# Graphs Based on BCK/BCI-Algebras

Authors: Young Bae Jun; Kyoung Ja Lee
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

The associated graphs of BCK/BCI-algebras will be studied. To do so, the
notions of (l-prime) quasi-ideals and zero divisors are first introduced and
related properties are investigated. The concept of associative graph of a
BCK/BCI-algebra is introduced, and several examples are displayed.


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## Body

## 1. Introduction

Many authors studied the graph theory in connection with (commutative) semigroups and (commutative and noncommutative) rings as we can refer to references. For example, Beck [1] associated to any commutative ring R its zero-divisor graph G(R) whose vertices are the zero-divisors of R (including 0), with two vertices a,b joined by an edge in case ab=0. Also, DeMeyer et al. [2] defined the zero-divisor graph of a commutative semigroup S with zero (0x=0∀x∈S).In this paper, motivated by these works, we study the associated graphs of BCK/BCI-algebras. We first introduce the notions of (l-prime) quasi-ideals and zero divisors and investigated related properties. We introduce the concept of associative graph of a BCK/BCI-algebra and provide several examples. We give conditions for a proper (quasi-) ideal of a BCK/BCI-algebra to be l-prime. We show that the associative graph of a BCK-algebra is a connected graph in which every nonzero vertex is adjacent to 0, but the associative graph of a BCI-algebra is not connected by providing an example.


## 2. Preliminaries

An algebra(X;*,0) of type (2,0) is called a  BCI-algebra if it satisfies the following axioms:  (I)
(∀x,y,z∈X)(((x*y)*(x*z))*(z*y)=0),(II)
(∀x,y∈X)((x*(x*y))*y=0),(III)
(∀x∈X)(x*x=0),(IV)
(∀x,y∈X)(x*y=0,y*x=0⇒x=y).If a BCI-algebraX satisfies the following identity:  (V)
(∀x∈X)(0*x=0),thenX is called a  BCK-algebra. Any BCK/BCI-algebra X satisfies the following conditions:  (a1)
(∀x∈X)(x*0=x),(a2)
(∀x,y,z∈X)(x*y=0⇒(x*z)*(y*z)=0,(z*y)*(z*x)=0),(a3)
(∀x,y,z∈X)((x*y)*z=(x*z)*y),(a4)
(∀x,y,z∈X)(((x*z)*(y*z))*(x*y)=0).We can define a partial ordering ≤ on a BCK/BCI-algebraX by x≤y if and only if x*y=0.A subsetA of a BCK/BCI-algebra X is called an ideal of X if it satisfies the following conditions:  (b1)
0∈A,(b2)
(∀x,y∈X)(x*y∈A,y∈A⇒x∈A).We refer the reader to the books [3, 4] for further information regarding BCK/BCI-algebras.


## 3. Associated Graphs

In what follows, letX denote a BCK/BCI-algebra unless otherwise specified.For any subsetA of X, we will use the notations r(A) and l(A) to denote the sets (3.1)r(A):={x∈X∣a*x=0,∀a∈A},l(A):={x∈X∣x*a=0,∀a∈A}.Proposition 3.1.
LetA and B be subsets of X, then  (1)
A⊆l(r(A)) and A⊆r(l(A)),(2)
IfA⊆B, then l(B)⊆l(A) and r(B)⊆r(A),(3)
l(A)=l(r(l(A))) and r(A)=r(l(r(A))).Proof.
Leta∈A and x∈l(A), then x*a=0, and so a∈r(l(A)). This says that A⊆r(l(A)). Dually, A⊆l(r(A)). Hence, (1) is valid.
Assume thatA⊆B and let x∈l(B), then x*b=0 for all b∈B, which implies from A⊆B that x*b=0 for all b∈A. Thus, x∈l(A), which shows that l(B)⊆l(A). Similarly, we have r(B)⊆r(A). Thus, (2) holds.
Using (1) and (2), we havel(r(l(A)))⊆l(A) and r(l(r(A)))⊆r(A). If we apply (1) to l(A) and r(A), then l(A)⊆l(r(l(A))) and r(A)⊆r(l(r(A))). Hence, l(A)=l(r(l(A))) and r(A)=r(l(r(A))).Definition 3.2.
A nonempty subsetI of X is called a quasi-ideal of X if it satisfies
(3.2)(∀x∈X)(∀y∈I)(x*y=0⟹x∈I).Example 3.3.
LetX={0,a,b,c,d} be a set with the *-operation given by Table 1, then (X;*,0) is a BCK-algebra (see [4]). The set I:={0,a,b} is a quasi-ideal of X.
Obviously, every quasi-idealI of a BCK-algebra X contains the zero element 0. The following example shows that there exists a quasi-ideal I of a BCI-algebra X such that 0∉I.Table 1
*-operation.
*0abcd000000aa0000bba0a0ccaa00ddbab0Example 3.4.
LetX={0,1,a,b,c} be a set with the *-operation given by Table 2, then (X;*,0) is a BCI-algebra (see [3]). The set I:={0,1,a} is a quasi-ideal of X containing the zero element 0, but the set J:={a,b,c} is a quasi-ideal of X which does not contain the zero element 0.
Obviously, every ideal ofX is a quasi-ideal of X, but the converse is not true. In fact, the quasi-ideal I:={0,a,b} in Example 3.3 is not an ideal of X. Also, quasi-ideals I and J in Example 3.4 are not ideals of X.Table 2
*-operation.
*01abc000cca110ccaaaa00cbba10ccccaa0Definition 3.5.
A (quasi-) idealI of X is said to be  l-prime if it satisfies  (i)
I is proper, that is, I≠X,(ii)
(∀x,y∈X)(l({x,y})⊆I⇒x∈Iory∈I).Example 3.6.
Consider the BCK-algebraX={0,a,b,c,d} with the operation * which is given by the Table 3, then the set I={0,a,c} is an l-prime ideal of X.Table 3
*-operation.
*0abcd000000aa0a00bbb0b0ccac0addddd0Theorem 3.7.
A proper (quasi-) idealI of X is l-prime  if and only if it satisfies
(3.3)l({x1,…,xn})⊆I⟹(∃i∈{1,…,n})(xi∈I),
for all x1,…,xn∈X.Proof.
Assume thatI is an l-prime (quasi-) ideal of X. We proceed by induction on n. If n=2, then the result is true. Suppose that the statement holds for n-1. Let x1,…,xn∈X be such that l({x1,…,xn-1,xn})⊆I. If y∈l({x1,…,xn-1}), then l({y,xn})⊆l({x1,…,xn-1,xn})⊆I. Assume that xn∉I, then y∈I by the l-primeness of I, which shows that l({x1,…,xn-1})⊆I. Using the induction hypothesis, we conclude that xi∈I for some i∈{1,…,n-1}. The converse is clear.For anyx∈X, we will use the notation Zx to denote the set of all elements y∈X such that l({x,y})={0}, that is,(3.4)Zx:={y∈X∣l({x,y})={0}},
which is called the set of zero divisors of x.Lemma 3.8.
IfX is a BCK-algebra, then l({x,0})={0} for all x∈X.Proof.
Letx∈X and a∈l({x,0}), then a*x=0=a*0=a, and so l({x,0})={0} for all x∈X.IfX is a BCI-algebra, then Lemma 3.8 does not necessarily hold. In fact, let X={0,1,2,a,b} be a set with the *-operation given by Table 4, then (X;*,0) is a BCI-algebra (see [4]). Note that l({x,0})={0} for all x∈{1,2} and l({x,0})=∅ for all x∈{a,b}.Table 4
*-operation.
*012ab0000aa1101ba2220aaaaaa00bbab10Corollary 3.9.
IfX is a BCI-algebra, then l({x,0})={0} for all x∈X with l({x,0})≠∅.Lemma 3.10.
IfX is a BCI-algebra, then l({x,0})={0} for all x∈X+, where X+ is the BCK-part of X.Proof.
Straightforward.Lemma 3.11.
For any elementsa and b of a BCK-algebra X, if a*b=0, then l({a})⊆l({b}) and Zb⊆Za.Proof.
Assume thata*b=0. Let x∈l({a}), then x*a=0, and so
(3.5)0=(x*b)*(x*a)=(x*b)*0=x*b.
Thus, x∈l({b}), which shows that l({a})⊆l({b}). Obviously, Zb⊆Za.Theorem 3.12.
For any elementx of a BCK-algebra X, the set of zero divisors of x is a quasi-ideal of X containing the zero element 0. Moreover, if Zx is maximal in {Za∣a∈X,Za≠X}, then Zx is l-prime.Proof.
By Lemma3.8, we have 0∈Zx. Let a∈X and b∈Zx be such that a*b=0. Using Lemma 3.11, we have
(3.6)l({x,a})=l({x})∩l({a})⊆l({x})∩l({b})=l({x,b})={0},
and so l({x,a})={0}. Hence, a∈Zx. Therefore, Zx is a quasi-ideal of X. Let a,b∈X be such that l({a,b})⊆Zx and a∉Zx, then l({a,b,x})={0}. Let 0≠y∈l({a,x}) be an arbitrarily element, then l({b,y})⊆l({a,b,x})={0}, and so l({b,y})={0}, that is, b∈Zy. Since y∈l({a,x}), we have y*x=0. It follows from Lemma 3.11 that Zx⊆Zy≠X so from the maximality of Zx it follows that Zx=Zy. Hence, b∈Zx, which shows that Zx is l-prime.Definition 3.13.
By theassociated graph of a BCK/BCI-algebra X, denoted Γ(X), we mean the graph whose vertices are just the elements of X, and for distinct x,y∈Γ(X), there is an edge connecting x and y, denoted by x-y if and only if l({x,y})={0}.Example 3.14.
LetX={0,a,b,c} be a set with the *-operation given by Table 5, then X is a BCK-algebra (see [4]). The associated graph Γ(X) of X is given by the Figure 1.Table 5
*-operation.
*0abc00000aa0aabbb0bcccc0Figure 1
Associated graphΓ(X) of X.Example 3.15.
LetX={0,a,b,c,d} be a set with the *-operation given by Table 6, then X is a BCK-algebra (see [4]). By Lemma 3.8, each nonzero point is adjacent to 0. Note that l({a,b})=l({a,d})=l({b,c})=l({c,d})={0}, l({a,c})={0,a}, and l({b,d})={0,b}. Hence the associated graph Γ(X) of X is given by the Figure 2.Table 6
*-operation.
*0abcd000000aa0a0abbb0b0ccac0cddddd0Figure 2
Associated graphΓ(X) of X.Example 3.16.
LetX={0,1,2,3,4} be a set with the *-operation given by Table 7, then X is a BCK-algebra (see [4]). By Lemma 3.8, each nonzero point is adjacent to 0. Note that l({1,2})={0,1}, that is, 1 is not adjacent to 2 and l({1,3})=l({1,4})=l({2,3})=l({2,4})=l({3,4})={0}. Hence, the associated graph Γ(X) of X is given by Figure 3.Table 7
*-operation.
*01234000000110011221022333303444440Figure 3
Associated graphΓ(X) of X.Example 3.17.
Consider a BCI-algebraX={0,1,2,a,b} with the *-operation given by Table 4, then
(3.7)l({1,a})=l({1,b})=l({2,a})=l({2,b})=∅,l({a,b})={a}, and l({1,2})={0}. Since X+={0,1,2}, we know from Lemma 3.10 that two points 1 and 2 are adjacent to 0. The associated graph Γ(X) of X is given by Figure 4.Figure 4
Associated graphΓ(X) of X.Theorem 3.18.
LetΓ(X) be the associated graph of a BCK-algebra X. For any x,y∈Γ(X), if Zx and Zy are distinct l-prime quasi-ideals of X, then there is an edge connecting x and y.Proof.
It is sufficient to show thatl({x,y})={0}. If l({x,y})≠{0}, then x∉Zy and y∉Zx. For any a∈Zx, we have l({x,a})={0}⊆Zy. Since Zy is l-prime, it follows that a∈Zy so that Zx⊆Zy. Similarly, Zy⊆Zx. Hence, Zx=Zy, which is a contradiction. Therefore, x is adjacent to y.Theorem 3.19.
The associated graph of a BCK-algebra is connected in which every nonzero vertex is adjacent to 0.Proof.
It follows from Lemma3.8.Example3.17 shows that the associated graph of a proper BCI-algebra may not be connected.


## 4. Conclusions

We have introduced the associative graph of a BCK/BCI-algebra with several examples. We have shown that the associative graph of a BCK-algebra is connected, but the associative graph of a BCI-algebra is not connected.Our future work is to study how to induce BCK/BCI-algebras from the given graph (with some additional conditions).



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