fuzzy \Gamma-ideals of \Gamma-semigroups

Applied Mathematical Sciences, Vol. 6, 2012, no. 101, 5013 - 5028

A Note on (∈, ∈ ∨qk )-Fuzzy Γ-Ideals of Γ-Semigroups Faisal Department of Mathematics COMSATS Institute of Information Technology Abbottabad, Pakistan [email protected] Naveed Yaqoob Department of Mathematics Quaid-i-Azam University Islamabad, Pakistan [email protected] Ronnason Chinram Department of Mathematics and Statistics Prince of Songkla University, Songkhla, Thailand Centre of Excellence in Mathematics, CHE Si Ayuthaya Road, Bangkok, 10400 Thailand [email protected] Abstract In this paper, we study some structural properties of Γ-semigroups by using a number of special fuzzy subsets. In this regard, we have characterized different classes of a Γ-semigroup in terms of (∈, ∈ ∨qk )fuzzy Γ-left (right, two-sided, bi-, generalized-bi-, quasi, interior) ideals. Also we introduce a method to construct a Γ-semigroup.

Mathematics Subject Classification: 03E72, 20M12, 20M99 Keywords: Γ-semigroups; fuzzy points; (∈, ∈ ∨qk )-fuzzy Γ-ideals

1

Introduction

In [15], Murali defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset. The idea of quasicoincidence of a fuzzy point with a fuzzy set is defined in [16], played a vital

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Faisal, N. Yaqoob and R. Chinram

role to generate some different types of fuzzy subgroups. It is worth pointing out that Bhakat and Das [2, 3] gave the concept of (α, β)-fuzzy subgroups by using the “belongs to” relation ∈ and “quasi-coincident with” relation q between a fuzzy point and a fuzzy subgroup, and introduced the concept of an (∈, ∈ ∨q)-fuzzy subgroups, where α, β ∈ {∈, q, ∈ ∨q, ∈ ∧q} and α =∈ ∧q. In particular, (∈, ∈ ∨q)-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. These fuzzy subgroups are further studied in [4, 5]. The concept of (∈, ∈ ∨qk )-fuzzy subgroups is a viable generalization of Rosenfeld’s fuzzy subgroups. (∈, ∈ ∨qk )-fuzzy subrings and ideals are defined in [6]. Davvaz [7], introduced the concept of (∈, ∈ ∨q)-fuzzy subnearrings and ideals. Jun and Song initiated the study of (α, β)-fuzzy interior ideals of a semigroup in [8] which is the generalization of fuzzy interior ideals [9]. Kuroki [13, 14], applied fuzzy set theory to ideals in semigroups. In [12], Kazanci and Yamak studied (∈, ∈ ∨qk )-fuzzy bi-ideals of a semigroup. In [19], regular semigroups are characterized by the properties of (∈, ∈ ∨qk )-fuzzy ideals. Generalizing the concept of the quasi-coincident of a fuzzy point with a fuzzy subset, Jun [10] defined (∈, ∈ ∨qk )-fuzzy subgroups and (∈, ∈ ∨qk )-fuzzy subalgebras in BCK/BCI-algebras, respectively. In [18], (∈, ∈ ∨qk )-fuzzy ideals, (∈, ∈ ∨qk )-fuzzy quasi-ideals and (∈, ∈ ∨qk )-fuzzy bi-ideals of a semigroup and characterize regular and intra-regular semigroups by the properties of these fuzzy ideals. The concept of a Γ-semigroup has been first introduced by Sen [17] in 1981 as follows: A nonempty set S is called a Γ-semigroup (S, Γ) if xαy ∈ S and (xαy)βz = xα(yβz) for all x, y, z ∈ S and α, β ∈ Γ. A Γ-semigroup is the generalization of a semigroup. Let (S, .) be a semigroup, γ a symbol (γ ∈ / S) and define aγb = a.b for all a, b ∈ S, then S is a {γ}-semigroup. Conversely, if F is a Γ-semigroup, and define a.b = aγb, then (F, .) is a semigroup. This means that if F is a {γ}-semigroup, then (F, .) is a semigroup [11]. In [1], Faisal et al. generalized Γ-semigroups (associative) and introduced the concept of ΓAG**-groupoids (non-associative) followed by the characterizations of a (2,2) regular class of an ordered Γ-AG**-groupoid in terms of different fuzzy ideals. In this paper, we have given the notions of different (∈, ∈ ∨qk )-fuzzy Γideals in Γ-semigroups which are infact the generalization of (∈, ∈ ∨qk )-fuzzy ideals in semigroups. Throughout in this paper SΓ will denote a Γ-semigroup unless otherwise specified.

2

Basic Definitions and results

A non-empty subset A of SΓ is called a Γ-subsemigroup of SΓ if AΓA ⊆ A. A non-empty subset A of SΓ is called a Γ-left (right) ideal of SΓ if SΓ ΓA ⊆ A (AΓSΓ ⊆ A). A is called a Γ-two-sided ideal or simply a Γ-ideal of SΓ if it is

A note on (∈, ∈ ∨qk )-fuzzy Γ-ideals of Γ-semigroups

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both Γ-left and Γ-right ideal of SΓ . A non-empty subset Q of SΓ is called a Γ-quasi-ideal of SΓ if QΓSΓ ∩SΓ ΓQ ⊆ Q. A non-empty subset B of SΓ is called a Γ-generalized bi-ideal of SΓ if BΓSΓ ΓB ⊆ B. A non-empty subset B of SΓ is called a bi-ideal of SΓ if it is both a Γ-subsemigroup and a Γ-generalized bi-ideal of SΓ . A Γ-subsemigroup I of SΓ is called a Γ-interior ideal of SΓ if SΓ ΓIΓSΓ ⊆ I. Obviously every Γ-one-sided ideal of SΓ is a Γ-quasi-ideal, every Γ-quasi-ideal is a Γ-bi-ideal and every Γ-bi-ideal is a Γ-generalized biideal. Also every Γ-ideal is a Γ-interior ideal. An element a of SΓ is called regular if there exists x ∈ SΓ such that a = aαxβa, where α, β ∈ Γ and SΓ is called regular if every element of SΓ is regular. It is easy to see that in a regular SΓ , the concepts of Γ-quasi-ideal, Γ-bi-ideal and Γ-generalized bi-ideal coincide. Also in a regular SΓ , every Γ-interior ideal is a Γ-ideal. A Γ-semigroup SΓ is called intra-regular if for each a ∈ SΓ there exist x, y ∈ SΓ such that a = xαaβaγy, where α, β, γ ∈ Γ. SΓ is called left quasi-regular if for each a ∈ SΓ there exist x, y ∈ SΓ such that a = xαaβyγa, where α, β, γ ∈ Γ. SΓ is called weakly regular if for each a ∈ S there exist x, y ∈ S such that a = aαxβaγy, where α, β, γ ∈ Γ. Definition 2.1. Let f and g be two fuzzy subsets of SΓ and k ∈ [0, 1), then the product f ◦k g is defined by (f ◦k g) (a) =

f (b) ∧ g(c) ∧ a=bαc

1−k 2

, if ∃ b, c ∈ SΓ a = bαc, α ∈ Γ.

0

otherwise.

Note that (f ◦k g) (a) = (f ◦ g) (a) ∧ where (f ◦ g) (a) =

a=bαc

1−k , 2

{f (b) ∧ g(c)} , if ∃ b, c ∈ SΓ a = bαc, α ∈ Γ.

0

otherwise.

Definition 2.2. For a fuzzy set f of SΓ and t ∈ (0, 1], the set UΓ (f ; t) = {x ∈ SΓ such that f (x) ≥ t} is called a level subset of f . Definition 2.3. A fuzzy subset f of SΓ of the form f (y) =

t ∈ (0, 1] if y = x 0 if y = x

is said to be a fuzzy point with support x and value t and is denoted by xt .

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A fuzzy point xt is said to be belong to a fuzzy set f , written as xt ∈ f if f (x) ≥ t. For k ∈ [0, 1), a fuzzy point xt is said to be k quasi-coincident with a fuzzy set f , written as xt qk f if f (x) + t + k > 1. In this case xt ∈ ∨qk f mean that xt ∈ f or xt qk f . Definition 2.4. A fuzzy subset f of SΓ is called an (∈, ∈ ∨qk )-fuzzy Γsubsemigroup of SΓ if for all x, y ∈ SΓ and t, r ∈ (0, 1], the following condition holds. xt ∈ f , yr ∈ f implies (xαy)min{t,r} ∈ ∨qk f , where α ∈ Γ. Definition 2.5. A fuzzy subset f of SΓ is called an (∈, ∈ ∨qk )-fuzzy Γ-left (right) ideal of SΓ if for all x, y ∈ SΓ and t, r ∈ (0, 1], the following condition holds. yr ∈ f implies (xαy)t ∈ ∨qk f (xt ∈ f implies (xαy)t ∈ ∨qk f ), where α ∈ Γ. Definition 2.6. A fuzzy subset f of SΓ is called an (∈, ∈ ∨qk )-fuzzy twosided ideal of SΓ if it is both (∈, ∈ ∨qk )-fuzzy Γ-left and (∈, ∈ ∨qk )-fuzzy Γ-right ideal of SΓ . Definition 2.7. A fuzzy subset f of SΓ is called an (∈, ∈ ∨qk )-fuzzy Γgeneralized bi-ideal of SΓ if for all x, y, z ∈ SΓ and t, r ∈ (0, 1], the following condition holds. xt ∈ f and zr ∈ f implies (xαyβz)min{t,r} ∈ ∨qk f , where α, β ∈ Γ. Definition 2.8. An (∈, ∈ ∨qk )-fuzzy Γ-subsemigroup f of SΓ is called an (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal of SΓ if for all x, y, z ∈ SΓ and t, r ∈ (0, 1], the following condition holds. xt ∈ f and zr ∈ f implies (xαyβz)min{t,r} ∈ ∨qk f , where α, β ∈ Γ. Definition 2.9. An (∈, ∈ ∨qk )-fuzzy Γ-subsemigroup f of SΓ is called an (∈, ∈ ∨qk )-fuzzy interior ideal of SΓ if for all x, y, z ∈ SΓ and t, r ∈ (0, 1], the following condition holds. yt ∈ f implies (xαyβz)t ∈ ∨qk f , where α, β ∈ Γ. ∨qk )-fuzzy Γ-quasi Definition 2.10. A fuzzy subset f of SΓ is called (∈, ∈1−k ideal of SΓ , if it satisfies, f (x) ≥ (f ◦k 1)(x) ∧ (1 ◦k f ) ∧ 2 , where 1 is the fuzzy subset of SΓ which maps every element of SΓ on 1. Lemmas 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18 and Theorems 2.19, 2.20, 2.21, 2.22, 2.23 are the generalizations of the work carried out in [18] and can be easily followed by keeping the same proof in generalized form. Therefore the proofs are omitted.


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Lemma 2.11. A fuzzy subset f of SΓ is an (∈, ∈ ∨qk)-fuzzy Γ-subsemigroup for all x, y ∈ SΓ and of SΓ if and only if f (xαy) ≥ min f (x) , f (y) , 1−k 2 k ∈ [0, 1), where α ∈ Γ. Lemma 2.12. A fuzzy subset f of SΓ is an (∈, ∈ ∨qk )-fuzzy Γ-left (right) f (xαy) ≥ min f (x) , 1−k ideal of SΓ if and only if f (xαy) ≥ min f (y) , 1−k 2 2 for all x, y ∈ SΓ and k ∈ [0, 1), where α ∈ Γ. Lemma 2.13. A fuzzy subset f of SΓ is an (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal of SΓ if and only if it satisfies the following conditions. 1−k (i) f (xαy) ≥ min f (x) , f (y) , 2 for all x, y ∈ SΓ and k ∈ [0, 1), where α ∈ Γ. (ii) f (xαyβz) ≥ min f (x) , f (z) , 1−k for all x, y, z ∈ SΓ and k ∈ [0, 1), 2 where α, β ∈ Γ. Lemma 2.14. A fuzzy subset f of SΓ is an (∈, ∈ ∨qk )-fuzzy Γ-generalized for all x, y, z ∈ bi-ideal of SΓ if and only if f (xαyβz) ≥ min f (x) , f (z) , 1−k 2 SΓ and k ∈ [0, 1), where α, β ∈ Γ. Lemma 2.15. A fuzzy subset f of SΓ is an (∈, ∈ ∨qk )-fuzzy interior ideal conditions. of SΓ if and only if it satisfies the following 1−k (i) f (xαy) ≥ min f (x) , f (y) , 2 for all x, y ∈ SΓ and k ∈ [0, 1), where α ∈ Γ. (ii) f (xαyβz) ≥ min f (y) , 1−k for all x, y, z ∈ SΓ and k ∈ [0, 1), where 2 α, β ∈ Γ. For a subset P of SΓ , CP denote the characteristic function of P . Lemma 2.16. A nonempty subset P of SΓ is a Γ-left (right, quasi) ideal of SΓ if and only if (CP )k is an (∈, ∈ ∨qk )-fuzzy Γ-left (right, quasi) ideal of SΓ . Lemma 2.17. Let A and B be any non-empty subsets of SΓ , then the following properties hold. (i) CA ∧k CB = (CA∩B )k . (ii) CA ◦k CB = (CAΓB )k . Lemma 2.18. Let f and g be any fuzzy subsets of SΓ , then the following properties hold. (i) f ∧k g = fk ∧ gk . (ii) f ◦k g = fk ◦ gk . Theorem 2.19. Every (∈, ∈ ∨qk )-fuzzy Γ-left (right) ideal f of SΓ is an (∈, ∈ ∨qk )-fuzzy Γ-interior ideal of SΓ . Theorem 2.20. Every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal f of SΓ is an (∈, ∈ ∨qk )fuzzy Γ-bi-ideal of SΓ .

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Theorem 2.21. A fuzzy subset f of SΓ is an (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal of ]. S if and only if UΓ (f ; t) is a Γ-bi-ideal of SΓ for all t ∈ (0, 1−k 2 Theorem 2.22. For SΓ , the following conditions are equivalent. (i) SΓ is regular. (ii) f ∧k g = f ◦k g for every (∈, ∈ ∨qk )-fuzzy Γ-right ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-left ideal g of SΓ . Theorem 2.23. For SΓ , the following are equivalent. (i) SΓ is intra-regular. (ii) f ∧k g ≤ f ◦k g for every (∈, ∈ ∨qk )-fuzzy Γ-left ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g of SΓ . Example 2.24. Consider the set S = {a, b, c, d, e} and let Γ = {α, β, γ} be the set of three binary operations on S in the tables below. α a b c d e

a a a a a a

b b d d d d

c d e a d d c d e a d d c d e a d d

β a b c d e

a a a a a a

b d d d d b

c d e c d e a d d c d e a d d a d d

γ a b c d e

a a a a a a

b d b d d d

c d e a d d a d d c d e a d d c d e

Since (aβb)γc = aβ(bγc) for all a, b, c ∈ S and all β, γ ∈ Γ, therefore (S, Γ) is a Γ-semigroup. Clearly, {a, b, d, e} is a Γ-bi-ideal of S. If we define a fuzzy subset f : S −→ [0, 1] by f (a) = 0.8, f (b) = 0.7, f (c) = 0.3, f (d) = 0.5, f (e) = 0.6. Then

UΓ (f ; t) =

S {a, b, d, e}

if t ∈ (0, 0.3], if t ∈ (0.3, 0.5].

], where k ∈ [0, 1), UΓ (f ; t) is a Γ-bi-ideal of S, therefore Since for all t ∈ (0, 1−k 2 by Theorem 2.21, f is an (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal of (S, Γ).

3

Main results Theorem 3.1. For SΓ , the following conditions are equivalent. (i) SΓ is regular. (ii) R ∩ L = RΓL for every Γ-right ideal R and Γ-left ideal L of SΓ . (iii) AΓSΓA = A for every Γ-quasi-ideal A of SΓ .

Proof. It is obvious.


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Theorem 3.2. The following conditions are equivalent for SΓ . (i) SΓ is regular. (ii) fk = f ◦k SΓ ◦k f for every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal f of SΓ . (iii) fk = f ◦k SΓ ◦k f for every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal f of SΓ . Proof. (i) ⇒ (iii) : Let f be an (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal of SΓ . Since SΓ is regular, therefore for each a ∈ SΓ there exists x ∈ SΓ such that a = aαxβa, where α, β ∈ Γ. Then, we have 1−k (f ◦k SΓ ◦k f ) (a) = (f ◦k SΓ ◦k f ) (a) ∧ 2

1−k 1−k ∧ {(f ◦k SΓ ) (p) ∧ f (q)} ∧ = 2 2 a=pβq 1−k ≥ (f ◦k SΓ ) (aαx) ∧ f (a) ∧ 2

1−k 1−k ∧ f (a) ∧ {f (a) ∧ SΓ (x)} ∧ = 2 2 aαx=aαx 1−k 2 1−k = {f (a) ∧ 1 ∧ f (a)} ∧ 2 1−k = f (a) ∧ = fk (a) . 2

≥ {f (a) ∧ SΓ (x) ∧ f (a)} ∧

Therefore f ◦k SΓ ◦k f ≥ fk and since f is an (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal of SΓ , therefore f ◦k SΓ ◦k f ≤ fk , implies fk = f ◦k SΓ ◦k f . (iii) ⇒ (ii) is obvious. (ii) ⇒ (i) : Let A be any Γ-quasi-ideal of SΓ , then AΓSΓ ΓA ⊆ AΓSΓ ΓSΓ ∩ SΓ ΓSΓ ΓA ⊆ AΓSΓ ∩ SΓ ΓA ⊆ A. Let a ∈ A. Now by Lemma 2.16, (CA )k is an (∈, ∈ ∨qk )-fuzzy Γ-interior ideal of SΓ , therefore

a=pαq

1−k (CA ◦k SΓ ) (p) ∧ CA (q) ∧ 2

= (CA ◦k SΓ ◦k CA ) (a) = (CA )k (a) = 1 ∧ =

1−k . 2

1−k 2

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1−k 2


Thus there exist b, c ∈ SΓ such that a = bαc for α ∈ Γ, therefore (CA ◦k SΓ ) (b) = and (CA )k (c) = 1−k . Now, we have 2

1−k 1−k . CA (y) ∧ SΓ (z) ∧ = (CA ◦k SΓ ) (b) = 2 2 b=yβz

This shows that there exist d, e ∈ SΓ such that b = dγe for γ ∈ Γ and (CA )k (d) = 1−k and SΓ (e) = 1−k . Therefore a = bαc = dγeαc ∈ AΓSΓ ΓA, 2 2 therefore A ⊆ AΓSΓ ΓA ⊆ A and A = AΓSΓ ΓA. Thus by Theorem 3.1, SΓ is regular.

Theorem 3.3. The following conditions are equivalent for SΓ . (i) SΓ is regular and intra-regular. (ii) f ◦k g ◦k h ≥ f ∧k g ∧k h for every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal f , every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g and every (∈, ∈ ∨qk )-fuzzy Γ-left ideal h of SΓ . (iii) f ◦k g ◦k h ≥ f ∧k g ∧k h for every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal f , every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g and every (∈, ∈ ∨qk )-fuzzy Γ-left ideal h of SΓ . (iv) f ◦k g ◦k h ≥ f ∧k g ∧k h for every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal f , every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g and every (∈, ∈ ∨qk )-fuzzy Γ-left ideal h of SΓ .

Proof. (i) ⇒ (iv) : Let f, g and h be any (∈, ∈ ∨qk )-fuzzy Γ-generalized biideal, (∈, ∈ ∨qk )-fuzzy Γ-left ideal and (∈, ∈ ∨qk )-fuzzy Γ-right ideal of SΓ , respectively. Since SΓ is regular and intra-regular so for each a ∈ SΓ , there exist x, y, z ∈ SΓ and α, β, γ, δ, ξ ∈ Γ such that a = aαxβa and a = yγaδaξz. Therefore

a = aαxβa = aαxβaαxβaαxβa = (aαxβyγa)δ(aξzαxβyγa)δ(aξzαxβa) = pδq.

Thus, we have


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1−k f (p) ∧ (g ◦k h) (q) ∧ (f ◦k g ◦k h) (a) = 2 a=pδq 1−k ≥ f (aαxβyγa) ∧ (g ◦k h) (q) ∧ 2

1−k 1−k ≥ f (a) ∧ g (b) ∧ h (c) ∧ ∧ 2 2 q=bδc ≥ f (a) ∧ g (b) ∧ h (c) ∧

1−k 1−k ∧ 2 2

= f (a) ∧ g (aξzαxβyγa) ∧ h (aξzαxβa) ∧ ≥ f (a) ∧ g (a) ∧ h (a) ∧ = (f ∧k g ∧k h) (a) .

1−k 2

1−k 2

Therefore f ◦k g ◦k h ≥ f ∧k g ∧k h. (iv) ⇒ (iii) ⇒ (ii) is clear. (ii) ⇒ (i) : Let f, g and h be any (∈, ∈ ∨qk )-fuzzy Γ-interior ideal, (∈, ∈ ∨qk )fuzzy Γ-right ideal and (∈, ∈ ∨qk )-fuzzy Γ-left ideal and of SΓ respectively. Then since f is an (∈, ∈ ∨qk )-fuzzy Γ-interior ideal of SΓ and SΓ itself is an (∈, ∈ ∨qk )-fuzzy Γ-left ideal of SΓ , therefore by assumption, we have f ∧k g = f ∧k g ∧k SΓ ≤ f ◦k g ◦k SΓ = f ◦k (g ◦k SΓ ) ≤ f ◦k g. Thus it follows from Theorem 2.23 that SΓ is intra-regular. Since S is intra-regular, therefore g is an (∈, ∈ ∨qk )-fuzzy Γ-interior ideal of SΓ and SΓ itself is an (∈, ∈ ∨qk )-fuzzy Γ-right ideal of SΓ , therefore we have g ∧k f = g ∧k SΓ ∧k f ≤ g ◦k SΓ ◦k f = (g ◦k SΓ ) ◦k f ≤ g ◦k f . But g ◦k f ⊆ g ∧k f always hold. Thus from Theorem 2.22, SΓ is regular. Theorem 3.4. The following conditions are equivalent for SΓ . (i) SΓ is left quasi-regular. (ii) fk = f ◦k f for every (∈, ∈ ∨qk )-fuzzy Γ-left ideal f of SΓ . Proof. (i) ⇒ (ii) : Let SΓ be a left quasi-regular semigroup and f be any (∈, ∈ ∨qk )-fuzzy Γ-left ideal of SΓ . Then for each a ∈ SΓ there exist x, y ∈ SΓ

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and α, β, γ ∈ Γ such that a = xαaβyγa, we have (f ◦k f ) (a) =

a=pβq

1−k f (p) ∧ f (q) ∧ 2

1−k 2 1−k ≥ f (a) ∧ f (a) ∧ 2 1−k = fk (a) . = f (a) ∧ 2 ≥ f (xa) ∧ f (ya) ∧

Thus f ◦k f ≥ fk but f ◦k f ≤ fk always hold, therefore fk = f ◦k f . (ii) ⇒ (i) : Let a ∈ SΓ and LΓ [a] be a Γ-left ideal of SΓ , therefore by Lemma 2.16, CLΓ [a] k is an (∈, ∈ ∨qk )-fuzzy Γ-left ideal of SΓ . Thus 1−k CLΓ [a] k (a) = CLΓ [a] ◦k CLΓ [a] (a) = CLΓ [a]ΓLΓ [a] k (a) = . 2 This implies that a ∈ LΓ [a] ΓLΓ [a] = ({a} ∪ SΓ Γa) ({a} ∪ SΓ Γa) = {aβa} ∪ aΓSΓ Γa ∪ SΓ Γaβa ∪ SΓ ΓaΓSΓ Γa = SΓ ΓaΓSΓ Γa, where β ∈ Γ. This shows that SΓ is left quasi-regular. Theorem 3.5. The following conditions are equivalent for SΓ . (i) SΓ is both intra-regular and left quasi-regular. (ii) f ◦k g ◦k h ≥ f ∧k g ∧k h for every (∈, ∈ ∨qk )-fuzzy Γ-left ideal f , every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g and every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal of SΓ . (iii) f ◦k g ◦k h ≥ f ∧k g ∧k h for every (∈, ∈ ∨qk )-fuzzy Γ-left ideal f , every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g and every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal of SΓ . (iv) f ◦k g ◦k h ≥ f ∧k g ∧k h for every (∈, ∈ ∨qk )-fuzzy Γ-left ideal f , every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g and every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal of SΓ . Proof. (i) ⇒ (iv) : Suppose that SΓ is both intra-regular and left quasi-regular. Let f, g and h be any (∈, ∈ ∨qk )-fuzzy Γ-left ideal, (∈, ∈ ∨qk )-fuzzy Γ-right ideal and (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal of SΓ , respectively. Then for each a ∈ SΓ , there exist x, y, u, v ∈ SΓ and α, β, γ, δ, ζ, ξ ∈ Γ such that a = xαaβaγy and a = uδaζvξa = uδ (xαaβaγy) ζvξa = ((uδx) αa) β ((aγ (yζv)) ξa).


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Therefore 1−k (f ◦k g ◦k h) (a) = (f ◦ g ◦ h) (a) ∧ 2

1−k = {f (p) ∧ (g ◦ h) (q)} ∧ 2 a=pβq 1−k ≥ {f ((uδx) αa) ∧ (g ◦ h) ((aγ (yζv)) ξa)} ∧ 2

1−k {g (b) ∧ h (c)} ∧ ≥ f (a) ∧ 2 (aγ(yζv))ξa=bξc

≥ f (a) ∧ g (aγ (yζv)) ∧ h (a) ∧ ≥ f (a) ∧ g (a) ∧ h (a) ∧ = (f ∧k g ∧k h) (a) .

1−k 2

1−k 2

(iv) ⇒ (iii) ⇒ (ii) is clear. Now (ii) ⇒ (i) : Let f be any (∈, ∈ ∨qk )-fuzzy Γ-left ideal of SΓ . Then since every (∈, ∈ ∨qk )-fuzzy Γ-left ideal is (∈, ∈ ∨qk )-fuzzy Γ-interior ideal, therefore f is (∈, ∈ ∨qk )-fuzzy Γ-interior ideal of SΓ and since SΓ itself is (∈, ∈ ∨qk )-fuzzy Γ-right ideal so, we have 1−k 1−k = f (a) ∧ SΓ (a) ∧ f (a) ∧ 2 2 = (f ∧k SΓ ∧k f ) (a) ≤ (f ◦k SΓ ◦k f ) (a) ≤ (f ◦k f ) (a) ≤ (f ◦k SΓ ) (a) ≤ fk (a) .

fk (a) = f (a) ∧

Therefore fk = f ◦k f . Now by using Theorem 3.4, SΓ is left quasi-regular. Again, since every (∈, ∈ ∨qk )-fuzzy Γ-right ideal h of SΓ is an (∈, ∈ ∨qk )fuzzy Γ-interior ideal of SΓ , SΓ itself is an (∈, ∈ ∨qk )-fuzzy Γ-right ideal of SΓ and f is any (∈, ∈ ∨qk )-fuzzy Γ-left ideal of SΓ , therefore we have 1−k 1−k = f (a) ∧ h (a) ∧ 2 2 1−k = f (a) ∧ SΓ (a) ∧ h (a) ∧ 2 = (f ∧k SΓ ∧k h) (a) ≤ (f ◦k SΓ ◦k h) (a) ≤ (f ◦k h) (a) .

(f ∧k h) (a) = (f ∧ h) (a) ∧

Thus f ∧k h ≤ f ◦k h. Now by using Theorem 2.23, SΓ is intra-regular. Theorem 3.6. The following conditions are equivalent for SΓ . (i) SΓ is regular and intra-regular.

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(ii) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-left ideal g of SΓ . (iii) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g of SΓ . (iv) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal f and g of SΓ . (v) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal g of SΓ . (vi) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal g of SΓ . (vii) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-left ideal g of SΓ . (viii) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g of SΓ . (ix) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal g of SΓ . (x) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal f and g of SΓ . (xi) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal g of SΓ . (xii) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-left ideal g of SΓ . (xiii) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g of SΓ . (xiv) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-interior ideal g of SΓ . (xv) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideal f and every (∈, ∈ ∨qk )-fuzzy Γ-bi-ideal g of SΓ . (xvi) f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideals f and g of SΓ . Proof. (i) ⇒ (xvi) : Let f and g be any (∈, ∈ ∨qk )-fuzzy Γ-generalized biideals of SΓ . Since SΓ is regular and intra-regular- regular, therefore for each a ∈ SΓ there exist x, y, z ∈ SΓ and α, β, γ, δ, ξ ∈ Γ such that a = aαxβa and a = yγaδaξz. Thus a = aαxβa = aαxβaαxβaαxβa = aαxβ(yγaδaξz)αxβ(yγaδaξz)αxβa = (aαxβyγa)δ(aξzαxβyγa)δ(aξzαxβa) = pδbδc.


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Since f and g are (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideals of SΓ so, we have (f ◦k g ◦k f ) (a) =

{f (p) ∧ (g ◦k f ) (q)}

a=pδq

∧

1−k 2

1−k ≥ {f (aαxβyγa) ∧ (g ◦k f ) ((b)δ(c))} ∧ 2

1−k {g (b) ∧ f (c)} ∧ ≥ f (a) ∧ 2 bδc=bδc ≥ {f (a) ∧ {g (aξzαxβyγa) ∧ f (aξzαxβa)}} ∧ ≥ {f (a) ∧ g (a) ∧ f (a)} ∧ = (f (a) ∧ g (a)) ∧

1−k 2

1−k 2

1−k = (f ∧k g) (a) . 2

Therefore f ◦k g ◦k f ≥ f ∧k g for all (∈, ∈ ∨qk )-fuzzy Γ-generalized bi-ideals f and g of SΓ . It is clear that (xi) ⇒ (vi) ⇒ (v) ⇒ (iv) ⇒ (iii) , (xvi) ⇒ (xv) ⇒ (xiv) ⇒ (xiii) ⇒ (viii) ⇒ (iii), (xiv) ⇒ (xii) ⇒ (vii) ⇒ (ii) and (xvi) ⇒ (xi) ⇒ (x) ⇒ (ix) ⇒ (viii) . (iii) ⇒ (i) : Let f be any (∈, ∈ ∨qk )-fuzzy Γ-interior ideal of SΓ , then f ◦k g ◦k f ≥ f ∧k g for every (∈, ∈ ∨qk )-fuzzy Γ-right ideal g of SΓ . Since SΓ itself is (∈, ∈ ∨qk )-fuzzy Γ-right ideal, therefore we have

f ◦k SΓ ◦k f ≥ f ∧k SΓ = (f ∧ SΓ ) ∧

1−k 1−k =f∧ = fk . 2 2

Thus f ◦k SΓ ◦k f = fk . Now by Theorem 3.2, SΓ is regular.

4

The construction of a Γ-semigroup

Let us consider a semigroup S = {a, b, c, d, e} under the binary operation “∗” in the table below. ∗ a b c d e

a a a a a a

b d b d d d

c d e a d d a d d c d e a d d c d e

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Let Γ = {α, β, γ, δ, } be the set of three operations defined on S in the following tables. α a b c d e

a a a a a a

b d b d d d

c d e a d d a d d c d e a d d c d e δ a b c d e

a a a a a a

b d d d b d

β a b c d e

a a a a a a

c d e a d d c d e a d d a d d c d e

b b d d d d

c a c a c a

d d d d d d

e d e d e d a b c d e

γ a b c d e a b c a d c a d a a b a a d c a d a

a b c a d c a d a a d c a d a a b a d d d d d d

d d d d d d

e e d e d d

e e d d e d

To form an α-table, replace “∗” by “α” in ∗-table. Now to form the β-table, copy the e-row (a d c d e) from α-table and paste into the d-row of β-table, copy the d-row (a d a d d) from α-table and paste into the c-row of β-table, copy the c-row (a d c d e) from α-table and paste into the b-row of β-table, copy the b-row (a b a d d) from α-table and paste into the a-row of β-table, copy the a-row (a d a d d) from α-table and paste into the e-row of β-table. The resulting table is β-table. In a similar way, we can get the γ-table. If we repeat the same process again, we will get back an α-table. It is easy to see that S = {a, b, c, d, e} is a Γ-semigroup with Γ = {α, β, γ, δ, }. Note that in the above example of a Γ-semigroup, both S and Γ have the same number of elements. In this method, the number of operations (at most) in Γ will be the same as that of the number of elements in S. In general, if S is a semigroup having n elements (n > 2), then the possible number of operations in Γ will also be n. Conclusion 4.1. In abstract algebra, Γ-semigroup is a semigroup together with a set of operations Γ. Since Γ-semigroups are the generalization of semigroups, therefore the results discussed in the paper can be obtained in a semigroup theory by omitting the generalization. The obtained results will be the extension of the work carried out in [18] and [19]. ACKNOWLEDGEMENTS. Ronnason Chinram is grateful to Faculty of Science, Prince of Songkla University for funding this research.

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[18] M. Shabir, Y.B. Jun and Y. Nawaz, Semigroups characterized by (∈, ∈ ∨qk )-fuzzy ideals, Comput. Math. Appl., 60 (2010),1473-1493. [19] M. Shabir, Y.B. Jun and Y. Nawaz, Characterizations of regular semigroups by (α, β)-fuzzy ideals, Comput. Math. Appl., 59 (2010), 161-175. Received: May, 2012