HESITANT FUZZY SETS APPROACH TO IDEAL THEORY IN TERNARY SEMIGROUPS

Volume 31 No. 4 2018, 527-539

ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi:_{http://dx.doi.org/10.12732/ijam.v31i4.2}

HESITANT FUZZY SETS APPROACH TO IDEAL THEORY IN TERNARY SEMIGROUPS

A. Fairooze Talee1§, M. Yahya Abbasi2, S. Ali Khan3

1,2,3_{Department of Mathematics}

Jamia Millia Islamia New Delhi – 110 025, INDIA

Abstract: In this paper, we introduce hesitant fuzzy ideals on ternary semi-groups and define hesitant fuzzy left (resp. lateral, right)ideals on ternary semigroups. Here we define hesitant fuzzy point on ternary semigroups and prove some interesting results on them. We also characterize regular ternary semigroups in terms of hesitant fuzzy left, hesitant fuzzy lateral and hesitant fuzzy right ideals using hesitant fuzzy points.

AMS Subject Classification: 20M12, 20N99, 08A72

Key Words: ternary semigroups, hesitant fuzzy ideals

1. Introduction and Preliminaries

The idea of investigation ofn-ary algebras, i.e. the sets with onen-ary operation was given by Kasner [4]. A ternary semigroup is a particular case of an n-ary semigroup for n= 3, [5]. The ternary semigroups are universal algebras with one associative operation. The ideal theory in ternary semigroup was studied by Sioson [7].

Zadeh [11] introduced the concept of fuzzy set as an extension of the classical notion of set. The fuzzy set theory can be used in a wide range of domains to manipulate different types of uncertainties. Torra [8, 9] introduced hesitant

Received: January 20, 2018 c 2018 Academic Publications

fuzzy sets (briefly HFSs) as a new generalization of fuzzy set. Hesitant fuzzy set is used to control hesitant situation which were difficult to handle by previous extensions. Jun et. al. [2] applied the HFSs in semigroups. Abbasi et al. [6] applied the notion of HFSs to ordered semigroups.

Kar and Sarkar [3] introduced fuzzy left (right, lateral) ideals in ternary semigroups and characterized regular and intra-regular ternary semigroups by using the concept of fuzzy ideals of ternary semigroups. Our main aim in this study is to use the idea of Jun and Kar to introduce the notion of hesitant fuzzy ideals in ternary semigroups and study their related properties. Here we define hesitant fuzzy left (right, lateral) ideals of ternary semigroups and characterize regular and intra-regular ternary semigroups by using the concept of hesitant fuzzy ideals of ternary semigroups.

Definition 1. [5] A non-empty set S with a ternary operation S×S×

S → S, written as (x1, x2, x3)7→[x1, x2, x3], is called a ternary semigroup if it

satisfies the following identity, for anyx1, x2, x3, x4, x5 ∈S,

[[x1x2x3]x4x5] = [x1[x2x3x4]x5] = [[x1x2[x3x4x5]].

Example 2. Let S=

(



0 0 a

0 b 0

c 0 0



: a, b, c ∈ N∪ {0}

)

. Then S is

a ternary semigroup under the usual multiplication of matrices over N ∪{0} while S is not a semigroup under the same operation.

If A = {a}, then we write [{a}BC] as [aBC] and similarly if B = {b} or C={c}, we write [AbC] and [ABc], respectively. For the sake of simplicity, we write [x1x2x3] asx1x2x3 and [ABC] as ABC.

Definition 3. A non-empty subsetT of a ternary semigroupS is called a ternary subsemigroup ofS if T T T ⊆T.

Definition 4. An element ain a ternary semigroup S is called regular if there exists an elementx inS such that axa=a.

Torra defined HFSs in terms of a function that returns a set of membership values for each element in the domain. We give some background on the basic notions on HFSs. For more details, we refer to references.

Hthat when applied toX returns a subset of [0,1]. For a HFS HonS andx, y ∈S, we use the notationsHx :=H(x) andHx =Hy ⇔ Hx ⊆ Hy,Hx ⊇ Hy. Let H and G be two HFSs on S. The hesitant union H ⊔ G and hesitant intersection H ⊓ G ofH and G are defined to be HFSs onS by:

H ⊔ G : S → P([0,1]), x 7→ Hx ∪ Gx

and

H ⊓ G : S → P([0,1]), x 7→ Hx ∩ Gx

For any HFSsHand G onS, we define

H ⊑ G if Hx ⊆ Gx for all x∈ S.

For a non-empty subsetAofS andε,δ∈P([0,1]) withε)δ, define a map [χ(ε,δ)_A ] as follows:

[χ(ε,δ)_A ] : S →P([0,1]), x7→ (

ε, if x∈A,

δ, otherwise .

Then [χ(ε,δ)_A ] is a HFS on S, which is called the (ε, δ)-characteristic HFS. The HFS [χ(ε,δ)_S ] is called the (ε, δ)-identity HFS onS. The (ε, δ)-characteristic HFS with ε= [0,1] and δ =∅ is called the characteristic HFS, and is denoted by [χA]. The (ε, δ)-identity HFS withε= [0,1] andδ =∅is called the identity

HFS, and is denoted by [χS].

2. Hesitant Fuzzy Ideals on Ternary Semigroups

Definition 5. A non-empty hesitant fuzzy subset H of S is called a hesitant fuzzy ternary subsemigroup on S if Huvw ⊇ Hu ∩ Hv ∩ Hw for all u, v, w∈ S.

Definition 6. A non-empty hesitant fuzzy subset H of S is called a hesitant fuzzy left (resp. lateral, right) ideal on S if Huvw ⊇ Hw (Huvw ⊇ Hv,

Example 7. LetS=

(

O=





0 0 0 0 0 0 0 0 0



, X = 



0 0 1 0 0 0 0 0 0



,

Y =





0 0 0 0 1 0 0 0 0



, Z = 



0 0 0 0 0 0 1 0 0



, W = 



0 0 1 0 1 0 1 0 0



 )

. ThenSis a ternary

semigroup under the usual multiplication of matrices.

ConsiderL ={O, Z}. ThenL is a left ideal of ternary semigroup S. Now we define a HFS Lon S by:

L:S→P[0,1], x 7→ (

(0.7,0.9], if x= O or Z,

∅, if x= X or Y or W

Hence we can easily show that Lis a hesitant fuzzy left ideal on S.

We say that H is a hesitant fuzzy two sided ideal if H is both a hesitant fuzzy left ideal and a hesitant fuzzy right ideal on S. H is called a hesitant fuzzy ideal ifH is a hesitant fuzzy left ideal, a hesitant fuzzy lateral ideal and a hesitant fuzzy right ideal onS.

Lemma 8. LetS be a ternary semigroup. IfAis a non-empty subset and

[χ(ε,δ)_A ]is a(ε, δ)-characteristic HFS on S. Then,

(1) A is a ternary subsemigroup on S if and only if [χ(ε,δ)_A ] is a hesitant fuzzy ternary subsemigroup onS.

(2) A is a left(resp. lateral, right) ideal on S if and only if [χ(ε,δ)_A ] is a hesitant fuzzy left(resp. lateral, right)ideal on S.

Proof. The proof is a straightforward.

Definition 9. Let H,I,K ∈ HF(S), whereHF(S) is the set of all HFS on S . The product of H,I, K denoted by H ˆ⋄ I ˆ⋄ K is defined as, for any x

∈S,

(Hˆ⋄ I ˆ⋄ K)x= 



 S

x=uvw{Hu∩ Iv∩ Kw}

if ∃ u, v, w ∈ S s.t. x = uvw

Lemma 10. Let H ∈HF(S) such that Hu ⊆εfor all u ∈S and [χ(ε,δ)_S ]

be a(ε, δ)-identity HFS on S. Then,

(1)His a hesitant fuzzy ternary subsemigroup on S if and only ifH ˆ⋄ Hˆ⋄ H ⊑ H.

(2) His a hesitant fuzzy left ideal on S if and only if [χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄ H ⊑ H.

(3)His a hesitant fuzzy lateral ideal onS if and only if[χ(ε,δ)_S ] ˆ⋄ Hˆ⋄[χ(ε,δ)_S ]

⊑ H.

(4)His a hesitant fuzzy right ideal on S if and only ifHˆ⋄ [χ(ε,δ)_S ] ˆ⋄[χ(ε,δ)_S ]

⊑ H.

Proof. (1) Let H be a hesitant fuzzy ternary subsemigroup on S and x ∈

S. Now we have following two cases:

Case 1: _{If h 6=uvwfor any u,v,w ∈S, then (Hˆ⋄ H ˆ⋄ H)h =∅ ⊆ Hh.} Case 2: _{If there exists h such thath =uvw for someu,v,w∈S, then}

(Hˆ⋄ H ˆ⋄ H)h = S h=uvw

{Hu∩ Hv∩ Hw}

⊆ S

h=uvw

{Huvw}

= S

h=uvw {Hh}

= Hh.

Hence in all the cases Hˆ⋄ H ˆ⋄ H ⊑ H.

Conversely, supposeHˆ⋄ Hˆ⋄ H ⊑ Hfor any H ∈HF(S) and letu,v,w∈

S. AsS is a ternary semigroup, we have uvw ∈S. Leth =uvw. Then,

Huvw = Hh

⊇ (Hˆ⋄ H ˆ⋄ H)h

= S

h=rst

{Hr∩ Hs∩ Ht}

⊇ Hu∩ Hv∩ Hw.

ThereforeH is a hesitant fuzzy ternary subsemigroup onS.

(2) Let H be a hesitant fuzzy left ideal on S and h ∈ S. Then we have following two cases:

Case 1: _{Ifh 6=uvw for anyu,v,w∈ S, then ([χ}(ε,δ)

S ] ˆ⋄[χ (ε,δ)

Case 2: _{If there exists} h such thath =uvw for someu,v,w∈S, then

([χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄ H)h = S h=uvw

{[χ(ε,δ)_S ]u∩[χ(ε,δ)S ]v∩ Hw}

= S

h=uvw

{ε ∩ ε ∩ Hw}

= S

h=uvw

{Hw} as ([χ(ε,δ)_S ]u = [χ(ε,δ)S ]v = ε)

⊆ S

h=uvw

{Huvw}

= S

h=uvw {Hh}

= Hh.

ThereforeH ˆ⋄ H ˆ⋄ H ⊑ H.

Conversely, suppose [χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄ H ⊑ H for anyH ∈HF(S) and let u, v, w ∈ S. As S is a ternary semigroup, we have uvw ∈ S. Let h = uvw. Then,

Huvw = Hh

⊇ ([χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄ H)h

= S

h=rst

{[χ(ε,δ)_S ]r∩[χ(ε,δ)_S ]s∩ Ht} ⊇ [χ(ε,δ)_S ]u∩[χ(ε,δ)S ]v∩ Hw

= Hw (As [χ(ε,δ)_S ]u = [χ(ε,δ)_S ]v = ε).

Hence His a hesitant fuzzy left ideal on S. (3) and (4) are analogous to (2).

Example 11. Consider S as given in the Example 7. Let [χ(ε,δ)_S ] be a (ε, δ)-identity HFS on S, where ε= [0,1], δ =∅. Now we define HFS Ron S by:

R:S →P[0,1], x 7→ (

(0.5,0.8], if x= O or X,

∅, if x= Y or Z or W

Then it is easy to verify that Rˆ⋄ [χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ]⊑ R. Hence Ris a hesitant fuzzy right ideal onS.

M:S →P[0,1], x 7→     

(0.7,0.8], if x= O , (0.5,0.65], if x= X or Y

∅, if x= Z or W .

ForZ ∈ S,MZ =∅. Now

Z =





0 0 0 0 0 0 1 0 0



= 



0 0 1 0 1 0 1 0 0



 



0 0 1 0 0 0 0 0 0



 



0 0 1 0 1 0 1 0 0



.

It impliesZ =W XW. Hence we have

([χ(ε,δ)_S ] ˆ⋄ Mˆ⋄ [χ(ε,δ)_S ])Z = S Z=U V P

{[χ(ε,δ)_S ]U ∩ MV ∩ [χ(ε,δ)_S ]P}

= {[χ(ε,δ)_S ]W ∩ MX ∩ [χ(ε,δ)S ]W}

= ε ∩ (0.5,0.65] ∩ ε = (0.5,0.65].

Then it can be seen that ([χ(ε,δ)_S ] ˆ⋄ Mˆ⋄[χ(ε,δ)_S ])Z *MZ. Hence [χ(ε,δ)S ] ˆ⋄ Mˆ⋄

[χ(ε,δ)_S ] not⊑ M. ThereforeMis not a hesitant fuzzy lateral ideal on S.

Lemma 13. IfH andG are two hesitant fuzzy ternary subsemigroups on

S, thenH ⊓ G is also a hesitant fuzzy ternary subsemigroup on S.

Proof. The proof is a straightforward.

Lemma 14. If H and G are two hesitant fuzzy left(resp. lateral, right) ideals on S and [χ(ε,δ)_S ] be a (ε, δ)-identity HFS on S. Then H ⊓ G is also a hesitant fuzzy left (resp. lateral, right) ideal onS.

Proof. The proof is a straightforward.

Definition 15. Let S be a non-empty set and h ∈S. Let ε⊆[0,1] such that ε 6= ∅. A hesitant fuzzy point ¯hε on S is a hesitant fuzzy subset of S

defined by:

(¯hε)x= (

ε, if x=h,

We will denote the set of all hesitant fuzzy points byHF P(S).

Proposition 16. Let α, ε,β ∈ P([0,1]) and g¯α, ¯hε,¯iβ ∈ HF P(S) such

thatα ∩ ε∩β 6=∅, theng¯α ˆ⋄¯hε ˆ⋄¯iβ =(ghi)α ∩ ε∩ β.

Proof. The proof is a straightforward.

Proposition 17. Let¯hα ∈ HF P(S) such that α ⊆ ε. Then for any x ∈

S,

(1)([χ(ε,δ)_S ] ˆ⋄[χ_S(ε,δ)] ˆ⋄ ¯hα)x = (

α, if x∈SSh,

∅, otherwise.

(2)([χ(ε,δ)_S ] ˆ⋄¯hα ˆ⋄ [χ(ε,δ)_S ])x = (

α, if x∈ShS,

∅, otherwise.

(3)(¯hα ˆ⋄[χ(ε,δ)S ] ˆ⋄ [χ (ε,δ) S ])x =

(

α, if x∈hSS,

∅, otherwise.

Proof. (1) Let x∈ S. We have following two cases:

Case 1: _{If x 6=rstfor any r,s,t∈ S, then ([χ}(ε,δ)

S ] ˆ⋄ [χ (ε,δ)

S ] ˆ⋄ ¯hα)x =∅.

Case 2: _{If x =rstfor somer,s,t ∈S, then}

([χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄h¯α)x = S x=rst

{[χ(ε,δ)_S ]r ∩ [χ_S(ε,δ)]s ∩ (¯hα)t}.

Subcase 1: _{Ifx 6=uvhfor anyu,v ∈S, then ([χ}(ε,δ)

S ] ˆ⋄[χ (ε,δ)

S ] ˆ⋄h¯α)x =∅.

Subcase 2: _{If there exist x such that x = uvh for some u, v ∈ S, then x}

∈SSh and hence

([χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄ ¯hα)x = S x=rst

{[χ(ε,δ)_S ]r ∩ [χS(ε,δ)]s ∩ (¯hα)t}.

= [χ(ε,δ)_S ]u ∩ [χ_S(ε,δ)]v ∩ (¯hα)h

= ε ∩ ε ∩ α = α.

Hence the result is proved.

Corollary 18. Let¯hα ∈HF P(S), then for any x∈ S,

(¯hα⊔([χ(ε,δ)S ] ˆ⋄ [χ (ε,δ)

S ] ˆ⋄¯hα))x = (

α, if x=h or x∈SSh,

∅, otherwise.

Proof. It is easy to show that ¯hα⊔([χ(ε,δ)S ] ˆ⋄[χ (ε,δ)

S ] ˆ⋄¯hα) is a hesitant fuzzy

left ideal on S containing ¯hα. Ifx = h, then (¯hα⊔([χ(ε,δ)S ] ˆ⋄ [χ (ε,δ)

S ] ˆ⋄ ¯hα))x =

(¯hα)h∪([χ(ε,δ)_S ] ˆ⋄ [χ_S(ε,δ)] ˆ⋄ ¯hα)h =α ∪ ∅=α.

If x = uvh ∈ S for some u, v ∈ S, then (¯hα⊔([χ(ε,δ)S ] ˆ⋄ [χ (ε,δ)

S ] ˆ⋄ ¯hα))x =

(¯hα)uvh∪([χ(ε,δ)S ] ˆ⋄[χ (ε,δ)

S ] ˆ⋄ ¯hα)uvh =∅ ∪α(by the Proposition 17(1)) =α.

Ifx6=h andx 6=uvh for anyu,v ∈S, then (¯hα⊔([χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄h¯α))x

=∅. Hence proof the result.

Definition 19. Let H ∈HF(S) and ¯hα ∈ HF P(S). Then we say ¯hα ∈ Hif (H)h ⊇α.

Proposition 20. For anyH ∈ HF(S) and¯hα,¯gβ ∈ HF P(S). Then,

(1)¯hα ∈ H if and only if ¯hα ⊑ H.

(2)H = S ¯ hα∈H

¯ hα.

(3)¯hα ⊑g¯β if and only if h =g andα ⊆ β.

(4)¯hα ˆ⋄ H ˆ⋄ ¯gβ = S ¯ zγ∈H

¯

hα ˆ⋄z¯γ ˆ⋄ ¯gβ.

Definition 21. (1) Let S be a ternary semigroup and ¯hα be a hesitant

fuzzy point onS. Then the hesitant intersection of all hesitant fuzzy left ideals on S containing ¯hα is a hesitant fuzzy left ideal on S containing ¯hα, denoted

by <¯hα >Land defined as

<¯hα>L = d ¯

hα∈H∈HF LL(S) H,

where HF LL(S) is the set of all hesitant fuzzy left ideals on S. < ¯hα >L is

said to be hesitant fuzzy left ideal generated by hesitant fuzzy point ¯hα.

Analogously, we can define<h¯α>M and<¯hα >Rrespectively, the hesitant

fuzzy lateral ideal and hesitant fuzzy right ideal generated by the hesitant fuzzy point ¯hα.

(<¯hα >L)x= (

α, if x∈< h >L, ∅, otherwise.

Proof. As we know that < ¯hα >L = d ¯

hα∈H∈HF LL(S)

H. Hence for any x ∈

S, (<¯hα>L)x = ( d ¯

hα∈H∈HF LL(S)

H)x = ( T ¯

hα∈H∈HF LL(S)

)Hx.

Ifx =h, then (<¯hα >L)x = (<¯hα >L)h = ( T ¯

hα∈H∈HF LL(S)

)(H)h =α (As

¯

hα ∈ H, (H)h ⊇ α).

Ifx=uvhfor someu,v ∈S, then for any hesitant fuzzy left idealHon S,

Huvh ⊇ Hh ⊇α and therefore (<¯hα>L)x = ( T ¯

hα∈H∈HF LL(S)

)(H)uvh = α.

Ifx 6= h and x 6=uvh for anyu,v ∈S, then by the Corollary 18, we have (<¯hα>L)x = ∅. Hence the result is proved.

Proposition 23. Let¯hα ∈HF P(S). For anyx ∈S,

(1)(<¯hα >M)x = (

α, if x∈< h >M, ∅, otherwise.

(2)(<¯hα >R)x = (

α, if x∈< h >R, ∅, otherwise.

Proof. Proofs of (1) and (2) are analogous to Proposition 22.

Proposition 24. Let¯hα ∈HF P(S). Then,

(1)<¯hα>L =¯hα⊔([χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄ ¯hα).

(2)<¯hα>M =¯hα⊔([χ(ε,δ)_S ] ˆ⋄h¯α ˆ⋄ [χ(ε,δ)_S ])⊔([χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄¯hα

ˆ

⋄ [χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ]).

(3)<¯hα>R =¯hα⊔(¯hα ˆ⋄[χ(ε,δ)S ] ˆ⋄ [χ (ε,δ) S ]).

Proof. It is easy to show that ¯hα⊔([χ(ε,δ)_S ] ˆ⋄[χ(ε,δ)_S ] ˆ⋄¯hα) is a hesitant fuzzy

left ideal on S containing ¯hα. Now it remains to show that it is the smallest

hesitant fuzzy left ideal onS containing ¯hα. For this let G be another hesitant

fuzzy left ideal onS containing ¯hα such that G ⊑¯hα⊔([χ(ε,δ)_S ] ˆ⋄[χ(ε,δ)_S ] ˆ⋄ ¯hα).

As ¯hα ∈ G, we have [χ(ε,δ)S ] ˆ⋄ [χ (ε,δ)

S ] ˆ⋄ h¯α ⊑ [χ(ε,δ)S ] ˆ⋄ [χ (ε,δ)

([χ(ε,δ)_S ] ˆ⋄ [χ(ε,δ)_S ] ˆ⋄ ¯hα) is a smallest hesitant fuzzy left ideal on S containing

hesitant fuzzy point ¯hα. Therefore<h¯α>L = ¯hα⊔([χ(ε,δ)S ] ˆ⋄[χ (ε,δ)

S ] ˆ⋄ ¯hα).

Proofs of (2) and (3) are analogous to (1).

Definition 25. A ternary semigroupSis called regular if for each element aof S, there exists an element x inS such that axa=a.

Theorem 26. Let S be a ternary semigroup. Then the following state-ments are equivalent:

(1)S is regular;

(2)Rˆ⋄ Mˆ⋄ L=R⊓ M⊓ L, for every hesitant fuzzy right idealR, hesitant fuzzy lateral idealM and hesitant fuzzy left ideal L onS and Ru,Mu,Lu ⊆

εfor all u ∈S;

(3)<r¯α >R ˆ⋄ <m¯α >M ˆ⋄ <¯lα >L = <r¯α>R⊓<m¯α >M ⊓<¯lα >L

for all r,m,l ∈S and α ⊆ε such thatα 6=∅;

(4)<¯hα >R ˆ⋄ <h¯α >M ˆ⋄ <¯hα >L =<¯hα >R⊓<h¯α >M ⊓<¯hα>L

for all h ∈S and α ⊆ε such thatα 6=∅.

Proof. (1)⇒ (2).

Let S be a regular ternary semigroup and let R, M, L be hesitant fuzzy right ideal, hesitant fuzzy lateral ideal and hesitant fuzzy left ideal onS respec-tively. Now

Rˆ⋄ M ˆ⋄ L ⊑ R ˆ⋄[χ_S(ε,δ)] ˆ⋄ [χ(ε,δ)_S ]⊑ R,

Rˆ⋄ M ˆ⋄ L ⊑ [χ(ε,δ)_S ] ˆ⋄ M ˆ⋄ [χ(ε,δ)_S ]⊑ M,

Rˆ⋄ Mˆ⋄ L ⊑ [χ(ε,δ)_S ] ˆ⋄[χ(ε,δ)_S ] ˆ⋄ L ⊑ L. It impliesRˆ⋄ Mˆ⋄ L ⊑ R⊓ M⊓ L. AsS is regular, hence for anyx ∈ S, there existh ∈ S such that x =xhx and sox =xhx= xhxhx. Now

(Rˆ⋄ M ˆ⋄ L)x = S x=uvw

{(R)u∩(M)v∩(L)w} ⊇ (R)x∩(M)hxh∩(L)x ⊇ (R)x∩(M)x∩(L)x

= (R ∩ M ∩ L)x.

ThereforeR ⊓ M ⊓ L ⊑ Rˆ⋄ M ˆ⋄ L. HenceR ˆ⋄ M ˆ⋄ L=R ⊓ M ⊓ L. It is easy to verify (2) ⇒ (3) and (3)⇒ (4).

Let h ∈ S. Then ¯hα ∈ < ¯hα >R ⊓ < ¯hα >M ⊓ < ¯hα >L. Given that

<¯hα>R ˆ⋄ <¯hα >M ˆ⋄ <¯hα >L =<¯hα >R⊓<¯hα >M ⊓<¯hα>L. Now

¯

hα ∈ <h¯α>R⊓<¯hα >M ⊓<¯hα >L

= <h¯α>R ˆ⋄ <h¯α >M ˆ⋄ <¯hα>L ⊑ ¯hα ˆ⋄ [χ(ε,δ)S ] ˆ⋄ ¯hα

= S

¯

zγ∈[χ(ε,δ)_S ]

(¯hα ˆ⋄ z¯γ ˆ⋄h¯α) (By the proposition20 ).

Hence from the above expression there exists ¯zα ∈ HF P(S) such that ¯hα ⊑

¯

hα ˆ⋄ z¯α ˆ⋄ ¯hα = (hzh)α. Thus by Proposition 20, h = hzh for some z ∈ S.

Hence S is regular.

Definition 27. LetS be a ternary semigroup andH ∈ HF(S). His said to be idempotent if H=H ˆ⋄ H ˆ⋄ H.

Lemma 28. LetS be a regular ternary semigroup. Then every hesitant fuzzy lateral ideal on S is idempotent.

Proof. The proof is a straightforward.

Definition 29. A ternary semigroupS is called left (resp. lateral, right) simple if S contains no proper left (resp. lateral, right) ideal of S.

Definition 30. A ternary semigroupS is called hesitant fuzzy left (resp. lateral, right) simple if every hesitant fuzzy left (resp. lateral, right) ideal onS is constant function.

Theorem 31. LetS be a ternary semigroup. ThenS is left (resp. lateral, right) simple if and only if S is hesitant fuzzy left (resp. lateral, right) simple ternary semigroup.

Proof. Suppose S is left simple and H is a hesitant fuzzy left ideal on S. Let h, k ∈ S. Then SSh and SSk are the left ideal of S. As S is left simple, it implies SSh =S and SSk =S. So there exists t,u, v, w ∈S s.t. k = tuh and h =vwk . It follows that Hk = Htuh ⊇ Hh = Hvwk ⊇ Hk. Hence Hh =

Hkfor allh,k∈S. Therefore His a constant function and henceS is hesitant fuzzy left simple.

S. By Lemma 8, [χ(ε,δ)_A ] is a hesitant fuzzy left ideal onS. Now by assumption, [χ(ε,δ)_A ] is a constant function. As A is a left ideal, it is non-empty. If a ∈ A, then [χ(ε,δ)_A ]a =ε. Therefore for any elementh ∈S, [χ(ε,δ)A ]h =ε, which implies

h ∈A. HenceS ⊆ A. ConsequentlyS is simple.

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