CUBIC STRUCTURES OF MEDIAL IDEAL ON BCI -ALGEBRAS by IJFLS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

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 / BCIalgebras 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 BCIalgebras, 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 â&#x20AC;&#x201C;ideals under homomorphism of BCI-algebras. DOI : 10.5121/ijfls.2015.5302

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

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 ( X ,∗,0) of type (2, 0) is called a BCI-algebra if it satisfying the following conditions: (BCI-1) (( x ∗ y ) ∗ ( x ∗ z )) ∗ ( z ∗ y ) = 0, (BCI-2) ( x ∗ ( x ∗ y )) ∗ y = 0, (BCI-3) x ∗ x = 0, (BCI-4) x ∗ y = 0 and y ∗ x = 0 imply x = y .

For all x, y and z ∈ X . In a BCI-algebra X, we can define a partial ordering” ≤ ” by x ≤ y if and only if x ∗ y = 0 . In what follows, X will denote a BCI-algebra unless otherwise specified. Definition 2.2( see [11]) A BCI-algebra ( X ,∗,0) of type (2, 0) is called a medial BCI-algebra if it satisfying the following condition: ( x ∗ y ) ∗ ( z ∗ u ) = ( x ∗ z ) ∗ ( y ∗ u ) ,for all

x, y, z and u ∈ X . 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) x ∗ ( y ∗ z ) = z ∗ ( y ∗ x) (ii) x ∗ 0 = x (iii) x ∗ x = 0 Lemma 2.4( see [11]) In a medial BCI-algebra X, the following holds:

x ∗ ( x ∗ y ) = y , for all x, y ∈ X . Lemma 2.5 Let X be a medial BCI-algebra, then 0 ∗ ( y ∗ x ) = x ∗ y , for all x, y ∈ X . Proof. Clear.

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Definition 2.6 A non empty subset S of a medial BCI-algebra X is said to be medial sub-

algebra of X, if x ∗ y ∈ S , for all x, y ∈ S . 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) x ∗ y ∈ I and y ∈ I implies x ∈ I for all x, y ∈ X . 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) z ∗ ( y ∗ x) ∈ M and y ∗ z ∈ M imply x ∈ M for all x, y and z ∈ X . 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) µ ( x ∗ y ) ≥ min{µ ( x), µ ( y )}, for all x, y ∈ X . 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) µ (0) ≥ µ ( x), (FI2) µ ( x) ≥ min{µ ( x ∗ y ), µ ( y )}, for all x, y and z ∈ X . 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) µ (0) ≥ µ ( x), (FM2) µ ( x) ≥ min{µ ( z ∗ ( y ∗ x)), µ ( y ∗ z )}, for all x, y and z ∈ X .

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

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 a~ = [ a L , aU ] , where 0 ≤ a L ≤ aU ≤ 1 .Let D[0, 1] denote the family of all closed subintervals of [0, 1], i.e.,

D[0,1] = {a~ = [a L , aU ] : a L ≤ aU for a L , aU ∈ I }. We define the operations ≤ , ≥ , = ,rmin and rmax in case of two elements in D[0, 1]. We

~ b = [bL , bU ] in D[0, 1].

consider two elements a~ = [ a L , aU ] and

Then

~ 1- a~ ≤ b iff a L ≤ bL , aU ≤ bU ; ~ 2- a~ ≥ b iff a L ≥ bL , aU ≥ bU ; ~ 3- a~ = b iff a L = bL , aU = bU ; ~ 4- rmim a~, b = [ min{a L , bL }, min{aU , bU } ] ;

{ } ~ 5- r max{a~, b } = [ max{a , b }, max{a L

, bU } ]

~ ~ Here we consider that 0 = [0,0] as least element and 1 = [1,1] as greatest element. Let a~i ∈ D[0,1] ,where i ∈ Λ .We define

    ~ ~     = inf( a ) , inf( a ) and = sup( a ) , sup( a ) r inf ai r sup ai i L i U i L i U  i∈Λ   i∈Λ  i∈Λ i∈Λ i∈Λ i∈Λ     An interval valued fuzzy set (briefly, i-v-f-set) µ~ on X is defined as

µ~ = { x , [µ L ( x), µU ( x)], x ∈ X

} , where µ~ : X → D[0,1] and µ

( x) ≤ µU ( x) ,for all

x ∈ X . Then the ordinary fuzzy sets µ L : X → [0,1] and µU : X → [0,1] are called a

lower fuzzy set and an upper fuzzy set of µ~ respectively.

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

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.

~ ( x)) in X is called cubic Definition 4.1[5 ] Let X be a BCI-algebra. A set Ω λ , µ = (λ ( x ), µ sub -algebra of X if it satisfies the following conditions:

(SC1) µ~ ( x ∗ y ) ≥ r min{µ~ ( x), µ~ ( y )} , (SC2) λ ( x ∗ y ) ≤ max{λ ( x), λ ( y )} ,for all x, y ∈ X . Example 4.2 Let X = {0,1,2,3} 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 ( X ,∗,0) is a BCI-algebra. Define µ~ ( x) as follows:

 [0.3,0.9] µ~ ( x) =   [0.1,0.6] X

λ (x)

if x = {0,1} otherwise 0 0.2

1 0.2

2 0.6

3 0.7

It is easy to check that Ω λ , µ = (λ , µ~ ) is cubic sub BCI –algebra.

Lemma 4.3 If Ω λ , µ = (λ ( x), µ~ ( x)) is is cubic BCI-sub-algebra of X, then µ~ (0) ≥ µ~ ( x) , λ (0) ≤ λ ( x) for all x ∈ X . Proof. For every x ∈ X , we have µ~ (0) = µ~ ( x ∗ x) ≥ r min{µ~ ( x), µ~ ( x)} = µ~ ( x) . And

λ ( x ∗ x) = λ (0) ≤ max{λ ( x), λ ( x)} = λ ( x) .This completes the proof.

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015 ~ Definition 4.4 Let X be a BCI-algebra. A cubic set Ω λ , µ = (λ ( x), µ~ ( x)) is called µ cubic

medial-ideal of X if it satisfies the following conditions:

(MC1) µ~ (0) ≥ µ~ ( x) , λ (0) ≤ λ ( x) (MC2) µ~ ( x) ≥ r min{µ~ ( z ∗ ( y ∗ x)), µ~ ( y ∗ z )} , for all x, y, z ∈ X . (MC3) λ ( x) ≤ max{λ ( z ∗ ( y ∗ x), λ ( y ∗ z )}, for all x, y, z ∈ X .

Example 4.5 Let X = {0, 1, 2, 3} as in example (4.2). Define µ~ ( x) as follows:

 [0.2,0.8] µ~ ( x) =   [0.1,0.5]

if x = {0,1} otherwise

λ (x)

0.1

0.2

0.3

0.4

~ ( x)) is cubic medial-idea of X. It is easy to check that Ω λ , µ = (λ ( x ), µ ~ ( x)) be cubic medial-ideal of X. If x ≤ y in X, Lemma 4.6 Let Ω λ , µ = (λ ( x ), µ ~ ( y ) ≥ µ~ ( x) and λ ( x) ≤ λ ( y ) , for all x, y ∈ X . then µ Proof. Let x, y ∈ X be such that x ≤ y , then x ∗ y = 0 . From (MC2), we have

µ~ ( x) ≥ r min{µ~ ( z ∗ ( y ∗ x)), µ~ ( y ∗ z )} = r min{µ~ (0 ∗ ( y ∗ x)), µ~ ( y ∗ 0)} = r min{µ~ ( x ∗ y )), µ~ ( y )} = = r min{µ~ (0)), µ~ ( y )} = µ~ ( y ) Similarly, form (M C3), we have λ ( x) ≤ max{λ (0 ∗ ( y ∗ x)), λ ( y ∗ 0)}, hence,

λ ( x) ≤ max{λ ( x ∗ y ), λ ( y )} = max{λ (0), λ ( y )} = λ ( y ) .This completes the proof Lemma 4.7 Let Ω λ , µ = (λ ( x ), µ~ ( x )) be cubic medial-ideal of X, if the

inequality x ∗ y ≤ z hold in X, then µ~ ( x) ≥ r min{µ~ ( z ), µ~ ( y )} , λ ( x) ≤ max{λ ( y ), λ ( z )} , for all x, y, z ∈ X .

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Proof. Let x, y, z ∈ X be such that x ∗ y ≤ z . Thus, put z = 0 in ( (MC2) Definition 3.4) and using lemma2.5 , lemma 4.6), we get, µ~ ( x) ≥ r min{µ~ ( z ∗ ( y ∗ x)), µ~ ( y ∗ z )} = r min{µ~ (0 ∗ ( y ∗ x)), µ~ ( y ∗ 0)} ~

ce µ ( x* y ) ≥ µ ( z ) 6sin4 47448 ~ ~ ≥ r min{µ ( x ∗ y ), µ ( y )} ≥ r r min{µ~ ( z ), µ~ ( y )} . Similarly we can prove that,

λ ( x) ≤ max{λ ( z ), λ ( y )} . This completes the proof

Theorem 4.8

Every cubic medial-ideal of X is a cubic sub-algebra of X.

Proof. Let Ω λ , µ = (λ ( x ), µ~ ( x )) be cubic medial ideal of X. Since x ∗ y ≤ x , for all x, y ∈ X , Lemma ( 4.6 ) 64 4744 8 ~ then µ ( x ∗ y ) ≥ µ~ ( x) , λ ( x ∗ y ) ≤ λ ( x) .Put z = 0 in (MC2), (MC3), we have µ~ ( x ∗ y ) ≥ µ~ ( x) ≥ r min{µ~ (0 ∗ ( y ∗ x)), µ~ ( y ∗ 0)} = r min{µ~ (0)), µ~ ( y )} = µ~ ( y ) . ~ ( x ∗ y ) ≥ r min{µ~ ( x), µ~ ( y )} -----------(a) Therefore µ

and

λ ( x ∗ y) ≤ λ ( x) ≤ max{λ (0 ∗ ( y ∗ x)), λ ( y ∗ 0)}= max{λ ( x ∗ y ), λ ( y )} ≤ max{λ ( x), λ ( y} ….. (b) From (a) and (b) , we get Ω λ , µ = (λ ( x ), µ~ ( x )) is cubic sub-algebra of X. This completes the proof.

Theorem 4.9

Let Ω

λ ,µ

= ( λ ( x ), µ~ ( x )) be a cubic subalgebra of X ,such that

µ~( x) ≥ r min{µ~( y ), µ~( z )}, λ ( x) ≤ max{λ ( y ), λ ( z )} , satisfying the inequality x ∗ y ≤ z for = ( λ ( x ), µ~ ( x )) is a cubic medial ideal of X. all x, y, z ∈ X . Then Ω λ ,µ

Proof. Let Ω λ , µ = (λ ( x ), µ~ ( x )) be a cubic subalgebra of X. Recall that

λ (0) ≤ λ ( x) and µ~ (0) ≥ µ~ ( x) , for all x ∈ X . Since, for all x, y, z ∈ X , we have x ∗ ( z ∗ ( y ∗ x)) = ( y ∗ x) ∗ ( z ∗ x) ≤ y ∗ z , it follows from Lemma 4.7 that , λ ( x) ≤ max{λ ( z ∗ ( y ∗ x), λ ( y * z )} , µ~ ( x) ≥ r min{µ~ ( z ∗ ( y ∗ x), µ~( y * z )} . Hence Ω λ , µ = (λ ( x ), µ~ ( x )) is a cubic medial ideal of X.

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Theorem 4.10 Let Ω Ω

λ ,µ

= ( λ ( x ), µ~ ( x )) be a cubic set. Then

= ( λ ( x ), µ~ ( x )) is bipolar cubic medial-ideal of X if and only if the non empty

~ ~ ( x) ≥ [δ , δ ], λ ( x) ≤ γ } is a medial-ideal of X , for every set U (Ω λ , µ~ ; [δ 1 , δ 2 ], γ ) := {x ∈ X | µ 1 2

[δ 1 , δ 2 ] ∈ D[0,1] , γ ∈ [−1,0] . Proof . Assume that Ω

λ ,µ

= ( λ ( x ), µ~ ( x )) is cubic medial-ideal of X. Let

~ [δ 1 , δ 2 ] ∈ D[0,1] , γ ∈ [0,1] be such that z ∗ ( y ∗ x ) , y ∗ z U (Ω λ , µ~ ; [δ1 , δ 2 ], γ ) , then µ~ ( x) ≥ r min{µ~( z ∗ ( y ∗ x )), µ~( y ∗ z )} ≥ r min{[δ 1 , δ 2 ], [δ 1 , δ 2 ]} = [δ 1 , δ 2 ] and ~ λ ( x) ≤ max{λ ( z ∗ ( y ∗ x), λ ( y ∗ z )}, = max{γ , γ } = γ , so x ∈U (Ω λ , µ~ ; [δ 1 , δ 2 ], γ ) . Thus

~ U (Ω λ , µ~ ; [δ1 , δ 2 ], γ ) 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 U (Ω λ , µ~ ; [δ1 , δ 2 ], γ ) (≠ φ ) is a medial-ideal of X . For every

[δ 1 , δ 2 ] ∈ D[0,1] , γ ∈ [0,1] suppose that there exist x0 , y 0 , z 0 ∈ X such that µ~ ( x 0 ) < r min{µ~ ( z 0 ∗ ( y 0 ∗ x 0 ), µ~( y 0 ∗ z 0 )} and λ ( x 0 ) > max{λ ( z 0 ∗ ( y 0 ∗ x 0 ), λ ( y 0 ∗ z 0 )} First, let µ~ ( z 0 ∗ ( y 0 ∗ x 0 )) = [γ 1 , γ 2 ] , µ~ ( y 0 ∗ z 0 ) = [γ 3 , γ 4 ] and µ~ ( x 0 ) = [δ 1 , δ 2 ] . Then

[δ 1 , δ 2 ] < r min{[γ 1 , γ 2 ], [γ 3 , γ 4 ]} . Taking [λ1 , λ 2 ] = [ 12 {µ~ ( x 0 ) + r min{µ~ (( z 0 ∗ ( y 0 ∗ x 0 )), µ~ ( y 0 ∗ z 0 )}]} = 1 2

([δ 1 , δ 2 ] + {min{γ 1 , γ 2 }, min{γ 3 , γ 4 }}) = 12 [(δ 1 + min{γ 1 , γ 3 }), (δ 2 + min{γ 2 , γ 4 })] .

It follows that min{γ 1 , γ 3 } > λ1 =

min{γ 2 , γ 4 } > λ 2 =

1 2

(δ 1 + min{γ 1 , γ 3 }) > δ 1 and

(δ 2 + min{γ 2 , γ 4 }) > δ 2 ,so that

~ [min{γ 1 , γ 3 }), min{γ 2 , γ 4 }] > [λ1 , λ 2 ] > [δ 1 , δ 2 ] = µ~ ( x0 ) .Therefore x0 ∉U (Ω λ , µ~ ;[δ1 , δ 2 ], γ ) . If (MC3) is false , let x0 , y 0 , z 0 ∈ X be such that λ ( x0 ) > max{λ ( z0 ∗ ( y0 ∗ x0 ), λ ( y0 ∗ z0 )} ,taking β0 = 1/2 { λ (x0) + min { λ (z0 * (y0 * x0)) , λ (y0*x0) } , we have β 0 ∈ [0,1] and

λ ( x0 ) > β 0 > max{λ ( z0 ∗ ( y0 ∗ x0 ), λ ( y0 ∗ z0 )} ,it follows that x0 * ( y 0 * z 0 ), ( y 0 * z 0 ) ~ ~ ∈ U (Ω λ , µ~ ; [δ1 , δ 2 ], γ ) and x0 ∉U (Ω λ , µ~ ;[δ1 , δ 2 ], γ ) , this is a contradiction and therefore λ is anti 22

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

fuzzy medial - ideal .On the other hand

µ~ ( z0 ∗ ( y0 ∗ x0 )) = [γ 1 , γ 2 ] ≥ [min{γ 1 , γ 3 }, min{γ 2 , γ 4 }] > [λ1 , λ 2 ] , µ~( y0 ∗ z0 ) = [γ 3 , γ 4 ] ≥ [min{γ 1 , γ 3 }, min{γ 2 , γ 4 }] > [λ1 , λ 2 ] , and so ~ ~ z0 ∗ ( y0 ∗ x0 ) , y0 ∗ z0 ∉ U (Ω λ , µ~ ; [δ1 , δ 2 ], γ ) . It contradicts that U (Ω λ , µ~ ; [δ1 , δ 2 ], γ ) is a medial-ideal of X. Hence µ~ ( x ) ≥ r min{µ~ ( z ∗ ( y ∗ x )), µ~ ( y ∗ z )} , and λ ( x) ≤ max{λ ( z * ( y ∗ x)), λ ( y * z )} for all x, y , z ∈ X . i.e Ω λ , µ = (λ ( x ), µ~ ( x )) is cubic medial-ideal of X. This completes the

proof

Theorem 4.10 A

cubic set Ω λ , µ = (λ ( x ), µ~ ( x )) is a cubic medial-ideal if and only if

µ~ ( x) is interval fuzzy medial-ideal and λ (x) is anti fuzzy medial-ideal of X . Proof. If µ~ ( x) is interval fuzzy medial-ideal and λ (x) is anti fuzzy medial-ideal of X. It is ~ (0) ≥ µ~ ( x) , clear that µ

λ (0) ≤ λ ( x) .Let µ~ ( x) interval fuzzy medial-ideal of X

and x, y, z ∈ X . Consider

µ~ ( x) = [ µ L ( x), µU ( x)] ≥ [min{µ L ( z ∗ ( y ∗ x)), µ L ( y ∗ z )}, min{µU ( z ∗ ( y ∗ x)), µU (( y ∗ z )}] = r min{[µ L ( z ∗ ( y ∗ x)), µU ( z ∗ ( y ∗ x))],[ µ L ( y ∗ z ), µU ( y ∗ z )]} = r min{µ~ ( z ∗ ( y ∗ x)), µ~ ( y ∗ z )}, If λ (x) is anti fuzzy medial-ideal of X, then

λ ( x) ≤ max{λ ( z ∗ ( y ∗ x), λ ( y ∗ z )}, for all x, y, z ∈ X . Hence Ω λ , µ = (λ ( x ), µ~ ( x )) is a cubic medial-ideal . Conversely, suppose Ω λ , µ = (λ ( x ), µ~ ( x )) is cubic medial-ideal of X. For any x, y, z ∈ X we

~ (0) ≥ µ~ ( x) , λ (0) ≤ λ ( x) , have µ

µ~ ( x) ≥ r min{µ~( z ∗ ( y ∗ x)), µ~ ( y ∗ z )} = = r min{[ µ L ( z ∗ ( y ∗ x)), µU ( z ∗ ( y ∗ x))],[ µ L ( y ∗ z ), µU ( y ∗ z )][= = [min{µ L ( z ∗ ( y ∗ x)), µ L ( y ∗ z )}, min{µU ( z ∗ ( y ∗ x)), µU (( y ∗ z )}] Therefore,

µ L ( x) ≥ min{µ L (( z ∗ ( y ∗ x)), µ L ( y ∗ z )} and µU ( x) ≥ min{µU (( z ∗ ( y ∗ x)), µU ( y ∗ z )} , (MC3) λ ( x) ≤ max{λ ( z ∗ ( y ∗ x), λ ( y ∗ z )}, for all x, y, z ∈ X . 23

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Hence we get that µ L (x) and µ U ( x ) are fuzzy medial-ideal of X and λ (x) is anti fuzzy medialideal of X. This completes the proof.

Proposition 4.11 Let (Μ i ) i∈Λ be a family of cubic medial-ideals of a BCI-algebra, then

I Μ i is a cubic medial-ideals of a BCI-algebra X.

i∈Λ

Proof. Let (Μ i ) i∈Λ be a family of cubic medial-ideals of a BCI-algebras , then for any x, y, z ∈ X ,we get (I µ~Μ i )(0) = r inf( µ~Μ i (0)) ≥ r inf( µ~Μ i ( x )) = (I µ~Μ i )( x ) ,

(I µ~Μ i )( x ) = r inf( µ~Μ i ( x )) ≥ r inf{r min{µ~Μ i ( z ∗ ( y ∗ x )), µ~Μ i ( y ∗ z )}} = r min{r inf{µ~Μ i ( z ∗ ( y ∗ x )), r inf{µ~Μ i ( y ∗ z )}}} = r min{(I µ~Μ i )( z ∗ ( y ∗ x )), (I µ~Μ i )( y ∗ z )}}} and (U λΜ i )(0) = sup( λΜ i (0)) ≤ sup( λΜ i ( x )) = ( U λΜ i )( x ) ,

(U λΜ i )( x) = sup(λΜ i ) ≤ sup{max{{λΜ i ( z ∗ ( y ∗ x)), λ Μ i ( y ∗ z )}} = max{sup{λΜ i ( z ∗ ( y ∗ x)), sup{λΜ i ( y ∗ z )}}} = max{(UλΜ i )( z ∗ ( y ∗ x)), (UλΜ i )( y ∗ z )}}}. This completes the proof.

5.Image (Pre-image) bipolar fuzzy medial ideal under homomorphism of BCI-algebras Definition 5.1 Let ( X ,∗,0) and (Y ,∗′,0′) be BCI-algebras. A mapping f : X → Y is said to be a homomorphism if f ( x ∗ y ) = f ( x) ∗′ f ( y ) for all x, y ∈ X . Definition 5.2. Let X and Y be two BCI-algebras, µ a fuzzy subset of X, β a fuzzy subset of Y and f : X → Y a BCI-homomorphism. The image of µ under f denoted by f ( µ ) is a fuzzy set of Y defined by

 sup µ ( x) if f −1 ( y ) ≠ φ f ( µ )( y ) =  x∈ f −1 ( y )  0 otherwise The pre-image of β under f denoted by f

x∈ X, f

−1

( β ) is a fuzzy set of X defined by: For all

( β )( x) = β ( f ( x)) .

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Let f : X → Y be a homomorphism of BCI-algebras for any cubic medial-ideals

Ω λ , µ = (λ ( x ), µ~ ( x )) in Y, we define new cubic medial-ideals Ω λ ,µ , f = (λ f ( x ), µ~ f ( x )) in X by λ ( x) := λ ( f ( x)) , and µ~ ( x) := µ~ ( f ( x)) for all x ∈ X . f

Theorem 5.3 Let f : X → Y be a homomorphism of BCI-algebras. If Ω λ , µ = (λ ( x ), µ~ ( x )) is cubic medial ideal of Y, then Ω = (λ ( x ), µ~ ( x )) is cubic medial ideal of X. λ ,µ , f

Proof. λ f ( x) := λ ( f ( x)) ≥ λ (0) = λ ( f (0)) = λ f (0) , and

µ~ f ( x) := µ~ ( f ( x)) ≤ µ~(0) = µ~ ( f (0)) = µ~ f (0) , for all x, y ∈ X . And µ~ f ( x) := µ~ ( f ( x)) ≥ r min{µ~ ( f ( z ) ∗ ( f ( y ) ∗ f ( x))), µ~( f ( y ) ∗ f ( z ))} = r min{µ~ ( f ( z ) ∗ f ( y ∗ x)), µ~ ( f ( y ∗ z ))} = r min{µ~ ( f ( z ∗ ( y ∗ x)), µ~ ( f ( y ∗ z ))}

= r min{µ~ f ( z ∗ ( y ∗ x)), µ~ f ( y ∗ z )} , and

λ f ( x) := λ ( f ( x)) ≤ max{λ ( f ( z ) ∗ ( f ( y ) ∗ f ( x)), λ ( f ( y ∗ z ))} = max{λ ( f ( z ) ∗ f ( y ∗ x)), λ ( f ( y ∗ z ))} = max{λ ( f ( z ∗ ( y ∗ x)), λ ( f ( y ∗ z ))} = max{λ f ( z ∗ ( y ∗ x)), λ f ( y ∗ z )} . Hence Ω λ ,µ , f = (λ f ( x ), µ~ f ( x )) is cubic medial ideal of X. Theorem5.4 Let f : X → Y be an epimorphism of BCI-algebras and let Ω λ , µ = (λ ( x ), µ~ ( x )) be an cubic set in Y. If Ω λ ,µ , f = (λ f ( x ), µ~ f ( x )) is cubic medial ideal of X , then Ω = (λ ( x), µ~ ( x )) is cubic medial ideal of Y. λ ,µ

Proof. For any a ∈ Y , there exists x ∈ X such that f ( x) = a .Then

µ~ (a) = µ~ ( f ( x)) = µ~ f ( x) ≤ µ~ f (0) = µ~ ( f (0)) = µ~ (0), λ (a) = λ ( f ( x)) = λ f ( x) ≥ λ f (0) = λ ( f (0)) = λ (0). Let a, b, c ∈ Y . Then f ( x) = a, f ( y ) = b, f ( z ) = c , for some x, y, z ∈ X . It follows that

µ~ (a) = µ~ ( f ( x)) = µ~ f ( x) ≥ r min{µ~ f ( z ∗ ( y ∗ x)), µ~ f ( y ∗ z )} = r min{µ~ ( f ( z ∗ ( y ∗ x)), µ~ ( f ( y ∗ z ))} = r min{µ~ ( f ( z ) ∗ f ( y ∗ x)), µ~ ( f ( y ) ∗ f ( z ))} = r min{µ~ ( f ( z ) ∗ ( f ( y ) ∗ f ( x))), µ~ ( f ( y ) ∗ f ( z ))} 25

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= r min{µ~ (c ∗ (b ∗ a )), µ~ (b ∗ c)} ,and

λ (a) = λ ( f ( x)) = λ f ( x) ≤ max{λ f ( z ∗ ( y ∗ x)), λ f ( y ∗ z )} = max{λ ( f ( z ∗ ( y ∗ x)), λ ( f ( y ∗ z ))} = max{λ ( f ( z ) ∗ f ( y ∗ x)), λ ( f ( y ) ∗ f ( z ))} = max{λ ( f ( z ) ∗ ( f ( y ) ∗ f ( x))), λ ( f ( y ) ∗ f ( z ))} = max{λ (c ∗ (b ∗ a )), λ (b ∗ c)} . This completes the proof.

6.PRODUCT OF CUBIC MEDIAL IDEALS ~ Definition 6.1 Let A = ( X , µ A , µ~A ) and B = ( X , λB , λB ) are two cubic set of X, the ~ Cartesian product A × B = ( X × X , µ A × λB , µ~A × λB ) is defined by ~

µ A × λB ( x, y ) = r min{ µ~A ( x), λB ( y )} and ~ µ A × λB ( x, y ) = max{ µ A ( x), λB ( y )} , where µ~A × λB : X × X → D[0,1] ,

µ A × λB : X × X → [0,1] for all x, y ∈ X . Remark 6.2 Let X and Y be medial BCI-algebras, we define* on X ×Y by: For every ( x, y ), (u , v) ∈ X × Y , ( x, y ) ∗ (u , v) = ( x ∗ u , y ∗ v) . Clearly ( X ×Y ;∗, (0,0)) is BCI-algebra. ~ Proposition 6.3 Let A = ( X , µ A , µ~A ) and B = ( X , λB , λB ) are two cubic medial ideal of X, then A × B is cubic medial ideal of X × X . ~ ~ ~ Proof. µ~A × λB (0,0) = r min{ µ~A (0), λB (0)} ≥ r min{ µ~A ( x), λB ( y )}} = µ~A × λB ( x, y ) , for all x, y ∈ X . And µ A × λB (0,0) = max{ µ A (0), λB (0)} ≤ max{ µ A ( x), λB ( y )}} = µ A × λB ( x, y ) , for all x, y ∈ X . Now let ( x1 , x 2 ), ( y1 , y 2 ), ( z1 , z 2 ) ∈ X × X , then

~ ~ r min{( µ~A × λB )(( z1 , z2 ) ∗ (( y1 , y2 ) ∗ ( x1 , x2 ))), ( µ~A × λB )(( y1 , y2 ) ∗ ( x1 , x2 ))} ~ × λ~ )(( z , z ) ∗ ( y ∗ x , y ∗ x )), ( µ~ × λ~ )( y ∗ x , y ∗ x )} = r min{( µ A B 1 2 1 1 2 2 A B 1 1 2 2 ~ × λ~ )( z ∗ ( y ∗ x ), z ∗ ( y ∗ x )), ( µ~ × λ~ )( y ∗ x , y ∗ x )} = min{( µ A B 1 1 1 2 2 2 A B 1 1 2 2 ~ ( z ∗ ( y ∗ x )), λ~ ( z ∗ ( y ∗ x ))}, r min{µ~ ( y ∗ x ), λ~ ( y ∗ x )}} = r min{r min{µ A 1 1 1 B 2 2 2 A 1 1 B 2 2 26

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

~ ( z ∗ ( y ∗ x )), µ~ ( y ∗ x )}, r min{λ ( z ∗ ( y ∗ x )), λ ( y ∗ x )} = r min{r min{µ A 1 1 1 A 1 1 B 2 2 2 B 2 2

~ ( z ∗ ( y ∗ x )), µ~ ( y ∗ x )}, r min{λ ( z ∗ ( y ∗ x )),λ ( y ∗ x )} = r min{r min{µ A 1 1 1 A 1 1 B 2 2 2 B 2 2 ~ ~ ≤ r min{µ~A ( x1 ), λB ( x2 ) = ( µ~A × λB )( x1 , x2 ) . and

max{(µ A × λB )(( z1 , z2 ) ∗ (( y1 , y2 ) ∗ ( x1 , x2 ))), ( µ A × λB )(( y1 , y2 ) ∗ ( x1 , x2 ))} = max{(µ A × λB )(( z1 , z2 ) ∗ ( y1 ∗ x1 , y2 ∗ x2 )), ( µ A × λB )( y1 ∗ x1 , y2 ∗ x2 )} = max{(µ A × λB )( z1 ∗ ( y1 ∗ x1 ), z2 ∗ ( y2 ∗ x2 )), ( µ A × λB )( y1 ∗ x1 , y2 ∗ x2 )} = max{max{µ A ( z1 ∗ ( y1 ∗ x1 )), λB ( z2 ∗ ( y2 ∗ x2 ))}, max{µ A ( y1 ∗ x1 ), λB ( y2 ∗ x2 )}} = max{max{µ A ( z1 ∗ ( y1 ∗ x1 )), µ A ( y1 ∗ x1 )}, max{λB ( z2 ∗ ( y2 ∗ x2 )), λB ( y2 ∗ x2 )}

= max{max { µ A ( z1 ∗ ( y1 ∗ x1 )), µ A ( y1 ∗ x1 )}, max { λB ( z2 ∗ ( y2 ∗ x2 )),λB ( y2 ∗ x2 )} ≥ max{µ A ( x1 ), λB ( x2 )} = ( µ A × λB )( x1 , x2 ) . This completes the proof.

~ Definition 6.4 Let A = ( X , µ A , µ~A ) and B = ( X , λB , λB ) are two cubic medial ideals of BCIalgebra X. for s, t ∈ [0,1] the set

~ ~ ~ U ( µ~A × λB , ~t ) := {( x, y ) ∈ X × X | ( µ~A × λB )( x, y ) ≥ ~t } ~ ~ × λ~ )( x, y ) , where ~ is called upper t -level of ( µ t ∈ D(0,1] and the set A B

L(µ A × λB , s) := {(x, y) ∈ X × X | (µ A × λB )(x, y) ≤ s} is called lower s-level of ( µ A × λB )( x, y ) ,where s ∈ [0,1]

~ Theorem 6.5 The sets A = ( X , µ AN , µ~A ) and B = ( X , λNB , λB ) are cubic medial ideals of BCI~

~ ~

~ × λ , t ) and the non-empty algebra X if and only if the non-empty set upper t -level cut U ( µ A B

lower s-level cut L( µ A × λB , s ) are medial ideals of X × X for any s ∈ [0,1], t ∈ D (0,1] .

~ Proof. Let A = ( X , µ A , µ~A ) and B = ( X , λB , λB ) are two cubic medial ideals of BCI-algebra X, therefore for any ( x, y ) ∈ X × X ,

~ ~ ~ ~ ( µ~A × λB )(0,0) = r min{µ~A (0), λB (0)} ≥ r min{µ~A ( x), λB ( y )} = ( µ~A × λB )( x, y ) and for ~ ~ ~ ~ t ∈ D(0,1] , if ( µ~A × λB )( x1 , x2 ) ≥ ~t , therefore ( x1 , x2 ) ∈ U ( µ~A × λB , ~ t ). 27

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Let ( x1 , x 2 ), ( y1 , y 2 ), ( z1 , z 2 ) ∈ X × X be such that

~ ~ ~ ~ (( z1 , z 2 ) ∗ (( y1 , y 2 ) ∗ ( x1 , x2 ))) ∈ U ( µ~A × λB , ~ t ) , and ( y1 , y 2 ) ∗ ( z1 , z 2 ) ∈ U ( µ~A × λB , ~ t ). Now

~ ( µ~A × λB )( x1 , x2 ) ≥

~ ~ r min{( µ~A × λB )(( z1 , z2 ) ∗ (( y1 , y2 ) ∗ ( x1 , x2 ))), ( µ~A × λB )(( y1 , y2 ) ∗ ( z1 , z2 ))} ~ × λ~ )(( z , z ) ∗ ( y ∗ x , y ∗ x )), ( µ~ × λ~ )( y ∗ z , y ∗ z )} = r min{( µ A B 1 2 1 1 2 2 A B 1 1 2 2 ~ ~ ~ × λ )( z ∗ ( y ∗ x ), z ∗ ( y ∗ x )), ( µ~ × λ )( y ∗ z , y ∗ z )} = r min{( µ A B 1 1 1 2 2 2 A B 1 1 2 2 ~~ ~ ≥ r min{t , t } = t , ~ ~ Therefore ( x1 , x2 ) ∈ U ( µ~A × λB , ~ t ) is a medial ideal of X × X . Similar, we can prove , L(( µ A × λB )( x, y ), s ) is a medial ideal of X × X .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 X = φ then go to (1.); End If If 0 ∉ X then go to (1.); End If 28

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Stop: =false;

i := 1 ; While i ≤ X and not (Stop) do If xi ∗ xi ≠ 0 then Stop: = true; End If

j := 1 While j ≤ X and not (Stop) do If ( xi ∗ ( xi ∗ y j )) ∗ y j ≠ 0, then Stop: = true; End If End If

k := 1 While k ≤ X and not (Stop) do If (( xi ∗ y j ) ∗ ( xi ∗ z k )) ∗ ( z k ∗ yi ) ≠ 0, 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, µ : X → [0,1] ); Output (“ A is a fuzzy subset of X or not”) Begin 29

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

Stop: =false;

i := 1 ; While i ≤ X and not (Stop) do If ( µ ( xi ) < 0 ) or ( µ ( xi ) > 1 ) 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 I = φ then go to (1.); End If If 0 ∉ I then go to (1.); End If Stop: =false;

i := 1 ; While i ≤ X and not (Stop) do

j := 1 While j ≤ X and not (Stop) do

k := 1 While k ≤ X and not (Stop) do If zk ∗ ( y j ∗ xi ) ∈ I and y j ∗ z k ∈ I then 30

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.2/3, July 2015

If xi ∉ I 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 ~ interval value of X ); Input ( X : BCI-algebra, ∗ : binary operation, λ anti fuzzy subsets and µ ~ ) is cubic medial ideal of X or not”) Output (“ Β = ( x, λ , µ Begin Stop: =false;

i := 1 ; While i ≤ X and not (Stop) do

~ (0) > µ~ ( x) and λ (0) < λ ( x) then If µ Stop: = true; End If

j := 1 While j ≤ X and not (Stop) do

k := 1 While k ≤ X and not (Stop) do

~ ( x) < r min{µ~ ( x ∗ y ), µ~ ( x)} then If λ ( x) > max{λ ( x ∗ y ), λ ( y )} , µ Stop: = true; 31

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End If End While End While End While If Stop then

~ ) is not bipolar cubic medial ideal of X ”) Output (“ Β = ( x, λ , µ Else

~ ) is bipolar cubic medial ideal of X ”) Output(“ Β = ( x, λ , µ End If. End.

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