In this paper, we introduce the concept of cubic medial-ideal and investigate several properties. Also, we give relations between cubic medial-ideal and cubic BCI-ideal .The image and the pre-image of cubic medial-ideal under homomorphism of BCI-algebras are defined and how the image and the pre-image of cubic medial-ideal under homomorphism of BCI-algebras become cubic medial-ideal are studied. Moreover, the Cartesian product of cubic medial-ideal in Cartesian product BCI-algebras is given. 2010 AMS Classification. 06F35, 03G25, 08A72

1. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
DOI : 10.5121/ijfls.2015.5302 15
CUBIC STRUCTURES OF MEDIAL IDEAL ON
BCI -ALGEBRAS
Samy M.Mostafa and Reham Ghanem
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo,
Egypt.
Abstract
In this paper, we introduce the concept of cubic medial-ideal and investigate several properties. Also, we
give relations between cubic medial-ideal and cubic BCI-ideal .The image and the pre-image of cubic
medial-ideal under homomorphism of BCI-algebras are defined and how the image and the pre-image of
cubic medial-ideal under homomorphism of BCI-algebras become cubic medial-ideal are studied.
Moreover, the Cartesian product of cubic medial-ideal in Cartesian product BCI-algebras is given.
2010 AMS Classification. 06F35, 03G25, 08A72
Key words
Medial BCI-algebra, fuzzy medial-ideal, (cubic) medial-ideal, the pre-image of cubic medial-ideal under
homomorphism of BCI-algebras, Cartesian product of cubic medial-ideal
1.INTRODUCTION
In 1966 Iami and Iseki[3,4,10] introduced the notion of BCK-algebras .Iseki [1,2] introduced the
notion of a BCI-algebra which is a generalization of BCK-algebra . Since then numerous
mathematical papers have been written investigating the algebraic properties of the BCK / BCI-
algebras and their relationship with other structures including lattices and Boolean algebras.
There is a great deal of literature which has been produced on the theory of BCK/BCI-algebras, in
particular, emphasis seems to have been put on the ideal theory of BCK/BCI-algebras . In 1956,
Zadeh [16] introduced the notion of fuzzy sets. At present this concept has been applied to many
mathematical branches. There are several kinds of fuzzy sets extensions in the fuzzy set theory,
for example, intuitionistic fuzzy sets, interval valued fuzzy sets, vague sets etc . In 1991, Xi [15]
applied the concept of fuzzy sets to BCI, BCK, MV-algebras. In [11] J.Meng and Y.B.Jun studied
medial BCI-algebras. S.M.Mostafa etal. [13] introduced the notion of medial ideals in BCI-
algebras, they state the fuzzification of medial ideals and investigate its properties. Jun et al. [5]
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 [6,7,8,9,12]. In this paper, we
introduce the notion of cubic medial-ideals of BCI-algebras and then we study the homomorphic
image and inverse image of cubic medial –ideals under homomorphism of BCI-algebras.

2. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
16
2. PRELIMINARIES
Now we review some definitions and properties that will be useful in our results.
Definition 2.1 (see [1,2]) An algebraic system )0,,( ∗X of type (2, 0) is called a BCI-algebra if it
satisfying the following conditions:
(BCI-1) ,0)())()(( =∗∗∗∗∗ yzzxyx
(BCI-2) ,0))(( =∗∗∗ yyxx
(BCI-3) ,0=∗ xx
(BCI-4) 0=∗ yx and 0=∗ xy imply yx = .
For all Xzyx ∈and, . In a BCI-algebra X, we can define a partial ordering”≤” by yx ≤ if and
only if 0=∗ yx .
In what follows, X will denote a BCI-algebra unless otherwise specified.
Definition 2.2( see [11]) A BCI-algebra )0,,( ∗X of type (2, 0) is called a medial BCI-algebra if
it satisfying the following condition: )()()()( uyzxuzyx ∗∗∗=∗∗∗ ,for all
Xuzyx ∈and,, .
Lemma 2.3( see [11]) An algebra (X,∗ , 0) of type (2, 0) is a medial BCI-algebra if and only if it
satisfies the following conditions:
(i) )()( xyzzyx ∗∗=∗∗
(ii) xx =∗0
(iii) 0=∗ xx
Lemma 2.4( see [11]) In a medial BCI-algebra X, the following holds:
yyxx =∗∗ )( , for all Xyx ∈, .
Lemma 2.5 Let X be a medial BCI-algebra, then yxxy ∗=∗∗ )(0 , for all Xyx ∈, .
Proof. Clear.

3. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
17
Definition 2.6 A non empty subset S of a medial BCI-algebra X is said to be medial sub-
algebra of X, if Syx ∈∗ , for all Syx ∈, .
Definition 2.7 ( see [1,2]) A non-empty subset I of a BCI-algebra X is said to be a BCI-ideal of X
if it satisfies:
(I1) ,0 I∈
(I2) Iyx ∈∗ and Iy ∈ implies Ix ∈ for all Xyx ∈, .
Definition 2.8( see [13]) A non empty subset M of a medial BCI-algebra X is said to be a medial
ideal of X if it satisfies:
(M1) ,0 M∈
(M2) Mxyz ∈∗∗ )( and Mzy ∈∗ imply Mx∈ for all Xzyx ∈and, .
Proposition 2.9(see [13]) Any medial ideal of a BCI-algebra must be a BCI- ideal but the
converse is not true.
Proposition 2.10 Any BCI- ideal of a medial BCI-algebra is a medial ideal.
3.Fuzzy medial BCI-ideals
Definition 3.1 ( see [15]) Let µ be a fuzzy set on a BCI-algebra X, then µ is called a fuzzy
BCI-subalgebra of X if
(FS1) )},(),(min{)( yxyx µµµ ≥∗ for all Xyx ∈, .
Definition 3.2 ( see [15]) Let X be a BCI-algebra. a fuzzy set µ in X is called a fuzzy BCI-ideal
of X if it satisfies:
(FI1) ),()0( xµµ ≥
(FI2) )},(),(min{)( yyxx µµµ ∗≥ for all Xzyx ∈and, .
Definition 3.3[ 13 ] Let X be a medial BCI-algebra. A fuzzy set µ in X is called a fuzzy medial ideal
of X if it satisfies:
(FM1) ),()0( xµµ ≥
(FM2) )},()),((min{)( zyxyzx ∗∗∗≥ µµµ for all Xzyx ∈and, .

4. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
18
Lemma 3.4 Any fuzzy medial-ideal of a BCI-algebra is a fuzzy BCI-ideal of X.
Proof. Clear.
Now, we begin with the concepts of interval-valued fuzzy sets.
An interval number is ],[~
UL aaa = , where 10 ≤≤≤ UL aa .Let D[0, 1]
denote the family of all closed subintervals of [0, 1], i.e.,
{ }IaaforaaaaaD ULULUL ∈≤== ,:],[~]1,0[ .
We define the operations =≥≤ ,, ,rmin and rmax in case of two elements in D[0, 1]. We
consider two elements ],[~
UL aaa = and ],[
~
UL bbb = in D[0, 1].
Then
1- UULL babaiffba ≤≤≤ ,
~~ ;
2- UULL babaiffba ≥≥≥ ,
~~ ;
3- UULL babaiffba === ,
~~ ;
4- { } { } { }[ ]UULL bababarmim ,min,,min
~
,~ = ;
5- { } { } { }[ ]UULL bababar ,max,,max
~
,~max =
Here we consider that [ ]0,00
~
= as least element and [ ]1,11
~
= as greatest element.
Let [ ]1,0~ Dai ∈ ,where Λ∈i .We define








=
Λ∈ Λ∈Λ∈ i i
UiLi
i
aa
iar )inf(,)inf(~inf and








=
Λ∈ Λ∈Λ∈ i i
UiLi
i
aa
iar )sup(,)sup(~sup
An interval valued fuzzy set (briefly, i-v-f-set) µ~ on X is defined as
[ ]{ }Xxxxx UL ∈= ,)(),(,~ µµµ , where [ ]1,0:~ DX →µ and )()( xx UL µµ ≤ ,for all
Xx ∈ . Then the ordinary fuzzy sets [ ]1,0: →XLµ and [ ]1,0: →XUµ are called a
lower fuzzy set and an upper fuzzy set of µ~ respectively.

5. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
19
4.CUBIC MEDIAL-IDEAL IN BCI-ALGEBRAS
In this section, we will introduce a new notion called cubic medial-ideal in BCI-algebras and
study several properties of it.
Definition 4.1[5 ] Let X be a BCI-algebra. A set ))(~),((, xx µλµλ =Ω in X is called cubic
sub -algebra of X if it satisfies the following conditions:
(SC1) )}(~),(~min{)(~ yxryx µµµ ≥∗ ,
(SC2) )}(),(max{)( yxyx λλλ ≤∗ ,for all Xyx ∈, .
Example 4.2 Let }3,2,1,0{=X be a set with a binary operation∗ 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
Using the algorithms in Appendix B , we can prove that )0,,( ∗X is a BCI-algebra. Define
)(~ xµ as follows:
)(~ xµ =


 =
otherwise
xif
]6.0,1.0[
}1,0{]9.0,3.0[
X 0 1 2 3
)(xλ 0.2 0.2 0.6 0.7
It is easy to check that )~,(, µλµλ =Ω is cubic sub BCI –algebra.
Lemma 4.3 If ))(~),((, xx µλµλ =Ω is is cubic BCI-sub-algebra of X, then
)(~)0(~ xµµ ≥ , )()0( xλλ ≤ for all Xx ∈ .
Proof. For every Xx ∈ , we have )(~)}(~),(~min{)(~)0(~ xxxrxx µµµµµ =≥∗= . And
)()}(),(max{)0()( xxxxx λλλλλ =≤=∗ .This completes the proof.

6. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
20
Definition 4.4 Let X be a BCI-algebra. A cubic set ))(~),((, xx µλµλ =Ω is called µ~
cubic
medial-ideal of X if it satisfies the following conditions:
(MC1) )(~)0(~ xµµ ≥ , )()0( xλλ ≤
(MC2) )}(~)),((~min{)(~ zyxyzrx ∗∗∗≥ µµµ , for all .,, Xzyx ∈
(MC3) )},(),((max{)( zyxyzx ∗∗∗≤ λλλ for all .,, Xzyx ∈
Example 4.5 Let X = {0, 1, 2, 3} as in example (4.2). Define )(~ xµ as follows:
)(~ xµ =


 =
otherwise
xif
]5.0,1.0[
}1,0{]8.0,2.0[
.
X 0 1 2 3
)(xλ 0.1 0.2 0.3 0.4
It is easy to check that ))(~),((, xx µλµλ =Ω is cubic medial-idea of X.
Lemma 4.6 Let ))(~),((, xx µλµλ =Ω be cubic medial-ideal of X. If yx ≤ in X,
then )(~)(~ xy µµ ≥ and )()( yx λλ ≤ , for all Xyx ∈, .
Proof. Let Xyx ∈, be such that yx ≤ , then 0=∗ yx . From (MC2), we have
)}(~)),((~min{)(~ zyxyzrx ∗∗∗≥ µµµ = )}0(~)),(0(~min{ ∗∗∗ yxyr µµ
= )}(~)),(~min{ yyxr µµ ∗ =
= )}(~)),0(~min{ yr µµ = )(~ yµ
Similarly, form (M C3), we have )},0()),(0(max{)( ∗∗∗≤ yxyx λλλ hence,
=∗≤ )}(),(max{)( yyxx λλλ )()}(),0(max{ yy λλλ = .This completes the proof
Lemma 4.7 Let ))(~),((, xx µλµλ =Ω be cubic medial-ideal of X, if the
inequality zyx ≤∗ hold in X, then )}(~),(~min{)(~ yzrx µµµ ≥ , )}(),(max{)( zyx λλλ ≤ ,
for all Xzyx ∈,, .

7. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
21
Proof. Let Xzyx ∈,, be such that zyx ≤∗ . Thus, put 0=z in ( (MC2) Definition 3.4) and
using lemma2.5 , lemma 4.6), we get,
)}(~)),((~min{)(~ zyxyzrx ∗∗∗≥ µµµ = )}0(~)),(0(~min{ ∗∗∗ yxyr µµ
≥∗≥ )}(~),(~min{ yyxr µµ r })(~),(~min{
)(~)*(~sin
44 844 76 zyxce
yzr
µµ
µµ
≥
. Similarly we can prove that,
)}(),(max{)( yzx λλλ ≤ . This completes the proof
Theorem 4.8 Every cubic medial-ideal of X is a cubic sub-algebra of X.
Proof. Let ))(~),((, xx µλµλ =Ω be cubic medial ideal of X. Since xyx ≤∗ , for all Xyx ∈, ,
then )(~)(~
)6.4(
44 844 76 Lemma
xyx µµ ≥∗ , )()( xyx λλ ≤∗ .Put z = 0 in (MC2), (MC3), we
have ≥≥∗ )(~)(~ xyx µµ )}0(~)),(0(~min{ ∗∗∗ yxyr µµ = )}(~)),0(~min{ yr µµ = )(~ yµ .
Therefore )}(~),(~min{)(~ yxryx µµµ ≥∗ -----------(a)
and
)}0()),(0(max{)()( ∗∗∗≤≤∗ yxyxyx λλλλ = )}(),(max{ yyx λλ ∗
}(),(max{ yx λλ≤ ….. (b)
From (a) and (b) , we get ))(~),((, xx µλµλ =Ω is cubic sub-algebra of X.
This completes the proof.
Theorem 4.9 Let ))(~),((, xx µλµλ =Ω be a cubic subalgebra of X ,such that
)},(~),(~min{)(~ zyrx µµµ ≥ )}(),(max{)( zyx λλλ ≤ , satisfying the inequality zyx ≤∗ for
all Xzyx ∈,, . Then ))(~),((, xx µλµλ =Ω is a cubic medial ideal of X.
Proof. Let ))(~),((, xx µλµλ =Ω be a cubic subalgebra of X. Recall that
)()0( xλλ ≤ and )(~)0(~ xµµ ≥ , for all Xx ∈ . Since, for all Xzyx ∈,, , we have
zyxzxyxyzx ∗≤∗∗∗=∗∗∗ )()())(( , it follows from Lemma 4.7 that ,
)}*(),((max{)( zyxyzx λλλ ∗∗≤ , )}*(~),((~min{)(~ zyxyzrx µµµ ∗∗≥ .
Hence ))(~),((, xx µλµλ =Ω is a cubic medial ideal of X.

8. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
22
Theorem 4.10 Let ))(~),((, xx µλµλ =Ω be a cubic set. Then
))(~),((, xx µλµλ =Ω is bipolar cubic medial-ideal of X if and only if the non empty
set })(],,[)(~|{:)],,[;(
~
2121~, γλδδµγδδµλ ≤≥∈=Ω xxXxU is a medial-ideal of X , for every
]1,0[],[ 21 D∈δδ , ]0,1[−∈γ .
Proof . Assume that ))(~),((, xx µλµλ =Ω is cubic medial-ideal of X. Let
]1,0[],[ 21 D∈δδ , ]1,0[∈γ be such that zyxyz ∗∗∗ ,)( )],,[;(
~
21~, γδδµλΩU , then
)}(~)),((~min{)(~ zyxyzrx ∗∗∗≥ µµµ ],[]},[],,min{[ 212121 δδδδδδ =≥ r and
)},(),((max{)( zyxyzx ∗∗∗≤ λλλ { } γγγ == ,max , so )],,[;(
~
21~, γδδµλΩ∈Ux . Thus
)],,[;(
~
21~, γδδµλΩU is medial-ideal of X.
Conversely, we only need to show that (MC2) and (MC3) of definition 3.4 are true . If (MC2) is
false , assume that )],,[;(
~
21~, γδδµλΩU )( φ≠ is a medial-ideal of X . For every
]1,0[],[ 21 D∈δδ , ]1,0[∈γ suppose that there exist Xzyx ∈000 ,, such that
)}(~),((~min{)(~
000000 zyxyzrx ∗∗∗< µµµ and )}(),((max{)( 000000 zyxyzx ∗∗∗> λλλ
First, let ],[))((~
21000 γγµ =∗∗ xyz , ],[)(~
4300 γγµ =∗ zy and ],[)(~
210 δδµ =x . Then
]},[],,min{[],[ 432121 γγγγδδ r< .
Taking )}]}(~)),(((~min{)(~{[],[ 0000002
1
21 zyxyzrx ∗∗∗+= µµµλλ =
}}),min{},,{min{],([ 4321212
1
γγγγδδ + = })],min{(}),,min{[( 4223112
1
γγδγγδ ++ .
It follows that 2
1
131 },min{ => λγγ }),min{( 311 γγδ + > 1δ and
2
1
242 },min{ => λγγ }),min{( 422 γγδ + > 2δ ,so that
)(~],[],[}],min{}),,[min{ 021214231 xµδδλλγγγγ =>> .Therefore )],,[;(
~
21~,0 γδδµλΩ∉Ux . If
(MC3) is false , let Xzyx ∈000 ,, be such that )}(),((max{)( 000000 zyxyzx ∗∗∗> λλλ ,taking
β0 = 1/2 {λ (x0) + min {λ (z0 * (y0 * x0)) , λ (y0*x0) } , we have 0β ∈ [0,1] and
)}(),((max{)( 0000000 zyxyzx ∗∗∗>> λλβλ ,it follows that )*(),*(* 00000 zyzyx
∈ )],,[;(
~
21~, γδδµλΩU and )],,[;(
~
21~,0 γδδµλΩ∉Ux , this is a contradiction and therefore λ is anti

9. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
23
fuzzy medial - ideal .On the other hand
≥=∗∗ ],[))((~
21000 γγµ xyz }],min{},,[min{ 4231 γγγγ > ],[ 21 λλ ,
],[)(~
4300 γγµ =∗ zy }],min{},,[min{ 4231 γγγγ≥ > ],[ 21 λλ , and so
00000 ,)( zyxyz ∗∗∗ ∉ )],,[;(
~
21~, γδδµλΩU . It contradicts that )],,[;(
~
21~, γδδµλΩU is a medial-ideal
of X. Hence )}(~)),((~min{)(~ zyxyzrx ∗∗∗≥ µµµ , and )}*()),(*(max{)( zyxyzx λλλ ∗≤
for all Xzyx ∈,, . i.e ))(~),((, xx µλµλ =Ω is cubic medial-ideal of X. This completes the
proof
Theorem 4.10 A cubic set ))(~),((, xx µλµλ =Ω is a cubic medial-ideal if and only if
)(~ xµ is interval fuzzy medial-ideal and )(xλ is anti fuzzy medial-ideal of X .
Proof. If )(~ xµ is interval fuzzy medial-ideal and )(xλ is anti fuzzy medial-ideal of X. It is
clear that )(~)0(~ xµµ ≥ , )()0( xλλ ≤ .Let )(~ xµ interval fuzzy medial-ideal of X
and Xzyx ∈,, . Consider
)}](()),((min{)},()),(([min{
)](),([)(~
zyxyzzyxyz
xxx
UULL
UL
∗∗∗∗∗∗≥
=
µµµµ
µµµ
)},(~)),((~min{
)]}(),([))],(()),((min{[
zyxyzr
zyzyxyzxyzr ULUL
∗∗∗=
∗∗∗∗∗∗=
µµ
µµµµ
If )(xλ is anti fuzzy medial-ideal of X, then
)},(),((max{)( zyxyzx ∗∗∗≤ λλλ for all .,, Xzyx ∈ Hence ))(~),((, xx µλµλ =Ω is a cubic
medial-ideal .
Conversely, suppose ))(~),((, xx µλµλ =Ω is cubic medial-ideal of X. For any Xzyx ∈,, we
have )(~)0(~ xµµ ≥ , )()0( xλλ ≤ ,
)}](()),((min{)},()),(([min{
)][(),([))],(()),((min{[
)}(~)),((~min{)(~
zyxyzzyxyz
zyzyxyzxyzr
zyxyzrx
UULL
ULUL
∗∗∗∗∗∗=
=∗∗∗∗∗∗=
=∗∗∗≥
µµµµ
µµµµ
µµµ
Therefore,
)}()),(((min{)( zyxyzx LLL ∗∗∗≥ µµµ and )}()),(((min{)( zyxyzx UUU ∗∗∗≥ µµµ ,
(MC3) )},(),((max{)( zyxyzx ∗∗∗≤ λλλ for all .,, Xzyx ∈

10. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
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Hence we get that )(xLµ and )(xUµ are fuzzy medial-ideal of X and )(xλ is anti fuzzy medial-
ideal of X. This completes the proof.
Proposition 4.11 Let Λ∈Μ ii )( be a family of cubic medial-ideals of a BCI-algebra, then
i
i
Μ
Λ∈
I is a cubic medial-ideals of a BCI-algebra X.
Proof. Let Λ∈Μ ii )( be a family of cubic medial-ideals of a BCI-algebras , then for any
Xzyx ∈,, ,we get ))(~())(~inf())0(~inf()0)(~( xxrr iiii ΜΜΜΜ =≥= µµµµ II ,
)}}})(~()),()(~min{()}}}(~inf{)),((~inf{min{
)}}(~)),((~min{inf{))(~inf())(~(
zyxyzrzyrxyzrr
zyxyzrrxrx
iiii
iiii
∗∗∗=∗∗∗
=∗∗∗≥=
ΜΜΜΜ
ΜΜΜΜ
µµµµ
µµµµ
II
I
and ))(())(sup())0(sup()0)(( xx iiii ΜΜΜΜ =≤= λλλλ UU ,
)}}}.)(()),()(max{()}}}(sup{)),((max{sup{
)}}()),((sup{max{{)sup())((
zyxyzzyxyz
zyxyzx
iiii
iiii
∗∗∗=∗∗∗
=∗∗∗≤=
ΜΜΜΜ
ΜΜΜΜ
λλλλ
λλλλ
UU
U
This completes the proof.
5.Image (Pre-image) bipolar fuzzy medial ideal under homomorphism
of BCI-algebras
Definition 5.1 Let )0,,( ∗X and )0,,( ′∗′Y be BCI-algebras. A mapping YXf →: is said to be
a homomorphism if )()()( yfxfyxf ∗′=∗ for all Xyx ∈, .
Definition 5.2. Let X and Y be two BCI-algebras, µ a fuzzy subset of X, β a fuzzy subset of Y
and YXf →: a BCI-homomorphism.
The image of µ under f denoted by )(µf is a fuzzy set of Y defined by



 ≠
=
−
∈ −
otherwise
yfifx
yf yfx
0
)()(sup
))((
1
)(1
φµ
µ
The pre-image of β under f denoted by )(1
β−
f is a fuzzy set of X defined by: For all
))(())((, 1
xfxfXx ββ =∈ −
.

11. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
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Let YXf →: be a homomorphism of BCI-algebras for any cubic medial-ideals
))(~),((, xx µλµλ =Ω in Y, we define new cubic medial-ideals ))(~),((,, xx fff µλµλ =Ω in X by
))((:)( xfxf λλ = , and ))((~:)(~ xfxf µµ = for all Xx ∈ .
Theorem 5.3 Let YXf →: be a homomorphism of BCI-algebras. If ))(~),((, xx µλµλ =Ω is
cubic medial ideal of Y, then ))(~),((,, xx fff µλµλ =Ω is cubic medial ideal of X.
Proof. )0())0(()0())((:)( ff fxfx λλλλλ ==≥= , and
)0(~))0((~)0(~))((~:)(~
ff fxfx µµµµµ ==≤= , for all Xyx ∈, . And
))}()((~))),()(()((~min{))((~:)(~ zfyfxfyfzfrxfxf ∗∗∗≥= µµµµ
))}((~)),(((~min{))}((~)),()((~min{ zyfxyzfrzyfxyfzfr ∗∗∗=∗∗∗= µµµµ
)}(~)),((~min{ zyxyzr ff ∗∗∗= µµ , and
))}(()),()(()((max{))((:)( zyfxfyfzfxfxf ∗∗∗≤= λλλλ
))}(()),(((max{))}(()),()((max{ zyfxyzfzyfxyfzf ∗∗∗=∗∗∗= λλλλ
)}()),((max{ zyxyz ff ∗∗∗= λλ .
Hence ))(~),((,, xx fff µλµλ =Ω is cubic medial ideal of X.
Theorem5.4 Let YXf →: be an epimorphism of BCI-algebras and let
))(~),((, xx µλµλ =Ω be an cubic set in Y. If ))(~),((,, xx fff µλµλ =Ω is cubic medial ideal
of X , then ))(~),((, xx µλµλ =Ω is cubic medial ideal of Y.
Proof. For any Ya ∈ , there exists Xx ∈ such that axf =)( .Then
),0(~))0((~)0(~)(~))((~)(~ µµµµµµ ==≤== fxxfa ff
).0())0(()0()())(()( λλλλλλ ==≥== fxxfa ff
Let Ycba ∈,, . Then czfbyfaxf === )(,)(,)( , for some Xzyx ∈,, . It follows that
)}(~)),((~min{)(~))((~)(~ zyxyzrxxfa fff ∗∗∗≥== µµµµµ
= ))}((~)),(((~min{ zyfxyzfr ∗∗∗ µµ
= ))}()((~)),()((~min{ zfyfxyfzfr ∗∗∗ µµ
))}()((~))),()(()((~min{ zfyfxfyfzfr ∗∗∗= µµ

12. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
26
)}(~)),((~min{ cbabcr ∗∗∗= µµ ,and
)}()),((max{)())(()( zyxyzxxfa fff ∗∗∗≤== λλλλλ
= ))}(()),(((max{ zyfxyzf ∗∗∗ λλ
= ))}()(()),()((max{ zfyfxyfzf ∗∗∗ λλ
))}()(())),()(()((max{ zfyfxfyfzf ∗∗∗= λλ
)}()),((max{ cbabc ∗∗∗= λλ . This completes the proof.
6.PRODUCT OF CUBIC MEDIAL IDEALS
Definition 6.1 Let )~,,( AAXA µµ= and )
~
,,( BBXB λλ= are two cubic set of X, the
Cartesian product )
~~,,( BABAXXBA λµλµ ×××=× is defined by
)}(
~
),(~min{),( yxryx BABA λµλµ =× and
)}(),(max{),( yxyx BABA λµλµ =× , where ]1,0[:
~~ DXXBA →×× λµ ,
]1,0[: →×× XXBA λµ for all Xyx ∈, .
Remark 6.2 Let X and Y be medial BCI-algebras, we define* on X ×Y by: For every
YXvuyx ×∈),(),,( , ),(),(),( vyuxvuyx ∗∗=∗ . Clearly ))0,0(,;( ∗×YX is BCI-algebra.
Proposition 6.3 Let )~,,( AAXA µµ= and )
~
,,( BBXB λλ= are two cubic medial ideal of X,
then BA× is cubic medial ideal of XX × .
Proof. ),(
~~)}}(
~
),(~min{)}0(),0(~min{)0,0(
~~ yxyxrr BABABABA λµλµλµλµ ×=≥=× , for all
Xyx ∈, . And ),()}}(),(max{)}0(),0(max{)0,0( yxyx BABABABA λµλµλµλµ ×=≤=× ,
for all Xyx ∈, .
Now let XXzzyyxx ×∈),(),,(),,( 212121 , then
))},(),)((
~~())),,(),((),)((
~~min{( 2121212121 xxyyxxyyzzr BABA ∗×∗∗× λµλµ
= )},)(
~~()),,(),)((
~~min{( 2211221121 xyxyxyxyzzr BABA ∗∗×∗∗∗× λµλµ
= )},)(
~~()),(),()(
~~min{( 2211222111 xyxyxyzxyz BABA ∗∗×∗∗∗∗× λµλµ
= )}}(
~
),(~min{))},((
~
)),((~min{min{ 2211222111 xyxyrxyzxyzrr BABA ∗∗∗∗∗∗ λµλµ

13. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
27
= )}(
~
)),((
~
min{)},(~)),((~min{min{ 2222211111 xyxyzrxyxyzrr BBAA ∗∗∗∗∗∗ λλµµ
= )}(
~
)),((
~
min{)},(~)),((~min{min{ 2222211111 xyxyzrxyxyzrr BBAA ∗∗∗∗∗∗ λλµµ
),)(
~~()(
~
),(~min{ 2121 xxxxr BABA λµλµ ×=≤ .
and
))},(),)((())),,(),((),)((max{( 2121212121 xxyyxxyyzz BABA ∗×∗∗× λµλµ
= )},)(()),,(),)((max{( 2211221121 xyxyxyxyzz BABA ∗∗×∗∗∗× λµλµ
= )},)(()),(),()(max{( 2211222111 xyxyxyzxyz BABA ∗∗×∗∗∗∗× λµλµ
= )}}(),(max{))},(()),((max{max{ 2211222111 xyxyxyzxyz BABA ∗∗∗∗∗∗ λµλµ
= )}()),((max{)},()),((max{max{ 2222211111 xyxyzxyxyz BBAA ∗∗∗∗∗∗ λλµµ
)}()),(({max)},()),(({{maxmax 2222211111 xyxyzxyxyz BBAA ∗∗∗∗∗∗= λλµµ
)}(),(max{ 21 xx BA λµ≥ = ),)(( 21 xxBA λµ × . This completes the proof.
Definition 6.4 Let )~,,( AAXA µµ= and )
~
,,( BBXB λλ= are two cubic medial ideals of BCI-
algebra X. for ]1,0[, ∈ts the set
}~),)(
~~(|),{(:)~,
~~(
~
tyxXXyxtU BABA ≥××∈=× λµλµ
is called upper t
~
-level of ),)(
~~( yxBA λµ × , where ]1,0(~ Dt ∈ and the set
}),)((|),{(:),( syxXXyxsL BABA ≤××∈=× λµλµ
is called lower s-level of ),)(( yxBA λµ × ,where ]1,0[∈s
Theorem 6.5 The sets )~,,( A
N
AXA µµ= and )
~
,,( B
N
BXB λλ= are cubic medial ideals of BCI-
algebra X if and only if the non-empty set upper t
~
-level cut )~,
~~(
~
tU BA λµ × and the non-empty
lower s-level cut ),( sL BA λµ × are medial ideals of XX × for any ]1,0(~],1,0[ Dts ∈∈ .
Proof. Let )~,,( AAXA µµ= and )
~
,,( BBXB λλ= are two cubic medial ideals of BCI-algebra X,
therefore for any ∈),( yx XX × ,
),)(
~~()}(
~
),(~min{)}0(
~
),0(~min{)0,0)(
~~( yxyxrr BABABABA λµλµλµλµ ×=≥=× and for
]1,0(~ Dt ∈ , if txxBA
~),)(
~~( 21 ≥× λµ , therefore ∈),( 21 xx )~,
~~(
~
tU BA λµ × .

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Let XXzzyyxx ×∈),(),,(),,( 212121 be such that
∈∗∗ ))),(),((),(( 212121 xxyyzz )~,
~~(
~
tU BA λµ × , and ∈∗ ),(),( 2121 zzyy )~,
~~(
~
tU BA λµ × .
Now
≥× ),)(
~~( 21 xxBA λµ
))},(),()(
~~())),,(),((),)((
~~min{( 2121212121 zzyyxxyyzzr BABA ∗×∗∗× λµλµ
= )},)(
~~()),,(),)((
~~min{( 2211221121 zyzyxyxyzzr BABA ∗∗×∗∗∗× λµλµ
= )},)(
~~()),(z),()(
~~min{( 2211222111 zyzyxyxyzr BABA ∗∗×∗∗∗∗× λµλµ
tttr
~
}
~
,
~
min{ =≥ ,
Therefore ∈),( 21 xx )~,
~~(
~
tU BA λµ × is a medial ideal of XX × . Similar, we can prove ,
)),,)((( syxL BA λµ × is a medial ideal of XX × .This completes the proof.
7.CONCLUSION
We have studied the cubic of medial-ideal in BCI-algebras. Also we discussed few results of
cubic of medial-ideal in BCI-algebras. The image and the pre- image of cubic of medial-ideal in
BCI-algebras under homomorphism are defined and how the image and the pre-image of cubic of
medial-ideal in BCI-algebras become cubic of medial-ideal are studied. Moreover, the product of
cubic of medial-ideal is established. Furthermore, we construct some algorithms applied to
medial-ideal in BCI-algebras. The main purpose of our future work is to investigate the cubic
foldedness of medial-ideal in BCI-algebras.
Acknowledgment
The authors are greatly appreciate the referees for their valuable comments and suggestions for
improving the paper.
Algorithm for BC I-algebras
Input ( :X set, :∗ binary operation)
Output (“ X is a BCI -algebra or not”)
Begin
If φ=X then go to (1.);
End If
If X∉0 then go to (1.);
End If

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Stop: =false;
1:=i ;
While Xi ≤ and not (Stop) do
If 0≠∗ ii xx then
Stop: = true;
End If
1:=j
While Xj ≤ and not (Stop) do
If ,0))(( ≠∗∗∗ jjii yyxx then
Stop: = true;
End If
End If
1:=k
While Xk ≤ and not (Stop) do
If ,0)())()(( ≠∗∗∗∗∗ ikkiji yzzxyx then
Stop: = true;
End If
End While
End While
End While
If Stop then
(1.) Output (“ X is not a BCI-algebra”)
Else
Output (“ X is a BCI -algebra”)
End If.
Algorithm for fuzzy subsets
Input ( :X BCI-algebra, ]1,0[: →Xµ );
Output (“ A is a fuzzy subset of X or not”)
Begin

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Stop: =false;
1:=i ;
While Xi ≤ and not (Stop) do
If ( 0)( <ixµ ) or ( 1)( >ixµ ) then
Stop: = true;
End If
End While
If Stop then
Output (“ µ is a fuzzy subset of X ”)
Else
Output (“ µ is not a fuzzy subset of X ”)
End If
End.
Algorithm for medial -ideals
Input ( :X BCI-algebra, :I subset of X );
Output (“ I is an medial -ideals of X or not”);
Begin
If φ=I then go to (1.);
End If
If I∉0 then go to (1.);
End If
Stop: =false;
1:=i ;
While Xi ≤ and not (Stop) do
1:=j
While Xj ≤ and not (Stop) do
1:=k
While Xk ≤ and not (Stop) do
If Ixyz ijk ∈∗∗ )( and Izy kj ∈∗ then

17. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
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If Ixi ∉ then
Stop: = true;
End If
End If
End While
End While
End While
If Stop then
Output (“ I is is an medial -ideals of X ”)
Else
(1.) Output (“ I is not is an medial -ideals of X ”)
End If
End .
Algorithm for cubic medial ideal of X
Input ( :X BCI-algebra, :∗ binary operation,λ anti fuzzy subsets and µ~ interval value of X );
Output (“ )~,,( µλx=Β is cubic medial ideal of X or not”)
Begin
Stop: =false;
1:=i ;
While Xi ≤ and not (Stop) do
If )(~)0(~ xµµ > and )()0( xλλ < then
Stop: = true;
End If
1:=j
While Xj ≤ and not (Stop) do
1:=k
While Xk ≤ and not (Stop) do
If )}(),(max{)( yyxx λλλ ∗> , )}(~),(~min{)(~ xyxrx µµµ ∗< then
Stop: = true;

18. International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015
32
End If
End While
End While
End While
If Stop then
Output (“ )~,,( µλx=Β is not bipolar cubic medial ideal of X ”)
Else
Output(“ )~,,( µλx=Β is bipolar cubic medial ideal of X ”)
End If.
End.
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