Generalized cubic relations in Hv -LA-semigroups

Journal of Discrete Mathematical Sciences and Cryptography

ISSN: 0972-0529 (Print) 2169-0065 (Online) Journal homepage: http://www.tandfonline.com/loi/tdmc20

Generalized cubic relations in Hv -LA-semigroups Xue-Ling Ma, Jianming Zhan, Madad Khan, Muhammad Gulistan & Naveed Yaqoob To cite this article: Xue-Ling Ma, Jianming Zhan, Madad Khan, Muhammad Gulistan & Naveed Yaqoob (2018) Generalized cubic relations in Hv -LA-semigroups, Journal of Discrete Mathematical Sciences and Cryptography, 21:3, 607-630, DOI: 10.1080/09720529.2016.1191174 To link to this article: https://doi.org/10.1080/09720529.2016.1191174

Published online: 23 May 2018.

Submit your article to this journal

Article views: 2

View related articles

View Crossmark data

Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tdmc20

Journal of Discrete Mathematical Sciences & Cryptography ISSN 0972-0529 (Print), ISSN 2169-0065 (Online) Vol. 21 (2018), No. 3, pp. 607–630 DOI : 10.1080/09720529.2016.1191174

Generalized cubic relations in Hv -LA-semigroups Xue-Ling Ma † Jianming Zhan * Department of Mathematics Hubei University for Nationalities Enshi Hubei Province 445000 China Madad Khan § Department of Mathematics COMSATS Institute of Information Technology Abbottabad Pakistan Muhammad Gulistan ‡ Department of Mathematics Hazara University Mansehra Pakistan Naveed Yaqoob ^ Department of Mathematics College of Science in Al-Zulfi Majmaah University Al-Zulfi Saudi Arabia

E-mail: *E-mail: § E-mail: ‡ E-mail: ^ E-mail: †

©

[email protected] [email protected] (Corresponding Author) [email protected] [email protected] [email protected]

608

X. L. MA, J. ZHAN, M. KHAN, M. GULISTAN AND N. YAQOOB

Abstract The main purpose of this paper is to explore the structural properties of Hv-LA-semigroups with respect to generalized cubic relations. Some generalized cubic equivalence relations in Hv-LA-semigroups are investigated. Furthermore, certain results on generalized cubic relations by using the images and pre-images of Hv-LA-semigroups are provided. Subject Classification: 06F35, 03G25, 08A72. Keywords: Hv-LA-semigroup, Cubic set, Generalized cubic equivalence relation, Image and preimage of the generalized cubic relation.

1. Introduction The classical theory of hyper structure was presented by Marty in 1934 [1]. In 1990, Vougiouklis [2] introduced the concept of Hv-structures. After that Spartalis [3] studied the idea of Hv-semigroups. Kazim and Naseeruddin [8], presented the idea of LA-semigroups. Hila et. al. [9], initiated the notion of LA-semihypergroups and further Yaqoob et. al. [10] characterized intra-regular left almost semihypergroups. Recently Gulistan et. al. [11] gave the idea of Hv-LA-semigroups by introducing weak left invertive law. Zadeh introduced the concept of fuzzy relations and fuzzy similarity relations which is the generalizations of crisp relations of a set [12, 13]. [14] Ali et. al. introduced the concept of (∈, ∈∨q) -fuzzy equivalence relations and indistinguishability operators. The fuzzification of hyperstructures was considered by many authors. For instance Fotea et. al. [15]. The fuzzification of Hv-structures was also considered by many mathematicians, see Davvaz et. al. [16, 17]. Jun et. al. [18] introduced the notion of cubic sets. Madad et. al. gave the idea of generalized version of jun’s cubic sets in semigroups recently in [19]. In the current paper, we study several structural properties of HvLA-semigroups with respect to generalized cubic relations. In particular, we investigate some generalized cubic equivalence relations in Hv-LAsemigroups. Moreover, certain results on generalized cubic relations by using the images and pre-images of Hv-LA-semigroups are given. 2. Hv-LA-semigroups Throughout this paper, unless otherwise mentioned, H will denote a Hv-LA-semigroup. For simplicity we use K = (k 1 , k 2 ), Γ = (γ 1 , γ 2 ) and ∆ = (δ 1 , δ ). 2

Recall first the basic definitions from the hyperstructure theory.

GENERALIZED CUBIC RELATIONS

609

Definition 2.1 A mapping  : S × S → P∗ (S) is called a hyperoperation or join operation on the set S, where S is a non-empty set and P∗ (S) = P(S)\{∅} denotes the set of all non-empty subsets of S. A hypergroupoid is a set S together with a (binary) hyperoperation. Definition 2.2 [(9, 10)] A hypergroupoid (S, ), which is left invertive (non-associative), that is (x  y )  z = ( z  y )  x , ∀x , y , z ∈ S, is called an LAsemihypergroup. Definition 2.3 ([11]) Let H be a non-empty set and * be a hyperoperation on H. Then, (H, *) is called an Hv-LA-semigroup if it satisfies the weak left invertive law i.e. for all x , y , z ∈ H , (x ∗ y ) ∗ z ∩ ( z ∗ y ) ∗ x ≠ ∅. Example 2.4 Consider H = {x, y, z} and define a hyperoperation * on H by the following table: ∗

x

y

z

x

x

{x, z}

H

y

{x, z}

x

x

z

{x, y}

z

{x, z}

Then (H, *) is an Hv-LA-semigroup. Definition 2.5 ([11]) Let H be an Hv-LA-semigroup and q an equivalence relation in H. Then we can extend this relation q to the non-empty subsets – A and B of H as follows: Aq B if and only if for all a ŒA there exist aq b such that and for all b ŒB and there exist a ŒA such that bq a. An equivalence – relation q is said to be regular if for all x, y, z ŒH, xq y ﬁ (x * z) q (y * z) and – (z * x) q (z * y). Jun et. al. [18] introduced the concept of cubic sets defined on a nonempty set X as objects having the form: Ξ =

{ x, µ (x), η (x) Ξ

Ξ

}

: x ∈X ,

which is briefly denoted by Ξ = µ Ξ , ηΞ , where the functions µ Ξ : X ÆD[0, 1] and ηΞ : X → 0,1]. Definition 2.6 Let Ξ = µ Ξ , ηΞ and Σ = µ Σ , ηΣ be two cubic subsets of H. Then for all x ŒH, (1) their intersection Ξ ∩ Σ is defined by Ξ ∩ Σ =

}

(x ) : x ∈ H , , where

{ x, µ

Ξ∩Σ

(x ), ηΞ∩Σ

610


µ Ξ∩Σ (x ) = ( µ Ξ ∧ µ Ξ ) (x ) = µ Ξ (x ) ∧ µ Ξ (x ) and

ηΞ∩Σ (x ) = (ηΞ ∨ ηΣ ) (x ) = ηΞ (x ) ∨ ηΣ (x ).

(2) their union Ξ ∪ Σ is defined by Ξ ∪ Σ = where

{ x, µ

Ξ∪Σ

}

(x ), ηΞ∪Σ (x ) : x ∈ H ,

µ Ξ∪Σ (x ) = ( µ Ξ ∨ µ Σ ) (x ) = µ Ξ (x ) ∨ µ Σ (x ) and

ηΞ∪Σ (x ) = (ηΞ ∧ ηΣ ) (x ) = ηΞ (x ) ∧ ηΣ (x ).

{

}

(3) their product Ξ  Σ is defined by Ξ  Σ = x , µ ΞΣ (x), ηΞΣ (x) : x ∈ H , where * rsup {* rmin {µ Ξ ( y ), µ Σ ( z)}} if x ∈ y  z x ∈y  z

µ ΞΣ (x) = 

[0, 0]

otherwise

and

 inf {max{ηΞ ( y ), ηΣ ( z)}} if x ∈ y  z

ηΞΣ (x) =  x∈yz 1

otherwise

Definition 2.7 Let H be an Hv-LA-subsemigroup. Then the cubic characteristic function χ Ξ = µ χΞ , ηχΞ

of Ξ = µ Ξ , ηΞ

χ Ξ = {〈 x , µ χ (x), ηχ (x)〉|x ∈ H }, where µ χ Ξ

Ξ

Ξ

and ηχ

is defined as are fuzzy sets

Ξ

respectively, defined as follows: 1,1] if x ∈Ξ and 0, 0] if x ∉Ξ

µ χ : H → D[0,1]|x → µ χ (x) :=  Ξ

Ξ

0 if x ∈Ξ .  1 if x ∉Ξ

ηχ : H → 0,1]|x → ηχ (x ) :=  Ξ

Ξ

Definition 2.8 Let H be an Hv-LA-subsemigroup. Then the (∈, ∈∨qK ) -cubic characteristic function χ Ξ = µ χ , ηχ Ξ

as χ Ξ = {〈 x , µ χ (x ), ηχ (x )〉|x ∈ H }, where µ χ Ξ

Ξ

respectively, defined as follows:

of Ξ = µ Ξ , ηΞ

Ξ

Ξ

is defined

and ηχ are fuzzy sets Ξ


611

 1 − k 1  if x ∈Ξ   µ χ : H → D [0,1]|x → µ χ (x) :=  2 and ηχ : H → 0,1]| Ξ Ξ Ξ  0, 0] if x ∉Ξ   1 − k2 if x ∈Ξ  . x → ηχ (x ) :=  2 Ξ  1 if x ∉Ξ 

Definition 2.9 Let H be an Hv-LA-subsemigroup. Then the (∈G , ∈G ∨qD ) -cubic characteristic function χ Ξ = µ χΞ , ηχΞ

χ Ξ = {〈 x , µ

χ∆

Ξ

G

(x ), η

χ∆

Ξ

G

of Ξ = µ Ξ , ηΞ is defined as

(x )〉|x ∈ H }, where µ χ ∆ Ξ and η ∆ are fuzzy sets χ Ξ G G

respectively, defined as follows: δ 1 = (1,1] if x ∈Ξ (x)    Ξ Ξ γ 1 = (0, 0] if x ∉Ξ δ = 0 if x ∈Ξ : H → 0,1]|x →|η ∆ (x ) ≤  2 , η∆ χ Ξ χ Ξ G G γ 2 = 1 if x ∉Ξ

µ

χ∆ G

: H → D[0,1]|x → µ

χ∆ G

where δ 1 , γ 1 ∈ D(0,1] such that γ 1  δ 1 and δ 2 , γ 2 ∈ 0,1) such that δ 2 < γ 2 . Definition 2.10 A cubic binary relation on H1 and H2 is the cubic set  : H × H → D(0,1] R = µ R , ηR of H1 × H2 such that µ and hR : H1 × H2 Æ 0, R 1 2 1). It is of the form R = µ R , ηR = {((x , y ); µ R (x , y ), ηR (x , y )|(x , y ) ∈ H1 × H 2 },  where the functions µ R : H1 × H 2 → D(0,1] and hR : H1 × H2 Æ 0, 1). By cubic binary relation on H is the cubic set R = µ R , ηR of H × H  such that µ R : H1 × H 2 → D(0,1] and hR : H1 × H2 Æ 0, 1). Definition 2.11 Let R is CR (H1 × H2) and S is CR (H2 × H3). Then the composition between R and S is CR (H1 × H3) defined as R  S = {(x , z), ∨ y∈Y {*rmin(µ R (x , y ), µ R ( y , z))}, ∧ y∈Y {max(ηR (x , y ), ηR ( y , z))}. Definition 2.12 A cubic relation R = µ R , ηR on a set H is (i) Reflexive if µ (x , x) = 1 and η (x , x) = 0 for x ŒH. (ii) Symmetric if µ (x , y) = 1 and R

R

R

612


ηR (x , y) = 0 implies that µ R ( y , x) = 1 and ηR ( y , x) = 0 for any x, y ŒH, or if µ R (x , y ) = µ R ( y , x ) and ηR (x , y ) = ηR ( y , x ), for all x, y ŒH. (iii) Transitive {*rinf( a ,b )∈( x , z ) µ R (a, b)}  *rmin {µ R (x , y ), µ R ( y , z)},

if (a)

{

}

(b)

{sup( a ,b )∈( x , z )ηR

(a, b)} ≤ max ηR (x , y ), ηR ( y , z) for all x, y, z, a, b ŒH. for all x, y, z, a, b ŒH. 3. Generalized Cubic Relations in Hv-LA-semigroups Fuzzy relations are the generalizations of crisp relations of a set. Zadeh [12], introduced the concept of fuzzy relations. [14] Ali et. al. introduced the concept of (∈, ∈∨q) -fuzzy equivalence relations and indistinguishability operators. Here we generalized the concept of [14] Ali et. al. and give the concept of generalized cubic relations in Hv-LAsemigroups. Definition 3.1 A cubic set Ξ = µ Ξ (x , y ) , ηΞ (x , y )t t 1

of H1 × H2 of the form

2

t ∈ D(0,1] if (x , y ) = ( a, b) and (0, 0] otherwise 

µ Ξ (x , y) t (a, b) =  1 1

t2 ∈(0,1] if (x , y) = ( a, b) otherwise. 1

ηΞ (x , y) t (a, b) =  2

~

is called a cubic ordered pair (COP) with support (x, y) and value (t1, t2)and it ~ is denoted by (x , y ) t ,t where t1 ŒD (0, 1] and t2 Œ0, 1) be such that 0  t1 and

( 1 2)

t2 < 1. For any cubic set Ξ = µ Ξ , ηΞ and for a cubic ordered pair (x , y ) t ,t , ( ) 1

2

( x, y )( t1 ,t2 ) ∈Ξ if µ Ξ (x , y)  t1 and (ii) ηΞ (x , y) ≤ t2 . (x , y)(t ,t ) q Ξ

we have (i)

1

2

if µ Ξ (x , y ) + t1  1 and ηΞ (x , y ) + t2 < 1. (iii) (x , y )  ∈∨q Ξ if (x , y )(t ,t ) ∈Ξ ( t ,t ) 1 2 1

2

or (x , y )(t ,t ) q Ξ. (iv) (x , y )(t ,t ) ∈∧q Ξ if (x , y )(t ,t ) ∈Ξ and (x , y )(t ,t ) q Ξ. 1

1

2

2

1

2

1

2

(v) (x , y )(t ,t ) qK Ξ if µ Ξ (x , y ) + t1 + k1  1 and ηΞ (x , y ) + t2 + k 2 < 1. (iv) 1

2

(x , y )(t ,t ) ∈G Ξ if µ Ξ (x , y )  t1  γ 1 and ηΞ (x , y ) ≤ t2 < γ 2 . (vii) (x , y )(t ,t ) q∆ Ξ 1

2

if µ Ξ (x , y ) + t1  2δ 1 and ηΞ (x , y ) + t2 < 2δ 2 .

1

2


613

Definition 3.2 A cubic relation R = µ R , ηR on H is called (Œ, Œ⁄q),(resp., (Œ, Œ⁄qK), (ŒG, ŒG⁄qD)-cubic reflexive relation on H if for x, y ŒH, (x , y ) 

(t1 ,t2 )

ŒR (resp., (x , y ) 

(t1 ,t2 )

∈ R,(x , y )(t ,t ) ∈G R) implies that (a, a)  1

(t1 ,t2 )

2

∈∨qR, (resp.,

(a, a)(t ,t ) ∈∨qK R, (a, a)(t ,t ) ∈G ∨q∆ R) for all (a, a) ∈(x , y ) where t1 ∈ D(0,1] 1

2

1

2

and t2 ∈ 0,1) be such that 0  t1 and t2 < 1. Remark 3.3 Every (Œ, Œ⁄q)-cubic reflexive relation is an (Œ, Œ⁄qK)-cubic reflexive relation and every (Œ, Œ⁄qK)-cubic reflexive relation is an (ŒG, ŒG⁄qK)-cubic reflexive relation, but not conversely as shown in the following example. Example 3.4 Consider the Hv-LA-semigroup defined in Example 2.4 and define the cubic relation R = µ R , ηR as ~

H

µR

hR

x

[0.5, 0.6)

0.3

y

[0.7, 0.8)

0.3

z

[0.9, 1)

0.3

then µ R (x , x ) = [0.5, 0.6), µ R ( y , y ) = [0.7, 0.8), µ R ( z , z) = [0.9,1), µ R (x , y ) = [0.5, 0.6) = µ R ( y , x ), µ R (x , z) = [0.5, 0.6) = µ R ( z , x ), µ R ( y , z) = [0.7, 0.8) = µ R ( z , y ) and ηR (x , x ) = 0.3, ηR ( y , y ) = 0.3, ηR ( z , z) = 0.3, ηR (x , y ) = 0.3 = ηR ( y , x ), ηR (x , z) = 0.3 = ηR ( z , x), ηR ( y , z) = 0.3 = ηR ( z , y). Let us define ~

t1 = [0.5, 0.56]

t2 = 0.3

d 1 = k1 = [0.41, 0.42]

d2 = k2 = 0.4

~

~

~

g 1 = [0.2, 0.25)

g2 = 0.41

such that γ 1 = [0.2, 0.25)  δ1 = [0.41, 0.42) and δ 2 = 0.4 < γ 2 = 0.41. Then R = µ R , ηR

on H × H is an (Œ, Œ⁄q)-cubic reflexive relation, as well as

(∈, ∈∨q( k 1 , k ) ) -cubic reflexive relation and an (∈ γ ,γ , ∈ γ ,γ ∨q δ 1 ,δ ) -cubic ( 1 2) ( 1 2) ( 2) 2 reflexive relation on H × H. If we define the cubic relation R = µ R , ηR as

614

X. L. MA, J. ZHAN, M. KHAN, M. GULISTAN AND N. YAQOOB ~

H

µR

hR

x

[0.3, 0.4)

0.43

y

[0.41, 0.5)

0.4

z

[0.51, 6)

0.3

then µ R (x , x ) = [0.3, 0.4), µ R ( y , y ) = [0.41, 0.5), µ R ( z , z) = [0.51, 0.6), µ R (x , y ) = [0.3, 0.4) = µ R ( y , x ), µ R (x , z) = [0.3, 0.4) = µ R ( z , x ), µ R ( y , z) = [0.41, 0.5) = µ R ( z , y )

and ηR (x , x ) = 0.43, ηR ( y , y ) = 0.4, ηR ( z , z) = 0.3, ηR (x , y ) = 0.43 = ηR ( y , x ),

ηR (x , x) = 0.43, ηR ( y , y) = 0.4, ηR ( z , z) = 0.3, ηR (x , y) = 0.43 = ηR ( y , x), with ~

t1 = [0.22, 0.23]

t2 = 0.75

d 1 = k1 = [0.18, 0.19]

d2 = k2 = 0.6

~

~

~ 1

g = [0.15, 0.18)

g2 = 0.7

is an (∈( 0.15,0.18),0.7 ) , ∈( 0.15,0.18),0.7 ) ∨q([0.18,0.19),0.6) ) -cubic reflexive relation on H × H. (ii) R = µ R , ηR is not an (∈, ∈∨q([0.18,0.19),0.6) ) Then (i) R = µ R , ηR

-cubic reflexive relation on H × H, as µ R (x , x ) = [0.3, 0.4) + [0.22, 0.23) + [0.18, η x , x + 0.75 + 0.6  1 for every (x , x) ∈ H × H. 0.19)  1 and R

( )

(iii) R = µ R , ηR

is not an (Œ, Œ⁄q)-cubic reflexive relation on H × H, as µ R (x , x) = [0.3, 0.4) + [0.22, 0.23)  1 and ηR (x , x) + 0.75  1 for every (x, x)

Œ H × H. Definition 3.5 A cubic relation R = µ R , ηR on H is called (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK), (ŒG, ŒG⁄qD))-cubic symmetric relation on H if for x, y ŒH, (x, y) (x , y ) 

(t1 ,t2 )

∈ R (resp., (x , y ) 

(t1 ,t2 )

∈∨qR, (resp., (a, a) 

(t1 ,t2 )

∈ R,(x , y )(t ,t ) ∈G R) ) implies that (a, a) t ,t ( ) 1 2 1

2

∈∨qK R,(a, a)(t ,t ) ∈G ∨q∆ R) ) for all (a, a) ∈( y , x ) 1

2

where t1 ∈ D(0,1] and t2 ∈ 0,1) be such that 0  t1 and t2 < 1. Remark 3.6 Every (Œ, Œ⁄q)-cubic symmetric relation is an (Œ, Œ⁄qK)-cubic symmetric relation and every (Œ, Œ⁄qK)-cubic symmetric relation is an (ŒG, ŒG⁄qK)-cubic symmetric relation, but not conversely as shown in the following example.


615

Example 3.7 By considering the same cubic relation defined in Example 3.4 it is easy to see that R = µ R , ηR

on H × H is an (Œ, Œ⁄q)-cubic

symmetric relation, as well as (∈, ∈∨q( k 1 , k ) ) -cubic symmetric relation 2

and an (∈ γ

(

1 ,γ 2

, ∈ γ

) (

1 ,γ 2

∨q δ

)

(

1 ,δ 2

)

) -cubic symmetric relation on H × H. But

for a different cubic relation R = µ R , ηR

as in Example 3.4, we observe

is an (∈( 0.15,0.18),0.7 ) , ∈( 0.15,0.18),0.7 ) ∨q([0.18,0.19),0.6) ) -cubic symmetric relation on H × H. (ii) R = µ R , ηR is not an (∈, ∈∨q([0.18,0.19),0.6) ) R = µ R , ηR

that (i)

-cubic symmetric relation on H × H, as (x , y )(t ,t ) ∈ R does not imply that 1

2

( y , x )(t ,t ) ∈∨q( k 1 , k ) R because µ R ( y , x ) = [0.3, 0.4) + [0.22, 0.23) + [0.18, 0.19) 1

2

2

 1 and ηR ( y , x ) = 0.43 + 0.75 + 0.6  1 for ( y , x ) ∈ H × H . (iii) R = µ R , ηR

is not an (Œ, Œ⁄q)-cubic symmetric relation on H × H, as (x , y )(t1 ,t2 ) ∈ R does ∈∨qR because µ ( y , x ) = [0.3, 0.4) + [0.22, 0.23)  1 not imply that ( y , x ) (t1 ,t2 )

R

and ηR (x , x ) = 0.43 + 0.75  1 for ( y , x ) ∈ H × H . Definition 3.8 A cubic relation R = µ R , ηR on H is called (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))-cubic transitive relation on H if for x, y ŒH, (x, y) (x , y) 

∈ R (resp., (x , y ) (t ,t ) ∈ R,(x , y )(t1 ,t2 ) ∈G R) and ( y , z) t3 ,t ∈ R (resp.,

( y , z) t

∈ R,( y , z) t

(t1 ,t2 )

(

3 , t4

(resp.,

(

1 2

)

(

(a, a) * rmin {t

(

3 , t4

)

∈G R)



1 , t 3 },max{ t3 , t4 }

)

(a, a) * rmin {t

implies that

(

∈∨qK R,(a, a) * rmin {t

(



1 , t 3 },max{ t3 , t4 }

4

)



1 , t 3 },max{ t3 , t4 }

)

∈G ∨q∆ R)

)

∈∨qR

for all

(a, a) ∈(x , z) where t1 ∈ D(0,1] and t2 ∈ 0,1) be such that 0  t1 and t2 < 1. Remark 3.9 Every (Œ, Œ⁄q)-cubic transitive relation is an (Œ, Œ⁄qK)-cubic transitive relation and every (Œ, Œ⁄qK)-cubic transitive relation is an (ŒG, ŒG⁄qD)-cubic transitive relation, but not conversely as shown in the following example. Example 3.10 By considering the same cubic relation defined in Example 3.4 it is easy to see that R = µ R , ηR

on H × H is an (Œ, Œ⁄q)-cubic

transitive relation, as well as (∈, ∈∨q( k 1 , k ) ) -cubic transitive relation and 2

an (∈ γ

(

1 ,γ 2

, ∈ γ

) (

1 ,γ 2

)

∨q δ

(

1 ,δ 2

)

) -cubic transitive relation on H × H. But for a

616


different cubic relation R = µ R , ηR

as in Example 3.4, we observe that

(i) R = µ R , ηR

is an (∈( 0.15,0.18),0.7 ) , ∈( 0.15,0.18),0.7 ) ∨q([0.18,0.19),0.6) ) -cubic transitive relation on H × H. (ii) R = µ R , ηR is not an (∈, ∈∨q([0.18,0.19),0.6) ) -cubic transitive relation on H × H, as (x , y ) t1 ,t ∈ R and ( y , z) t3 ,t ∈ R do not imply

(

that (x , z) * rmin {t

(



1 , t 3 }, max{ t3 , t4 }

∈∨q( k 1 , k ) R

)

2

2

(

)

4

)

µ R (x , z) = [0.3, 0.4) + [0.22,

because

0.23) + [0.18, 0.19)  1 and ηR (x , z) = 0.43 + 0.75 + 0.6  1 for (x , z) ∈ H × H .

(iii) R = µ R , ηR is not an (Œ, Œ⁄q)-cubic transitive relation on H × H, as (x , y ) t

(

1 , t2

)

∈ R and ( y , z)  t

(

3 , t4

)

∈ R do not imply that (x , z) * rmin {t

(



1 , t 3 },max{ t3 , t4 }

)

∈∨qR

 because µ R (x , z) = [0.3, 0.4) + [0.22, 0.23)  1 and ηR (x , z) = 0.43 + 0.75  1 for every (x , x ) ∈ H × H . Lemma 3.11 A cubic relation R = µ R , ηR on H is called cubic reflexive (resp., symmetric, transitive) if and only if it is (Œ, Œ) (resp., (ŒG, ŒG)-cubic reflexive (resp., symmetric, transitive) relation on H. Proof: Straightforward.  Theorem 3.12 For an Hv-LA-semigroup H, the following hold: (i) A cubic relation R = µ R , ηR

on H is an (Œ, Œ⁄q)-cubic reflexive

relation on H if and only if (a) { *rinf µ R (a, a)}  * rmin {µ R (x , y ),(0.5, 0.5]}, ( a , a )∈( x , y )

{

}

(b) {sup( a , a )∈( x , y )ηR (a, a)} ≤ max ηR (x , y ), 0.5 for all x, y, a ŒH. (ii) A cubic relation R = µ R , ηR relation on H if and only if

on H is an (Œ, Œ⁄qK)-cubic reflexive

 1 − k 1  (a) { *rinf µ R (a, a)}  *rmin µ R (x , y ), , ( a , a )∈( x , y ) 2  

 1 − k2  (b) {sup( a , a )∈( x , y )ηR (a, a)} ≤ max ηR (x , y ),  for all x, y, a ŒH and 2   k1 ∈ D(0,1] and k 2 ∈ 0,1).


617

(iii) A cubic relation R = µ R , ηR on H is an (ŒG, ŒG⁄qD)-cubic reflexive relation if and only if (a) *rmax {*rinf( a , a )∈( x , y ) µ R (a, a), γ 1 }  *rmin {µ R (x , y ), δ1 },

{

}

(b) min{sup( a , a )∈( x , y )ηR (a, a), γ 2 } ≤ max ηΞ (x , y ), δ 2 for all x, y, a ŒH, where δ1 , γ 1 ∈ D(0,1] such that γ 1  δ1 , and δ 2 , γ 2 ∈ 0,1) such that d2 < g2.

Proof: (i) Let R = µ R , ηR be an (Œ, Œ⁄q)-cubic reflexive relation on H and let x, y, a ŒH such that (a) and (b) do not hold. So we have {*rinf( a , a )∈( x , y )

µ R (a, a)}  *rmin{µ R (x , y ),(0.5, 0.5] and {sup( a , a)∈( x , y )ηR (a, a)} > max {ηR (x , y), 0.5}

for any x, y, a ŒH. Now if µ R (x , y )  (0.5, 0.5] and ηR (x , y ) > 0.5. Then *rinf µ R (a, a)  µ R (x , y ) and {sup( a , a )∈( x , y )ηR (a, a)} > ηR (x , y ). Thus there exist t1 ∈ D(0,1] and t2 ∈ 0,1) such that *rinf µ R (a, a)  t1  µ R (x , y ) and ( a , a )∈( x , y )

( a , a )∈( x , y )

{sup( a , a )∈( x , y )ηR (a, a)} ≥ t2 > ηR (x , y ). This implies that (x , y )(t ,t ) ∈ R but 1

2

(a, a)(t ,t ) ∈∨qR, for all (a, a) ∈(x , y ), which contradicts the hypothesis. Now 1

2

if µ R (x , y )  (0.5, 0.5] and ηR (x , y ) ≤ 0.5. Then *rinf µ R (a, a)  (0.5, 0.5] and ( a , a )∈( x , y )

{sup( a , a )∈( x , y )ηR (a, a)} > 0.5. Since µ R (x , y )  (0.5, 0.5] and ηR (x , y ) ≤ 0.5 so (x , y )((0.5,0.5],0.5) ∈ R but (a, a)((0.5,0.5],0.5) ∈∨qR, for all (a, a) ∈(x , y ), which again contradicts the hypothesis. Hence (a) and (b) hold. Conversely, assume that (a) and (b) hold. Let (x , y )(t ,t ) ∈ R. Then µ R (x , y )  t1 and ηR (x , y ) ≤ t2 . Now 1 2 if t1  (0.5, 0.5] and t2 ≥ 0.5, then {*rinf( a , a )∈( x , y ) µ R (a, a)}  *rmin {µ R (x , y ),(0.5,

{

}

0.5]}  t1 and {sup( a , a )∈( x , y )ηR (a, a)} ≤ max ηR (x , y ), 0.5 ≤ t2 . This implies that (a, a) ∈ R, for all (a, a) ∈(x , y ). On the other hand if t  (0.5, 0.5] and (t1 ,t2 )

1

t2 < 0.5, then {*rinf( a , a )∈( x , y ) µ R (a, a)}  *rmin {µ R (x , y ),(0.5, 0.5]} = (0.5, 0.5]

{

and

}

{sup( a , a )∈( x , y )ηR (a, a)} ≤ max ηR (x , y ), 0.5 ≤ 0.5.

Then

µ Ξ (x , y) + t1 

(0.5, 0.5] + (0.5, 0.5] = (1,1] and ηΞ (x , y ) + t2 < 0.5 + 0.5 = 1. This implies that (a, a)(t ,t ) qR, for all (a, a) ∈(x , y ). Hence we have (a, a)(t ,t ) ∈∨qR, for all 1

2

(a, a) ∈(x , y ). (ii) and (iii) are straightforward. 

1

2

618


Theorem 3.13 Let R = µ R , ηR

be an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG,

ŒG⁄qD))-cubic reflexive (resp., symmetric, transitive) relation on H. Then the set Ξ = {(x , y ) ∈ H × H|µ (x , y )  0 and ηΞ (x, y) < 1} is an reflexive (resp.,  0,1

Ξ

symmetric, transitive) relation on H. Proof: Straightforward.  Theorem 3.14 Let R = µ R , ηR be a reflexive (resp., symmetric, transitive) relation on H and Ξ = µ Ξ , ηΞ be a cubic set of H1 × H2. ≤  (0.5, 0.5] if (x , y ) ∈ R  0.5 if (x , y ) ∈ R . (i) If µ Ξ (x , y ) =  and ηΞ (x ) =   1 otherwise  (0, 0] otherwise Then Ξ = µ Ξ , ηΞ is an (Œ, Œ⁄q)-cubic reflexive (resp., symmetric, transitive) relation of H.   1 − k 1 1 − k 1   1 − k2 ,    if (x , y ) ∈ R if (x , y ) ∈ R ≤  (ii) If µΞ (x , y) =   2 and ηΞ (x , y) =  . 2  2   1 otherwise  (0, 0] otherwise 

Then Ξ = µ Ξ , ηΞ

is an (Œ, Œ⁄qK)-cubic reflexive (resp., symmetric,

transitive) relation of H. ≤  (δ 1 , δ 1 ] if (x , y ) ∈ R  δ 2 if (x , y ) ∈ R . (iii) If µ Ξ (x , y ) =  and ηΞ (x , y ) =   γ 2 otherwise   (γ 1 , γ 1 ] otherwise Then Ξ = µ Ξ , ηΞ is an (ŒG, ŒG⁄qD)-cubic reflexive (resp., symmetric, transitive) relation of H. Proof. (i) Let x, y ŒH and t1 ∈ D(0,1] and t2 ∈ 0,1) be such that (x , y )(t ,t ) 1

2

∈Ξ = µ Ξ , ηΞ . Then (x , y ) ∈ R. But R = µ , η is a reflexive relation on H, R R

so (t , t) ∈ R for all t ŒH. This implies that µ Ξ (t , t)  (0.5, 0.5] and ηΞ (t , t) ≤ 0.5 if (x , y ) ∈ R. Let r1  (0.5, 0.5] and r2 ≥ 0.5, then µ Ξ (t , t)  (0.5, 0.5]  r1 and ηΞ (t , t) ≤ 0.5 ≤ r2 . Therefore (t , t) r , r ∈Ξ = µ Ξ , ηΞ . On the other hand if r1  (0.5, 0.5] and ( 1 2) r2 < .05, then µ Ξ (t , t) + r1  (0.5, 0.5] + (0.5, 0.5] = (1,1] and ηΞ (t , t) + r2 < 0.5 +


619

0.5 = 1. So (t , t) r , r q Ξ = µ Ξ , ηΞ . Hence ( 1 2) Similarly other cases can easily be seen. (ii) and (iii) are straightforward. 

(t , t) r

(

1 , r2

)

∈∨q Ξ = µ Ξ , ηΞ .

Theorem 3.15 For an Hv-LA-semigroup, the following hold: (i) A cubic relation R = µ R , ηR relation on H if and only if

on H is an (Œ, Œ⁄q)-cubic symmetric

(a) {*rinf( a ,b )∈( y , x ) µ R (a, b)}  *rmin {µ R (x , y ),(0.5, 0.5]},

{

}

(b) {sup( a ,b )∈( y , x )ηR (a, b)} ≤ max ηR (x , y ), 0.5 for all x, y, a ŒH. (ii) A cubic relation R = µ R , ηR on H is an (Œ, Œ⁄qK)-cubic symmetric relation on H if and only if  1 − k 1  (a) {*rinf( a ,b )∈( y , x ) µ R (a, b)}  *rmin µ R (x , y ), , 2  

 1 − k2  (b) {sup( a ,b )∈( y , x )ηR (a, b)} ≤ max ηR (x , y ),  for all x, y, a ŒH and 2   k1 ∈ D(0,1] and k 2 ∈ 0,1).

(iii) A cubic relation R = µ R , ηR on H is an (ŒG, ŒG⁄qD)-cubic symmetric relation if and only if (a) *rmax {*rinf( a ,b )∈( y , x ) µ R (a, b), γ 1 }  *rmin {µ R (x , y ), δ1 },

{

}

(b) min{sup( a ,b )∈( y , x )ηR (a, b), γ 2 } ≤ max ηΞ (x , y ), δ 2 , for all x, y, a ŒH, where δ , γ ∈ D(0,1] such that γ  δ , and δ , γ ∈ 0,1) such 1

1

1

1

2

2

that δ 2 < γ 2 . Proof: Same as 3.12.  Theorem 3.16 For an Hv-LA-semigroup, the following hold: (i) A cubic relation R = µ R , ηR relation on H if and only if

on H is an (Œ, Œ⁄qK)-cubic transitive

620


(a) {*rinf( a ,b )∈( x , z ) µ R (a, b)}  *rmin {µ R (x , y ), µ R ( y , z),(0.5, 0.5]},

{

}

(b) {sup( a ,b )∈( x , z )ηR (a, b)} ≤ max ηR (x , y ), ηR ( y , z), 0.5 for all x, y, a ŒH. (ii) A cubic relation R = µ R , ηR on H is an (ŒG, ŒG⁄qD)-cubic transitive relation on H if and only if  1 − k 1  (a) {*rinf( a ,b )∈( x , z ) µ R (a, b)}  *rmin µ R (x , y ), µ R ( y , z), , 2  

 1 − k2  (b) {sup( a ,b )∈( x , z )ηR (a, b)} ≤ max ηR (x , y ), ηR ( y , z),  for all x, y, a 2   ŒH and k1 ∈ D(0,1] and k 2 ∈ 0,1).

(iii) A cubic relation R = µ R , ηR on H is an (ŒG, ŒG⁄qD)-cubic transitive relation if and only if (a) *rmax {*rinf( a ,b )∈( x , z ) µ R (a, b), γ 1 }  *rmin {* rmax {µ R (x , y ), µ R ( y , z)}, δ1 },

(b) min{sup( a ,b )∈( x , z )ηR (a, b), γ 2 } ≤ max {min{ηΞ (x , y), ηR ( y , z)}, δ 2 } , for all x, y, a ŒH where δ1 , γ 1 ∈ D(0,1] such that γ 1  δ1 , and δ 2 , γ 2 ∈ 0,1) such that δ 2 < γ 2 .

Proof: Same as 3.12.  Definition 3.17 A cubic relation R = µ R , ηR equivalence relation if

on H is an (a, b)-cubic

(i) R = µ R , ηR is an (a, b)-cubic reflexive relation on H, (ii) R = µ R , ηR is an (a, b)-cubic symmetric relation on H, (iii) R = µ R , ηR

is an (a, b)-cubic transitive relation on H, where

α ∈ {∈, ∈G } and β ∈ {∈∨q, ∈∨qK , ∈G ∨qD }. Theorem 3.18 A cubic relation R = µ R , ηR on H is an (Œ, Œ⁄q)-cubic equivalence relation on H if and only if it satisfy the following conditions for all x, y, a ŒH/ (i) (a) {*rinf( a , a )∈( x , y ) µ R (a, a)}  *rmin {µ R (x , y ),(0.5, 0.5]},


{

621

}

(b) {sup( a , a )∈( x , y )ηR (a, a)} ≤ max ηR (x , y ), 0.5 . (ii) (a) {*rinf( a , a )∈( y , x ) µ R (a, a)}  *rmin {µ R (x , y ),(0.5, 0.5]},

{

}

(b) {sup( a , a )∈( y , x )ηR (a, a)} ≤ max ηR (x , y ), 0.5 . (iii) (a) {*rinf( a , a )∈( x , z ) µ R (a, a)}  *rmin {µ R (x , y ), µ R ( y , z),(0.5, 0.5]},

{

}

(b) {sup( a ,b )∈( x , z )ηR (a, b)} ≤ max ηR (x , y ), ηR ( y , z), 0.5 . Proof: It follows from 3.12, 3.15, 3.16.  Theorem 3.19 A cubic relation R = µ R , ηR on H is an (Œ, Œ⁄qK)-cubic equivalence relation on H if and only if it satisfy the following conditions for all x, y, a ŒH.  1 − k 1   (i) (a) {*rinf( a , a )∈( x , y ) µ R (a, a)}  *rmin µ R (x , y ),  , 2      1 − k2  (b) {sup( a , a )∈( x , y )ηR (a, a)} ≤ max ηR (x , y ), . 2    1 − k 1  (ii) (a) {*rinf( a ,b )∈( y , x ) µ R (a, a)}  *rmin µ R (x , y ), , 2    1 − k2  (b) {sup( a ,b )∈( y , x )ηR (a, a)} ≤ max ηR (x , y ), . 2    1 − k 1  (iii) (a) {* rinf( a , a )∈( x , z ) µ R (a, a)}  *rmin µ R (x , y ), µ R ( y , z), , 2    1 − k2  (b) {sup( a ,b )∈( x , z )ηR (a, b)} ≤ max ηR (x , y ), ηR ( y , z), . 2   Proof: It follows from 3.12, 3.15, 3.16.  Theorem 3.20 A cubic relation R = µ R , ηR on H is an (ŒG, ŒG⁄qD)-cubic equivalence relation on H if and only if it satisfy the following conditions for all x, y, a ŒH. (i) (a) *rmax {inf ( a , a )∈( x , y ) µ R (a, a), γ 1 }  *rmin {µ R (x , y ), δ1 },

622


(b) min{sup( a , a )∈( x , y )ηR (a, a), γ 2 } ≤ max{ηR (x , y ), δ 2 }.  (ii) (a) *rmax {inf ( a , a )∈( y , x ) µ R (a, a), γ 1 }  *rmin {µ R (x , y ), δ 1 }, (b) min{sup( a , a )∈( x , y )ηR (a, a)} ≤ max{ηR (x , y ), δ 2 }. (iii) (a) *rmax {inf ( a , a )∈( x , z ) µ R (a, a), γ 1 }  *rmin {* rmax {µ R (x , y ), µ R ( y , z)}, δ1 },

(b) min{sup( a ,b )∈( x , z )ηR (a, b), γ 2 } ≤ max {min{ηΞ (x , y), ηR ( y , z)}, δ 2 } , where δ , γ ∈ D(0,1] such that γ  δ , and δ , γ ∈ 0,1) such that 1

1

1

1

2

2

δ2 < γ 2. Proof: It follows from 3.12, 3.15, 3.16.  Theorem 3.21 Let R = µ R , ηR be an equivalence relation on H and Ξ = µ Ξ , ηΞ be a cubic set of H1 × H2. ≤ 0.5 if (x , y ) ∈ R  (0.5, 0.5] if (x , y ) ∈ R . (i) If µ Ξ (x , y ) =  and ηΞ (x ) =   1 otherwise  (0, 0] otherwise Then Ξ = µ Ξ , ηΞ is an (Œ, Œ⁄q)-cubic equivalence relation of H.   1 − k 1 1 − k 1   1 − k2 ,    if (x , y ) ∈ R if (x , y ) ∈ R ≤  η ( x , y ) = . µ x y ( , ) = 2 2 (ii) If Ξ and Ξ  2      1 otherwise  (0, 0] otherwise 

Then Ξ = µ Ξ , ηΞ is an (Œ, Œ⁄qK)-cubic equivalence relation of H. ≤  (δ 1 , δ 1 ] if (x , y ) ∈ R  δ 2 if (x , y ) ∈ R . (iii) If µ Ξ (x , y ) =  and ηΞ (x , y ) =   γ 2 otherwise   (γ 1 , γ 1 ] otherwise Then Ξ = µ Ξ , ηΞ is an (ŒG, ŒG⁄qD)-cubic equivalence relation of H. Proof: Straightforward.  Remark 3.22 It is clear by Examples 3.4, 3.7 and 3.10, that every (Œ, Œ⁄q)cubic equivalence relation is an (Œ, Œ⁄qK)-cubic equivalence relation and every (Œ, Œ⁄qK)-cubic equivalence relation is an (ŒG, ŒG⁄qD)-cubic equivalence relation, but not conversely.


623

Theorem 3.23 Let R = µ R , ηR be an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))cubic equivalence relation on H. Then the set Ξ = {(x , y ) ∈ H × H|µ (x , y )  0  0,1

Ξ

and ηΞ (x , y ) < 1} is an equivalence relation on H. Proof: Straightforward.  Theorem 3.24 Let R = µ R , ηR be an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))cubic equivalence relation on H. Then the set Ξ  = U (Ξ; t , s) = {(x , y ) ∈ H × H : t ,s

1

2

µ Ξ (x , y)  t , ηΞ (x , y) ≤ s} is equivalence relation on H, where t ∈ D(0,1] and

s ∈ 0,1).

Proof. Straightforward.  Theorem 3.25 If R = µ R , ηR is an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))cubic equivalence relation on H. Then R  R = µ RR , ηRR is also an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))-cubic equivalence relation on H.

Proof. Let R = µ R , ηR is an (Œ, Œ⁄q)-cubic equivalence relation on H. Let x, y, z, t ŒH, consider * rmin {µ RR (x , z),(0.5, 0.5]} = * rmin {∨ y∈H {* rmin(µ R (x , y ), µ R ( y , z))},(0.5, 0.5]} = ∨ y∈H {{* rmin(µ R (x , y ), µ R ( y , z))} * rmin(0.5, 0.5]} = ∨ y∈H {* rmin[{* rmin(µ R (x , y ),(0.5, 0.5]}, * rmin{µ R ( y , z),(0.5, 0.5]}]}

 ∨ t∈H {*rmin(µ R (t , t), µ R (t , t))} = µ RR (t , t)

and max{ηRR (x , z), 0.5}

= max{∧ y∈H {max(ηR (x , y ), ηR ( y , z))}, 0.5} = ∧ y∈H {{max(ηR (x , y ), ηR ( y , z))} max 0.5} = ∧ y∈H {max[{max(ηR (x , y ), 0.5}, max{ηR ( y , z), 0.5}]}

≥ ∧t∈H {max(ηR (t , t), ηR (t , t))} = ηRR (t , t)

This shows that R  R = µ RR , ηRR relation on H. Now

is an (Œ, Œ⁄q)-cubic reflexive

624


*rmin {µ RR (x , y ),(0.5, 0.5]} = *rmin {∨ z∈H {* rmin(µ R (x , z), µ R ( z , y ))},(0.5, 0.5]} = ∨ y∈H {{*rmin(µ R (x , z), µ R ( z , x ))} * rmin(0.5, 0.5]}

= ∨ y∈H {*rmin[{* rmin(µ R (x , z),(0.5, 0.5]}, * rmin {µ R ( z , x ),(0.5, 0.5]}]}  ∨ t∈H {*rmin(µ R ( z , x ), µ R ( y , z))} = ∨ t∈H {*rmin(µ R ( y , z), µ R ( z , x ))} = µ RR ( y , x )

and

max{ηRR (x , y ), 0.5}

= max{∧ z∈H {max(ηR (x , z), ηR ( z , y ))}, 0.5} = ∧ z∈H {{max(ηR (x , z), ηR ( z , y ))} max 0.5}

= ∧ y∈H {max[{max(ηR (x , z), 0.5}, max{ηR ( z , y ), 0.5}]} ≥ ∧t∈H {max(ηR ( z , x), ηR ( y , z))}

= ∧t∈H {max(ηR ( y , z), ηR ( z , x ))} = ηRR ( y , x )

This shows that R  R = µ RR , ηRR relation on H. Now again

is an (Œ, Œ⁄q)-cubic symmetric

*rmin {µ RR (x , y ), µ RR ( y , z),(0.5, 0.5]} = *rmin {[∨ p∈H {* rmin(µ R (x , p), µ R ( p, y ))}] ∧ [∨ q∈H {* rmin(µ R ( y , q), µ R (q, z))}],(0.5, 0.5]} = ∨ p ,q∈H {* rmin[µ R (x , p), µ R ( p, y ), µ R ( y , q), µ R (q, z)] * rmin(0.5, 0.5]} = ∨ p ,q∈H {*rmin[* rmin {µ R (x , p), µ R ( p, y ),(0.5, 0.5]}

, *rmin {µ R ( y , q), µ R (q, z),(0.5, 0.5]}   µ R (x , y ) ∧ µ R ( y , z)  ∨ y∈H {* rmin(µ R (x , y ), µ R ( y , z))} = µ RR (x , z)

and max{ηRR (x , y ), ηRR ( y , z), 0.5}

= max{[∧ p∈H {max(ηR (x , p), ηR ( p, y ))}] ∨ ∧ q∈H {max(ηR ( y , q), ηR (q, z))}], 0.5}


625

= ∧ p ,q∈H {max[ηR (x , p), ηR ( p, y ), ηR ( y , q), ηR (q, z)]max 0.5} = ∧ p ,q∈H {max[max{ηR (x , p), ηR ( p, y ), 0.5} , max{ηR ( y , q), ηR (q, z), 0.5}

 ηR (x , y ) ∨ ηR ( y , z) ≥ ∨ y∈H {max(ηR (x , y ), ηR ( y , z))} = ηRR (x , z) This shows that R  R = µ RR , ηRR

is an cubic transitive relation on

H. Hence R  R = µ RR , ηRR is also an (Œ, Œ⁄q)-cubic equivalence relation on H. Similarly other cases can be seen.  Theorem 3.26 Let {Ri }i∈I = µ R , ηR i

i

be an family of (Œ, Œ⁄q) (resp., (Œ,

Œ⁄qK) (ŒG, ŒG⁄qD))-cubic equivalence relation on H. Then R = µ R , ηR = ∩i∈I {Ri }i∈I =

∧ µ , ∨ η i ∈I

R

i ∈I

R

is also an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))-cubic

equivalence relation on H. Proof. Straightforward.  4. Images and Preimages of Generalized Cubic Relations In this section we present some results on images and preimages of generalized cubic relations of Hv-LA-semigroups. Let us denote by C( H1 × H1 ) the family of cubic sets in a set H1 × H1. Let H1 and H2 be given classical sets. A mapping h : H1 × H1 → H 2 × H 2 induces two mappings Ch : C( H1 × H1 ) → C( H 2 × H 2 ), R  Ch (R), and −1 −1 Ch : C( H 2 × H 2 ) → C( H1 × H1 ), R  Ch (R), where Ch(R) is given by

 *rsup µ R (a, b) if h −1 (x , y ) ≠ ∅  h (µ R )(x , y ) = ( x , y )∈h( a ,b ) , (0, 0] otherwise

 inf ηR (a, b) if h − 1(x , y ) ≠ ∅ , h (ηR )(x , y) = ( x , y )= h( a ,b ) otherwise 1

626


for all (x , y ) ∈ H 2 × H 2 and Ch−1 (R) is defined by Ch−1 (µ R )(x , y ) = µ R (h(x , y )) and Ch−1 (ηR )(x , y ) = ηR (h(x , y )) for all (x , y ) ∈ H1 × H1 . Then the mapping Ch (resp., Ch–1) is called an cubic transformation (resp., inverse cubic transformation) induced by H. A cubic set in H1 has the cubic property if for any subset T of H1 there exists (x0 , y0 ) ∈ T such that µ R (x0 , y0 ) = *rsup µ R (x , y ) and ηR (x0 , y0 ) = inf ηR (x , y ). ( x , y )∈T

( x , y )∈T

Theorem 4.1 For a hyper homomorphism h : H1 × H1 → H 2 × H 2 of Hv-LAsemigroups, let Ch : C(H1 × H1 ) → C(H 2 × H 2 ) and Ch−1 : C(H 2 × H 2 ) → C(H1 × H1 ) be cubic transformation and inverse cubic transformation, respectively, induced by h. (i) If R = µ R , ηR is an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))-cubic equivalence relation on H1 which has the cubic property, then Ch(R) is an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))-cubic equivalence relation on H2. (ii) If R = µ R , ηR is an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))-cubic equivalence relation on H2, then Ch–1(R) is an (Œ, Œ⁄q) (resp., (Œ, Œ⁄qK) (ŒG, ŒG⁄qD))-cubic equivalence relation on H1. Proof. (i) Given h(x1 , x2 ), h( y1 , y2 ) ∈ h( H1 × H1 ), let (x0 , x/0 ) ∈ h −1 (h(x1 , x2 )) and ( y0 , y/0 ) ∈ h −1 (h( y1 , y2 )) be such that

µ R (x0 , x/0 ) =

*rsup

µ R (a, b), ηR (x0 , x/0 ) =

*rsup

µ R (c , d), ηR ( y0 , y/0 ) =

( a , b )∈h−1 ( h ( x1 , x2 ))

inf

ηR (a, b),

inf

ηR (c , d),

( a , b )∈h−1 ( h ( x1 , x2 ))

and

µ R ( y0 , y/0 ) =

( c , d )∈h−1 ( h ( y1 , y2 ))

( c , d )∈h−1 ( h ( y1 , y2 ))

respectively. Then

h (µ R )(h(x , x )) =

* rsup

(µ R )( z , z)  (µ R )(a, b)

( z , z )∈h−1 ( h ( x ) h ( y ))


627

 * rmin {(µ R )(a, b),(0.5, 0.5]} = *rmin {h (µ R )(h(u, v )),(0.5, 0.5]},

and h (ηR )(h(x , x )) =

inf

(ηR )( z , z)  (ηR )(a, b)

( z , z )∈h−1 ( h ( x , x ))

≤ max{(ηR )(a, b), 0.5}

= max{h (ηR )(h(u, v )), 0.5}.

Thus Ch(R) is an (Œ, Œ⁄q)-cubic reflexive relation. Also h (µ R )(h(v , u)) =

*rsup (µ R )( z , z)  (µ R )(b, a)

( z , z )∈h−1h ( v , u )

 * rmin {(µ R )(a, b),(0.5, 0.5]} = * rmin {h (µ R )(h(u, v )),(0.5, 0.5]},

and

h (ηR )(h(v , u)) =

inf

(ηR )( z , z)  (ηR )(b, a)

( z , z )∈h−1h ( v , u )

≤ max{(ηR )(a, b), 0.5}

= max{h (ηR )(h(u, v )), 0.5}.

Thus Ch(R) is an (Œ, Œ⁄q)-cubic symmetric relation. Finally, h (µ R )(h(u, w)) =

*rsup

(µ R )( z , z)  * rmin {(µ R )(a, b),(µ R )(b, c),(0.5, 0.5]}

( z , z )∈h−1h ( u , w )

= * rmin {h (µ R )(h(u, v )), h (µ R )(h(v , w)),(0.5, 0.5]},

and

h (ηR )(h(u, w)) =

inf (ηR )( z , z) ≤ max{(ηR )(a, b),(ηR )(b, c), 0.5} h(u,w) = max{h (ηR )(h(u, v )), h (ηR )(h(v , w)), 0.5}. ( z , z )∈h −1

628


Thus Ch(R) is an (Œ, Œ⁄q)-cubic transitive relation. Hence Ch(R) is an (Œ, Œ⁄q)-cubic equivalence relation on H2. Similarly other cases can be seen. (ii) For any (x1 , x2 ) ∈ H1 × H1 , we have h−1 (µ R )((x1 , x2 )) = µ R (h)((x1 , x2 )) = µ R (h(x1 , x2 ))  *rmin(µ R )(h(a, b)),(0.5, 0.5]) = *rmin(h−1 (µ R )(u, v )),(0.5, 0.5])

and

h−1 (ηR )((x1 , x2 )) = ηR (h)((x1 , x2 )) = ηR (h(x1 , x2 )) ≤ max(ηR )(h(a, b)), 0.5)

= max(h−1 (ηR )(u, v )), 0.5)

Hence Ch–1(R) is an (Œ, Œ⁄q)-cubic reflexive relation on H1. Now h−1 (µ R )((x1 , x2 )) = µ R (h)((x1 , x2 )) = µ R (h(x1 , x2 ))  *rmin(µ R )(h(b, a)),(0.5, 0.5]) = *rmin(h−1 (µ R )(x2 , x1 )),(0.5, 0.5])

and

h−1 (ηR )((x1 , x2 )) = ηR (h)((x1 , x2 )) = ηR (h(x1 , x2 )) ≤ max(ηR )(h(b, a)), 0.5)

= max(h−1 (ηR )(x2 , x1 )), 0.5)

Hence Ch– 1(R) is an (Œ, Œ⁄q)-cubic symmetric relation on H1. Also h−1 (µ R )((x1 , x3 )) = µ R (h)((x1 , x3 )) = µ R (h(x1 , x3 ))  *rmin((µ R )(h(a, b)),(µ R )(h(b, c)),(0.5, 0.5])

= *rmin(h−1 (µ R )(x1 , x2 )), h−1 (µ R )(x2 , x3 )),(0.5, 0.5])

and

h−1 (ηR )((x1 , x3 )) = ηR (h)((x1 , x3 )) = ηR (h(x1 , x3 )) ≤ max((ηR )(h(a, b)),(ηR )(h(b, c)), 0.5)

= max(h−1 (ηR )(x1 , x2 )), h−1 (ηR )(x2 , x3 )), 0.5).


629

Hence Ch–1(R) is an (Œ, Œ⁄q)-cubic transitive relation on H1. Thus Ch (R) is an (Œ, Œ⁄q)-cubic equivalence relation on H1. Similarly other cases can be seen.  –1

Acknowledgement This research is partially supported by a grant of National Natural Science Foundation of China (11461025) and Natural Science Foundation of Hubei Province (2014CFC1125). References [1] Marty, F. 1934. Sur une generalization de la notion de groupe, 8iemCongres des Mathematicians Scandinaves Tenua Stockholm, 45-49. [2] Vougiouklis, T. 1990. The fundamental relation in hyperrings. The general hyperfield, Algebraic hyperstructures and applications, 203211. [3] Spartalis, S. 2002. On Hv-semigroups. Italian Journal of Pure and Applied Mathematics, 11:165-174. [4] Davvaz, B. 2003. A brief survey of the theory of Hv-structures. Proc. 8th International Congress on Algebraic Hyperstructures and Applications, (2002)1-9, Samothraki, Greece, Spanidis Press, 39-70. [5] Nezhad, A.D. and B. Davvaz. 2009. An introduction to the theory of Hv-semilattices. Bulletin of the Malaysian Mathematical Sciences Society, 32(3):375-390. [6] Hedayati, H. and B. Davvaz. 2013. Regular relations and hyperideals in Hv-G-semigroups. Utilitas Mathematica, 75:33-46. [7] Hedayati, H. and I. Cristea. 2011. Fundamental G-semigroups through Hv-G-semigroups. U.P.B. Scientific Bulletin, Series A, 73(4):71-78. [8] Kazim, M. A. and M. Naseerudin. (1972). On almost-semigroup. Aligarh Bulletin of Mathematics, 2:1-7. [9] Hila, K. and J. Dine. 2011. On hyperideals in left almost semihypergroups. ISRN Algebra, Article ID 953124, 8 pages. [10] Yaqoob, N., P. Corsini and F. Yousafzai. 2013b. On Intra-regular left almost semihypergroups with pure left identity. Journal of Mathematics, Article ID 510790, 10 pages. [11] Gulistan, M., N. Yaqoob and M. Shahzad. 2015. A note on Hv-LAsemigroups. U.P.B. Scientific Bulletin, Series A, 77(3): 93-106.

630


[12] Zadeh, L.A. 1965. Fuzzy Sets. Information and Control, 8:339-353. [13] Zadeh, L.A. 1971. Similarity relation and fuzzy ordering. Information Sciences 3:177-200. [14] Ali, M. I., F. Feng and M. Shabir. (2011). A note on (Œ, Œ⁄q)-fuzzy equivalence relations and indistinguishability operators. Hacettepe Journal of Mathematics and Statistics, 40(3):383-400. [15] Leoreanu-Fotea, V. 2009. Fuzzy hypermodules. Computers and Mathematics with Applications, 57:466-475. [16] Davvaz, B. 1999. Fuzzy Hv-groups. Fuzzy Sets and Systems, 101:191195. [17] Davvaz, B. and P. Corsini. 2008. On (a, b)-fuzzy Hv-ideals of Hvrings. , 5(2):35-47. [18] Jun, Y. B., C. S. Kim and K. O. Yang. 2012, Cubic sets. Annals of Fuzzy Mathematics and Informatics. 4(1):83-98. [19] Khan, M., Y. B. Jun, M. Gulistan and N. Yaqoob. 2015. The generalized version of Jun’s cubic sets in semigroups. Journal of Intelligent & Fuzzy Systems, 28(2):947-960.

Received December, 2015