NEUTROSOPHIC CUBIC IDEALS

NEUTROSOPHIC CUBIC IDEALS MAJID KHAN 1 AND MUHAMMAD GULISTAN 2 Abstract. Operational properties of neutrosophic cubic sets are investigated.The notion of neutrosophic cubic subsemigroups and neutrosophic cubic left(resp.right) ideals are introduced, and several properties are investigated. Relations between neutrosophic cubic subsemigroups and neutrosophic cubic left(resp.right) ideals are discussed. Characterizations of neutrosophic cubic left(resp. right)ideals are considered, and how the images or inverse images of neutrosophic cubic subsemigroups and cubic left (resp. right) ideals become neutrosophic cubic subsemigroups and neutrosophic cubic left (resp. right)ideals, respectively, are studied..

1. Introduction Fuzzy sets are initiated by Zadeh [1]. In [2], Zadeh made an extension of the concept of a fuzzy set by an interval-valued fuzzy set, i.e., a fuzzy set with an interval-valued membership function. In traditional fuzzy logic, to represent, e.g., the expert’s degree of certainty in di¤erent statements, numbers from the interval [0, 1] are used. It is often di¢ cult for an expert to exactly quantify his or her certainty,therefore, instead of a real number, it is more adequate to represent this degree of certainty by an interval or even by a fuzzy set. In the …rst case, we get an interval-valued fuzzy set. In the second case, we get a second-order fuzzy set.

Interval-valued fuzzy sets have been actively used in real-life applications. For example, Sambuc [10] in Medical diagnosis in thyroidian pathology, Kohout [18] also in Medicine, in a system CLINAID, Gorzalczany in Approximate reasoning, Turksen [10, 11] in Interval-valued logic, in preferences modelling [12], etc. These works and others show the importance of these sets. Fuzzy sets deal with possibilistic uncertainty, connected with imprecision of states, perceptions and preferences. Using a fuzzy set and an interval-valued fuzzy set, Jun et al. [4] introduced a new notion, called a cubic set, and investigated several properties. Cubic set theory is applied to BCK/BCI-algebras (see [6, 7, 8, 9]) and -semihypergroups (see [2]). Jun et al.[5] introduced the concept of cubic ideals in semigroup. They investigated operational properties of cubic sets, the notion of cubic subsemigroups and cubic left (resp. right) ideals, and investigate several properties. The concept of neutrosophic set (NS) developed by Smarandache[16], is a more general platform which extends the concepts of the classic set and fuzzy set[17], Neutrosophic set theory is applied to various parts (refer to the site http://fs.gallup.unm.edu/neutrosophy.htm).Haiban Date : December 8, 2016 .

Thanks for Author One. Thanks for Author Two. This paper is in …nal form and no version of it will be submitted for publication elsewhere.

1

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

2

extended the idea of neutrosophic sets to interval neutrosophic sets[14, 15]. Jun introduced the notion of neutrosophic cubic set[19].

In this paper we introduce the concept of neutrosophic cubic ideals in semigroup.

We investigate some operational properties of neutrosophic cubic sets. The notion of neutrosophic cubic subsemigroups and neutrosophic cubic left (resp.right) ideals are introduced, and several properties are investigated. Relations between neutrosophic cubic subsemigroups and neutrosophic cubic left (resp. right) ideals are discussed.

Characterizations of neutrosophic cubic left (resp. right)ideals are considered, and how the images or inverse images of neutrosophic cubic subsemigroups and cubic left (resp. right) ideals become neutrosophic cubic subsemigroups and neutrosophic cubic left (resp. right) ideals, respectively, are studied.

2. Preliminaries A non-empty set S together with an associative binary operation “ ” is called a semigroup. A non-empty subset A of a semigroup S is called a subsemigroup if AA A: A non empty subset A of S is left(right) ideal of S if SA A (AS A) :Jun et al. [4] have de…ned the cubic set as follows: Let X be a non-empty set. A cubic set in X is a structure of the form: C = fhx; e(x); (x)i jx 2 Xg where e is an interval-valued fuzzy set in X and is a fuzzy set in X. Let X be a non-empty set. A neutrosophic set (NS) in X (see [16,17]) is a structure of the form: = fh T (x); I (x); F (x)i jx 2 Xg where T : X ! [0; 1]is a truth membership function, I : X ! [0; 1] is an indeterminate membership function, and F : X ! [0; 1] is a false membership function. Let X be a non-empty set. An interval neutrosophic set (INS) in X (see [14, 15]) is a structure of the form: e = fheT (x); eI (x); eF (x)i jx 2 Xg where eT ; eI and eF are interval-valued fuzzy sets[23] in X, which are called an interval truth membership function, an interval indeterminacy membership function and an interval falsity membership function, respectively. Jun et al.[19] considered the notion of neutrosophic cubic sets. Let X be a non-empty set. A neutrosophic cubic set (NCS) in X is a pair A = (e; ) where e = fheT (x); eI (x); eF (x)i jx 2 Xg is an interval neutrosophic set in X and = fh T (x); I (x); F (x)i jx 2 Xg is a neutrosophic set in X. Jun et al.[5] inrodeced the idea of cubic ideals in semigroup and investigated several properties.

3. Operational Properties of neutrosophic cubic sets De…nition 1. Let X be nonempty set. A NC set A in X is structure A = fhx; eT ; eI ; eF ;

T;

I;

Fi

: x 2 Xg

For any nonempty subset G of set X,the characteristic NC set of G in to be a structure G

= fhx; eT G ; eIG ; eF G ;

T G;

IG ;

F Gi

: x 2 Xg

which is brie‡y denoted by G

where

= heT G ; eIG ; eF G ;

T G;

IG ;

F Gi

de…ned

NC IDEALS

3

e 1 if x 2 G 0 if x 2 G TG = e 1 otherwise 0 otherwise e 1 if x 2 G 0 if x 2 G eIG = e IG = 1 otherwise 0 otherwise e 0 if x 2 G 1 if x 2 G eF G = e FG = 1 otherwise 0 otherwise The whole NC set S in semigroup S is de…ned to be structure nD E o S= x; e 1T S ; e 1IS ; e 1F S ; 0T S ; 0IS ; 0F S : x 2 S eT G =

with

e 1T S = [1 1]; e 1IS = [1 1]; e 1F S = [1 1]; and 0T S = 0; 0IS = 0; 0F S = 0 for all x 2 X: It will brie‡y denoted by

E D 1T S ; e 1IS ; e 1F S ; 0T S ; 0IS ; 0F S ; S= e

For two NC set A = hx; eT ; eI ; eF ; semigroup S.We de…ne

T;

I;

Fi

and B=heT ; eI ; eF ;

A v B () eT eT ; eI eI ; eF and T T; I I; F F

and the NC product of A=hx; eT ; eI ; eF ; is de…ne to be NC set A

x; (eT

B=

(

which brie‡y denoted by A

B=

T

T;

I;

F iand

eF ;

T ; I; F i

B=heT ; eI ; eF ;

in

T ; I; F i

eT ) (x); (eI eI ) (x); (eF eF ) (x) I )(x); ( F F )(x) : x 2 S

T )(x); ( I

(eT eT ) (x); (eI eI ) (x); (eF ( T T )(x); ( I I )(x); ( F

eF ) (x) F )(x)

where eT eT ; eI eI ; eF eF and T T; I I; F F are de…ne as follows, respectively ( r sup [r min feT (y); eT (z)g] if x=yz for some y,z 2 S x=yz (eT eT )(x) = and [0 0] otherewise, ( max f T (y); T (z)g if x=yz for some y,z 2 S x=yz ( T T )(x) = 1 otherewise, ( r sup [r min feI (y); eI (z)g] if x=yz for some y,z 2 S x=yz (eI eI )(x) = and [0 0] otherewise, ( max f I (y); I (z)g if x=yz for some y,z 2 S x=yz ( I I )(x) = 1 otherewise, ( r sup [r min feF (y); eF (z)g] if x=yz for some y,z 2 S x=yz (eF eF )(x) = and [0 0] otherewise, ( max f F (y); F (z)g if x=yz for some y,z 2 S x=yz ( F F )(x) = 1 otherewise,

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

4

for all x2 S we also de…ne the cap and union of two NC sets as follows. Let A and B be a twoNC setsin x. The intersection of A and B, denoted by A\B,is the NC set e eT ; e I \ e eI ; e F \ e eF ; A u B = eT \

T

_

T;

I

_

I;

F

_

F

e eT (x) = r minfeT (x); eT (x)g; eI \ e eI (x) = r minfeI (x); eI (x)g; eF \ e eF (x) = where eT \ r minfeF (x); eF (x)g and ( T _ T )(x) = max f T (x); T (x)g ; ( I _ I )(x) = max f I (x); I (x)g ; ( F _ )(x) = max f F (x); F (x)g : F The union of A and B, denoted by A [ B,is the NC set e eT ; e I [ e eI ; e F [ e eF ; A t B = eT [

T

^

T;

I

^

I;

F

^

F

e eT (x) = r maxfeT (x); eT (x)g; eI [ e eI (x) = r maxfeI (x); eI (x)g; eF [ e eF (x) = where eT [ r maxfeF (x); eF (x)g and ( T ^ T )(x) = min f T (x); T (x)g ; ( I ^ I )(x) = min f I (x); I (x)g ; ( F ^ )(x) = min f F (x); F (x)g : F Proposition 1. For any ENC sets A=hx; eT ; eI ; eF ; D e e e C= T ; I ; F ; T ; I ; F in semigroup S. We have [1] [2] [3] [4]

A t (B u C) A u (B t C) A (B t C) A (B u C)

T;

I;

F i,

B=heT ; eI ; eF ;

= (A t B) u (A t C) = (A u B) t (A u C) = (A B) t (A C) v (A B) u (A C)

Proof. (1) and (2) are straightforwart.

(3) let x be any elemen of s. if x is not expressed as x=yz , then

(eT (eI (eF and (

T

( (

F

I

e eT ))(x) = [0 0] = ((eT eT )[ e (eT eT ))(x); (eT [ e eI ))(x) = [0 0] = ((eI eI )[ e (eI eI ))(x); (eI [ e eF ))(x) = [0 0] = ((eF eF )[ e (eF eF ))(x) (eF [

( T^ ( I^ ( F^

T ))(x)

= 1 = (( I ))(x) = 1 = (( F ))(x) = 1 = ((

T I F

T)

^( T I) ^ ( I F) ^ ( F

T ))(x); I ))(x); F ))(x):

T ; I ; F iand

NC IDEALS

therefore A Then

(eT

(B t C) = (A

e eT ))(x) (eT [

B) t (A

5

C):Assume that xis expressed as x=yz.

h n oi e eT )(z) = r sup r min eT (y); (eT [ x=yz h n n ooi = r sup r min eT (y); r max eT (z); eT (z) x=yz h n n ooi = r sup r max r minfeT (y); eT (z)gr min eT (y); eT (z) x=yz

n n oo = r max r sup [r minfeT (y); eT (z)gr min eT (y); eT (z) ] x=yz

e (eT eT ))(x) eT )[ n oi e eI ))(x) = r sup r min eI (y); (eI [ e eI )(z) (eI [ x=yz h n n ooi = r sup r min eI (y); r max eI (z); eI (z) x=yz h n n ooi = r sup r max r minfeI (y); eI (z)gr min eI (y); eI (z) =

(eI

((eT

h

x=yz

n n oo = r max r sup [r minfeI (y); eI (z)gr min eI (y); eI (z) ] x=yz

e (eI eI ))(x) eI )[ oi h n e eF ))(x) = r sup r min eF (y); (eF [ e eF )(z) (eF [ x=yz h n n ooi = r sup r min eF (y); r max eF (z); eF (z) x=yz h n n ooi = r sup r max r minfeF (y); eF (z)gr min eF (y); eF (z) =

(eF

((eI

x=yz

n n oo = r max r sup [r minfeF (y); eF (z)gr min eF (y); eF (z) ] x=yz

=

and

((eF

e (eF eF )[

e ))(x) F

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

6

(

(

T

T

^

T ))(x)

= = =

(

(

I

I

(

F

T (y); ( T

x=yz

[max f

T (y); min( T (z); T (z))g]

x=yz

[max fmin(

min

=

min

x=yz

x=yz

fmax( max(

T (y); T (z));

x=yz

[max f

x=yz

[max fmin(

min

=

min

x=yz

x=yz

T )(z)g]

T (y); T (z)); max( T (y); T (z))g

x=yz

=

^

T (y); T (z)); min( T (y); T (z))g]

= min [( T T; T = (( T T) ^ ( T ^ I ))(x) = [max f I (y); ( =

F

[max f

=

=

(

x=yz

x=yz

max(

T (y); T (z))

T ) (x)] T )) (x) I

^

I )(z)g]

I (y); min( I (z); I (z))g] I (y); I (z)); min( I (y); I (z))g]

fmax( max(

I (y); I (z)); max( I (y); I (z))g I (y); I (z));

x=yz

max(

I (y); I (z))

= min [( I I ) (x)] I; I = (( I ) ^ ( I I )) (x) I ^ F ))(x) = [max f F (y); ( F ^ F )(z)g] x=yz

= =

x=yz

[max f

x=yz

[max fmin(

=

min

=

min

x=yz

x=yz

= min [( = (( F

F (y); min( F (z); F (z))g] F (y); F (z)); min( F (y); F (z))g]

fmax( max(

F F)

F;

^(

F (y); F (z)); max( F (y); F (z))g F (y); F (z));

F F

x=yz

max(

F ) (x)] F )) (x)

hence (3 ) holds.

(4) let x2 S:If x isnot expressed asx=yz , then it is clear . As

F (y); F (z))

NC IDEALS

7

sume that there exsist y,z2 S such that x=yz .Then

(eT

e eT ))(x) (eT \

h n oi e eT )(z) = r sup r min eT (y); (eT \ x=yz h n n ooi = r sup r min eT (y); r min eT (z); eT (z) x=yz h n n ooi = r sup r max r minfeT (y); eT (z)gr min eT (y); eT (z) x=yz

n o r min r sup[r minfeT (y); eT (z)g]; r sup[r min eT (y); eT (z) ] x=yz

e (eT eT )\ n oi e eI ))(x) = r sup r min eI (y); (eI \ e eI )(z) (eI \ x=yz h n n ooi = r sup r min eI (y); r min eI (z); eI (z) x=yz h n n ooi = r sup r max r minfeI (y); eI (z)gr min eI (y); eI (z) =

(eI

x=yz

e ))(x) T

((eT

h

x=yz

n o r min r sup[r minfeI (y); eI (z)g]; r sup[r min eI (y); eI (z) ] x=yz

e (eI eI ))(x) eI )\ oi h n e eF ))(x) = r sup r min eF (y); (eF \ e eF )(z) (eF \ x=yz h n n ooi = r sup r min eF (y); r min eF (z); eF (z) x=yz h n n ooi = r sup r max r minfeF (y); eF (z)gr min eF (y); eF (z) =

(eF

x=yz

((eI

x=yz

n o r min r sup[r minfeF (y); eF (z)g]; r sup[r min eF (y); eF (z) ] x=yz

=

((eF

e (eF eF )\

x=yz

e ))(x) F

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

8

and (

(

T

T

_

T ))(x)

= =

x=yz

max f

x=yz

=

(

(

I

I

= (( _ I ))(x) =

T

x=yz

=

F

(

F

I

x=yz

=

=

((

F

maxf

T (y); T (z)g

I

_

I )(z)g

I (y); maxf I (z); I (z)gg

maxf

I (y); I (z)g; maxf I (y); I (z)gg

I (y); I (z)g;

x=yz

maxf

I (y); I (z)g

I)

_( I I ))(x): max f F (y); ( F _ F )(z)g max f

x=yz

max

x=yz

T ))(x):

max fmaxf

x=yz

x=yz

=

T (y); T (z)g;

_( T max f I (y); (

max

T )(z)g

T (y); T (z)g; maxf T (y); T (z)gg

T)

x=yz

= (( _ F ))(x) =

maxf

max f

_

T (y); maxf T (z); T (z)gg

max fmaxf

x=yz

x=yz

=

(

max f

x=yz

max

T (y); ( T

F (y); maxf F (z); F (z)gg

max fmaxf

x=yz F)

maxf _(

F (y); F (z)g; maxf F (y); F (z)gg

F (y); F (z)g;

x=yz

maxf

F (y); F (z)g

F ))(x):

F

Hence (4) holds.

Proposition 2. For any NC sets D E A=hx; eT ; eI ; eF ; T ; I ; F i, B=heT ; eI ; eF ; T ; I ; and C= e ; e ; e ; T ; I ; F in semigroup S, if Av B; then A C v B C and T

C AvC

I

Fi

F

B:

Proof. Strraitforward.

Proposition 3. For any nonempty subsets G and H of semigroup S, we have [1]

eT

eT

G

G

eT

G ee T

\ eT

[ eT

H ; eI

H ; eI G

H ; eI G

G ee I

\ eI

[ eI

H ; eF

H ; eF G

H ; eF G

G ee F

\ eF

[ eF

Proof. (1) Let a2 S: If a2 GH; then eT [11]; T GH (a) = 0; I GH (a) = 0; F

H;

T G

T

H;

I G

I

H;

F G

[2] H;

H;

T G

T G

GH (a) GH (a)

_

T H;

^

T H;

I G

I G

_

I H;

^

I H;

G

F

[3] F

H F H

G\ H G_ F H

G[ H ^ G F H

= [1 1]; eI GH (a) = [11]; eF GH (a) = = 0 and a=bc for some b2 G and

= = = = = =

GH; i:e: eT GH ; eI

GH ; ; e

G \ H; i:e: eT G\H ; eF

G [ H; i:e: eT G[H ; eF

G\H

G[H

NC IDEALS

9

c2 H Thus

(eT

(eI

(eF

eT

G

= r sup r min eT

H )(a)

a=xy

r min eT

G (b); e T H (c)

= [1 1] eI H )(a) = r sup r min eI

G

a=xy

eF

G

r min eI a=xy

=

r min eF [1 1]

G (x); e I H (y)

G (b); e I H (c)

= [1 1] = r sup r min eF

H )(a)

G (x); e T H (y)

G (x); e F H (y)

G (b); e F H (c)

and

(

(

(

T H )(a)

T G

I G

I

F G

F

=

a=xy

[max f

T G (x);

max f T G (b); T H (c)g = 0 [max f I G (x); I H (y)g] H )(a) = a=xy

max f I G (b); I H (c)g = 0 )(a) = [max f F G (x); F H (y)g] H a=xy

max f = 0

F G (b);

F H (c)g

It follows that (eT G eT H )(a) = [1 1]; (eI G eI eF H )(a) = [1 1] and ( T G T H )(a) = 0; ( I G F H )(a) = 0:Therefore eT

G

eI

eF

G

G

eT

eI

eF

T H (y)g]

H; H; H;

T G

T H

=

I G

I H

=

F G

F H

=

eT

eI

eF

H )(a)

= [1 1]; (eF = 0; ( F

I H )(a)

GH ;

T GH

GH ;

I GH

GH ;

F GH

G G

that is , G H = GH: Assume that a2 = GH: Then eT GH (a) = [0 0]; eI GH (a) = [0 0]; eF GH (a) = [0 0] and T GH (a) = 1; I GH (a) = 1; F GH (a) = 1: Let y,z2 S be such that a= yz. Then we know that y2 = G or z2 = H:Assume that

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

10

y2 = G:Then (eT

G

eT

H )(a)

= r sup[r min eT a=yz

= r sup[r min [0 0]; eT a=yz

= (eI

eI

G

H )(a)

G

a=yz

eF

H )(a)

T G

a=yz

a=yz

= =

(

I G

I

(

F G

F

[0 0] = eF a=yz

[max f

a=yz

[max f1;

T G (y);

[max f1;

= 1 = I GH (a) [max f F H )(a) = a=yz

= =

a=yz

1=

]

[max f1;

T H (z)g]

T H (z)g]

a=yz

a=yz

H (z)

]

GH (a)

= 1 = T GH (a) [max f I G (y); H )(a) = =

]

G (y); e F H (z)

= r sup[r min [0 0]; eF

T H )(a)

H (z)

GH (a)

= r sup[r min eF

and

]

G (y); e I H (z)

= r sup[r min [0 0]; eI [0 0] = eI

H (z)

]

GH (a)

a=yz

=

(

[0 0] = eT

= r sup r min eI

= (eF

G (y); e T H (z)

I H (z)g]

I H (z)g]

G (y);

F H (z)g]

F H (z)g]

F GH (a)

Similarly, if z2 = H ,then (eT G eT H )(a) = [0 0] = eT GH (a); (eI G eI H )(a) = [0 0] = eI GH (a); (eF G eF H )(a) = [0 0] = eF GH (a) and ( T G T H )(a) = 1 = T GH (a); ( I G I H )(a) = 1 = I GH (a); ( F G F H )(a) = 1 = F GH (a): Therefore G H = GH:(2) and (3) are straightforward.

4. Neutrosophic cubic subsemigroups and ideals De…nition 2. A NC set A=heT ; eI ; eF ; subsemigroup of S if it satis…es: (8x; y 2 S) Example : a a c b c c c

1. b c c b

eT (xy) T (xy)

T;

I;

Fi

in semigroug S is called a NC

r min feT (x) ; eT (y)g ; eI (xy) r min feI (x) ; eI (y)g ; eF (xy) r min feF (x) ; eF max f T (x); T (y)g ; I (xy) max f I (x); I (y)g ; F (xy) f F (x); F (y)g

Consider a semigroup S=fa; b; cg with the following Caylay’s table. c c a c

NC IDEALS

11

De…ns a NC set A=heT ; eI ; eF ; T ; I ; F i in S by S eT eI eF T I F a [0:3; 0:6] [0:5; 0:7] [0:6; 0:7] 0:4 0:6 0:8 b [0:2; 0:4] [0:3; 0:4] [0:2; 0:5] 0:6 0:7 0:9 c [0:7; 0:9] [0:8; 0:9] [0:7; 0:8] 0:2 0:3 0:4 Theorem 1. A NC set A=heT ; eI ; eF ; semigroup of S if and only if A A v A:

T;

I;

Fi

in semigroug S is a NC sub-

Proof. straightforward.

De…nition 3. A NC set A=heT ; eI ; eF ; right(lef t) NC ideal of S if it satis…es: and

(8a; b T (xy)

2

T;

I;

Fi

in a semigroug S is called a

eT (xy) eT (x) (eT (xy) T (x) ( T (xy) T (y)) ; I (xy)

S)

eT (y)) ; eI (xy) I (x) ( I (xy)

eI (x) (eI (xy) I (y)) ; F (xy)

By a (two sided) NC ideal we mean a left and right NC ideal.

Example 2. Consider a semigroup S=fa; b; cg with the following Caylay’s table. : a b c a a a a b a a a c a a c De…ns a NC set A=heT ; eI ; eF ; T ; I ; F i in S by S eT eI eF T I F a [0:7; 0:9] [0:8; 0:9] [0:7; 0:8] 0:2 0:3 0:4 b [0:2; 0:4] [0:3; 0:4] [0:2; 0:5] 0:6 0:7 0:9 c [0:1; 0:3] [0:5; 0:6] [0:2; 0:4] 0:7 0:5 0:8 It is easy to verify that A = heT ; eI ; eF ; T ; I ; F i is a cubic ideal of S. Obviously, every left (resp. right) cubic ideal is a cubic subsemigroup.

But the converse may not be true as seen in the following example Example 3. Consider a semigroup S=fa; b; c; dg with the following Caylay’s table. : a b c d a a a a a b a a a a c a a a b d a a b c De…ns a NC set A=heT ; eI ; eF ; T ; I ; F i in S by S eT eI eF T I F a [0:5; 0:8] [0:7; 0:9] [0:6; 0:8] 0:2 0:1 0:3 b [0:3; 0:6] [0:4; 0:7] [0:3; 0:5] 0:6 0:7 0:4 c [0:5; 0:8] [0:5; 0:6] [0:5; 0:6] 0:4 0:5 0:4 d [0:2; 0:4] [0:2; 0:4] [0:1; 0:3] 0:6 0:8 0:5 It is easy to verify that A = heT ; eI ; eF ; T ; I ; F i is a NC subsemigroup of S; but it is not a left NC ideal of S since eT (dc) = eT (b) =[0:3; 0:6] [0:5; 0:8] = eT (c) and/or T (dc) = T (b) = 0:6 > 0:4 = T (c) : Theorem 2. For NC set A = heT ; eI ; eF ; are equivalent:

T;

I;

Fi

in semigroug S , the following

eI (y)) ; eF (xy) F (x)( F (xy) F(

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

12

(1) A = heT ; eI ; eF ; (2) S A v A:

T;

I;

Fi

is a left NC ideal of S.

Proof. Assume that A = heT ; eI ; eF ; T ; I ; F i is a left NC ideal of S. Let a2 S: If (S A) (a) = h[0; 0]; 1i ; then it is clear that S A v A:Otherewise, there exsists x; y 2 S such that a = xy. Since A = heT ; eI ; eF ; T ; I ; F i is a left NC ideal of S, we have 1S

1S

1S

eT (a)

h n oi = r sup r min 1 S (x); eT (y) a=xy

r sup [r min f[1; 1] ; eT (xy)g] a=xy

eI

= r sup eT (a) = eT (a) h n oi (a) = r sup r min 1 S (x); eI (y)

eF

= r sup eI (a) = eI (a) h n oi (a) = r sup r min 1 S (x); eF (y)

a=xy

r sup [r min f[1; 1] ; eI (xy)g] a=xy

a=xy

r sup [r min f[1; 1] ; eF (xy)g] a=xy

= r sup eF (a) = eF (a)

and (0S

(0S

(0S

T ) (a)

=

= I ) (a) =

= F ) (a) =

= Therfore S

A v A:

a=xy

max f0S (x) ;

a=xy

max f

T (y)g

T (xy)g

T (a) a=xy

= T (a) max f0S (x) ;

a=xy

max f

I (y)g

I (xy)g

I (a) a=xy

= I (a) max f0S (x) ;

a=xy

max f

F (a)

=

F (xy)g F (a)

F (y)g

NC IDEALS

Conversely, suppose that S a = xy: Then eT (xy)

eI (xy)

eF (xy)

and T

(xy)

13

A v A. For any elements x and y of S, let

eT (a) 1 S eT (a) h n oi = r sup r min 1 S (b); eT (c) a=bc n o r min 1 S (x); eT (y) = eT (y) =

eI (a) 1 S eI (a) h n oi = r sup r min 1 S (b); eI (c) a=bc n o r min 1 S (x); eI (y) = eI (y) =

eF (a) 1 S eF (a) h n oi = r sup r min 1 S (b); eF (c) a=bc n o r min 1 S (x); eF (y) = eF (y) =

= =

T

(a) (0S T ) (a) max f0S (b) ; T (c)g

a=bc

max f0S (x) ; T (y)g = (0S I (xy) = I (a) I ) (a) = max f0S (b) ; I (c)g

T (y)

a=bc

F

max f0S (x) ; I (y)g = I (y) (xy) = (0S F (a) F ) (a) = max f0S (b) ; F (c)g a=bc

Hence A = heT ; eI ; eF ;

max f0S (x) ; T;

I;

Fi

F (y)g

=

F (y)

is a left NC ideal of S.

Similarly, we can induces the following theorem.

Theorem 3. For a NS cubic st A = heT ; eI ; eF ; T ; I ; F i in asemigroup S, the following are equivalent: (1) A = heT ; eI ; eF ; T ; I ; F i is a right NS cubic ideal of S. (2) A S v A:

Theorem 4. If A = heT ; eI ; eF ; T ; I ; F i is a NS cubic set in a semigroup S, then S A (resp. A S) is a left (resp: right) NS cubic ideal of S.

Proof. Since S (S A) = (S S) A v S A; it follows from theorem 2 that S A is a left NS cubic ideal of S. Similarly A S is a right NS cubic ideal of S.

Now we will consider conditions for a left (resp. right) NS cubic ideal to be constant.

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

14

Proposition 4. Let U be a left zero subsemigroup of a semigroup S. If A = heT ; eI ; eF ; T ; I ; F i is a left NS cubic ideal of , then A (x) = A (y) for all x; y 2 U .

Proof. Let x; y 2 U: Then xy = x and yx = y. Thus

and

eT (x) = eI (x) = eF (x) =

(x) = I (x) = F (x) = T

eT (xy) eI (xy) eF (xy)

(xy) (xy) I F (xy)

T

eT (y) = eT (yx) eT (x) eI (y) = eI (yx) eI (x) eF (y) = eF (yx) eF (x) (y) = (y) = I F (y) = T

(xy) (xy) I F (xy) T

(x) (x) I F (x) T

Therefore A (x) = A (y) for all x; y 2 U: Similarly, we have the following proposition.

Proposition 5. Let U be a right zero subsemigroup of a semigroup S. If A = heT ; eI ; eF ; T ; I ; F i is a right NS cubic ideal of , then A (x) = A (y) for all x; y 2 U .

Theorem 5. Let A = heT ; eI ; eF ; T ; I ; F i is a left NC ideal of a semigroup S. If the set of all idempotent elements of S forms a left zero subsemigroup of S, then A (u) = A (v) for all idempotent elements u and v of S. Proof. Let Idm (S) be the set of all idempotent elements of S and assume that Idm (S) is a left zero subsemigroup of S. For any u; v 2 Idm (S) ; we have uv = u and uv = v: Hence

and

eT (u) = eI (u) = eF (u) =

(u) = I (u) = F (u) = T

eT (uv) eI (uv) eF (uv)

(uv) I (uv) F (uv) T

eT (v) = eT (uv) eT (u) eI (v) = eI (uv) eI (u) eF (v) = eF (uv) eF (u) (v) = I (v) = F (v) = T

(uv) I (uv) F (uv) T

(u) I (u) F (u) T

Therefore A (u) = A (v) for all u; v 2 Idm (S).

Similarly, we have the following theorem.

Theorem 6. Let A = heT ; eI ; eF ; T ; I ; F i is a right NC ideal of a semigroup S. If the set of all idempotent elements of S forms a right zero subsemigroup of S, then A (u) = A (v) for all idempotent elements u and v of S. Theorem 7. Let S be a semigroup. Then the following properties hold.

(1) The intersection of two NC subsemigroups of S is a NC subsemigroup of S. (2) The intersection of two left (resp. right) NC ideals of S is a left (resp. right) NC ideals of S

NC IDEALS

15

Proof. (1) Let A = heT ; eI ; eF ; T ; I ; F i and B = eT ; eI ; eF ; T; ; two NC subsemigroups of S. Let x and y be any elements of S. Then

I; F

be

(eT \ eT ) (xy)

= r min feT (xy) ; eT (xy)g r min fr min feT (x) ; eT (y)g ; r min feT (x) ; eT (y)gg = r min fr min feT (x) ; eT (x)g ; r min feT (y) ; eT (y)gg = r min f(eT \ eT ) (x) ; (eT \ eT ) (y)g (eI \ eI ) (xy) = r min feI (xy) ; eI (xy)g r min fr min feI (x) ; eI (y)g ; r min feI (x) ; eI (y)gg = r min fr min feI (x) ; eI (x)g ; r min feI (y) ; eI (y)gg = r min f(eI \ eI ) (x) ; (eI \ eI ) (y)g (eF \ eF ) (xy) = r min feF (xy) ; eF (xy)g r min fr min feF (x) ; eF (y)g ; r min feF (x) ; eF (y)gg = r min fr min feF (x) ; eF (x)g ; r min feF (y) ; eF (y)gg = r min f(eF \ eF ) (x) ; (eF \ eF ) (y)g

and

(

(

(

T

I

F

_

T ) (xy)

=

= = _ I ) (xy) = = = _ F ) (xy) = = =

max f T (xy) ; T (xy)g max fmax f T (x) ; T (y)g ; max f T (x) ; T (y)gg max fmax f T (x) ; T (x)g ; max f T (y) ; T (y)gg max f( T _ T ) (x) ; ( T _ T ) (y)g max f I (xy) ; I (xy)g max fmax f I (x) ; I (y)g ; max f I (x) ; I (y)gg max fmax f I (x) ; I (x)g ; max f I (y) ; I (y)gg max f( I _ I ) (x) ; ( I _ I ) (y)g max f F (xy) ; F (xy)g max fmax f F (x) ; F (y)g ; max f F (x) ; F (y)gg max fmax f F (x) ; F (x)g ; max f F (y) ; F (y)gg max f( F _ F ) (x) ; ( F _ F ) (y)g

Therefore A u B = heT \ eT ; eI \ eI ; eF \ eF ; T _ T ; NC subsemigroup of S. The second property can be proved in a similar manner.

Proposition 6. If A = heT ; eI ; eF ; T ; I ; is a left NC ideal of a semigroup S, then A

I

_

I;

F

_

Fi

is a

is right NC ideal and B = eT ; eI ; eF ; B v A u B:

Fi

Proof. Let A = heT ; eI ; eF ; T ; I ; F i is right NC ideal and B = eT ; eI ; eF ; T; ; I ; is any NC ideal of S. Then by Theorem (2) and Theorem (3) we have A B v A S v A and A B v S B v B: Thus A B v A u B: Proposition 7. If S is a regular semigroup, then A B = AuB for every right NC ideal A = heT ; eI ; eF ; T ; I ; F i and every left NC ideal B = eT ; eI ; eF ; T; ; I ; F of S.

T; ; I ; F

F

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

16

Proof. Let a be any element of S. Since S is regular, there exist an element x 2 S such that a = axa: Hence we have eT ) (a)

(eT

= r sup fr min feT (y) ; eT (z)gg a=yz

r min feT (ax) ; eT (a)g r min feT (a) ; eT (a)g = (eT \ eT ) (a) eI ) (a) = r sup fr min feI (y) ; eI (z)gg

(eI

a=yz

r min feI (ax) ; eI (a)g r min feI (a) ; eI (a)g = (eI \ eI ) (a) eF ) (a) = r sup fr min feF (y) ; eF (z)gg

(eF

a=yz

and (

T

r min feF (ax) ; eF (a)g r min feF (a) ; eF (a)g = (eF \ eF ) (a)

T ) (a)

=

^

max f

T

(y) ;

T

(z)g

a=yz

(

I

max f T (ax) ; T (a)g max f T (a) ; T (a)g = ( T \ T ) (a) ^ max f I (y) ; I (z)g I ) (a) = a=yz

(

F

max f I (ax) ; I (a)g max f I (a) ; I (a)g = ( I \ I ) (a) ^ max f F (y) ; F (z)g F ) (a) = a=yz

and so A

max f max f = ( F\

(ax) ; F (a)g F (a) ; F (a)g F ) (a)

F

B w A u B: It follows from Proposition (6) that A

B = A u B:

We now discuss the converse of Proposition (7) : We …rst consider the following lemmas.

Lemma 1. [20] : For a semigroup S, the following conditions are equivalent.

(1) S is regular.

(2) R \ L = RL for every right ideal R of S and every left ideal L of S. Lemma 2. For a non-empty subset G of a semigroup S, we have (1) G is a subsemigroup of S if and only if the characteristic NC set eT ; eI ; eF ; T ; I ; F of G in S is a NC subsemigroup of S.

=

NC IDEALS

17

(2) G is a left (right) ideal of S if and only if the characteristic NC set eT ; eI ; eF ; T ; I ; F of G in S is a left (resp. right) NC ideal of S.

=

Proof. Straight forward

Theorem 8. For every right NC ideal A = heT ; eI ; eF ; T ; I ; F i and every left NC ideal B = eT ; eI ; eF ; T; ; I ; F of a semigroup S, if A B = A \ B; then S is regular.

Proof. Assume that A B = A\B for every right NC ideal A = heT ; eI ; eF ; T ; I ; every left NC ideal B = eT ; eI ; eF ; T; ; I ; F of a semigroup S. Let R and L be any right ideal and left ideal of S, respectively. In order to see that R \ L RL D holds, let a be any element of RE\ L: Then the D characteristic NC sets R =

eT R ; eI R ; eF R ; T R ; I R ; F R and L = eT L ; eI L ; eF L ; T L ; I L ; a right NC ideal and a left NC ideal of S, respectively, by Lemma (2) : It follows from the hypothesis and proposition (3) that eT

RL

(a)

= = =

eT

RL

(a)

= = =

eT

RL

(a)

= = =

and T

RL

(a)

RL

(a)

RL

(a)

eT

eT

eT

eT

eT

eT

eT

T

=

T T I

=

I I

eT

F

L

\ eT

R

R\L

L

eT

L

\ eT

R

R\L

\

(a)

L

(a) (a)

L

T

L

(a) (a)

(a) = [1; 1] I

R

\

(a)

L

I

L

(a)

(a) = [1; 1] F

R

R\L

(a)

(a) = [1; 1]

T

R

(a)

(a) = [1; 1]

R

R\L

=

L

(a)

(a) = [1; 1]

R

R

F

F

R\L

R

L

\ eT

R

R

=

=

eT

R

R\L

=

= F

eT

=

= I

eT

\

F

L

L

(a) (a)

(a) = [1; 1]

Fi

F

L

and

E

are

18

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

and so that a 2 RL: Thus R \ L RL: Since the inclusion in the otherdirection always holds, we obtain that R \ L = RL: It follows from Lemma (1) that S is regular.

Let A = heT ; eI ; eF ; T ; I ; F i be a NC set in X. For any s; o; g 2 [0; 1] and e e t; i; fe 2 D [0; 1] ; we de…ne U (A; [s; t] ; r) as follows: D E n D U A; e t; ei; fe; s; o; g = x 2 Xj eT (x) e t; eI (x) ei; eF (x) fe; T (x) s; I (x) and we say it is a NC level set of A = heT ; eI ; eF ;

T;

I;

Fi

(see [8]) :

Theorem 9. For a NC set A = heT ; eI ; eF ; T ; I ; F i in a semigroup S, the following are equivalent: (1) A = heT ; eI ; eF ; T ; I ; F i is a NC subsemigroup of S. (2) Every non-empty NC level set of A = heT ; eI ; eF ; T ; I ; F i is a subsemigroup of S.

Proof. Assume D that A = E heT ; eI ; eF ; T ; I ; F i is a NC subsemigroup of S. let x,y2 U A; e t; ei; fe; s; o; g for all s; o; g 2 [0; 1] and e t; ei; fe 2 D[0; 1]: Then eT (x) e t; eI (x) ei; eF (x) fe and T (x) s; I (x) o; F (x) g; eT (y) e t; eI (y) ei; eF (y) fe and s; I (y) o; F (y) g: It follows form De…nition T (y) e (2) that eT (xy) r min feT (x) ; eT (y)g t; eI (xy) r min feI (x) ; eI (y)g ei; eF (xy) r min feF (x) ; eF (y)g fe and (xy) maxf T (x) ; T (y)g T s; I (xy) maxf (x) ; (y)g o; (xy) maxf (x) ; F (y)g g:Hence I F FE D EI D e e x; y 2 U A; e t; ei; f ; s; o; g and thus U A; e t; ei; f ; s; o; g is a subsemigroup of S. D E Conversly, let s; o; g 2 [0; 1] and e t; ei; fe 2 D[0; 1] be a such that U A; e t; ei; fe; s; o; g 6= D E ;; and U A; e t; ei; fe; s; o; g is a subsemigroup of S. Suppose that De…nition (2) is false. Then there exist a; b 2 S such that eT (ab) r min feT (a) ; eT (b)g or eI (ab) r min feI (a) ; eI (b)gor eF (ab) r min feF (a) ; eF (b)g or T (ab) maxf T (a) ; T (b)g:or I (ab) maxf I (a) ; I (b)g or F (ab) maxf F (a) ; F (b)g: If eT (ab) r min feT (a) ; eT (b)g ; D then eT (ab) E e t r min feT (a) ; eT (b)g for some e t 2 D[0; 1]: Hence a; b 2 U A; e t; ei; fe; s; o; g ; but D E ab 2 = U A; e t; ei; fe; s; o; g

ThisD gives a contradiction.

Similar results can be deduced for any component E U A; e t; ei; fe; s; o; g : Hence De…nition (2) is valid, and therefore A = heT ; eI ; eF ; T ; I ; F i is a NC subsemigroup of S.

Theorem 10. For a NC set A = heT ; eI ; eF ; T ; I ; F i in a semigroup S, the following are equivalent: (1) A = heT ; eI ; eF ; T ; I ; F i is a left (resp. right) NC ideal of S. (2) Every non-empty NC level set of A = heT ; eI ; eF ; T ; I ; F i is a left (resp. right) NC ideal of S. Proof. It can be easily veri…ed by the similar way to the proof of Theorem(9) :

o;

F

(x)

g

Eo

NC IDEALS

19

Denote by N (X) the family of NC sets in a sets X. Let X and Y be given classical sets. A mapping h : X ! Y induces two mapping Nh : N (X) !

N (Y ); A 7 ! Nh (A); and Nh 1 : N (Y ) ! N (X); B 7 ! Nh 1 (B); where Nh (A) is given by

Nh (eT )(y)

=

Nh (eI )(y)

=

Nh (eF )(y)

Nh (

T )(y)

Nh ( Nh (

=

=

I )(y)

=

F )(y)

=

(

r sup eT (x)

1

if h

(y) 6= 0

y=h(x)

[0; 0]

(

otherewise

r sup eI (x)

1

if h

y=h(x)

[0; 0]

(

y=h(x)

(

y=h(x)

otherewise

r sup eF (x) T

1

if h

(y) 6= 0

otherewise

inf

I

y=h(x)

(x)

1

(

(y) 6= 0

otherewise

(x)

1

(

1

if h

[0; 0] inf

(y) 6= 0

if h

1

(y) 6= 0

otherewise

inf

F

y=h(x)

(x)

1

if h

1

(y) 6= 0

otherewise

for all y 2 Y ; and Nh 1 (B) is de…ned by Nh 1 (eT )(x) = eT (h(x)); Nh 1 (eI )(x) = eI (h(x)); Nh 1 (eF )(x) = eF (h(x)) and Nh 1 ( T )(x) = T (h(x)); Nh 1 ( I )(x) = 1 1 I (h(x)); Nh ( F )(x) = F (h(x)) for all x 2 X: Then the mapping Nh (resp: Nh ) is called a NC transformation (resp. inverse NC transformation) induced by h. A NC set A = heT ; eI ; eF ; T ; I ; F i in X has the NC property if for any subset C of X there exists x0 2 C such that eT (x0 ) = r supeT (x) ; eI (x0 ) = r supeI (x) ; eF (x0 ) = x2C

r supeF (x) and

T (x0 )

= inf

x2C

x2C

T

(x) ;

x2C

I (x0 )

= inf

x2C

I

(x) ;

F (x0 )

= inf

x2C

F

(x) :

Theorem 11. For a homomorphsim h : X ! Y of semigroups, let Nh : N (X) !

N (Y ) and Nh 1 : N (Y ) ! N (X) be the NC transformation and inverse NC transformation, respectively, induced by h.

(1) If A = heT ; eI ; eF ; T ; I ; F i 2 N (X) is a NC subsemigroup of X which has the NC property, then Nh (A) is a NC subsemigroup of Y. (2) If B = eT ; eI ; eF ; T; ; I ; F 2 N (Y ) is a NC subsemigroup of Y, then Nh 1 (B) is a NC subsemigroup of X. Proof. (1) Given h(x):h(y) 2 h(X); let x0 2 h such that eT (x0 )

T (x0 )

=

r sup a2h

=

a2h

1 (h(x))

inf 1

(h(x))

eT (a) ; eI (x0 ) = T

(a) ;

I (x0 )

=

1

(h(x)) and y0 2 h

r sup a2h a2h

1 (h(x))

inf 1

(h(x))

eI (a) ; eF (x0 ) = I

(a) ;

F (x0 )

=

1

(h(y)) be

r sup a2h a2h

1 (h(x))

inf 1

(h(x))

eF (a) F

(a)

M AJID KHAN 1 AND M UHAM M AD GULISTAN 2

20

and eT (y0 )

T (y0 )

=

r sup b2h

=

1 (h(y))

T

(b) ;

Nh (eT ) (h (x) h (y))

=

b2h

inf 1

eT (b) ; eI (y0 ) =

(h(y))

I (y0 )

=

r sup b2h b2h

1 (h(y))

inf 1

(h(y))

eI (b) ; eF (y0 ) = I

(b) ;

F (y0 )

=

r sup 1 (h(y))

b2h b2h

inf 1

(h(y))

eF (b) F

(b)

respectively. Then r sup z2h

1 (h(x)h(y))

eT (x0 y0 ) (

= r min

eT (z)

r min feT (x0 ) ; eT (y0 )g

r sup

a2h

1 (h(x))

eT (a) ;

r sup

b2h

1 (h(y))

eT (b)

= r min fNh (eT ) (h (x)) ; Nh (eT ) (h (y))g Nh (eI ) (h (x) h (y)) = r sup eI (z) z2h

1 (h(x)h(y))

eI (x0 y0 ) (

= r min

r min feI (x0 ) ; eI (y0 )g

r sup

a2h

Nh (eF ) (h (x) h (y))

1 (h(x))

eI (a) ;

r sup

b2h

1 (h(y))

)

eI (b)

= r min fNh (eI ) (h (x)) ; Nh (eI ) (h (y))g = r sup eF (z) z2h

)

1 (h(x)h(y))

eF (x0 y0 ) (

= r min

r min feF (x0 ) ; eF (y0 )g

r sup

a2h

1 (h(x))

eF (a) ;

r sup

b2h

1 (h(y))

)

eF (b)

= r min fNh (eF ) (h (x)) ; Nh (eF ) (h (y))g References

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NC IDEALS

21

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[19] Y. B.Jun "Neutrosophic Cubic sets"ovember 9, 2015 8:41 WSPC/INSTRUCTION FILE JSK151001R0-1108 [20] K. Iseki, A characterization of regular semigroups, Proc. Japan Acad., 32 (1965),676-677.

(1) Majid Khan, Department Of Mathematics Hazara University, Mansehra, KP, Pakistan E-mail address, 1: majid_swati@yahoo.com Current address, 2: Muhammad Gulistan, Department Of Mathematics Hazara University, Mansehra,KP, Pakistan E-mail address, 2: gulistanm21@yahoo.com