# On Cubic KU-Ideals of KU-Algebras

Authors: Naveed Yaqoob; Samy M. Mostafa; Moin A. Ansari
Journal: ISRN Algebra
 (2013)

---

## Abstract

We introduce the notion of cubic KU-ideals of KU-algebras and several results are presented in this regard. The image, preimage, and cartesian product of cubic KU-ideals of KU-algebras are defined.


---

## Body

## 1. Introduction

BCK-algebras form an important class of logical algebras introduced by Iséki and were extensively investigated by several researchers. The class of all BCK-algebras is quasivariety. Iséki and Tanaka introduced two classes of abstract algebras, BCK-algebras and BCI-algebras [1–3]. In connection with this problem, Komori [4] introduced a notion of BCC-algebras.Prabpayak and Leerawat [5] introduced a new algebraic structure which is called KU-algebra. They gave the concept of homomorphisms of KU-algebras and investigated some related properties in [5, 6].Zadeh [7] introduced the notion of fuzzy sets. At present this concept has been applied to many mathematical branches, such as groups, functional analysis, probability theory, and topology. Mostafa et al. [8] introduced the notion of fuzzy KU-ideals of KU-algebras and then they investigated several basic properties which are related to fuzzy KU-ideals, also see [9]. Abdullah et al. [10, 11] introduced the concept of direct product of intuitionistic fuzzy sets in BCK-algebras.Jun et al. [12] introduced the notion of cubic subalgebras/ideals in BCK/BCI-algebras, and then they investigated several properties. They discussed the relationship between a cubic subalgebra and a cubic ideal. Also, they provided characterizations of a cubic subalgebra/ideal and considered a method to make a new cubic subalgebra from an old one, also see [13–17].In this paper, we introduce the notion of cubic KU-ideals of KU-algebras and then we study the homomorphic image and inverse image of cubic KU-ideals.


## 2. Preliminaries

In this section we will recall some concepts related to KU-algebra and cubic sets.Definition 1 (see [5]).
By a KU-algebra we mean an algebra(X,*,0) of type (2,0) with a single binary operation * that satisfies the following identities: for any x,y,z∈X,(ku1)
(x*y)*[(y*z)*(x*z)]=0,(ku2)
x*0=0,(ku3)
0*x=x,(ku4)
x*y=0=y*x implies x=y.In what follows, let(X,*,0) denote a KU-algebra unless otherwise specified. For brevity we also call X a KU-algebra. In X we can define a binary relation ≤ by: x≤y if and only if y*x=0.Definition 2 (see [5]).
(
X
,
*
,
0
) is a KU-algebra if and only if it satisfies(ku5)
(y*z)*(x*z)≤(x*y),(ku6)
0≤x,(ku7)
x≤y,y≤x implies x=y,(ku8)
x≤y if and only if y*x=0.Definition 3 (see [8]).
In a KU-algebra, the following identities are true:(1)
z*z=0,(2)
z*(x*z)=0,(3)
x≤y imply y*z≤x*z,(4)
z*(y*x)=y*(z*x), for all x,y,z∈X,(5)
y*[(y*x)*x]=0.Example 4 (see [8]).
LetX={0,1,2,3,4} in which * is defined by Table 1.Table 1
·
0
1
2
3
4
0
0
1
2
3
4
1
0
0
2
3
4
2
0
1
0
3
3
3
0
0
2
0
2
4
0
0
0
0
0It is easy to see thatX is a KU-algebra.Definition 5 (see [6]).
A subsetS of a KU-algebra X is called subalgebra of X if x*y∈S, whenever x,y∈S.Definition 6 (see [6]).
A nonempty subsetA of a KU-algebra X is called a KU-ideal of X if it satisfies the following conditions:(1)
0∈A,(2)
x*(y*z)∈A,y∈A implies x*z∈A, for all x,y,z∈X.Example 7.
LetX={0,1,2,3,4,5} in which * is defined by Table 2.Table 2
·
0
1
2
3
4
5
0
0
1
2
3
4
5
1
0
0
2
2
4
5
2
0
0
0
1
4
5
3
0
0
0
0
4
5
4
0
0
0
1
0
5
5
0
0
0
0
0
0Clearly(X,*,0) is a KU-algebra. It is easy to show that A={0,1} and B={0,1,2,3,4} are KU-ideals of X.Definition 8 (see [6]).
Let(X,*,0) and (X′,*′,0′) be KU-algebras. A homomorphism is a map f:X→X′ satisfying f(x*y)=f(x)*′f(y), for all x,y∈X.Theorem 9 (see [6]).
Letf be a homomorphism of a KU-algebra X into a KU-algebra X′, then(1)
if0 is the identity in X, then f(0) is the identity in X′;(2)
ifS is a KU-subalgebra of X, then f(S) is a KU-subalgebra of X′;(3)
ifI is a KU-ideal of X, then f(I) is a KU-ideal in X′;(4)
ifS is a KU-subalgebra of f(X′), then f-1(S) is a KU-subalgebra algebra of X;(5)
ifB is a KU-ideal in f(X′), then f-1(B) is a KU-ideal in X;(6)
iff is a homomorphism from KU-algebra X to a KU-algebra X′ then f is one to one if and only if kerf={0}.Proposition 10 (see [8]).
Supposef:X→X′ is a homomorphism of KU-algebras, then:(1)
f(0)=0′,(2)
ifx≤y implies f(x)≤f(y).Proposition 11 (see [8]).
Let(X,*,0) and (X′,*′,0′) be KU-algebras and let f:X→X′ be a homomorphism, then kerf is a KU-ideal of X.Now we will recall the concept of interval-valued fuzzy sets.An interval number isa~=[a-,a+], where 0⩽a-⩽a+⩽1. Let D[0,1] denote the family of all closed subintervals of [0,1], that is,
(1)D[0,1]={a~=[a-,a+]:a-⩽a+,fora-,a+∈I}.
We define the operations “≽,” “⪯,” “=,” “rmin,” and “rmax” in case of two elements in D[0,1]. We consider two elements a~=[a-,a+] and 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 ai~∈D[0,1] where i∈Λ. We define
(2)rinfi∈Λa~i=[infi∈Λai-,infi∈Λai+],rsupi∈Λa~i=[supi∈Λai-,supi∈Λai+].
An interval-valued fuzzy set (briefly, IVF-set) μ~A on X is defined as
(3)μ~A={〈x,[μA-(x),μA+(x)]〉:x∈X},
where μA-(x)⩽μA+(x), for all x∈X. Then the ordinary fuzzy sets μA-:X→[0,1] and μA+:X→[0,1] are called a lower fuzzy set and an upper fuzzy set of μ~, respectively. Let μ~A(x)=[μA-(x),μA+(x)], then
(4)A={〈x,μ~A(x)〉:x∈X},
where μ~A:X→D[0,1].Jun et al. [12] introduced the concept of cubic sets defined on a nonempty set X as objects having the form:
(5)Ξ={〈x,μ~Ξ(x),λΞ(x)〉:x∈X},
which is briefly denoted by Ξ=〈μ~Ξ,λΞ〉, where the functions μ~Ξ:X→D[0,1] and λΞ:X→[0,1].Denote𝒞(X) by family of all cubic sets in X.


## 3. Cubic KU-Ideals of KU-Algebras

In this section, we will introduce a new notion called cubic KU-ideal of KU-algebras and study several properties of it.Definition 12.
LetX be a KU-algebra. A cubic set Ξ=〈μ~Ξ,λΞ〉 in X is called cubic KU-subalgebra of X if it satisfies the following conditions:(S1)
μ~Ξ(0)≽μ~Ξ(x) and λΞ(0)≤λΞ(x),(S2)
μ~Ξ(x)≽rmin{μ~Ξ(x*y),μ~Ξ(y)} and λΞ(x)≤max{λΞ(x*y),λΞ(y)}, for all x,y∈X.Definition 13.
LetX be a KU-algebra. A cubic set Ξ=〈μ~Ξ,λΞ〉 in X is called cubic KU-ideal of X if it satisfies the following conditions:(A1)
μ~Ξ(0)≽μ~Ξ(x) and λΞ(0)≤λΞ(x),(A2)
μ~Ξ(x*z)≽rmin{μ~Ξ(x*(y*z)),μ~Ξ(y)} and λΞ(x*z)≤max{λΞ(x*(y*z)),λΞ(y)}, for all x,y,z∈X.Example 14.
LetX={0,1,2,3,4} in which * is defined by Table 3.Table 3
·
0
1
2
3
4
0
0
1
2
3
4
1
0
0
0
0
1
2
0
3
0
3
4
3
0
1
2
0
1
4
0
0
0
0
0Clearly(X,*,0) is a KU-algebra. Define a cubic set Ξ=〈μ~Ξ,λΞ〉 in X as follows:
(6)μ~Ξ(x)={[0.6,0.7],ifx=0,[0.4,0.5],ifx∈{1,3},[0.1,0.3],ifx∈{2,4},λΞ(x)={0.1,ifx=0,0.3,ifx∈{1,3},0.6,ifx∈{2,4}.
By routine calculations it can be seen that the cubic set Ξ=〈μ~Ξ,λΞ〉 is a cubic KU-ideal of X.Lemma 15.
LetΞ=〈μ~Ξ,λΞ〉 be a cubic KU-ideal of KU-algebra X. If the inequality x*y≤z holds in X, then μ~Ξ(y)≽rmin{μ~Ξ(x),μ~Ξ(z)} and λΞ(y)≤max{λΞ(x),λΞ(z)}.Proof.
Assume that the inequalityx*y≤z holds in X, then z*(x*y)=0 and by (A2)μ~Ξ(z*y)≽rmin{μ~Ξ(z*(x*y)),μ~Ξ(x)}, if we put z=0 then
(i)μ~Ξ(0*y)=μ~Ξ(y)≽rmin{μ~Ξ(0*(x*y)),μ~Ξ(x)}=rmin{μ~Ξ(x*y),μ~Ξ(x)}
but(ii)μ~Ξ(x*y)≽rmin{μ~Ξ(x*(z*y)),μ~Ξ(z)}=rmin{μ~Ξ(z*(x*y)),μ~Ξ(z)}=rmin{μ~Ξ(0),μ(z)}=μ~Ξ(z).
From (i) and (ii), we get μ~Ξ(y)≽rmin{μ~Ξ(x),μ~Ξ(z)}. Similarly we can show that λΞ(y)≤max{λΞ(x),λΞ(z)}. This completes the proof.Lemma 16.
IfΞ=〈μ~Ξ,λΞ〉 is a cubic KU-ideal of KU-algebra X and if x≤y then μ~Ξ(x)≽μ~Ξ(y) and λΞ(x)≤λΞ(y).Proof.
Ifx≤y then y*x=0. This together with 0*x=x and μ~Ξ(0)≽μ~Ξ(y) also λΞ(0)≤λΞ(y), we get
(7)μ~Ξ(0*x)=μ~Ξ(x)≽rmin{μ~Ξ(0*(y*x)),μ~Ξ(y)}=rmin{μ~Ξ(0*0),μ~Ξ(y)}=rmin{μ~Ξ(0),μ~Ξ(y)}=μ~Ξ(y).
Also
(8)λΞ(0*x)=λΞ(x)≤max{λΞ(0*(y*x)),λΞ(y)}=max{λΞ(0*0),λΞ(y)}=max{λΞ(0),λΞ(y)}=λΞ(y).
This completes the proof.LetΞ=〈μ~Ξ,λΞ〉 and ℱ=〈μ~ℱ,λℱ〉 be two cubic sets in a KU-algebra X, then
(9)Ξ∩ℱ={〈x,rmin{μ~Ξ(x),μ~ℱ(x)},max{λΞ(x),λℱ(x)}〉:x∈X〈x,rmin{μ~Ξ(x),μ~ℱ(x)},max{λΞ(x),λℱ(x)}〉}={〈x,μ~Ξ(x)∩μ~ℱ(x),λΞ(x)∪λℱ(x)〉:x∈X}.Proposition 17.
Let{Ξi}i∈I be a family of cubic KU-ideals of a KU-algebra X, then ⋂i∈IΞi is a cubic KU-ideal of X.Proof.
Let{Ξi}i∈I be a family of cubic KU-ideals of a KU-algebra X, then for any x,y,z∈X,
(10)(⋂‍μ~Ξi)(0)=rinf(μ~Ξi(0))≽rinf(μ~Ξi(x))=(⋂‍μ~Ξi)(x),(⋂‍μ~Ξi)(x*z)=rinf(μ~Ξi(x*z))≽rinf(rmin{μ~Ξi(x*(y*z)),μ~Ξi(y)})=rmin{rinf(μ~Ξi(x*(y*z))),rinf(μ~Ξi(y))}=rmin{(⋂‍μ~Ξi)(x*(y*z)),(⋂‍μ~Ξi)(y)}.
Also
(11)(⋃‍λΞi)(0)=sup(λΞi(0))≤sup(λΞi(x))=(⋃‍λΞi)(x),(⋃‍λΞi)(x*z)=sup(λΞi(x*z))≤sup(max{λΞi(x*(y*z)),λΞi(y)})=max{sup(λΞi(x*(y*z))),sup(λΞi(y))}=max{(⋃‍λΞi)(x*(y*z)),(⋃‍λΞi)(y)}.
This completes the proof.For anyt~∈D[0,1] and s∈[0,1], let Ξ=〈μ~Ξ,λΞ〉 be a cubic set in a KU-algebra X, then the set
(12)U(Ξ;t~,s)={x∈X:μ~Ξ(x)≽t~,λΞ(x)≤s}
is called the cubic level set of Ξ=〈μ~Ξ,λΞ〉.Theorem 18.
LetΞ=〈μ~Ξ,λΞ〉 be a cubic subset in X then Ξ=〈μ~Ξ,λΞ〉 is a cubic KU-ideal of X if and only if for all t~∈D[0,1] and s∈[0,1], the set U(Ξ;t~,s) is either empty or a KU-ideal of X.Proof.
Assume thatΞ=〈μ~Ξ,λΞ〉 is a cubic KU-ideal of X, let t~∈D[0,1] and s∈[0,1], be such that U(Ξ;t~,s)≠∅, and let x,y∈X be such that x∈U(Ξ;t~,s), then μ~Ξ(x)≽t~ and λΞ(x)≤s. By (A2) we get
(13)μ~Ξ(y*0)=μ~Ξ(0)≽rmin{μ~Ξ(y*(x*0)),μ~Ξ(x)}=rmin{μ~Ξ(y*0),μ~Ξ(x)}=rmin{μ~Ξ(0),μ~Ξ(x)}=t~,
also
(14)λΞ(y*0)=λΞ(0)≤max{λΞ(y*(x*0)),λΞ(x)}=max{λΞ(y*0),λΞ(x)}=max{λΞ(0),λΞ(x)}=s.
Thus, 0∈U(Ξ;t~,s). Now letting x*(y*z)∈U(Ξ;t~,s),y∈U(Ξ;t~,s), it implies that
(15)μ~Ξ(x*z)≽rmin{μ~Ξ(x*(y*z)),μ~Ξ(y)}=t~,
also
(16)λΞ(x*z)≤max{λΞ(x*(y*z)),λΞ(y)}=s.
So that x*z∈U(Ξ;t~,s). Hence, U(Ξ;t~,s) is a KU-ideal of X.
Conversely, suppose thatU(Ξ;t~,s) is a KU-ideal of X and let x,y,z∈X be such that
(17)μ~Ξ(x*z)≺rmin{μ~Ξ(x*(y*z)),μ~Ξ(y)},λΞ(x*z)>max{λΞ(x*(y*z)),λΞ(y)},
taking
(18)β1~=12{μ~Ξ(x*z)+rmin{μ~Ξ(x*(y*z)),μ~Ξ(y)}},
and taking(19)β2=12{λΞ(x*z)+max{λΞ(x*(y*z)),λΞ(y)}},
we have β1~∈D[0,1] and β2∈[0,1], and
(20)μ~Ξ(x*z)≺β1~≺rmin{μ~Ξ(x*(y*z)),μ~Ξ(y)},λΞ(x*z)>β2>max{λΞ(x*(y*z)),λΞ(y)}.
It follows that x*(y*z)∈U(Ξ;β1~,β2) and x*z∉U(Ξ;β1~,β2). This is a contradiction and therefore Ξ=〈μ~Ξ,λΞ〉 is cubic KU-ideal of X.Proposition 19.
IfΞ=〈μ~Ξ,λΞ〉 is a cubic KU-ideal of KU-algebra X, then
(21)μ~Ξ(x*(x*y))≽μ~Ξ(y),λΞ(x*(x*y))≤λΞ(y).Proof.
Takingz=x*y in (A2) and using (ku2) and (A1), we get
(22)μ~Ξ(x*(x*y))≽rmin{μ~Ξ(x*(y*(x*y))),μ~Ξ(y)}=rmin{μ~Ξ(x*(x*(y*y))),μ~Ξ(y)}=rmin{μ~Ξ(x*(x*0)),μ~Ξ(y)}=rmin{μ~Ξ(0),μ~Ξ(y)}=μ~Ξ(y),λΞ(x*(x*y))≤max{λΞ(x*(y*(x*y))),λΞ(y)}=max{λΞ(x*(x*(y*y))),λΞ(y)}=max{λΞ(x*(x*0)),λΞ(y)}=max{λΞ(0),λΞ(y)}=λΞ(y).
Thus, we get μ~Ξ(x*(x*y))≽μ~Ξ(y) and λΞ(x*(x*y))≤λΞ(y). This completes the proof.


## 4. Image and Preimage of Cubic KU-Ideals

In this section we will present some results on images and preimages of cubic KU-ideals in KU-algebras.Definition 20.
Letf be a mapping from a set X to a set Y. If Ξ=〈μ~Ξ,λΞ〉 is a cubic subset of X, then the cubic subset β=〈μ~β,λβ〉 of Y is defined by
(23)f(μ~Ξ)(y)=μ~β(y)={rsupx∈f-1(y)μ~Ξ(x),iff-1(y)={x∈X,f(x)=y}≠∅,0,otherwise,f(λΞ)(y)=λβ(y)={infx∈f-1(y)λΞ(x),iff-1(y)={x∈X,f(x)=y}≠∅,1,otherwise,is said to be the image of Ξ=〈μ~Ξ,λΞ〉 under f.Similarly ifβ=〈μ~β,λβ〉 is a cubic subset of Y, then the cubic subset Ξ=β∘f in X (i.e., the cubic subset defined by μ~Ξ(x)=μ~β(f(x)) and λΞ(x)=λβ(f(x)) for all x∈X) is called the preimage of β under f.Theorem 21.
An onto homomorphic preimage of cubic KU-ideal is also cubic KU-ideal.Proof.
Letf:X→X′ be an onto homomorphism of KU-algebras, β=〈μ~β,λβ〉 a cubic KU-ideal of X′, and Ξ=〈μ~Ξ,λΞ〉 the preimage of β under f, then μ~Ξ(x)=μ~β(f(x)) and λΞ(x)=λβ(f(x)) for all x∈X. Let x∈X, then
(24)μ~Ξ(0)=μ~β(f(0))≽μ~β(f(x))=μ~Ξ(x),λΞ(0)=λβ(f(0))≤λβ(f(x))=λΞ(x).
Now let x,y,z∈X, then
(25)μ~Ξ(x*z)=μ~β(f(x*z))=μ~β(f(x)*′f(z))≽rmin{μ~β(f(x)*′(f(y)*′f(z))),μ~β(f(y))}=rmin{μ~β(f(x*(y*z))),μ~β(f(y))}=rmin{μ~Ξ(x*(y*z)),μ~Ξ(y)},λΞ(x*z)=λβ(f(x*z))=λβ(f(x)*′f(z))≤max{λβ(f(x)*′(f(y)*′f(z))),λβ(f(y))}=max{λβ(f(x*(y*z))),λβ(f(y))}=max{λΞ(x*(y*z)),λΞ(y)}.
This completes the proof.Definition 22.
A cubic subsetΞ=〈μ~Ξ,λΞ〉 of X has sup and inf properties if for any subset T of X, there exist t,s∈T such that μ~Ξ(t)=rsupt∈Tμ~Ξ(t) and λΞ(s)=inft∈TλΞ(s).Theorem 23.
Letf:X→Y be a homomorphism between KU-algebras X and Y. For every cubic KU-ideal Ξ=〈μ~Ξ,λΞ〉 in X,f(Ξ) is cubic KU-ideal of Y.Proof.
By definitionμ~β(y′)=f(μ~Ξ)(y′)=rsupx∈f-1(y′)μ~Ξ(x) and λβ(y′)=f(λΞ)(y′)=infx∈f-1(y′)λΞ(x) for all y′∈Y and rsup∅=[0,0] and inf∅=0. We have to prove that μ~β(x′*z′)≽rmin{μ~β(x′*(y′*z′)),μ~β(y′)} and λβ(x′*z′)≤max{λβ(x′*(y′*z′)),λβ(y′)} for all x′,y′,z′∈Y. Let f:X→Y be an onto homomorphism of KU-algebras, Ξ=〈μ~Ξ,λΞ〉 a cubic KU-ideal of X with sup and inf properties, and β=〈μ~β,λβ〉 the image of Ξ=〈μ~Ξ,λΞ〉 under f. Since Ξ=〈μ~Ξ,λΞ〉 is cubic KU-ideal of X, we have
(26)μ~Ξ(0)≽μ~Ξ(x),λΞ(0)≤λΞ(x)∀x∈X.
Note that 0∈f-1(0′) where 0,0′ are the zero of X and Y, respectively. Thus,
(27)μ~β(0′)=rsupt∈f-1(0′)μ~Ξ(t)=μ~Ξ(0)≽μ~Ξ(x)∀x∈X,λβ(0′)=inft∈f-1(0′)λΞ(t)=λΞ(0)≤λΞ(x)∀x∈X,
which implies that μ~β(0′)≽rsupt∈f-1(x′)μ~Ξ(t)=μ~β(x′) and λβ(0′)≤inft∈f-1(x′)λΞ(t)=λβ(x′) for any x′∈Y. For any x′,y′,z′∈Y, let x0∈f-1(x′),y0∈f-1(y′), and z0∈f-1(z′) be such that
(28)μ~Ξ(x0*z0)=rsupt∈f-1(x′*z′)μ~Ξ(t),μ~Ξ(y0)=rsupt∈f-1(y′)μ~Ξ(t),μ~Ξ(x0*(y0*z0))=μ~β{f(x0*(y0*z0))}=μ~β(x′*(y′*z′))=rsup(x0*(y0*z0))∈f-1(x′*(y′*z′))μ~Ξ(x0*(y0*z0))=rsupt∈f-1(x′*(y′*z′))μ~Ξ(t).
Also
(29)λΞ(x0*z0)=inft∈f-1(x′*z′)λΞ(t),λΞ(y0)=inft∈f-1(y′)λΞ(t),λΞ(x0*(y0*z0))=λβ{f(x0*(y0*z0))}=λβ(x′*(y′*z′))=inf(x0*(y0*z0))∈f-1(x′*(y′*z′))λΞ(x0*(y0*z0))=inft∈f-1(x′*(y′*z′))λΞ(t).
Then
(30)μ~β(x′*z′)=rsupt∈f-1(x′*z′)μ~Ξ(t)=μ~Ξ(x0*z0)≽rmin{μ~Ξ(x0*(y0*z0)),μ~Ξ(y0)}=rmin{rsupt∈f-1(x′*(y′*z′))μ~Ξ(t),rsupt∈f-1(y′)μ~Ξ(t)}=rmin{μ~β(x′*(y′*z′)),μ~β(y′)},λβ(x′*z′)=inft∈f-1(x′*z′)λΞ(t)=λΞ(x0*z0)≤max{λΞ(x0*(y0*z0)),λΞ(y0)}=max{inft∈f-1(x′*(y′*z′))λΞ(t),inft∈f-1(y′)λΞ(t)}=max{λβ(x′*(y′*z′)),λβ(y′)}.
Hence, β is cubic KU-ideal of Y.


## 5. Cartesian Product of Cubic KU-Ideals

In this section we will provide some new definitions on cartesian product of cubic KU-ideals in KU-algebras.Definition 24.
LetΞ1=〈μ~Ξ1,λΞ1〉 and Ξ2=〈μ~Ξ2,λΞ2〉 be two cubic subsets of KU-algebras X1 and X2, respectively. Then cartesian product of cubic subsets Ξ1 and Ξ2 is denoted by Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 and is defined as
(31)μ~Ξ1×Ξ2(x,y)=rmin{μ~Ξ1(x),μ~Ξ2(y)},λΞ1×Ξ2(x,y)=max{λΞ1(x),λΞ2(y)},
for all (x,y)∈X1×X2.Remark 25.
LetX and Y be KU-algebras. We define * on X×Y by (x,y)*(u,v)=(x*u,y*v) for every (x,y),(u,v) belong to X×Y, then clearly (X×Y,*,(0,0)) is a KU-algebra.Definition 26.
A cubic subsetΞ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 of X1×X2 is called a cubic KU-subalgebra of X1×X2 if(CP1)
μ~Ξ1×Ξ2(0,0)≽μ~Ξ1×Ξ2(x,y) and λΞ1×Ξ2(0,0)≤λΞ1×Ξ2(x,y),(CP2)
μ~Ξ1×Ξ2(x1,y1)≽rmin{μ~Ξ1×Ξ2((x1,y1)*(x2,y2)),μ~Ξ1×Ξ2(x2,y2)},(CP3)
λΞ1×Ξ2(x1,y1)≤max{λΞ1×Ξ2((x1,y1)*(x2,y2)),λΞ1×Ξ2(x2,y2)},for all (x1,y1),(x2,y2)∈X1×X2.Definition 27.
A cubic subsetΞ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 of X1×X2 is called a cubic KU-ideal of X1×X2 if(CP4)
μ~Ξ1×Ξ2(0,0)≽μ~Ξ1×Ξ2(x,y) and λΞ1×Ξ2(0,0)≤λΞ1×Ξ2(x,y),(CP5)
μ~Ξ1×Ξ2((x1,y1)*(x3,y3))≽rmin{μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),μ~Ξ1×Ξ2(x2,y2)},(CP6)
λΞ1×Ξ2((x1,y1)*(x3,y3))≤max{λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),λΞ1×Ξ2(x2,y2)},for all (x1,y1),(x2,y2),(x3,y3)∈X1×X2.Theorem 28.
LetΞ1=〈μ~Ξ1,λΞ1〉 and Ξ2=〈μ~Ξ2,λΞ2〉 be two cubic KU-subalgebras of KU-algebras X1 and X2, respectively. Then Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-subalgebra of KU-algebra X1×X2.Proof.
For any(x,y)∈X1×X2,
(32)μ~Ξ1×Ξ2(0,0)=rmin{μ~Ξ1(0),μ~Ξ2(0)}≽rmin{μ~Ξ1(x),μ~Ξ2(y)}=μ~Ξ1×Ξ2(x,y),λΞ1×Ξ2(0,0)=max{λΞ1(0),λΞ2(0)}≤max{λΞ1(x),λΞ2(y)}=λΞ1×Ξ2(x,y).
For any (x1,y1),(x2,y2)∈X1×X2. Then
(33)μ~Ξ1×Ξ2(x1,y1)=rmin{μ~Ξ1(x1),μ~Ξ2(y1)}≽rmin{rmin{μ~Ξ1(x1*x2),μ~Ξ1(x2)},rmin{μ~Ξ2(y1*y2),μ~Ξ2(y2)}}=rmin{rmin{μ~Ξ1(x1*x2),μ~Ξ2(y1*y2)},rmin{μ~Ξ1(x2),μ~Ξ2(y2)}}=rmin{μ~Ξ1×Ξ2(x1*x2,y1*y2),μ~Ξ1×Ξ2(x2,y2)}≽rmin{μ~Ξ1×Ξ2((x1*y1)(x2*y2)),μ~Ξ1×Ξ2(x2,y2)},λΞ1×Ξ2(x1,y1)=max{λΞ1(x1),λΞ2(y1)}≤max{max{λΞ1(x1*x2),λΞ1(x2)},max{λΞ2(y1*y2),λΞ2(y2)}}=max{max{λΞ1(x1*x2),λΞ1(y1*y2)},max{λΞ2(x2),λΞ2(y2)}}=max{λΞ1×Ξ2(x1*x2,y1*y2),λΞ1×Ξ2(x2,y2)}≤max{λΞ1×Ξ2((x1*y1)(x2*y2)),λΞ1×Ξ2(x2,y2)}.
Hence, for all (x1,y1),(x2,y2)∈X1×X2,Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-subalgebra of KU-algebra X1×X2.Theorem 29.
LetΞ1=〈μ~Ξ1,λΞ1〉 and Ξ2=〈μ~Ξ2,λΞ2〉 be two cubic KU-ideals of KU-algebras X1 and X2, respectively. Then Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-ideal of KU-algebra X1×X2.Proof.
For any(x,y)∈X1×X2,
(34)μ~Ξ1×Ξ2(0,0)=rmin{μ~Ξ1(0),μ~Ξ2(0)}≽rmin{μ~Ξ1(x),μ~Ξ2(y)}=μ~Ξ1×Ξ2(x,y),λΞ1×Ξ2(0,0)=max{λΞ1(0),λΞ2(0)}≤max{λΞ1(x),λΞ2(y)}=λΞ1×Ξ2(x,y).
Now for any (x1,y1),(x2,y2),(x3,y3)∈X1×X2,
(35)μ~Ξ1×Ξ2((x1,y1)*(x3,y3))=μ~Ξ1×Ξ2(x1*x3,y1*y3)=rmin{μ~Ξ1(x1*x3),μ~Ξ2(y1*y3)}≽rmin{rmin{μ~Ξ1(x1*(x2*x3)),μ~Ξ1(x2)},rmin{μ~Ξ2(y1*(y2*y3)),μ~Ξ2(y2)}}=rmin{rmin{μ~Ξ1(x1*(x2*x3)),μ~Ξ2(y1*(y2*y3))},rmin{μ~Ξ1(x2),μ~Ξ2(y2)}}=rmin{μ~Ξ1×Ξ2((x1*(x2*x3)),(y1*(y2*y3))),μ~Ξ1×Ξ2(x2,y2)}≽rmin{μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),μ~Ξ1×Ξ2(x2,y2)},λΞ1×Ξ2((x1,y1)*(x3,y3))=λΞ1×Ξ2(x1*x3,y1*y3)=max{λΞ1(x1*x3),λΞ2(y1*y3)}≤max{max{λΞ1(x1*(x2*x3)),λΞ1(x2)},min{λΞ2(y1*(y2*y3)),λΞ2(y2)}}=max{max{λΞ1(x1*(x2*x3)),λΞ2(y1*(y2*y3))},max{λΞ1(x2),λΞ2(y2)}}=max{λΞ1×Ξ2((x1*(x2*x3)),(y1*(y2*y3))),λΞ1×Ξ2(x2,y2)}≤max{λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),λΞ1×Ξ2(x2,y2)}.
Hence, for all (x1,y1),(x2,y2),(x3,y3)∈X1×X2,Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-ideal of KU-algebra X1×X2.Lemma 30.
IfΞ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-ideal of KU-algebra X1×X2 and if (x1,y1)≤(x2,y2), we have μ~Ξ1×Ξ2(x2,y2)⪯μ~Ξ1×Ξ2(x1,y1) and λΞ1×Ξ2(x2,y2)≥λΞ1×Ξ2(x1,y1), for all (x1,y1),(x2,y2)∈X1×X2.Proof.
Let(x1,y1),(x2,y2)∈X1×X2, such that (x1,y1)≤(x2,y2)⇒(x2,y2)*(x1,y1)=(0,0). This together with (0,0)*(x1,y1)=(x1,y1) and μ~Ξ1×Ξ2(x2,y2)⪯μ~Ξ1×Ξ2(0,0) also λΞ1×Ξ2(x2,y2)≥λΞ1×Ξ2(0,0). Consider
(36)μ~Ξ1×Ξ2((0,0)*(x1,y1))=μ~Ξ1×Ξ2(x1,y1)≽rmin{μ~Ξ1×Ξ2((0,0)*((x2,y2)*(x1,y1))),μ~Ξ1×Ξ2(x2,y2)}=rmin{μ~Ξ1×Ξ2((0,0)*(0,0)),μ~Ξ1×Ξ2(x2,y2)}=rmin{μ~Ξ1×Ξ2(0,0),μ~Ξ1×Ξ2(x2,y2)}=μ~Ξ1×Ξ2(x2,y2),λΞ1×Ξ2((0,0)*(x1,y1))=λΞ1×Ξ2(x1,y1)≤max{λΞ1×Ξ2((0,0)*((x2,y2)*(x1,y1))),λΞ1×Ξ2(x2,y2)}=max{λΞ1×Ξ2((0,0)*(0,0)),λΞ1×Ξ2(x2,y2)}=max{λΞ1×Ξ2(0,0),λΞ1×Ξ2(x2,y2)}=λΞ1×Ξ2(x2,y2).
This shows that μ~Ξ1×Ξ2(x2,y2)⪯μ~Ξ1×Ξ2(x1,y1) and λΞ1×Ξ2(x2,y2)≥λΞ1×Ξ2(x1,y1), for all (x1,y1),(x2,y2)∈X1×X2.Lemma 31.
IfΞ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-ideal of KU-algebra X1×X2 and if (x1,y1)*(x2,y2)≤(x3,y3) holds in X1×X2, then we have μ~Ξ1×Ξ2(x2,y2)≽rmin{μ~Ξ1×Ξ2(x1,y1),μ~Ξ1×Ξ2(x3,y3)} and λΞ1×Ξ2(x2,y2)≤max{λΞ1×Ξ2(x1,y1),λΞ1×Ξ2(x3,y3)}, for all (x1,y1),(x2,y2),(x3,y3)∈X1×X2.Proof.
Let(x1,y1),(x2,y2),(x3,y3)∈X1×X2 and let (x1,y1)*(x2,y2)≤(x3,y3) holds in X1×X2, then
(37)(x3,y3)*((x1,y1)*(x2,y2))=(0,0).
Now for (x3,y3)=(0,0) and from (CP5)(38)μ~Ξ1×Ξ2((x3,y3)*(x2,y2))≽rmin{μ~Ξ1×Ξ2((x3,y3)*((x1,y1)*(x2,y2))),μ~Ξ1×Ξ2(x1,y1)},
we have
(39)μ~Ξ1×Ξ2((0,0)*(x2,y2))=μ~Ξ1×Ξ2(x2,y2)≽rmin{μ~Ξ1×Ξ2((x1,y1)*(x2,y2)),μ~Ξ1×Ξ2(x1,y1)}≽rmin{rmin{μ~Ξ1×Ξ2((x1,y1)*((x3,y3)(x2,y2))),μ~Ξ1×Ξ2(x3,y3)},μ~Ξ1×Ξ2(x1,y1)}=rmin{rmin{μ~Ξ1×Ξ2((x3,y3)*((x1,y1)(x2,y2))),μ~Ξ1×Ξ2(x3,y3)},μ~Ξ1×Ξ2(x1,y1)}=rmin{rmin{μ~Ξ1×Ξ2(0,0),μ~Ξ1×Ξ2(x3,y3)},μ~Ξ1×Ξ2(x1,y1)}=rmin{μ~Ξ1×Ξ2(x3,y3),μ~Ξ1×Ξ2(x1,y1)}=rmin{μ~Ξ1×Ξ2(x1,y1),μ~Ξ1×Ξ2(x3,y3)},
and from (CP6)(40)λΞ1×Ξ2((x3,y3)*(x2,y2))≤max{λΞ1×Ξ2((x3,y3)*((x1,y1)*(x2,y2))),λΞ1×Ξ2(x1,y1)},
we have
(41)λΞ1×Ξ2((0,0)*(x2,y2))=λΞ1×Ξ2(x2,y2)≤max{λΞ1×Ξ2((x1,y1)*(x2,y2)),λΞ1×Ξ2(x1,y1)}≤max{max{λΞ1×Ξ2((x1,y1)*((x3,y3)(x2,y2))),λΞ1×Ξ2(x3,y3)},λΞ1×Ξ2(x1,y1)}=max{max{λΞ1×Ξ2((x3,y3)*((x1,y1)(x2,y2))),λΞ1×Ξ2(x3,y3)},λΞ1×Ξ2(x1,y1)}=max{max{λΞ1×Ξ2(0,0),λΞ1×Ξ2(x3,y3)},λΞ1×Ξ2(x1,y1)}=max{λΞ1×Ξ2(x3,y3),λΞ1×Ξ2(x1,y1)}=max{λΞ1×Ξ2(x1,y1),λΞ1×Ξ2(x3,y3)}.
This completes the proof.Definition 32.
LetΞ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 be a cubic subset of KU-algebra X1×X2, and for any t~∈D[0,1] and s∈[0,1], the set(42)U(Ξ1×Ξ2;t~,s)={(x,y)∈X1×X2:μ~Ξ1×Ξ2(x,y)≽t~,λΞ1×Ξ2(x,y)≤s}
is called the cubic level set of Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉.Theorem 33.
LetΞ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 be a cubic subset of KU-algebra X1×X2. Then Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-subalgebra of KU-algebra X1×X2 if and only if for any t~∈D[0,1] and s∈[0,1], the set U(Ξ1×Ξ2;t~,s) is either empty or a KU-subalgebra of X1×X2.Proof.
The proof is straightforward.Theorem 34.
LetΞ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 be a cubic subset of KU-algebra X1×X2. Then Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-ideal of KU-algebra X1×X2 if and only if for any t~∈D[0,1] and s∈[0,1], the set U(Ξ1×Ξ2;t~,s) is either empty or a KU-ideal of X1×X2.Proof.
LetΞ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 be a cubic KU-ideal of KU-algebra X1×X2. For any t~∈D[0,1] and s∈[0,1], define the sets
(43)U(Ξ1×Ξ2;t~,s)={(x,y)∈X1×X2:μ~Ξ1×Ξ2(x,y)≽t~,λΞ1×Ξ2(x,y)≤s}.
Since U(Ξ1×Ξ2;t~,s)≠∅, let (x,y)∈U(Ξ1×Ξ2;t~,s). This implies μ~Ξ1×Ξ2(x,y)≽t~ and λΞ1×Ξ2(x,y)≤s. So
(44)μ~Ξ1×Ξ2(0,0)≽μ~Ξ1×Ξ2(x,y)≽t~⇒μ~Ξ1×Ξ2(0,0)≽t~,λΞ1×Ξ2(0,0)≤λΞ1×Ξ2(x,y)≤s⇒λΞ1×Ξ2(0,0)≤s.
This shows that (0,0)∈U(Ξ1×Ξ2;t~,s). Let (x1,y1)*((x2,y2)*(x3,y3))∈U(Ξ1×Ξ2;t~,s) and (x2,y2)∈U(Ξ1×Ξ2;t~,s). This implies
(45)μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3)))≽t~,μ~Ξ1×Ξ2(x2,y2)≽t~,λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3)))≤s,λΞ1×Ξ2(x2,y2)≤s.
Since
(46)μ~Ξ1×Ξ2((x1,y1)*(x3,y3))≽rmin{μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),μ~Ξ1×Ξ2(x2,y2)}≽rmin{t~,t~}=t~,λΞ1×Ξ2((x1,y1)*(x3,y3))≤max{λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),λΞ1×Ξ2(x2,y2)}≤max{s,s}=s.
This implied that (x1,y1)*(x3,y3)∈U(Ξ1×Ξ2;t~,s). Hence, U(Ξ1×Ξ2;t~,s) is a KU-ideal of X1×X2.
Conversely, supposeU(Ξ1×Ξ2;t~,s) is a KU-ideal of X1×X2, for any t~∈D[0,1] and s∈[0,1]. Assume (x1,y1)∈X1×X2, such that
(47)μ~Ξ1×Ξ2(0,0)≺μ~Ξ1×Ξ2(x1,y1),λΞ1×Ξ2(0,0)>λΞ1×Ξ2(x1,y1).
Put
(48)t∘~=12{μ~Ξ1×Ξ2(0,0)+μ~Ξ1×Ξ2(x1,y1)}⇒μ~Ξ1×Ξ2(0,0)≺t∘~≺μ~Ξ1×Ξ2(x1,y1),s∘=12{λΞ1×Ξ2(0,0)+λΞ1×Ξ2(x2,y2)}⇒λΞ1×Ξ2(0,0)>t∘>λΞ1×Ξ2(x1,y1).
This implies (x1,y1)∈U(Ξ1×Ξ2;t~,s), but (0,0)∉U(Ξ1×Ξ2;t~,s), which is contradiction. Therefore, μ~Ξ1×Ξ2(0,0)≽μ~Ξ1×Ξ2(x,y) and λΞ1×Ξ2(0,0)≤λΞ1×Ξ2(x,y) for all (x,y)∈X1×X2. Assume (x1,y1),(x2,y2),(x3,y3)∈X1×X2, such that
(49)μ~Ξ1×Ξ2((x1,y1)*(x3,y3))<rmin{μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),μ~Ξ1×Ξ2(x2,y2)}.
Let
(50)t∘~=12{{μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),μ~Ξ1×Ξ2(x2,y2)}μ~Ξ1×Ξ2((x1,y1)*(x3,y3))12k+rmin{μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),12kaaaaaaμ~Ξ1×Ξ2(x2,y2)}},
then
(51)μ~Ξ1×Ξ2((x1,y1)*(x3,y3))≺t∘~≺rmin{μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),μ~Ξ1×Ξ2(x2,y2)}.
Also
(52)λΞ1×Ξ2((x1,y1)*(x3,y3))>max{λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),λΞ1×Ξ2(x2,y2)}.
Let
(53)s∘=12{{λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),λΞ1×Ξ2(x2,y2)}λΞ1×Ξ2((x1,y1)*(x3,y3))12k+max{λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),12xxxxxkλΞ1×Ξ2(x2,y2)}},
then
(54)λΞ1×Ξ2((x1,y1)*(x3,y3))>s∘>max{λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),λΞ1×Ξ2(x2,y2)}.
This shows that
(55)(x1,y1)*((x2,y2)*(x3,y3))∈U(Ξ1×Ξ2;t~,s),(x2,y2)∈U(Ξ1×Ξ2;t~,s).
But (x1,y1)*(x3,y3)∉U(μ~Ξ1×Ξ2;t∘~,s∘), which is a contradiction; therefore,
(56)μ~Ξ1×Ξ2((x1,y1)*(x3,y3))≽rmin{μ~Ξ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),μ~Ξ1×Ξ2(x2,y2)}.
Similarly
(57)λΞ1×Ξ2((x1,y1)*(x3,y3))≤max{λΞ1×Ξ2((x1,y1)*((x2,y2)*(x3,y3))),λΞ1×Ξ2(x2,y2)}.
Hence, Ξ1×Ξ2=〈μ~Ξ1×Ξ2,λΞ1×Ξ2〉 is a cubic KU-ideal of KU-algebra X1×X2.



---

*Source: 935905-2013-02-26.xml*