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1. *Corresponding Author: Areej Tawfeeq Hameed, Email: areej238@gmail.com
RESEARCH ARTICLE
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Cubic KUS-Ideals of KUS-Algebras
Areej Tawfeeq Hameed1
, Samy M. Mostafa2
, Ahmed Hamza Abed3
1,3
Department of Mathematics, Faculty of Education for Girls, University of Kufa , Najaf, Iraq.
2
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt
Received on: 16/02/2017, Revised on: 01/03/2017, Accepted on: 07/03/2017
ABSTRACT
In this paper , the notion of cubic KUS-ideals in KUS-algebras is introduced and several properties are
investigated. The image and inverse image of cubic KUS-ideals in KUS-algebras are defined.
Keywords: KUS-algebras, cubic KUS-algebras, homomorphism of KUS-algebras.
2010 Mathematics Subject classification: 06F35, 03G25
INTRODUCTION
Is’eki K. and Tanaka S. ([3]) studied ideals and congruence of BCK-algebras. S.M. Mostafa et al.
([1],[7]) introduced a new algebraic structure which is called KUS-algebras and investigated some related
properties. The concept of a fuzzy set, was introduced by L.A. Zadeh[4]. O.G. Xi[6] applied the concept
of fuzzy set to BCK-algebras and gave some of its properties Jun et al. [8] introduced the notion of cubic
sub algebras(ideals) in BCK-algebras, and then they discussed the relationship between a cubic sub
algebra and a cubic ideal. In this paper, we introduce the notion of cubic KUS-ideals of a KUS-algebra,
and then we study the homomorphic image and inverse image of cubic KUS-ideals.
PRELIMINARIES
In this section, we include some elementary aspects necessary for this paper.
Definition.1 ([7]). Let (X;
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Asian Journal of Mathematical Sciences 2017; 1(2):36-43
∗,0) be an algebra with a single binary operation (∗). X is called a KUS-
algebra if it satisfies the following identities:
(kus1): (z∗y) ∗ (z∗x) = y∗x,
(kus2): 0 ∗ x = x,
(kus3): x ∗ x = 0,
(kus4): x ∗ (y ∗z) = y∗ (x∗z), for any x, y, z ∈ X,
In what follows, let (X; ∗,0) be denote a KUS-algebra unless otherwise specified.
For brevity we also call X a KUS-algebra. In X we can define a binary relation (
≤) by: x ≤ y if and only
if y ∗ x = 0.
Example.2 Let X = {0, a, b, c, d} in which (∗) is defined by the following table:
It is easy to show that (X; ∗,0) is a KUS-algebra .
∗ 0 a b c d
0 0 a b c d
a d 0 a b c
b c d 0 a b
c b c d 0 a
d a b c d 0

2. Arrej Tawfeeq Hameed et al. Cubic KUS-Ideals of KUS-Algebras
37
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Lemma.3 ([7]). In any KUS-algebra (X; ∗,0), the following properties hold: for all x, y, z ∈X;
• x∗ y = 0 and y∗ x = 0 imply x = y,
• y ∗ [(y∗z) ∗z] = 0,
• (0∗x) ∗ (y∗x) = y∗0 ,
• x ≤ y implies that y ∗z ≤ x ∗z and z ∗x ≤ z ∗y ,
• x ≤ y and y ≤ z imply x ≤ z ,
• x ∗y ≤ z implies that z∗y ≤ x .
Definition.4 ([7]). Let (X; ∗,0) be a KUS-algebra and let S be a nonempty set of X. S is called a KUS-
sub-algebra of X if x∗y ∈ S whenever x, y ∈ S.
Definition.5 ([7]). A nonempty subset I of a KUS-algebra X is called a KUS-ideal of X if it satisfies: for
all x , y, z ∈ X,
(Ikus1) (0 ∈ I),
(Ikus2) (z∗y)∈ I and (y∗x)∈ I imply (z∗x)∈ I.
Definition.3 ([3]). Let X be a nonempty set, a fuzzy subset μ in X is a function μ: X → [0, 1].
Definition.4 ([5]). Let X be a set and μ be a fuzzy subset of X , for t ∈ [0,1] , the set μ t ={ x ∈ X |
μ(x) ≥ t} is called a level subset of μ .
Definition.5 ([2]) .Let (X ; ∗,0) and (Y; ∗`,0`) be nonempty sets . The mapping
f : (X; ∗,0) → (Y; ∗`,0`) is called a homomorphism if it satisfies
f (x∗y) = f (x) ∗` f (y) for all x , y ∈X. The set {x∈X | f (x) = 0'} is called the Kernel of f and
is denoted by Ker f .
Definition.6 ([2]). Let f : (X; ∗,0) →(Y; ∗',0') be a mapping from the set X to a set Y. If μ is a
fuzzy subset of X, then the fuzzy subset β of Y defined by:


 ≠
=
∈
=
∈
=
−
−
otherwise
y
x
f
X
x
y
f
if
y
f
x
x
y
f
0
}
)
(
,
{
)
(
)}
(
:
)
(
sup{
)
)(
(
1
1
φ
µ
µ
is said to be the image of μ under f .
Similarly if β is a fuzzy subset of Y , then the fuzzy subset μ = (β
о f ) in X ( i.e the fuzzy
subset defined by μ (x) = β ( f (x)) for all x ∈ X ) is called the pre-image of β under f .
Theorem.7 ([1]). Let f : (X; ∗,0) → (Y; ∗`,0`) be an into homomorphism of a KUS-algebras, then :
(F1) f (0) = 0’.
(F2) If S is a KUS-sub-algebra of X, then f (S) is a KUS-sub-algebra of Y.
(F3) If I is a KUS-ideal of X, then f (I) is a KUS-ideal in Y.
(F4) If B is a KUS-sub-algebra of Y, then 1
−
f (B) is a KUS-sub-algebra of X.
(F5) If J is a KUS- ideal in Y, then 1
−
f (J) is a KUS-ideal in X.
(F6) f is injective if and only if Ker f = {0}.
Now, we will recall the concept of interval-valued fuzzy subsets.
An interval number is a
~ = [
−
a ,
+
a ] , where 0 ≤
−
a ≤
+
a ≤ 1 . Let I be a closed unit interval, i.e., I
= [0, 1]. Let D [0, 1] denote the family of all closed subintervals of I = [0, 1] , that is ,
D [0, 1] = { a
~ = [
−
a ,
+
a ] |
−
a ≤
+
a , for
−
a ,
+
a ∈ I} .
Now we define what is known as refined minimum (briefly, rmin) of two element in D[0, 1] . We
also define the symbols (≽) , (≼) , (=) , "rmin " and "rmax " in case of two elements in D[0, 1] . Consider
two interval numbers (elements numbers)
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a
~ = [
−
a ,
+
a ] , b
~
= [
−
b ,
+
b ] in D[0, 1] : Then
(1) a
~ ≽ b
~
if and only if
−
a ≥
−
b and
+
a ≥
+
b ,
(2) a
~ ≼ b
~
if and only if
−
a ≤
−
b and
+
a ≤
+
b ,
(3) a
~ = b
~
if and only if
−
a =
−
b and
+
a =
+
b ,
(4) rmin { a
~ , b
~
} = [min {
−
a ,
−
b }, min {
+
a ,
+
b }] ,
(5) rmax { a
~ , b
~
} = [max {
−
a ,
−
b }, max {
+
a ,
+
b }] ,
It is obvious that (D[0, 1] , ≼ , ∨ , ∧ ) is a complete lattice with 0
~
= [0, 0] as its least element and 1
~
=
[1, 1] as its greatest element. Let i
a
~ ∈ D[0, 1] where i ∈ Λ .
We define
i
i
a
r ~
inf
Λ
∈
= [ i
i
a
r
−
Λ
∈
inf , i
i
a
r
+
Λ
∈
inf ], i
i
a
r ~
sup
Λ
∈
= [ i
i
a
r
−
Λ
∈
sup , i
i
a
r
+
Λ
∈
sup ].
An interval-valued fuzzy subset A
µ
~ on X is defined as
A
µ
~ = {< x, [ A
−
µ (x) , A
+
µ (x) ]> | x∈ X} .
Where A
−
µ (x) ≤ A
+
µ (x), for all x∈ X. Then the ordinary fuzzy subsets A
−
µ : X → [0, 1] and A
+
µ : X →
[0, 1] are called a lower fuzzy subset and an upper fuzzy subset of A
µ
~ respectively . Let A
µ
~ (x) = [
A
−
µ (x) , A
+
µ (x) ] , A
µ
~ :X → D[0, 1], then A = {< x, A
µ
~ (x) > | x∈ X} .
Cubic KUS-ideals of KUS-algebras
In this section, we will introduce a new notion called cubic KUS-ideal of KUS-algebras and study several
properties of it.
Definition.1 ([8]). Let X be a non-empty set. A cubic set Ω in a structure Ω = {< x, Ω
µ
~ (x), Ω
λ (x)
> | x∈ X} which is briefly denoted by Ω =< Ω
µ
~ , Ω
λ > , where Ω
µ
~ :X → D[0, 1] , Ω
µ
~ is an
interval-valued fuzzy subset of X and Ω
λ :X → [0, 1], Ω
λ is a fuzzy subset of X.
Definition.2 Let X be a KUS-algebra. A cubic set Ω =< Ω
µ
~ , Ω
λ > of X is called cubic KUS-sub-
algebra of X if
Ω
µ
~ (x ∗y) ≽ rmin{ Ω
µ
~ (x), Ω
µ
~ (y)}, and
Ω
λ (x ∗y) ≤ max{ Ω
λ (x), Ω
λ (y)}, for all x, y∈ X.
Proposition.3 Let Ω =< Ω
µ
~ , Ω
λ > be a cubic KUS-sub-algebra of a KUS-algebra X , then Ω
µ
~
(0) ≽ Ω
µ
~ (x) and Ω
λ (0) ≤ Ω
λ (x), for all x∈ X .
Proof. For all x∈ X, we have
Ω
µ
~ (0) = Ω
µ
~ (x * x) ≽ rmin{ Ω
µ
~ (x), Ω
µ
~ (x)}
= rmin {[ Ω
−
µ (x), Ω
+
µ (x)], [ Ω
−
µ (x), Ω
+
µ (x)]}
= rmin {[ Ω
−
µ (x), Ω
+
µ (x)]} = Ω
µ
~ (x) .
Similarly, we can show that Ω
λ (0) ≤ max {[ Ω
λ (x), Ω
λ (x)]} = Ω
λ (x) . ⌂
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Proposition.4 Let Ω =< Ω
µ
~ , Ω
λ > be a cubic KUS-sub-algebra of a KUS-algebra X , if there exist a
sequence { Xn} in X such that )
(
~
lim n
n
x
Ω
→∞
µ = [1,1] , then Ω
µ
~ (0) = [1, 1].
Proof. By proposition (3.3), we have Ω
µ
~ (0) ≽ Ω
µ
~ (x) , for all x∈ X . Then
Ω
µ
~ (0) ≽ Ω
µ
~ (xn) , for every positive integer n, Consider the inequality
[1,1] ≽ Ω
µ
~ (0) ≽ )
(
~
lim n
n
x
Ω
→∞
µ = [1,1]. Hence Ω
µ
~ (0) = [1,1] . ⌂
Definition .5 Let X be a KUS-algebra. A cubic set Ω = < Ω
µ
~ , Ω
λ > of X is called cubic KUS-
ideal if it satisfies the following conditions:
(A1) Ω
µ
~ (0) ≽ Ω
µ
~ (x) , and Ω
λ (0) ≤ Ω
λ (x)
(A2) Ω
µ
~ (z∗x) ≽ rmin{ Ω
µ
~ (z∗y), Ω
µ
~ (y∗x)}, and
Ω
λ (z∗x) ≤ max{ Ω
λ (z∗y), Ω
λ (y∗x)}, for all x, y, z ∈ X.
Example .6 Let X = {0, 1, 2, 3} in which the operation as in example (∗) be define by the following
table:
∗ 0 1 2 3
0 0 1 2 3
1 1 0 3 2
2 2 3 0 1
3 3 2 1 0
Then (X; ∗,0) is a KUS-algebra.
Define a cubic set Ω =< Ω
µ
~ , Ω
λ > of X as follows:
fuzzy subset μ: X→ [0,1] by
Ω
µ
~ (x) =


 =
otherwise
x
if
]
6
.
0
,
1
.
0
[
}
1
,
0
{
]
9
.
0
,
3
.
0
[
Ω
λ =


 =
otherwise
x
if
6
.
0
}
1
,
0
{
1
.
0
.
The cubic set Ω = < Ω
µ
~ , Ω
λ > is a cubic KUS-ideal of X .
Theorem .7 Let X be a KUS-algebra. A cubic set Ω =< Ω
µ
~ , Ω
λ > of X . A cubic set Ω of X
is a cubic KUS-ideal of X if and only if Ω
−
µ , Ω
+
µ and Ω
λ are cubic KUS-ideals of X.
Proof. If Ω
−
µ and Ω
+
µ are cubic KUS-ideals of X . For any x, y, z ∈ X . Observe Ω
µ
~ (z∗x) = [
Ω
−
µ (z∗x), Ω
+
µ (z∗x)] ≽ [ min { Ω
−
µ (z∗y), Ω
−
µ (y∗x)} , min { Ω
+
µ (z∗y), Ω
+
µ (y∗x)}]
= rmin {[ Ω
−
µ (z∗y), Ω
+
µ (z∗y)], [ Ω
−
µ (y∗x), Ω
+
µ (y∗x)]}
= rmin { Ω
µ
~ (z∗y), Ω
µ
~ (y∗x)] .
Similarly, we can show that Ω
λ (z∗x) ≤ max {[ Ω
λ (z∗y), Ω
λ (y∗x)]}.
From what was mentioned above we can conclude that Ω is a cubic KUS-ideal of X.
Conversely, suppose that Ω is a cubic KUS-ideal of X . For all x, y, z ∈ X we have [ Ω
−
µ (z∗x),
Ω
+
µ (z∗x)]= Ω
µ
~ (z∗x) ≽ rmin{ Ω
µ
~ (z∗y), Ω
µ
~ (y∗x)}
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= rmin{[ Ω
−
µ (z∗y), Ω
+
µ (z∗y)], [ Ω
−
µ (y∗x), Ω
+
µ (y∗x)]}
= [ min { Ω
−
µ (z∗y), Ω
−
µ (y∗x) } , min { Ω
+
µ (z∗y), Ω
+
µ (y∗x) }] .
There fore, Ω
−
µ (z∗x) ≥ min{ Ω
−
µ (z∗y), Ω
−
µ (y∗x)} and Ω
+
µ (z∗x) ≥ min{ Ω
+
µ (z∗y), Ω
+
µ (y∗x)}.
Similarly, we can show that Ω
λ (z∗x) ≤ max{ Ω
λ (z∗y), Ω
λ (y∗x)}
Hence, we get that Ω
−
µ , Ω
+
µ and Ω
λ are cubic KUS-ideals of X. ⌂
Theorem .8 Let { i
Ω | i∈ Λ} be family of cubic KUS-ideals of a KUS-algebra X. Then 
Λ
∈
Ω
i
i is a cubic
KUS-ideal of X.
Proof. Let { i
Ω | i∈ Λ} be family of cubic KUS-ideals of X, then for any x, y, z ∈ X,
(  i
Ω
µ
~ )(0) = rinf ( i
Ω
µ
~ (0))≽ rinf ( i
Ω
µ
~ (x)) = (  i
Ω
µ
~ )(x)
(  i
Ω
µ
~ (z∗x)) = rinf ( i
Ω
µ
~ (z∗x))
≽ rinf ( rmin { i
Ω
µ
~ (z∗y), i
Ω
µ
~ (y∗x)} )
= rmin { rinf ( i
Ω
µ
~ (z∗y)), rinf ( i
Ω
µ
~ (y∗x)) }
= rmin { (  i
Ω
µ
~ ) (z∗y) , (  i
Ω
µ
~ ) (y∗x) }
Also,
( i
Ω
λ )(0) = sup( i
Ω
λ (0))≤ sup ( i
Ω
λ (x)) = ( i
Ω
λ )(x)
( i
Ω
λ (z∗x)) = sup ( i
Ω
λ (z∗x))
≤ sup ( max { i
Ω
λ (z∗y), i
Ω
λ (y∗x)} )
= max { sup ( i
Ω
λ (z∗y)), sup( i
Ω
λ (y∗x)) }
= max { ( i
Ω
λ ) (z∗y) , ( i
Ω
λ ) (y∗x) } . ⌂
Theorem .9 Let X be a KUS-algebra. A cubic subset Ω =< Ω
µ
~ , Ω
λ > of X , then Ω is a cubic
KUS-ideal of X if and only if for all t
~
∈ D[0, 1] and
s ∈ [0, 1] , the set U
~
( Ω ; t
~
, s ) is either empty or a KUS-ideal of X, where
U
~
( Ω ; t
~
, s ) = :={ x∈ X | Ω
µ
~ (x)≽ t
~
, Ω
λ (x) ≤ s} .
Proof. Assume that Ω =< Ω
µ
~ , Ω
λ > is a cubic KUS-ideal of X and let t
~
∈ D[0, 1] and s ∈ [0,
1] ,be such that U
~
( Ω ; t
~
, s ) ≠ ∅ , and let x, y, z ∈ X such that (z∗y), (y∗x) ∈ U
~
( Ω ; t
~
, s ) , then
Ω
µ
~ (z∗y) ≽ t
~
, Ω
µ
~ (y∗x) ≽ t
~
and Ω
λ (z∗y) ≤ s , Ω
λ (y∗x) ≤ s. . By (A2) , we get
Ω
µ
~ (z∗x) ≽ rmin { Ω
µ
~ (z∗y), Ω
µ
~ (y∗x) } ≽ t
~
, and
Ω
λ (z∗x) ≤ max { Ω
λ (z∗y), Ω
λ (y∗x) } ≤ s .
Hence the set U
~
( Ω ; t
~
, s ) is a KUS-ideal of X .
Conversely, suppose that U
~
( Ω ; t
~
, s) is a KUS-ideal of X and let x, y, z ∈ X be such that
Ω
µ
~ (z∗x) ≺ rmin { Ω
µ
~ (z∗y), Ω
µ
~ (y∗x)}, and
Ω
λ (z∗x) > max { Ω
λ (z∗y), Ω
λ (y∗x) } .
Consider
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β
~
=
2
1
{ Ω
µ
~ (z0 ∗ x0) + rmin{ Ω
µ
~ (z0 ∗ y0)), Ω
µ
~ ( y0 ∗ x0)} } and
β =
2
1
{ Ω
λ (z0 ∗ x0) + max{ Ω
λ (z0 ∗ y0)), Ω
λ ( y0 ∗ x0)} }
We have β
~
∈ D[0, 1] and β ∈ [0, 1], and
Ω
µ
~ (z∗x) ≺ β
~
≺ rmin { Ω
µ
~ (z∗y), Ω
µ
~ (y∗x) }, and
Ω
λ (z∗x) > β > max { Ω
λ (z∗y), Ω
λ (y∗x) } .
It follows that (z∗y), (y∗x) ∈ U
~
( Ω ; t
~
, s ) , and (z∗x) ∉ U
~
( Ω ; t
~
, s ) . This is a contradiction and
therefore Ω =< Ω
µ
~ , Ω
λ > is a cubic KUS-ideal of X . ⌂
Homomorphism of Cubic KUS-ideals of KUS-algebras
In this section, we will present some results on images and preimages of cubic KUS-ideals of KUS-
algebras.
Definition 1. Let f : (X; ∗,0) →(Y; ∗',0') be a mapping from the set X to a set Y. If Ω = < Ω
µ
~ ,
Ω
λ > is a cubic subset of X, then the cubic subset β = < β
µ
~ , β
λ > of Y defined by:




 ≠
=
∈
=
=
=
−
Ω
∈
Ω
−
otherwise
y
x
f
X
x
y
f
if
x
r
y
y
f y
f
x
0
}
)
(
,
{
)
(
)
(
~
sup
)
(
~
)
)(
~
(
1
)
(
1
φ
µ
µ
µ β




 ≠
=
∈
=
=
=
−
Ω
∈
Ω
−
otherwise
y
x
f
X
x
y
f
if
x
y
y
f y
f
x
1
}
)
(
,
{
)
(
)
(
inf
)
(
)
)(
(
1
)
(
1
φ
λ
λ
λ β
is said to be the image of Ω = < Ω
µ
~ , Ω
λ > under f .
Similarly if β = < β
µ
~ , β
λ > is a cubic subset of Y , then the cubic subset Ω = (β о f ) in X
( i.e the cubic subset defined by Ω
µ
~ (x) = β
µ
~ ( f (x)) , Ω
λ (x) = β
λ ( f (x)) for all x ∈ X ) is
called the preimage of β under f .
Theorem.2 An onto homomorphic preimage of cubic KUS-ideal is also cubic KUS-ideal.
Proof. Let f : (X; ∗,0) →(Y; ∗',0') be onto homomorphism from a KUS-algebra X into a KUS-algebra
Y.
If β = < β
µ
~ , β
λ > is a cubic KUS-ideal of Y and Ω = < Ω
µ
~ , Ω
λ > the preimage of β under f
, then Ω
µ
~ (x) = β
µ
~ ( f (x)) , Ω
λ (x) = β
λ ( f (x)) , for all x ∈ X . Let x ∈X, then
( Ω
µ
~ )(0) = β
µ
~ ( f (0)) ≽ β
µ
~ ( f (x)) = Ω
µ
~ (x), and
( Ω
λ )(0) = β
λ ( f (0)) ≤ β
λ ( f (x)) = Ω
λ (x).
Now, let x, y, z ∈ X, then
Ω
µ
~ (z∗x) = β
µ
~ ( f (z∗x)) ≽ rmin { β
µ
~ ( f (z∗y)) , β
µ
~ ( f (y∗x)) }
= rmin { Ω
µ
~ (z ∗y) , Ω
µ
~ (y ∗x)}, and
Ω
λ (z∗x) = β
λ ( f (z∗x)) ≤ max { β
λ ( f (z ∗y)) , β
λ ( f (y ∗x))}
= max { Ω
λ (z ∗y) , Ω
λ (y ∗x)} . ⌂
Definition.3 Let f : (X; ∗,0) →(Y; ∗',0') be a mapping from a set X into a set Y . Ω = < Ω
µ
~ ,
Ω
λ > is a cubic subset of X has sup and inf properties if for any subset T of X, there exist t, s ∈ T such
that
AJMS,
Mar-April,
2017,
Vol.
1,
Issue
2

7. Arrej Tawfeeq Hameed et al. Cubic KUS-Ideals of KUS-Algebras
42
© 2017, AJMS. All Rights Reserved.
)
(
~
sup
)
(
~ t
r
t
T
t
Ω
∈
Ω = µ
µ and )
(
inf
)
( s
s
T
s
Ω
∈
Ω = λ
λ .
Theorem.4 Let f : (X; ∗,0) →(Y; ∗',0') be a homomorphism from a KUS-algebra X into a KUS-
algebra Y . For every cubic KUS-ideal
Ω = < Ω
µ
~ , Ω
λ > of X, then f ( Ω ) is a cubic KUS-ideal of Y.
Proof. By definition )
(
~
sup
)
'
)(
~
(
)
'
(
~
)
'
(
1
x
r
y
f
y
y
f
t
Ω
∈
Ω
−
=
= µ
µ
µβ and
)
(
inf
)
'
)(
(
)
'
(
)
'
(
1
x
y
f
y
y
f
t
Ω
∈
Ω −
=
= λ
λ
λ β for all y' ∈ Y and rsup(∅) = [0, 0] and inf (∅) = 0 .
We have prove that
Ω
µ
~ (z'∗x') ≽ rmin { Ω
µ
~ (z' ∗y') , Ω
µ
~ (y' ∗x')}, and
Ω
λ (z'∗x') ≤ max{ Ω
λ (z' ∗y') , Ω
λ (y' ∗x')}, for all x', y' , z' ∈ Y .
Let f : (X; ∗,0) →(Y; ∗',0') be a homomorphism of KUS-algebras,
Ω = < Ω
µ
~ , Ω
λ > is a cubic KUS-ideal of X has sup and inf properties, and
β = < β
µ
~ , β
λ > the image of Ω = < Ω
µ
~ , Ω
λ > under f .
Since Ω = < Ω
µ
~ , Ω
λ > is a cubic KUS-ideal of X, we have
( Ω
µ
~ )(0) ≽ Ω
µ
~ (x), and ( Ω
λ )(0) ≤ Ω
λ (x), for all x ∈ X .
Note that 0 ∈
1
−
f (0') where 0, 0' are the zero of X and Y, respectively. Thus
)
0
(
~
)
(
~
sup
)
'
0
(
~
)
'
0
(
1
Ω
Ω
∈
=
=
−
µ
µ
µβ t
r
f
t
≽ Ω
µ
~ (x) = )
'
(
~
)
(
~
sup
)
'
(
1
x
t
r
x
f
t
β
µ
µ =
Ω
∈ −
,
)
(
)
0
(
)
(
inf
)
'
0
(
)
'
0
(
1
x
t
f
t
Ω
Ω
Ω
∈
≤
=
= −
λ
λ
λ
λ β = )
'
(
)
(
inf
)
'
(
1
x
t
x
f
t
β
λ
λ =
Ω
∈ −
,
for all x ∈ X , which implies that
β
µ
~ (0') ≽ )
'
(
~ x
β
µ , and )
'
(
)
'
0
( x
β
β λ
λ ≤ , for all x' ∈ Y .
For any x' , y' , z' ∈ Y, let x 0 ∈
1
−
f (x') , y0 ∈
1
−
f (y') , and z0 ∈
1
−
f (z')be such that
)
(
~
sup
)
(
~
)
'
'
(
0
0
1
t
r
y
z
y
z
f
t
Ω
∗
∈
Ω
−
=
∗ µ
µ , )
(
~
sup
)
(
~
)
'
'
(
0
0
1
t
r
x
y
x
y
f
t
Ω
∗
∈
Ω
−
=
∗ µ
µ
and )
(
~
0
0 x
z ∗
Ω
µ = )}
(
{
~
0
0 x
z
f ∗
β
µ
= )
'
'
(
~ x
z ∗
β
µ )
(
~
sup 0
0
)
'
'
(
)
( 1
0
0
x
z
r
x
z
f
x
z
∗
= Ω
∗
∈
∗ −
µ
)
(
~
sup
)
'
'
(
1
t
r
x
z
f
t
Ω
∗
∈ −
= µ
Also,
)
(
inf
)
(
)
'
'
(
0
0 1
t
y
z
y
z
f
t
Ω
∗
∈
Ω −
=
∗ λ
λ , )
(
inf
)
(
)
'
'
(
0
0 1
t
x
y
x
y
f
t
Ω
∗
∈
Ω −
=
∗ λ
λ
and
)}
(
{
)
( 0
0
0
0 x
z
f
x
z ∗
=
∗
Ω β
λ
λ )}
'
'
(
{ x
z
f ∗
= β
λ
)
(
inf 0
0
)
'
'
(
)
( 1
0
0
x
z
x
z
f
x
z
∗
= Ω
∗
∈
∗ −
λ
)
(
inf
)
'
'
(
1
t
x
z
f
t
Ω
∗
∈ −
= λ
Then
)
'
'
(
~ x
z ∗
β
µ )
(
~
sup
)
'
'
(
1
t
r
x
z
f
t
Ω
∗
∈ −
= µ = )
(
~
0
0 x
z ∗
Ω
µ
AJMS,
Mar-April,
2017,
Vol.
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2

8. Arrej Tawfeeq Hameed et al. Cubic KUS-Ideals of KUS-Algebras
43
© 2017, AJMS. All Rights Reserved.
rmin { Ω
µ
~ (z0 ∗y0) , Ω
µ
~ (y0 ∗x0)},
= rmin { )
(
~
sup
)
'
'
(
1
t
r
y
z
f
t
Ω
∗
∈ −
µ , )
(
~
sup
)
'
'
(
1
t
r
x
y
f
t
Ω
∗
∈ −
µ }
= rmin { )
'
'
(
~ y
z ∗
β
µ , )
'
'
(
~ x
y ∗
β
µ }
and
=
∗ )
'
'
( x
z
β
λ )
(
inf
)
'
'
(
1
t
x
z
f
t
Ω
∗
∈ −
λ )
'
'
( x
z ∗
= Ω
λ
≤ Max { )
( 0
0 y
z ∗
Ω
λ , )
( 0
0 x
y ∗
Ω
λ }
= max { )
(
inf
)
'
'
(
1
t
y
z
f
t
Ω
∗
∈ −
λ , )
(
inf
)
'
'
(
1
t
x
y
f
t
Ω
∗
∈ −
λ }
= max { )
'
'
( y
z ∗
β
λ , )
'
'
( x
y ∗
β
λ }
Hence, β is a cubic KUS-ideal of Y. ⌂
REFERENCES
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University, Egypt ,2015.
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no .3 (2009), 25-31.
3. K. Is´eki and S.Tanaka, Ideal theory of BCK-algebras, Math. Japon. , vol.21 (1976), 351-366.
4. L. A. Zadeh, Fuzzy sets, Inform. And Control, vol. 8 (1965) 338-353.
5. L. A. Zadeh, The concept of a linguistic variable and its application to approximate I,
Information Sci. And Control , vol.8 (1975) , 199-249.
6. O. G. Xi, Fuzzy BCK-algebra, Math. Japon. , vol.36 (1991) 935-942.
7. S.M. Mostafa, M.A. Abdel Naby , F. Abdel Halim and A.T. Hameed , On KUS-algebras , Int.
Journal of Algebra, vol. 7 , no. 3 (2013) , 131-144.
8. Y.B. Jun, C.S. Kim, and M.S. Kang , Cubic subalgebras and ideals of BCK/BCI-algebras, Far
East Journal of Math. Sciences, vol. 44 , no. 2 (2010) , 239-250.
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2017,
Vol.
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Issue
2