# Fuzzyn-Fold BCI-Positive Implicative Ideals in BCI-Algebras

Authors: S. F. Tebu; C. Lele; J. B. Nganou
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We consider the notion of fuzzyn-fold positive implicative ideals in BCI-algebras. We analyse many properties of fuzzy n-fold positive implicative ideals. We also establish an extension properties for fuzzy n-fold positive implicative ideals in BCI-algberas. This work generalizes the corresponding results in the crisp case.


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## Body

## 1. Introduction

Liu and Zhang [1], Wei and Jun [2] independently introduced BCI-positive implicative ideals and used these to completely describe positive implicative BCI-algebras (namely, weakly positive implicative BCI-algebras). Tebu et al. [3] introduced the notion of n-fold positive implicative ideals in BCI-algebra. Fuzzy ideals are useful tool to obtain results on BCI-algebras. The sets of provable formulas in the corresponding inference systems from the point of view of uncertain information can be described by fuzzy ideals of those algebraic semantics. So in this paper, we discuss the notion of fuzzy n-fold positive implicative ideals in BCI-algebras. We show that every fuzzy n-fold positive implicative ideal is a fuzzy ideal, and give a condition for a fuzzy ideal to be a fuzzy n-fold positive implicative ideal. Using the level set, we provide a characterization of fuzzy n-fold positive implicative ideals. Finally, we establish an extension property for fuzzy n-fold positive implicative ideal. This paper generalizes the corresponding results in BCK/BCI-algebras [4].


## 2. Preliminaries

We recollect some definitions and results which will be used in the following, and we shall not cite them every time they are used.Definition 2.1 (see [5]).
An algebraX=〈X;⋆,0〉 of type 〈2,0〉, is said to be a BCI-algebra if it satisfies the following conditions for all x,y,z∈X: (i)
BCI1-((x⋆y)⋆(x⋆z))⋆(z⋆y)=0;(ii)
BCI2-(x⋆(x⋆y))⋆y=0;(iii)
BCI3-x⋆x=0;(iv)
BCI4-x⋆y=0 and y⋆x=0 imply x=y.If a BCI-algebraX satisfies the condition 0⋆x=0 for all x∈X; then X is called a BCK-algebra. Hence, BCK-algebra form a subclass of BCI-algebra.Proposition 2.2 (see [5]).
On every BCI-algebraX, there is a natural order called the BCI-ordering defined by x≤y if and only if x⋆y=0. Under this order, the following axioms hold for all x,y,z∈X.(a)
x⋆0=x;(b)
ifx≤y, then x⋆z≤y⋆z and z⋆y≤z⋆x;(c)
(x⋆y)⋆z=(x⋆z)⋆y;(d)
x⋆(x⋆(x⋆y))=x⋆y;(e)
(x⋆z)⋆(y⋆z)≤x⋆y;(f)
0⋆(x⋆y)=(0⋆x)⋆(0⋆y);(g)
0⋆(x⋆y)=0⋆(0⋆(y⋆x));(h)
0⋆x=0⋆(y⋆(y⋆x));(i)
(0⋆(x⋆y))⋆(y⋆x)=0;(j)
ifx≤y, then 0⋆x=0⋆y;(k)
ifx≤0⋆y, then x=0⋆y.Letn be a positive integer. Throughout this paper we appoint that X:=(X,⋆,0) denotes a BCI-algebra; x⋆yn:=(⋯((x⋆y)⋆y)⋆⋯)⋆y, in which y occurs n times; x⋆y0:=x and x⋆∏i=1n‍yi denotes (⋯((x⋆y1)⋆y2)⋆⋯)⋆yn, where x,y,yi∈X.Definition 2.3 (see [5]).
A nonempty subsetI of a BCI-algebra is called an ideal of X if it satisfies:
(I1) 0∈I,
(I2) x⋆y∈I and y∈I imply x∈I.Definition 2.4 (see [5]).
A nonempty subsetI of a BCI-algebra is called a BCI-positive implicative ideal of X if it satisfies (I1) and
(I3) (x⋆y2)⋆(z⋆y)∈I and z∈I imply x⋆y∈I; for all x,y, z∈X.We recall [6, 7] that a fuzzy set of a set X is a function μ:X→[0;1]. For a fuzzy set μ in X and t∈[0;1], define μt to be the set
(2.1)μt={x∈X∣μ(x)≥t}.Definition 2.5 (see [8]).
(a) LetX be a BCI-algebra. A fuzzy set in X is said to be a fuzzy ideal of X if:
(i)μ(0)≥μ(x) for all x∈X;
(ii)μ(x)≥Min{μ(x⋆y),μ(y)}; for all x,y∈X.
Note that every idealμ is order reversing, that is, if x≤y then μ(x)≥μ(y).(b)
A fuzzy setμ in X is called a fuzzy positive implicative ideal of X if it satisfies:
(i)
μ(0)≥μ(x) for all x∈X.(ii)
μ(x⋆y)≥Min{μ((x⋆y2)⋆(z⋆y)),μ(z)}; for all x,y,z∈X.Definition 2.6.
LetX be a BCK-algebra and n be a positive integer. Then the fuzzy set μ of X is called a fuzzy n-fold positive implicative ideal in BCK-algebra if it satisfies the following conditions:(i)
μ(0)≥μ(x), for all x∈X;(ii)
μ(x⋆yn)≥Min{μ((x⋆yn+1)⋆z);μ(z)}; for all x,y,z∈X.Definition 2.7 (see [9]).
LetX be a BCK-algebra and n be a positive integer. Then X is called an n-fold positive implicative BCK-algebra if x⋆yn+1=x⋆yn; for all x,y∈X.Definition 2.8 (see [3]).
LetX be a BCI-algebra and n be a positive integer. Then X is called an n-fold positive implicative BCI-algebra if (x⋆yn+1)⋆(0⋆y)=x⋆yn; for all x,y∈X.Definition 2.9 (see [3]).
A nonempty subsetI of a BCI-algebra is called an n-fold BCI-positive implicative ideal of X if it satisfies (I1) and
(
I
5
)
(
x
⋆
y
n
+
1
)
⋆
(
z
⋆
y
)
∈
I and z∈I imply x⋆yn∈I; for all x,y, z∈X.


## 3. Fuzzyn-Fold Positive Implicative Ideals in BCI-Algebras

In this section, we introduce the notion of fuzzyn-fold positive implicative ideal in a BCI-algebra and study some important properties.Definition 3.1.
A fuzzy setμ of a BCI-algebra X is called a fuzzy n-fold positive implicative ideal of X if it satisfies the following conditions:(i)
μ(0)≥μ(x) for all x∈X;(ii)
Min{μ((x⋆yn+1)⋆(z⋆y));μ(z)}≤μ(x⋆yn); for all x,y, z∈X.Remark 3.2.
(a) Notice that fuzzy1-fold positive implicative ideal in BCI-algebra is a fuzzy positive implicative ideal in BCI-algebra.
(b) The notion of fuzzyn-fold positive implicative ideal in a BCI-algebras generalizes the notion of fuzzy n-fold positive implicative ideal in BCK-algebras. This is because if X is a BCK-algebra, for every x∈X, 0⋆x=0.
(c) Every fuzzyn-fold positive implicative ideal in BCI-algebra is a fuzzy ideal.The following example shows that the converse of Remark3.2 (c) may not be true.Example 3.3.
Consider a BCI-algebraX={0,1,2,3} with Cayley table as follows:
(3.1)⋆012300032110322220333320.
We defineμ:X→[0;1] in X given by μ(0)=μ(1)=1, μ(2)=μ(3)=1/2. μ is a fuzzy ideal, but not a fuzzy 2-fold positive implicative ideal of X. As μ((2⋆23)⋆(3⋆2))=1, but μ(2⋆22)=1/2.Remark 3.4.
In ann-fold positive implicative BCI-algebra, every fuzzy ideal is a fuzzy n-fold positive implicative.Theorem 3.5.
Letμ be a fuzzy ideal of a BCI-algebra X. Then the following conditions are equivalent:(i)
μ((x⋆yn+1)⋆(z⋆y))≤μ((x⋆yn)⋆z), for all x,y, z∈X;(ii)
μ is a fuzzy n-fold positive implicative ideal;(iii)
μ((x⋆yn+1)⋆(0⋆y))≤μ(x⋆yn), for all x,y∈X.Proof.
(i)⇒(ii). Let x,y,z∈X. We will prove that μ(x⋆yn)≥Min{μ((x⋆yn+1)⋆(z⋆y));μ(z)}
Sinceμ is a fuzzy ideal, we have μ(x⋆yn)≥Min{μ((x⋆yn)⋆z; μ(z)}≥Min{μ((x⋆yn+1)⋆(z⋆y));μ(z)}.
(ii)⇒(iii). It is easy to obtain by setting z=0.
(iii)⇒(i). Let x,y,z∈X with v=((x⋆yn+1)⋆(z⋆y)) and u=x⋆z, it suffices to show that μ(u⋆yn+1)≥μ(v).(3.2)μ(((u⋆yn+1)⋆(0⋆y))⋆v)=μ(((((x⋆yn+1)⋆(0⋆y))⋆((x⋆yn+1)⋆(z⋆y)))⋆z))≥μ(((z⋆y)⋆(0⋆y))⋆z).≥μ((z⋆0)⋆z)=μ(0).
Thus,μ[(((u⋆yn+1)⋆(0⋆y))⋆v)]=μ(0), it follows μ((u⋆yn)=Min{μ[((u⋆yn+1)⋆(0⋆y))⋆v]; μ(v)}. Hence, μ((x⋆z)⋆yn)≥μ(v).Theorem 3.6.
LetX be a BCI-algebra and μ be a fuzzy set in X.
Ifμ is a fuzzy n-fold positive implicative ideal of X, then μ((x⋆yn)⋆(0⋆(0⋆y)))≤μ(x⋆yn+1).Proof.
Suppose thatμ is a fuzzy n-fold positive implicative ideal of X. putting s=x⋆y, then μ((x⋆yn+1)⋆(0⋆y))≤μ(x⋆yn), for all x,y∈X. 							(3.3)μ((((s⋆yn+1)⋆(0⋆y)))⋆((x⋆yn)⋆(0⋆(0⋆y))))=μ(((((x⋆yn)⋆(0⋆y))⋆y)⋆y)⋆((x⋆yn)⋆(0⋆(0⋆y))))=μ(((((x⋆yn+1)⋆(0⋆y))⋆((x⋆yn)⋆(0⋆(0⋆y))))⋆y)⋆y)≥μ((((0⋆(0⋆y))⋆(0⋆y))⋆y)⋆y)=μ((((0⋆(0⋆y))⋆y)⋆(0⋆y))⋆y)=μ((0⋆(0⋆y))⋆y)=μ(0).
We haveμ(((x⋆y)⋆yn+1)⋆(0⋆y))=μ((s⋆yn+1)⋆(0⋆y))=μ(0), since μ is a fuzzy n-fold positive implicative ideal, we obtain μ(x⋆yn+1)=μ((x⋆y)⋆yn)=μ(u⋆yn)≥μ((x⋆yn)⋆(0⋆(0⋆y))). The proof is complete.However the following example proves that the converse of Theorem3.6 is not true.Example 3.7.
Consider a BCI-algebraX={0,1,a,b,c} with the following Cayley table:
(3.4)⋆01abc000aaa110aaaaaa000bba10acca110.
We defineμ: X→[0;1] in X by μ(0)=1 and for all x≠0  μ(x)=0,3. It is clear that for all b,c∈X  μ((c⋆b2)⋆(0⋆c))≤μ(c⋆b); but μ is not a fuzzy 1-fold positive implicative ideal because μ((c⋆b2)⋆(0⋆c))=1, but μ(c⋆b)=0,3.Lemma 3.8.
Letμ be a fuzzy ideal in X. If μ is a fuzzy n-fold positive implicative ideal, then μ is a fuzzy (n+1)-fold positive implicative ideal.Proof.
letμ be a fuzzy n-fold positive implicative ideal of X, let x,y∈X. Then μ((x⋆yn+1)⋆(0⋆y))≤μ(x⋆yn). Therefore, μ(((x⋆y)⋆yn+1)⋆(0⋆y))≤μ((x⋆y)⋆yn). Hence, μ is a fuzzy (n+1)-fold positive implicative ideal.Using this lemma and a simple induction argument, we obtain the following proposition.Proposition 3.9.
Letμ be a fuzzy ideal of a BCI-algebra X and k∈ℕ∖{0}. If μ is a fuzzy n-fold positive implicative ideal, then μ is a fuzzy (n+k)-fold positive implicative ideal.However there exists fuzzy(n+k)-fold positive implicative ideals which are not fuzzy n-fold positive implicative ideals.Example 3.10.
Consider the setX2={0,1,2,π}, and define the operation ⋆ on X2 by:
(3.5)k⋆l=max(0,k-l)if0≤k,l≤n,k⋆π=π⋆k=πifk=0,1,2,π⋆π=0.
Consider the fuzzy idealμ define by μ(0)=0,4 and for all x≠0, μ(x)=0. Then μ is a fuzzy 2-fold positive implicative ideal, but not a fuzzy positive implicative ideal of X2 because μ((2⋆12)⋆(0⋆1))=0,4 and μ(2⋆1)=0.Now, we established some transfer principle for fuzzyn-fold positive implicative ideals in BCI-algebras.Proposition 3.11.
Letμ be a fuzzy ideal and I be an ideal of a BCI-algebra X.
ThenI is an n-fold positive implicative ideal if and only if it‘s characteristic function μI is a fuzzy n-fold positive implicative ideal.Proof.
Assume thatI is an n-fold positive implicative ideal. Let x,y∈X. If x⋆yn∈I, then μ(x⋆yn)≥μ((x⋆yn+1)⋆(0⋆y)). If x⋆yn∉I, then (x⋆yn+1)⋆(0⋆y)∉I because I is an n-fold positive implicative ideal. So, we also have μ(x⋆yn)≥μ((x⋆yn)⋆(0⋆y)).
Conversely, suppose thatμI is a fuzzy n-fold positive implicative ideal. Let x,y∈X such that (x⋆yn+1)⋆(0⋆y)∈I. Then μI((x⋆yn+1)⋆(0⋆y))=1; since μI is a fuzzy n-fold positive implicative ideal, μ(x⋆yn)≥μI((x⋆yn+1)⋆(0⋆y))=1. We conclude that x⋆yn∈I.Theorem 3.12.
Letμ be a fuzzy ideal in a BCI-algebra X. Then μ is a fuzzy n-fold positive implicative if and only if the t-cut set μt:={x∈X/μ(x)≥t} either is empty or is an n-fold positive implicative ideal.Proof.
Suppose thatμ is a fuzzy n-fold positive implicative ideal of X and μt≠∅, for all t∈[0;1]. We will show that μt is an n-fold positive implicative ideal. Since μt is nonempty, there exists x0∈X such that x0∈μt; since μ(0)≥μ(x), for all x∈X, we have: μ(0)≥μ(x0)≥t; thus 0∈μt.
Letx,y,z∈X; such that (x⋆yn+1)⋆(0⋆y)∈μt. Then μ(((x⋆yn+1)⋆(0⋆y))≥t, since μ is fuzzy n-fold positive implicative ideal, we have μ(x⋆yn)≥μ(((x⋆yn+1)⋆(0⋆y)))≥t. Hence x⋆yn∈μt and μt is an n-fold BCI-positive implicative ideal.
Conversely, assume thatμt≠∅ for all t∈[0;1] and μt is an n-fold BCI-positive implicative ideal. We will show that μ is a fuzzy n-fold positive implicative ideal.
It is easy to see thatμ(0)≥μ(x), for all x∈X.
Now assume that there exista,b∈X such that μ(a⋆bn)<μ((a⋆bn+1)⋆(0⋆b)).
settingt0=(1/2)[μ(a⋆bn)+μ((a⋆bn+1)⋆(0⋆b))],then μ(a⋆bn)<t0<μ((a⋆bn+1)⋆(0⋆b)); it follows that (a⋆bn+1)⋆(0⋆b)∈μt0, but a⋆bn∉μt0. This is a contradiction. Hence μ is a fuzzy n-fold positive implicative ideal.Theorem 3.13.
Ifμ is an n-fold fuzzy positive implicative ideal of a BCI-algebra X, then the set Xμ={x∈X,μ(x)=μ(0)} is an n-fold positive implicative ideal of X.Proof.
Letμ be an n-fold fuzzy positive implicative ideal of X. Clearly 0∈Xμ.
Letx,y,z∈X such that (x*yn+1)*(0*y)∈Xμ. Then μ(x*yn)≥μ((x*yn+1)*(0*y))=μ(0). It follows by μ(0)≥μ(x) for all x∈X, that μ(x*yn)=μ(0) so that x*yn∈Xμ. Hence Xμ is an n-fold positive implicative ideal of X.Theorem 3.14 (extension property for fuzzyn-fold positive implicative ideals in BCI-algebra).
Letμ, ν two fuzzy ideals of a BCI-algebra X such that μ(0)=ν(0) and μ⊂ν, that is, μ(x)≤ν(x) for all x∈X. If μ is a fuzzy n-fold positive implicative ideal then ν is also fuzzy n-fold positive implicative ideal.Proof.
Using Theorem3.5, it is sufficient to show that ν satisfies the inequality ν(x⋆yn)≥ν((x⋆yn+1)⋆(0⋆y)) for all x,y∈X.
Letx,y∈X; setting r=(x⋆yn+1)⋆(0⋆y), we have
(3.6)ν(0)=μ(0)=μ(((x⋆yn+1)⋆(0⋆y))⋆r)=μ(((x⋆yn+1)⋆r)⋆(0⋆y))=μ(((x⋆r)⋆yn+1)⋆(0⋆y))≤μ((x⋆yn)⋆r)≤ν((x⋆yn)⋆r).
Since ν is a fuzzy ideal, ν(x⋆yn)≥Min{ν((x⋆r)⋆yn);ν(r)}≥Min{ν(0);ν(r)}=ν(r).Hence by Theorem3.5(iii), ν is a fuzzy n-fold positive implicative ideal of X.Definition 3.15 (see [10]).
LetX,Y be two BCI-algebras.
A mapf:X→Y is called a BCI-homomorphism if: f(x⋆y)=f(x)⋆f(y) for all x,y∈X.Definition 3.16.
LetX and Y be two BCI-algebras, μ a fuzzy subset of X, ν a fuzzy subset of Y and f:X→Y a BCI-homomorphism.
The image ofμ under f denoted by f(μ) is a fuzzy set of Y defined by: For all y∈Y, f(μ)(y)=Sup{μ(x):x∈f-1(Y)} if f-1(Y)≠∅ and f(μ)(y)=0 if f-1(Y)=∅.
The preimage ofν under f denoted by f-1(ν) is a fuzzy set of X defined by: For all x∈X,f-1(ν)(x)=ν(f(x)). Let f:X→Y be an onto BCI-homomorphism, It is easy to prove that the preimage of a fuzzy n-fold positive implicative ideal ν under f is also a fuzzy n-fold positive implicative ideal.Definition 3.17.
A fuzzy subsetμ of X has a sup property if for any nonempty subset A of X, there exists a0∈A such that μ(a0)=Sup{μ(a):a∈A}. Using this fact, we can prove the following result:Proposition 3.18.
Letf:X→Y be an onto BCI-homomorphism, the image f(μ) of a fuzzy n-fold positive implicative ideal μ with a sup property is also a fuzzy n-fold positive implicative ideal.


## 4. Conclusion

To investigate the structure of an algebraic system, it is clear that ideals with special properties play an important role. The present paper introduced and studied the notion of fuzzyn-fold positive implicative ideal in BCI-algebra. The extension property of fuzzy n-fold positive implicative ideals was established. The main purpose of future work is to investigate the fuzzy foldness of other types of ideals in BCI-algebras and the relation diagram between them similar to the one in [8, 11, 12]. We are also doing some investigations on logic whose algebraic semantics is n-fold positive implicative BCI-algebras.



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