposition 3(ii) it follows that f(x)=f(y). Similarly, f(a)=f(b). Consequently, f((x*a)*(y*b))=(f(x)*f(a))*(f(y)*f(b))=(f(x)*f(a))*(f(x)*f(a))=0B, and hence, (x*a)*(y*b)∈ker⁡f.LetI be a normal ideal of A. For x∈A, we write x/I={y∈A:x~Iy}; that is, x/I=[x]~I. We note that
(16)x~Iy⟺xI=yI.
Denote A/I={x/I:x∈A} and set x/I*′y/I=x*y/I. The operation *′ is well-defined, since ~I is a congruence of A. It is easy to see that A/I=(A/I,*′,0/I) is a BN-algebra. The algebra A/I is called thequotient BN-algebra ofAmoduloI.Example 33.
LetA and θ be given as in Example 24. We know that A is a BN-algebra and θ is a normal congruence of A. Since [0]θ={0,1}, from Proposition 28 we see that I={0,1} is a normal ideal of A. We have 0/I=1/I={0,1} and 2/I=3/I={2,3}. Then A/I={0/I,2/I}. Clearly, 0/I*′0/I=0/I=2/I*′2/I and 0/I*′2/I=2/I=2/I*′0/I.Theorem 34.
LetA be a BN-algebra and let B be a BM-algebra. Let f:A→B be a homomorphism from A onto B. Then A/ker⁡f is isomorphic to B.Proof.
By Proposition32, I=ker⁡f is a normal ideal of A. Define a mapping g:A/I→B by g(x/I)=f(x) for all x∈I. Let x/I=y/I. Then, x~Iy; that is, x*y∈I. Hence, f(x)*f(y)=0B. By Proposition 3(ii) we have f(x)=f(y). Consequently, g(x/I)=g(y/I). This means that g is well-defined. It is easy to check that g is a homomorphism from A/I onto B. Observe that g is one-to-one. Let g(x/I)=g(y/I). Then f(x)=f(y) and hence f(x*y)=0B; that is, x*y∈I. Therefore, x~Iy and consequently, x/I=y/I. Thus g is an isomorphism from A/I onto B.


## 3. Fuzzy Ideals

We now review some fuzzy logic concepts. First, forΓ⊆0;1 we define ⋀Γ=inf⁡Γ and ⋁Γ=sup⁡Γ. Obviously, if Γ=α,β, then α∧β=min⁡α,β and α∨β=max⁡α,β. Recall that afuzzy set in A is a function μ:A→0;1.For any fuzzy setsμ and ν in A, we define
(17)μ≤ν⟺μx≤νx∀x∈A.
A trivial verification shows that this relation is an order relation in the set of fuzzy sets in A.LetA and B be any two sets, μ any fuzzy set in A, and f:A→B any function. Set f←(y)={x∈A:f(x)=y} for y∈B. The fuzzy set ν in B defined by
(18)νy=⋁μx:x∈f←y,iff←y≠⌀,0,otherwise,
for all y∈B, is called theimage of μ under f and is denoted by fμ.LetA and B be any two sets, f:A→B any function, and ν any fuzzy set in fA. The fuzzy set μ in A defined by
(19)μx=νfx,∀x∈A
is called thepreimage of ν under f and is denoted by f←ν.Now, we give the definition of a fuzzy ideal in a BN-algebra.Definition 35.
A fuzzy setμ in A is called afuzzy ideal of a BN-algebra A if it satisfies, for all x,y∈A, (d1)
μ(0)≥μ(x),(d2)
μ(x)≥μ(x*y)∧μ(y).Proposition 36.
Letμ be a fuzzy ideal of A. Then, for any x,y∈A, if x≤y, then μ(x)≤μ(y).Proof.
Ifx≤y, then x*y=0. From Proposition 8(iv) we obtain y*x=0. Hence, by (d2) and (d1), we have μ(y)≥μ(y*x)∧μ(x)=μ(0)∧μ(x)=μ(x).Denote byFIdA the set of all fuzzy ideals of a BN-algebra A.Example 37.
LetA=(A,*,0) be the BN-algebra given in Example 6. Let 0≤α3<α2<α1≤1. Define a fuzzy set μ in A by
(20)μx=α1,ifx=0,α2,ifx=2,α3,ifx∈1,3.
It is easily checked that μ satisfies (d1) and (d2). Thus μ∈FId(A).Example 38.
LetI be an ideal of a BN-algebra A and let α,β∈[0;1] with α≥β. Define μIα,β as follows:
(21)μIα,βx=α,ifx∈I,β,otherwise.
We denote μIα,β=μ. Since 0∈I, μ(0)=α≥μ(x) for all x∈A. To prove (d2), let x,y∈A. If x∈I, then μ(x)=α≥μ(x*y)∧μ(y). Suppose now that x∉I. By the definition of an ideal, x*y∉I or y∉I. Therefore, μ(x*y)∧μ(y)=β=μ(x). Thus μ is a fuzzy ideal of A.
In particular, the characteristic functionχI of I:
(22)χIx=1,ifx∈I,0,otherwise,
is a fuzzy ideal of A.Proposition 39.
A fuzzy setμ in A is a fuzzy ideal of A if and only if it satisfies (d1) and (d3)
for allx,y,z∈A, if (z*y)*x=0, then μ(z)≥μ(x)∧μ(y).Proof.
Letμ∈FIdA and let x,y,z∈A. Suppose that (z*y)*x=0. Since μ is a fuzzy ideal, we have μ(z*y)≥μ((z*y)*x)∧μ(x)=μ(0)∧μ(x)=μ(x) and μ(z)≥μ(z*y)∧μ(y). Therefore, μ(z)≥μ(x)∧μ(y).
Conversely, letμ satisfy (d3). From (B1) we have (x*y)*(x*y)=0. By (d3), μ(x)≥μ(x*y)∧μ(y). Then μ satisfies (d2) and hence μ∈FIdA.Theorem 40.
Letμ be a fuzzy set in A. Then μ∈FIdA if and only if its nonempty level subset
(23)Uμ;α=x∈A:μx≥α
is an ideal of A for all α∈[0;1].Proof.
Assume thatμ∈FIdA. Let α∈[0;1] and U(μ;α)≠∅. Then μ(x0)≥α for some x0∈A. Since μ(0)≥μ(x0), we have 0∈U(μ;α). Let x,y∈A such that x*y,y∈U(μ;α). Then μ(x*y)≥α and μ(y)≥α. It follows from (d2) that
(24)μx≥μx*y∧μy≥α
so that x∈U(μ;α). Therefore U(μ;α) is an ideal of A.
Conversely, suppose that, for eachα∈[0;1],U(μ;α)=∅ or U(μ;α) is an ideal of A. If (d1) is not valid, then there exists x0∈A such that μ(0)<μ(x0)=β. Then U(μ;β)≠∅ and, by assumption, U(μ;β) is an ideal of A. Hence 0∈U(μ;β) and consequently μ(0)≥β. This is a contradiction and (d1) is valid. Now assume that (d2) does not hold. Then there are a,b∈A such that μ(a)<μ(a*b)∧μ(b). Taking
(25)γ=12μa+μa*b∧μb,
we get μ(a)<γ<μ(a*b)∧μ(b)≤μ(a*b) and γ<μ(b). Therefore a*b,b∈U(μ;γ) but a∉U(μ;γ). This is impossible, and μ is a fuzzy ideal of A.By Theorem40, we have the following.Corollary 41.
Ifμ is a fuzzy ideal of a BN-algebra A, then the set
(26)Aμ=x∈A:μx=μ0
is an ideal of A.The following example shows that the converse of Corollary41 does not hold.Example 42.
LetA be a BN-algebra. Define a fuzzy set μ in A by
(27)μx=0.4,ifx=0,0.6,ifx≠0.
Then Aμ=0 is the ideal of A but μ∉FIdA (because μ does not satisfy (d1)).Example 43.
Consider the BN-algebraA=(A,*,0) given in Example 6. Let μ be defined as in Example 37. It is easy to check that for all α∈0;1 we have
(28)Uμ;α=∅,ifα>α1,0,ifα2<α≤α1,0,2,ifα3<α≤α2,A,ifα≤α3.
Since 0,0,2 and A are ideals of A, this is another proof (by Theorem 40) that μ is a fuzzy ideal of A.Lemma 44.
LetI1⊂I2⊂⋯⊂In⊂⋯ be a strictly ascending sequence of ideals in a BN-algebra A and let (tn) be a strictly decreasing sequence in 0;1. Let μ be the fuzzy set in A defined by
(29)μx=0,ifx∉Inforeachn∈N,tn,ifx∈In-In-1forsomen∈N,
where I0=∅. Then μ is a fuzzy ideal of A.Proof.
LetI=⋃n∈NIn. By Remark 14, I is an ideal of A. Obviously, μ(0)=t1≥μ(x) for all x∈A; that is, (d1) holds. Now we show that μ satisfies (d2). Let x,y∈A. We have two cases.
Case 1
(
x
∉
I
)
. Then x*y∉I or y∉I. Therefore μ(x*y)∧μ(y)=0=μx.
Case 2
(
x
∈
I
n
-
I
n
-
1 for some n∈N). Then x*y∉In-1 or y∉In-1. Hence μ(x*y)≤tn or μ(y)≤tn. Therefore μ(x*y)∧μ(y)≤tn=μx.
Thus (d2) is also satisfied and consequentlyμ is a fuzzy ideal of A.LetT be a nonempty set of indexes. Let μt∈FId(A) for t∈T. The meet ⋀t∈Tμt of fuzzy ideals μt of A is defined as follows:
(30)⋀t∈Tμtx=⋀μtx:t∈T.Theorem 45.
Letμt∈FId(A) for t∈T. Then ⋀t∈Tμt∈FId(A).Proof.
Letμ=⋀t∈Tμt. Then, by (d1),
(31)μ0=⋀μt0:t∈T≥⋀μtx:t∈T=μx
for all x∈A. Let x,y∈A. Since μt∈FId(A), we have μt(x)≥μt(x*y)∧μt(y). Hence
(32)⋀μtx:t∈T≥⋀μtx*y∧μty:t∈T=⋀μtx*y:t∈T∧⋀μty:t∈T.
Consequently, μ(x)≥μ(x*y)∧μ(y) and therefore μ∈FId(A).Letν be a fuzzy set in A. A fuzzy ideal μ of A is said to begenerated by ν if ν≤μ and, for any fuzzy ideal ρ of A, ν≤ρ implies μ≤ρ. The fuzzy ideal generated by ν will be denoted by (ν]. The fuzzy ideal (ν] can be defined equivalently as follows:
(33)ν=⋀ρ∈FIdA:ρ≥ν.Forμ,ν∈FId(A) let μ∨ν denote the join of μ and ν; that is, μ∨ν=(ρ], where ρ is the fuzzy set in A defined by ρ(x)=μ(x)∨ν(x) for all x∈A.From Theorem45 we obtain the following theorem.Theorem 46.
LetA be a BN-algebra. Then (FIdA;∧,∨) is a complete lattice.The following two theorems give the homomorphic properties of fuzzy ideals.Theorem 47.
LetA and B be BN-algebras and let f:A→B be a homomorphism and ν∈FIdB. Then f←ν∈FIdA.Proof.
Letx∈A. Since f(x)∈B and ν∈FIdB, we have ν(0)≥ν(f(x))=(f←(ν))(x), but ν(0)=ν(f(0))=(f←(ν))(0). Thus we get (f←(ν))(0)≥(f←(ν))(x) for any x∈A; that is, f←ν satisfies (d1).
Now letx,y∈A. Since ν∈FIdB, we obtain
(34)νfx≥νfx*fy∧νfy=νfx*y∧νfy
and hence (f←(ν))(x)≥(f←(ν))(x*y)∧(f←(ν))(y). Consequently, f←(ν)∈FId(A).Lemma 48.
LetA and B be BN-algebras and let f:A→B be a homomorphism and μ∈FIdA. Then, if μ is constant on ker⁡f=f←(0), then f←(f(μ))=μ.Proof.
Letx∈A and fx=y. Hence
(35)f←fμx=fμfx=fμy=⋁μa:a∈f←y.
For all a∈f←y, we have fx=fa. Hence f(a*x)=0; that is, a*x∈ker⁡f. Thus μ(a*x)=μ(0). Therefore, μ(a)≥μ(a*x)∧μ(x)=μ(0)∧μ(x)=μ(x). Similarly, μx≥μa. Hence μx=μa. Thus
(36)f←fμx=⋁μa:a∈f←y=μx;
that is, f←fμ=μ.Theorem 49.
LetA and B be BN-algebras and let f:A→B be a surjective homomorphism and μ∈FIdA such that Aμ⊇ker⁡f. Then fμ∈FIdB.Proof.
Sinceμ is a fuzzy ideal of A and 0∈f←0, we have
(37)fμ0=⋁μa:a∈f←0=μ0≥μx
for any x∈A. Hence
(38)fμ0≥⋁μx:x∈f←y=fμy
for any y∈B. Thus fμ satisfies (d1). Suppose that
(39)fμxB<fμxB*yB∧fμyB
for some xB,yB∈B. Since f is surjective, there are xA,yA∈A such that f(xA)=xB and f(yA)=yB. Hence
(40)fμfxA<fμfxA*yA∧fμfyA.
Therefore
(41)f←fμxA<f←fμxA*yA∧f←fμyA.
Since Aμ⊇ker⁡f, μ is constant on ker⁡f. Then, by Lemma 48, we get
(42)μxA<μxA*yA∧μyA,
which is a contradiction with the fact that μ is a fuzzy ideal. Thus, we obtain fμ∈FIdB.


## 4. Fuzzy Characterizations of Noetherian and Artinian BN-Algebras

In this section we characterize Noetherian BN-algebras and Artinian BN-algebras using some fuzzy concepts, in particular, fuzzy ideals.A BN-algebraA is calledNoetherian if for every ascending sequence I1⊆I2⊆⋯ of ideals of A there exists k∈N such that In=Ik for all n≥k. A BN-algebra A is calledArtinian if for every descending sequence I1⊇I2⊇⋯ of ideals of A there exists k∈N such that In=Ik for all n≥k.Theorem 50.
LetA be a BN-algebra. The following statements are equivalent: (a)
A is Noetherian,(b)
for each fuzzy idealμ of A, Im⁡μ={μx:x∈A} is a well-ordered set.Proof.
(a)⇒ (b): Assume that A is Noetherian and μ is a fuzzy ideal of A such that Im⁡μ is not a well-ordered subset of 0;1. Then there exists a strictly decreasing sequence (μxn), where xn∈A. Let tn=μxn and Un=Uμ;tn={x∈A:μx≥tn}. Then, by Theorem 40, Un is an ideal of A for every n∈N. So U1⊂U2⊂⋯ is a strictly ascending sequence of ideals of A. This is a contradiction with the assumption that A is Noetherian. Therefore Im⁡μ is a well-ordered set for each fuzzy ideal μ of A.
(b)⇒ (a): Assume that (b) is true. Suppose that A is not Noetherian. Then there exists a strictly ascending sequence I1⊂I2⊂⋯⊂In⊂⋯ of ideals of A. Let μ be a fuzzy set in A such that
(43)μx=0,ifx∉Infor eachn∈N,1n,ifx∈In-In-1for somen∈N,
where I0=∅. By Lemma 44, μ∈FId(A), but Im⁡μ is not a well-ordered set, which is impossible. Therefore A is Noetherian.Corollary 51.
LetA be a BN-algebra. If, for every fuzzy ideal μ of A, Im⁡μ is a finite set, then A is Noetherian.Theorem 52.
LetA be a BN-algebra and let T={t1,t2,…}∪0, where (tn) is a strictly decreasing sequence in 0;1. Then the following conditions are equivalent: (a)
A is Noetherian,(b)
for each fuzzy idealμ of A, if Im⁡μ⊆T, then there exists k∈N such that Im⁡μ⊆{t1,t2,…,tk}∪0.Proof.
(a)⇒ (b): Assume that A is Noetherian. Let μ be a fuzzy ideal of A such that Im⁡μ⊆T. From Theorem 50 we know that Im⁡μ is a well-ordered subset of 0;1. Then, since 1>t1>t2>⋯>tn>⋯>0 and Im⁡μ⊆{t1,t2,…}∪0, there exists k∈N such that Im⁡μ⊆{t1,t2,…,tk}∪0.
(b)⇒ (a): Assume that (b) is true. Suppose that A is not Noetherian. Then there exists a strictly ascending sequence I1⊂I2⊂⋯ of ideals of A. Define a fuzzy set μ in A by
(44)μx=0,ifx∉Infor eachn∈N,tn,ifx∈In-In-1for somen∈N,
where I0=∅. By Lemma 44, μ is a fuzzy ideal of A. This is a contradiction with our assumption. Thus A is Noetherian.Theorem 53.
LetA be a BN-algebra and let T={t1,t2,…}∪0,1, where (tn) is a strictly increasing sequence in 0;1. Then the following conditions are equivalent: (a)
A is Artinian,(b)
for each fuzzy idealμ of A, if Im⁡μ⊆T, then there exists k∈N such that Im⁡μ⊆{t1,t2,…,tk}∪0,1.Proof.
(a)⇒ (b): Suppose that ti1<ti2<⋯<tim<⋯ is a strictly increasing sequence of elements of Im⁡(μ). Let Um=Uμ;tim for m∈N. It is immediately seen that U1⊃U2⊃⋯⊃Um⊃⋯ is a strictly descending sequence of ideals of A. This contradicts the assumption that A is Artinian.
(b)⇒ (a): Assume that (b) is true. Suppose that A is not Artinian. Then there exists a strictly descending sequence I1⊃I2⊃⋯⊃In⊃⋯ of ideals of A. Define a fuzzy set μ in A by
(45)μx=0,ifx∉I1,tn,ifx∈In-In+1forn=1,2,…,1,ifx∈⋂In:n∈N.
Obviously, μ(0)=1≥μ(x) for all x∈A; that is, (d1) holds. Now we show that μ satisfies (d2). Let x,y∈A. We have three cases.
Case 1
(
x
∉
I
1
)
. Then x*y∉I1 or y∉I1. Therefore μ(x*y)∧μ(y)=0=μx.
Case 2
(
x
∈
I
n
-
I
n
+
1 for some n∈N). Then x*y∉In+1 or y∉In+1. Hence μ(x*y)≤tn or μ(y)≤tn. Therefore μ(x*y)∧μ(y)≤tn=μx.
Case 3
(
x
∈
⋂
I
n
:
n
∈
N
)
. It is obvious.
Thusμ is a fuzzy ideal of A. This contradicts our assumption. Thus A is Artinian.Corollary 54.
LetA be a BN-algebra. If, for every fuzzy ideal μ of A, Im⁡μ is a finite set, then A is Artinian.The following example shows that the converse of Corollary54 does not hold.Example 55.
Letp be a prime number. Set A={z∈C:zpn=1 for some n≥0}. It is known that A;·,1 is the p-quasicyclic group. Define x*y=xy-1 for all x,y∈A. By Example 7, A=(A;*,1) is a BN-algebra. Let In={z∈C:zpn=1} for n∈N∪{0}. It follows easily that I is an ideal of A if and only if I=A or I=In for some n≥0. We have I0={1}⊂I1⊂⋯⊂In⊂⋯⊂A and hence A is Artinian. Define μ by
(46)μx=1n+1ifx∈In-In-1forsomen∈N∪0,
where I-1=∅. Since A=⋃n∈NIn, μ is a fuzzy set in A. By the proof of Lemma 44, μ∈FId(A). However, Im⁡μ={1/n:n∈N} is not a finite set.


## 5. Conclusions

This paper begins by considering the notion of ideals in BN-algebras. We give its characterizations and introduce the concept of normal ideals investigating its properties. We also define the notion of normal congruences proving that there is a one-to-one correspondence between normal ideals and normal congruences of a BN-algebra. Moreover we obtain the isomorphism theorem for BN-algebras. Next, we define the notion of fuzzy ideals of BN-algebras giving its characterizations and providing conditions for a fuzzy set to be a fuzzy ideal. We give the relationships between ideals and fuzzy ideals of a BN-algebra and also provide the homomorphic properties of fuzzy ideals. Finally, we display characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals.The next step in studying fuzzy ideals in BN-algebras may be introducing and investigating the notions of fuzzy maximal ideals and fuzzy prime ideals of BN-algebras.



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