Research Article Semigroups Characterized by Their Generalized

Hindawi Publishing Corporation Journal of Mathematics Volume 2013, Article ID 592708, 7 pages http://dx.doi.org/10.1155/2013/592708

Research Article Semigroups Characterized by Their Generalized Fuzzy Ideals Madad Khan and Saima Anis Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan Correspondence should be addressed to Madad Khan; [email protected] Received 14 January 2013; Accepted 27 February 2013 Academic Editor: Feng Feng Copyright © 2013 M. Khan and S. Anis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We have characterized right weakly regular semigroups by the properties of their (∈, ∈ ∨𝑞𝑘 )-fuzzy ideals.

1. Introduction Usually the models of real world problems in almost all disciplines like in engineering, medical science, mathematics, physics, computer science, management sciences, operations research, and artificial intelligence are mostly full of complexities and consist of several types of uncertainties while dealing with them in several occasion. To overcome these difficulties of uncertainties, many theories had been developed such as rough sets theory, probability theory, fuzzy sets theory, theory of vague sets, theory of soft ideals, and the theory of intuitionistic fuzzy sets. Zadeh discovered the relationships of probability and fuzzy set theory in [1] which has appropriate approach to deal with uncertainties. Many authors have applied the fuzzy set theory to generalize the basic theories of Algebra. The concept of fuzzy sets in structure of groups was given by Rosenfeld [2]. The theory of fuzzy semigroups and fuzzy ideals in semigroups was introduced by Kuroki in [3, 4]. The theoretical exposition of fuzzy semigroups and their application in fuzzy coding, fuzzy finite state machines, and fuzzy languages was considered by Mordeson. The concept of belongingness of a fuzzy point to a fuzzy subset by using natural equivalence on a fuzzy subset was considered by Murali [5]. By using these ideas, Bhakat and Das [6, 7] 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 [8, 9]. The concept of (∈, ∈ ∨𝑞𝑘 )-fuzzy subgroups is a viable generalization of Rosenfeld’s fuzzy subgroups. Davvaz defined (∈, ∈ ∨𝑞𝑘 )-fuzzy subnearrings and ideals of a near ring in [10]. Jun and Song initiated the study of (𝛼, 𝛽)-fuzzy interior ideals of a semigroup in [11] which is the generalization of fuzzy interior ideals [12]. In [13], Kazanci and Yamak studied (∈, ∈ ∨𝑞𝑘 )-fuzzy bi-ideals of a semigroup. In this paper we have characterized right regular semigroups by the properties of their right ideal, bi-ideal, generalized bi-ideal, and interior ideal. Moreover we characterized right regular semigroups in terms of their (∈, ∈ ∨𝑞𝑘 )fuzzy right ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy bi-ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy generalized bi-ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy bi-ideal, and (∈, ∈ ∨𝑞𝑘 )fuzzy interior ideals. Throughout this paper 𝑆 denotes a semigroup. A nonempty subset 𝐴 of 𝑆 is called a subsemigroup of 𝑆 if 𝐴2 ⊆ 𝐴. A nonempty subset 𝐽 of 𝑆 is called a left (right) ideal of 𝑆 if 𝑆𝐽 ⊆ 𝐼 (𝐽𝑆 ⊆ 𝐼). 𝐽 is called a two-sided ideal or simply an ideal of 𝑆 if it is both left and right ideal of 𝑆. A nonempty subset 𝐵 of 𝑆 is called a generalized bi-ideal of 𝑆 if 𝐵𝑆𝐵 ⊆ 𝐵. A nonempty subset 𝐵 of 𝑆 is called a bi-ideal of 𝑆 if it is both a subsemigroup and a generalized bi-ideal of 𝑆. A subsemigroup 𝐼 of 𝑆 is called an interior ideal of 𝑆 if 𝑆𝐼𝑆 ⊆ 𝐼. An semigroup 𝑆 is called a right weakly regular if for every 𝑎 ∈ 𝑆 there exist 𝑥, 𝑦 ∈ 𝑆 such that 𝑎 = 𝑎𝑥𝑎𝑦. Definition 1. For a fuzzy set 𝑓 of a semigroup 𝑆 and 𝑡 ∈ (0, 1], the crisp set 𝑈(𝑓; 𝑡) = {𝑥 ∈ 𝑆 such that 𝑓(𝑥) ≥ 𝑡} is called level subset of 𝑓.

2

Journal of Mathematics

Definition 2. A fuzzy subset 𝑓 of a semigroup 𝑆 of the form 𝑡 ∈ (0, 1] 𝑓 (𝑦) = { 0

if 𝑦 = 𝑥, if 𝑦 ≠ 𝑥

(1)

is said to be a fuzzy point with support 𝑥 and value 𝑡 and is denoted by 𝑥𝑡 . A fuzzy point 𝑥𝑡 is said to belong to (resp., quasi-coincident with) a fuzzy set 𝑓, written as 𝑥𝑡 ∈ 𝑓 (resp., 𝑥𝑡 𝑞𝑓), if 𝑓(𝑥) ≥ 𝑡 (resp., 𝑓(𝑥) + 𝑡 > 1). If 𝑥𝑡 ∈ 𝑓 or 𝑥𝑡 𝑞𝑓, then we write 𝑥𝑡 ∈ ∨𝑞𝑓. The symbol ∈ ∨𝑞 means ∈ ∨𝑞 does not hold. For any two fuzzy subsets 𝑓 and 𝑔 of 𝑆, 𝑓 ≤ 𝑔 means that, for all 𝑥 ∈ 𝑆, 𝑓(𝑥) ≤ 𝑔(𝑥). Generalizing the concept of 𝑥𝑡 𝑞𝑓, Jun [12, 14] defined 𝑥𝑡 𝑞𝑘 𝑓, where 𝑘 ∈ [0, 1), as 𝑓(𝑥) + 𝑡 + 𝑘 > 1. 𝑥𝑡 ∈ ∨𝑞𝑘 𝑓 if 𝑥𝑡 ∈ 𝑓 or 𝑥𝑡 𝑞𝑘 𝑓.

Definition 11. A fuzzy subsemigroup 𝑓 of a semigroup 𝑆 is called an (∈, ∈ ∨𝑞𝑘 )-fuzzy interior ideal of 𝑆 if for all 𝑥, 𝑦, 𝑧 ∈ 𝑆 and 𝑡, 𝑟 ∈ (0, 1] the following condition holds: 𝑦𝑡 ∈ 𝑓 imply (𝑥𝑦𝑧)𝑡 ∈ ∨𝑞𝑘 𝑓. Lemma 12 (see [15]). A fuzzy subset 𝑓 of 𝑆 is an (∈, ∈ ∨𝑞𝑘 )fuzzy interior ideal of 𝑆 if and only if it satisfies the following condition: (i) 𝑓(𝑥𝑦) ≥ min{𝑓(𝑥), 𝑓(𝑦), (1−𝑘)/2} for all 𝑥, 𝑦 ∈ 𝑆 and 𝑘 ∈ [0, 1); (ii) 𝑓(𝑥𝑦𝑧) ≥ min{𝑓(𝑦), (1 − 𝑘)/2} for all 𝑥, 𝑦, 𝑧 ∈ 𝑆 and 𝑘 ∈ [0, 1). Example 13. Let 𝑆 = {1, 2, 3, } be a semigroup with binary operation “⋅,” as defined in the following Cayley table:

2. (∈,∈ ∨𝑞𝑘 )-Fuzzy Ideals in Semigroups Definition 3. A fuzzy subset of 𝑆 is called an (∈, ∈ ∨𝑞𝑘 )fuzzy subsemigroup of 𝑆 if for all 𝑥, 𝑦 ∈ 𝑆 and 𝑡, 𝑟 ∈ (0, 1] the following condition holds: 𝑥𝑡 ∈ 𝑓 and 𝑦𝑟 ∈ 𝑓 imply (𝑥𝑦)min{𝑡,𝑟} ∈ ∨𝑞𝑘 𝑓. Lemma 4 (see [15]). Let 𝑓 be a fuzzy subset of 𝑆. Then 𝑓 is an (∈, ∈ ∨𝑞𝑘 )-fuzzy subsemigroup of 𝑆 if and only if 𝑓(𝑥𝑦) ≥ min{𝑓(𝑥), 𝑓(𝑦), (1 − 𝑘)/2}. Definition 5. A fuzzy subset 𝑓 of 𝑆 is called an (∈, ∈ ∨𝑞𝑘 )fuzzy left (right) ideal of 𝑆 if for all 𝑥, 𝑦 ∈ 𝑆 and 𝑡, 𝑟 ∈ (0, 1] the following condition holds: 𝑦𝑟 ∈ 𝑓 implies (𝑥𝑦)𝑡 ∈ ∨𝑞𝑘 𝑓(𝑥𝑡 ∈ 𝑓 implies (𝑥𝑦)𝑡 ∈ ∨𝑞𝑘 𝑓). Lemma 6 (see [15]). Let 𝑓 be a fuzzy subset of 𝑆. Then 𝑓 is an (∈, ∈ ∨𝑞𝑘 )-fuzzy left (right) ideal of 𝑆 if and only if 𝑓(𝑥𝑦) ≥ min{𝑓(𝑦), (1 − 𝑘)/2} (𝑓(𝑥𝑦) ≥ min{𝑓(𝑥), (1 − 𝑘)/2}). Definition 7. A fuzzy subsemigroup 𝑓 of a semigroup 𝑆 is called an (∈, ∈ ∨𝑞𝑘 )-fuzzy bi-ideal of 𝑆 if for all 𝑥, 𝑦, 𝑧 ∈ 𝑆 and 𝑡, 𝑟 ∈ (0, 1] the following condition holds: 𝑥𝑡 ∈ 𝑓 and 𝑧𝑡 ∈ 𝑓 imply (𝑥𝑦𝑧)𝑡 ∈ ∨𝑞𝑘 𝑓. Lemma 8 (see [15]). A fuzzy subset 𝑓 of 𝑆 is an (∈, ∈ ∨𝑞𝑘 )fuzzy bi-ideal of 𝑆 if and only if it satisfies the following conditions: (i) 𝑓(𝑥𝑦) ≥ min{𝑓(𝑥), 𝑓(𝑦), (1−𝑘)/2} for all 𝑥, 𝑦 ∈ 𝑆 and 𝑘 ∈ [0, 1); (ii) 𝑓(𝑥𝑦𝑧) ≥ min{𝑓(𝑥), 𝑓(𝑧), (1 − 𝑘)/2} for all 𝑥, 𝑦, 𝑧 ∈ 𝑆 and 𝑘 ∈ [0, 1). Definition 9. A fuzzy subset 𝑓 of a semigroup 𝑆 is called an (∈, ∈ ∨𝑞𝑘 )-fuzzy generalized bi-ideal of 𝑆 if for all 𝑥, 𝑦, 𝑧 ∈ 𝑆 and 𝑡, 𝑟 ∈ (0, 1] the following condition holds: 𝑥𝑡 ∈ 𝑓 and 𝑧𝑡 ∈ 𝑓 imply (𝑥𝑦𝑧)𝑡 ∈ ∨𝑞𝑘 𝑓. Lemma 10 (see [15]). A fuzzy subset 𝑓 of 𝑆 is an (∈, ∈ ∨𝑞𝑘 )-fuzzy generalized bi-ideal of 𝑆 if and only if 𝑓(𝑥𝑦𝑧) ≥ min{𝑓(𝑥), 𝑓(𝑧), (1 − 𝑘)/2} for all 𝑥, 𝑦, 𝑧 ∈ 𝑆 and 𝑘 ∈ [0, 1).

⋅ 1 2 3

1 1 2 3

2 1 2 3

3 1 2 3

(2)

Clearly (𝑆, ⋅) is regular semigroup and {1}, {2}, and {3} are left ideals of 𝑆. Let us define a fuzzy subset 𝛿 of 𝑆 as 𝛿 (1) = 0.9,

𝛿 (2) = 0.6,

𝛿 (3) = 0.5.

(3)

Then clearly 𝛿 is an (∈, ∈ ∨𝑞)-fuzzy ideal of 𝑆. Lemma 14 (see [15]). A nonempty subset 𝑅 of a semigroup 𝑆 is right (left) ideal if and only if (𝐶𝑅 )𝑘 is an (∈, ∈ ∨𝑞𝑘 )-fuzzy right (left) ideal of 𝑆. Lemma 15. A nonempty subset 𝐼 of a semigroup 𝑆 is an interior ideal if and only if (𝐶𝐼 )𝑘 is an (∈, ∈ ∨𝑞𝑘 )-fuzzy interior ideal of 𝑆. Lemma 16. A nonempty subset 𝐵 of a semigroup 𝑆 is bi-ideal if and only if (𝐶𝐵 )𝑘 is an (∈, ∈ ∨𝑞𝑘 )-fuzzy bi-ideal of 𝑆. Lemma 17. Let 𝑓 and 𝑔 be any fuzzy subsets of semigroup 𝑆. Then following properties hold: (i) (𝑓∧𝑘 𝑔) = (𝑓𝑘 ∧ 𝑔𝑘 ), (ii) (𝑓∘𝑘 𝑔) = (𝑓𝑘 ∘ 𝑔𝑘 ). Proof. It is straightforward. Lemma 18. Let 𝐴 and 𝐵 be any nonempty subsets of a semigroup 𝑆. Then the following properties hold: (i) (𝐶𝐴 ∧𝑘 𝐶𝐵 ) = (𝐶𝐴∩𝐵 )𝑘 , (ii) (𝐶𝐴 ∘𝑘 𝐶𝐵 ) = (𝐶𝐴𝐵 )𝑘 . Proof. It is straightforward.


3

3. Characterizations of Regular Semigroups Theorem 19. For a semigroup 𝑆, the following conditions are equivalent:

(iii) ⇒ (i): As 𝑎 ∪ 𝑎𝑆 is right ideal and 𝑎 ∪ 𝑆𝑎 is left ideal of 𝑆 generated by 𝑎, respectively, thus by assumption we have (𝑎 ∪ 𝑆𝑎) ∩ (𝑎 ∪ 𝑎𝑆) ∩ (𝑎 ∪ 𝑎𝑆) ⊆ (𝑎 ∪ 𝑎𝑆) (𝑎 ∪ 𝑆𝑎) (𝑎 ∪ 𝑆𝑎)

(i) 𝑆 is regular;

⊆ (𝑎 ∪ 𝑎𝑆) 𝑆 (𝑎 ∪ 𝑆𝑎)

(ii) 𝐿 1 ∩ 𝐿 2 ∩ 𝐵 ⊆ 𝐵𝐿 1 𝐿 2 for left ideals 𝐿 1 , 𝐿 2 , and bi-ideal 𝐵 of a semigroup 𝑆.

⊆ (𝑎 ∪ 𝑎𝑆) (𝑎 ∪ 𝑆𝑎)

(iii) 𝐿[𝑎] ∩ 𝐿[𝑎] ∩ 𝐵[𝑎] ⊆ 𝐵[𝑎]𝐿[𝑎]𝐿[𝑎], for some 𝑎 in 𝑆;

⊆ 𝑎2 ∪ 𝑎𝑆𝑎.

Proof. (i) ⇒ (ii): Let 𝑆 be regular semigroup, then for an element 𝑎 ∈ 𝑆 there exists 𝑥 ∈ 𝑆 such that 𝑎 = 𝑎𝑥𝑎. Let 𝑎 ∈ 𝐿 1 ∩ 𝐿 2 ∩ 𝐵, where 𝐵 is a bi-ideal and 𝐿 1 and 𝐿 2 are left ideals of 𝑆. So 𝑎 ∈ 𝐿 1 , 𝑎 ∈ 𝐿 2 , and 𝑎 ∈ 𝐵. As 𝑎 = 𝑎𝑥𝑎 = 𝑎𝑥𝑎𝑥𝑎 = 𝑎𝑥𝑎𝑥𝑎𝑥𝑎 ∈ (𝐵𝑆𝐵)𝑆𝐿 1 𝑆𝐿 2 ⊆ 𝐵𝐿 1 𝐿 2 . Thus 𝐿 1 ∩ 𝐿 2 ∩ 𝐵 ⊆ 𝐵𝐿 1 𝐿 2 . (ii) ⇒ (iii) is obvious. (iii) ⇒ (i): As 𝑎 ∪ 𝑆𝑎 and 𝑎 ∪ 𝑎2 ∪ 𝑎𝑆𝑎 are left ideal and bi-ideal of 𝑆 generated by 𝑎, respectively, thus by assumption we have (𝑎 ∪ 𝑆𝑎) ∩ (𝑎 ∪ 𝑆𝑎) ∩ (𝑎2 ∪ 𝑎𝑆𝑎) ⊆ (𝑎 ∪ 𝑎 ∪ 𝑎𝑆𝑎) (𝑎 ∪ 𝑆𝑎) (𝑎 ∪ 𝑆𝑎)

⊆ (𝑎 ∪ 𝑎2 ∪ 𝑎𝑆𝑎) (𝑎 ∪ 𝑆𝑎)

= 𝑎2 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑆𝑎

Thus 𝑎 = 𝑎2 or 𝑎 = 𝑎𝑥𝑎, for some 𝑥 in 𝑆. Hence 𝑆 is regular semigroup. Theorem 21. For a semigroup 𝑆, the following conditions are equivalent: (i) 𝑆 is regular; (ii) 𝑓∘𝑘 𝑔∘𝑘 ℎ ≥ 𝑓∧𝑘 𝑔∧𝑘 ℎ for every (∈, ∈ ∨𝑞𝑘 )-fuzzy right ideal 𝑓, (∈, ∈ ∨𝑞𝑘 )-fuzzy left ideals 𝑔, and ℎ of a semigroup 𝑆. Proof. (i) ⇒ (ii): Let 𝑓 be (∈, ∈ ∨𝑞𝑘 )-fuzzy right ideal, 𝑔 and ℎ any (∈, ∈ ∨𝑞𝑘 )-fuzzy left ideals of 𝑆. Since 𝑆 is regular, therefore for each 𝑎 ∈ 𝑆 there exists 𝑥 ∈ 𝑆 such that

2

⊆ (𝑎 ∪ 𝑎2 ∪ 𝑎𝑆𝑎) 𝑆 (𝑎 ∪ 𝑆𝑎)

(5)

𝑎 = 𝑎𝑥𝑎 = 𝑎𝑥𝑎𝑥𝑎.

(4)

= 𝑎2 ∪ 𝑎𝑆𝑎 ∪ 𝑎3 ∪ 𝑎2 𝑆𝑎 ∪ 𝑎𝑆𝑎2 ∪ 𝑎𝑆𝑎𝑆𝑎 ⊆ 𝑎2 ∪ 𝑎3 ∪ 𝑎𝑆𝑎.

(6)

Thus (𝑓∘𝑘 𝑔∘𝑘 ℎ) (𝑎) = (𝑓 ∘ 𝑔 ∘ ℎ) (𝑎) ∧

1−𝑘 2

Thus 𝑎 = 𝑎2 = 𝑎𝑎 = 𝑎2 𝑎 = 𝑎𝑎𝑎 or 𝑎 = 𝑎3 = 𝑎𝑎𝑎 or 𝑎 = 𝑎𝑥𝑎, for some 𝑥 in 𝑆. Hence 𝑆 is regular semigroup.

= (𝑎=𝑝𝑞 {𝑓 (𝑝) ∧ (𝑔 ∘ ℎ) (𝑞)}) ∧

Theorem 20. For a semigroup 𝑆, the following conditions are equivalent:

≥ 𝑓 (𝑎) ∧ (𝑔 ∘ ℎ) ((𝑥𝑎) (𝑥𝑎)) ∧

(i) 𝑆 is regular; (ii) 𝑅∩𝐿 1 ∩𝐿 2 ⊆ 𝑅𝐿 1 𝐿 2 for every right ideal 𝑅 and bi-ideal 𝐵 of a semigroup 𝑆;

1−𝑘 2

≥ 𝑓 (𝑎) ∧ ((𝑥𝑎)(𝑥𝑎)=𝑏𝑐 {𝑔 (𝑏) ∧ ℎ (𝑐)}) ∧ ≥ 𝑓 (𝑎) ∧ 𝑔 (𝑥𝑎) ∧ ℎ (𝑥𝑎) ∧

1−𝑘 2

(7)

1−𝑘 2

≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎) ∧ ℎ (𝑎) ∧

1−𝑘 2

≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎) ∧ ℎ (𝑎) ∧

1−𝑘 . 2

(iii) 𝑅[𝑎] ∩ 𝐿[𝑎] ∩ 𝐿[𝑎] ⊆ 𝑅[𝑎]𝐿[𝑎]𝐿[𝑎], for some 𝑎 in 𝑆. Proof. (i) ⇒ (ii): Let 𝑆 be regular semigroup, then for an element 𝑎 ∈ 𝑆 there exists 𝑥 ∈ 𝑆 such that 𝑎 = 𝑎𝑥𝑎. Let 𝑎 ∈ 𝑅 ∩ 𝐿 1 ∩ 𝐿 2 , where 𝑅 is right ideal and 𝐿 1 , and 𝐿 2 are left ideals of 𝑆. So 𝑎 ∈ 𝑅, 𝑎 ∈ 𝐿 1 and 𝑎 ∈ 𝐿 2 . As 𝑎 = 𝑎𝑥𝑎𝑥𝑎 ∈ (𝑅𝑆)𝐿 1 𝑆𝐿 2 ⊆ 𝑅𝐿 1 𝐿 2 . Thus 𝑅 ∩ 𝐿 1 ∩ 𝐿 2 ⊆ 𝑅𝐿 1 𝐿 2 . (ii) ⇒ (iii) is obvious.

1−𝑘 2

(ii) ⇒ (i): Let 𝑅[𝑎] be right ideal, and let 𝐿 1 [𝑎] and 𝐿 2 [𝑎] be any two left ideals of 𝑆 generated by 𝑎, respectively. Then (𝐶𝑅[𝑎] )𝑘 is any (∈, ∈ ∨𝑞𝑘 )-fuzzy right ideal, and (𝐶𝐿 1 [𝑎] )𝑘 and (𝐶𝐿 2 [𝑎] )𝑘 are any (∈, ∈ ∨𝑞𝑘 )-fuzzy left ideals of

4


semigroup 𝑆, respectively. Let 𝑎 ∈ 𝑆 and 𝑏 ∈ 𝑅[𝑎] ∩ 𝐿 1 [𝑎] ∩ 𝐿 2 [𝑎]. Then 𝑏 ∈ 𝑅[𝑎], 𝑏 ∈ 𝐿 1 [𝑎], and 𝑏 ∈ 𝐿 2 [𝑎]. Now 1−𝑘 ≤ (𝐶𝑅[𝑎]∩𝐿 1 [𝑎]∩𝐿 2 [𝑎] )𝑘 (𝑏) 2

Theorem 23. For a semigroup 𝑆, the following conditions are equivalent: (i) 𝑆 is right weakly regular;

= ((𝐶𝑅[𝑎] )𝑘 ∧𝑘 (𝐶𝐿 1 [𝑎] )𝑘 ∧𝑘 (𝐶𝐿 2 [𝑎] )𝑘 ) (𝑏)

(ii) 𝑓 ∧ 𝑔 ∧ ℎ ≤ 𝑓 ∘ 𝑔 ∘ ℎ for every fuzzy right ideal, fuzzy left ideal, and fuzzy interior ideal of 𝑆, respectively. (8)

≤ ((𝐶𝑅[𝑎] )𝑘 ∘𝑘 (𝐶𝐿 1 [𝑎] )𝑘 ∘𝑘 (𝐶𝐿 2 [𝑎] )𝑘 ) (𝑏) = (𝐶𝑅[𝑎]𝐿 1 [𝑎]𝐿 2 [𝑎] )𝑘 (𝑏) . Thus 𝑏 ∈ 𝑅[𝑎]𝐿 1 [𝑎]𝐿 2 [𝑎]. Therefore 𝑅[𝑎] ∩ 𝐿 1 [𝑎] ∩ 𝐿 2 [𝑎] ⊆ 𝑅[𝑎]𝐿 1 [𝑎]𝐿 2 [𝑎]. So by Theorem 20, 𝑆 is regular.

4. Characterizations of Right Weakly Regular Semigroups in Terms of (∈, ∈ ∨𝑞𝑘 )-Fuzzy Ideals

Proof. (i) ⇒ (ii): Let 𝑓, 𝑔, and ℎ be any (∈, ∈ ∨𝑞𝑘 )-fuzzy right ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy generalized bi-ideal, and (∈, ∈ ∨𝑞𝑘 )-fuzzy interior ideal of 𝑆. Since 𝑆 is right weakly regular therefore for each 𝑎 ∈ 𝑆 there exist 𝑥, 𝑦 ∈ 𝑆 such that 𝑎 = 𝑎𝑥𝑎𝑦 = (𝑎𝑥𝑎𝑦) (𝑥𝑎𝑦) = (𝑎𝑥) (𝑎𝑦) (𝑥𝑎𝑦) = (𝑎𝑥) (𝑎𝑥𝑎𝑦𝑦) (𝑥𝑎𝑦) = (𝑎𝑥) (𝑎𝑥𝑎) (𝑦𝑦𝑥𝑎𝑦) . Then (𝑓∘𝑘 𝑔∘𝑘 ℎ) (𝑎)

Theorem 22. For a semigroup 𝑆, the following conditions are equivalent:

= (𝑓 ∘ 𝑔 ∘ ℎ) (𝑎) ∧

1−𝑘 2 1−𝑘 2

(i) 𝑆 is right weakly regular;

= (𝑎=𝑝𝑞 {𝑓 (𝑝) ∧ (𝑔 ∘ ℎ) (𝑞)}) ∧

(ii) 𝑅 ∩ 𝐿 ∩ 𝐼 ⊆ 𝑅𝐿𝐼 for every right ideal, left ideal, and interior ideal of 𝑆, respectively;

≥ 𝑓 (𝑎𝑥) ∧ (𝑔 ∘ ℎ) ((𝑎𝑥𝑎) (𝑦𝑦𝑥𝑎𝑦)) ∧

(iii) 𝑅[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎] ⊆ 𝑅[𝑎]𝐿[𝑎]𝐼[𝑎]. Proof. (i) ⇒ (ii): Let 𝑆 be right weakly regular semigroup, and let 𝑅, 𝐿, and 𝐼 be right ideal, left ideal, and interior ideal of 𝑆, respectively. Let 𝑎 ∈ 𝑅 ∩ 𝐿 ∩ 𝐼 then 𝑎 ∈ 𝑅, 𝑎 ∈ 𝐿, and 𝑎 ∈ 𝐼. Since 𝑆 is right weakly regular semigroup so for 𝑎 there exist 𝑥, 𝑦 ∈ 𝑆 such that 𝑎 = 𝑎𝑥𝑎𝑦 = 𝑎𝑥𝑎𝑥𝑎𝑦𝑦 = 𝑎𝑥𝑎𝑥𝑎𝑥𝑎𝑦𝑦𝑦 ∈ (𝑅𝑆) (𝑆𝐿) (𝑆𝐼𝑆) ⊆ 𝑅𝐿𝐼.

(9)

Therefore 𝑎 ∈ 𝑅𝐿𝐼. So 𝑅 ∩ 𝐿 ∩ 𝐼 ⊆ 𝑅𝐿𝐼. (ii) ⇒ (iii) is obvious. (iii) ⇒ (i): As 𝑎 ∪ 𝑎𝑆, 𝑎 ∪ 𝑆𝑎, and 𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆 are right ideal, left ideal, and interior ideal of 𝑆 generated by an element 𝑎 of 𝑆, respectively, thus by assumption, we have (𝑎 ∪ 𝑎𝑆) ∩ (𝑎 ∪ 𝑆𝑎) ∩ (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆) ⊆ (𝑎 ∪ 𝑎𝑆) (𝑎 ∪ 𝑆𝑎) (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆) = (𝑎2 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑆𝑎) (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆) 3

4

2

2

3

(10)

= 𝑎 ∪ 𝑎 ∪ 𝑎 𝑆𝑎𝑆 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑎𝑆𝑎𝑆 ∪ 𝑎𝑆𝑆𝑎2 ∪ 𝑎𝑆𝑆𝑎3 ∪ 𝑎𝑆𝑆𝑎𝑆𝑎𝑆 3

(11)

4

⊆ 𝑎 ∪ 𝑎 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑎𝑆. Thus 𝑎 = 𝑎3 = 𝑎𝑎𝑎 = 𝑎𝑎𝑎3 = 𝑎𝑎𝑎2 𝑎 or 𝑎 = 𝑎𝑥𝑎 = 𝑎𝑥𝑎𝑥𝑎 or 𝑎 = 𝑎𝑢𝑎V, for some 𝑥, 𝑢, V in 𝑆. Hence 𝑆 is right weakly regular semigroup.

1−𝑘 2

≥ 𝑓 (𝑎) ∧ ((𝑎𝑥𝑎)(𝑦𝑦𝑥𝑎𝑦)=𝑏𝑐 {𝑔 (𝑏) ∧ ℎ (𝑐)}) ∧ ≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎𝑥𝑎) ∧ ℎ (𝑦𝑦𝑥𝑎𝑦) ∧ ≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎) ∧ ℎ (𝑎) ∧

1−𝑘 2

≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎) ∧ ℎ (𝑎) ∧

1−𝑘 . 2

1−𝑘 2

(12)

1−𝑘 2

Therefore 𝑓 ∧𝑘 𝑔 ∧𝑘 ℎ ≤ 𝑓∘𝑘 𝑔∘𝑘 ℎ. Now (ii) ⇒ (i) (ii) ⇒ (i): Let 𝑅[𝑎], 𝐿[𝑎], and 𝐼[𝑎] be right ideal, left ideal, and interior ideal of 𝑆 generated by 𝑎, respectively. Then (𝐶𝑅[𝑎] )𝑘 , (𝐶𝐿[𝑎] )𝑘 , and (𝐶𝐼[𝑎] )𝑘 are (∈, ∈ ∨𝑞𝑘 )-fuzzy right ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy left ideal, and (∈, ∈ ∨𝑞𝑘 )-fuzzy interior ideal of semigroup 𝑆. Let 𝑎 ∈ 𝑆 and 𝑏 ∈ 𝑅[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎]. Then 𝑏 ∈ 𝑅[𝑎], 𝑏 ∈ 𝐿[𝑎], and 𝑏 ∈ 𝐼[𝑎]. Now 1−𝑘 ≤ (𝐶𝑅[𝑎]∩𝐿[𝑎]∩𝐼[𝑎] )𝑘 (𝑏) 2 = ((𝐶𝑅[𝑎] )𝑘 ∧𝑘 (𝐶𝐿[𝑎] )𝑘 ∧𝑘 (𝐶𝐼[𝑎] )𝑘 ) (𝑏)

(13)

≤ ((𝐶𝑅[𝑎] )𝑘 ∘𝑘 (𝐶𝐿[𝑎] )𝑘 ∘𝑘 (𝐶𝐼[𝑎] )𝑘 ) (𝑏) = (𝐶𝑅[𝑎]𝐿[𝑎]𝐼[𝑎] )𝑘 (𝑏) . Thus 𝑏 ∈ 𝑅[𝑎]𝐿[𝑎]𝐼[𝑎]. Therefore 𝑅[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎] ⊆ 𝑅[𝑎]𝐿[𝑎]𝐼[𝑎]. Hence by Theorem 22, 𝑆 is right weakly regular semigroup.


5

Theorem 24. For a semigroup 𝑆, the following conditions are equivalent: (i) 𝑆 is right weakly regular; (ii) 𝐵∩𝐿∩𝐼 ⊆ 𝐵𝐿𝐼 for every bi-ideal, left ideal, and interior ideal of 𝑆, respectively;

Proof. (i) ⇒ (ii): Let 𝑆 be right weakly regular semigroup, and 𝐵, 𝐿, and 𝐼 be bi-ideal, left ideal, and interior ideal of 𝑆, respectively. Let 𝑎 ∈ 𝐵 ∩ 𝐿 ∩ 𝐼 then 𝑎 ∈ 𝐵, 𝑎 ∈ 𝐿, and 𝑎 ∈ 𝐼. Since 𝑆 is right weakly regular semigroup so for 𝑎 there exist 𝑥, 𝑦 ∈ 𝑆 such that

= (𝑎𝑥𝑎) (𝑥𝑎) (𝑥𝑎𝑦𝑦𝑦) ∈ (𝐵𝑆) (𝑆𝐿) (𝑆𝐼𝑆) ⊆ 𝐵𝐿𝐼.

𝑎 = 𝑎𝑥𝑎𝑦 = (𝑎𝑥𝑎𝑦) (𝑥𝑎𝑦) = (𝑎𝑥) (𝑎𝑦) (𝑥𝑎𝑦) = (𝑎𝑥) (𝑎𝑥𝑎𝑦𝑦) (𝑥𝑎𝑦) = (𝑎𝑥𝑎) (𝑥𝑎) (𝑦𝑦𝑥𝑎𝑦) .

(iii) 𝐵[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎] ⊆ 𝐵[𝑎]𝐿[𝑎]𝐼[𝑎].

𝑎 = 𝑎𝑥𝑎𝑦 = 𝑎𝑥𝑎𝑥𝑎𝑥𝑎𝑦𝑦𝑦

Proof. (i) ⇒ (iii): Let 𝑓, 𝑔, and ℎ be any (∈, ∈ ∨𝑞𝑘 )-fuzzy generalized bi-ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy left ideal, and (∈, ∈ ∨𝑞𝑘 )-fuzzy interior ideal of 𝑆. Since 𝑆 is right weakly regular for each 𝑎 ∈ 𝑆 there exist 𝑥, 𝑦 ∈ 𝑆 such that

Then (𝑓∘𝑘 𝑔∘𝑘 ℎ) (𝑎) = (𝑓 ∘ 𝑔 ∘ ℎ) (𝑎) ∧

1−𝑘 2

= (𝑎=𝑝𝑞 {𝑓 (𝑝) ∧ (𝑔 ∘ ℎ) (𝑞)}) ∧ (14)

Therefore 𝑎 ∈ 𝐵𝐿𝐼.So 𝐵 ∩ 𝐿 ∩ 𝐼 ⊆ 𝐵𝐿𝐼. (ii) ⇒ (iii) is obvious. (iii) ⇒ (i): As 𝑎 ∪ 𝑎2 ∪ 𝑎𝑆𝑎, 𝑎 ∪ 𝑆𝑎, and 𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆 are bi-ideal, left ideal, and interior ideal of 𝑆 generated by an element 𝑎 of 𝑆, respectively, thus by assumption we have

(16)

1−𝑘 2

≥ 𝑓 (𝑎𝑥𝑎) ∧ (𝑔 ∘ ℎ) ((𝑥𝑎) (𝑦𝑦𝑥𝑎𝑦)) ∧

1−𝑘 2

≥ 𝑓 (𝑎) ∧ ((𝑥𝑎)(𝑦𝑦𝑥𝑎𝑦)=𝑏𝑐 {𝑔 (𝑏) ∧ ℎ (𝑐)}) ∧ ≥ 𝑓 (𝑎) ∧ 𝑔 (𝑥𝑎) ∧ ℎ (𝑦𝑦𝑥𝑎𝑦) ∧

(𝑎 ∪ 𝑎2 ∪ 𝑎𝑆𝑎) ∩ (𝑎 ∪ 𝑆𝑎) ∩ (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆)

≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎) ∧ ℎ (𝑎) ∧

1−𝑘 2

⊆ (𝑎 ∪ 𝑎2 ∪ 𝑎𝑆𝑎) (𝑎 ∪ 𝑆𝑎) (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆)

≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎) ∧ ℎ (𝑎) ∧

1−𝑘 . 2

1−𝑘 2

1−𝑘 2

(17)

= (𝑎2 ∪ 𝑎𝑆𝑎 ∪ 𝑎3 ∪ 𝑎2 𝑆𝑎 ∪ 𝑎𝑆𝑎2 ∪ 𝑎𝑆𝑎𝑆𝑎) × (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆) = 𝑎3 ∪ 𝑎4 ∪ 𝑎2 𝑆𝑎𝑆 ∪ 𝑎𝑆𝑎2 ∪ 𝑎𝑆𝑎3 ∪ 𝑎𝑆𝑎𝑆𝑎𝑆 ∪ 𝑎4 ∪ 𝑎5 ∪ 𝑎3 𝑆𝑎𝑆 ∪ 𝑎2 𝑆𝑎2 ∪ 𝑎2 𝑆𝑎3 ∪ 𝑎2 𝑆𝑎𝑆𝑎𝑆 ∪ 𝑎𝑆𝑎3 ∪ 𝑎𝑆𝑎4 ∪ 𝑎𝑆𝑎2 𝑆𝑎𝑆𝑎𝑆𝑎𝑆𝑎2

Therefore 𝑓∧𝑘 𝑔∧𝑘 ℎ ≤ 𝑓∘𝑘 𝑔∘𝑘 ℎ. (iii) ⇒ (ii) is obvious. (ii) ⇒ (i): Let 𝐵[𝑎], 𝐿[𝑎], and 𝐼[𝑎] be bi-ideal, left ideal, and interior ideal of 𝑆 generated by 𝑎, respectively. Then (𝐶𝐵[𝑎] )𝑘 , (𝐶𝐿[𝑎] )𝑘 , and (𝐶𝐼[𝑎] )𝑘 are (∈, ∈ ∨𝑞𝑘 )-fuzzy bi-ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy left ideal, and (∈, ∈ ∨𝑞𝑘 )-fuzzy interior ideal of semigroup 𝑆. Let 𝑎 ∈ 𝑆 and 𝑏 ∈ 𝐵[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎]. Then 𝑏 ∈ 𝐵[𝑎], 𝑏 ∈ 𝐿[𝑎], and 𝑏 ∈ 𝐼[𝑎]. Now 1−𝑘 ≤ (𝐶𝐵[𝑎]∩𝐿[𝑎]∩𝐼[𝑎] )𝑘 (𝑏) 2

∪ 𝑎𝑆𝑎𝑆𝑎3 ∪ 𝑎𝑆𝑎𝑆𝑎𝑆𝑎𝑆 ⊆ 𝑎3 ∪ 𝑎4 ∪ 𝑎5 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑎𝑆. (15) Thus 𝑎 = 𝑎4 = 𝑎𝑎𝑎𝑎 or 𝑎 = 𝑎3 = 𝑎𝑎𝑎 = 𝑎𝑎𝑎3 = 𝑎𝑎𝑎2 𝑎 or 𝑎 = 𝑎𝑥𝑎 = 𝑎𝑥𝑎𝑥𝑎 or 𝑎 = 𝑎𝑢𝑎V, for some 𝑥, 𝑢, V in 𝑆. Hence 𝑆 is right weakly regular semigroup. Theorem 25. For a semigroup 𝑆, the following conditions are equivalent: (i) 𝑆 is right weakly regular; (ii) 𝑓 ∧ 𝑔 ∧ ℎ ≤ 𝑓 ∘ 𝑔 ∘ ℎ for every fuzzy bi-ideal, fuzzy left ideal and fuzzy interior ideal of 𝑆, respectively; (iii) 𝑓 ∧ 𝑔 ∧ ℎ ≤ 𝑓 ∘ 𝑔 ∘ ℎ for every fuzzy generalized bi-ideal, fuzzy left ideal, and fuzzy interior ideal of 𝑆, respectively.

= ((𝐶𝐵[𝑎] )𝑘 ∧𝑘 (𝐶𝐿[𝑎] )𝑘 ∧𝑘 (𝐶𝐼[𝑎] )𝑘 ) (𝑏)

(18)

≤ ((𝐶𝐵[𝑎] )𝑘 ∘𝑘 (𝐶𝐿[𝑎] )𝑘 ∘𝑘 (𝐶𝐼[𝑎] )𝑘 ) (𝑏) = (𝐶𝐵[𝑎]𝐿[𝑎]𝐼[𝑎] )𝑘 (𝑏) . Thus 𝑏 ∈ 𝐵[𝑎]𝐿[𝑎]𝐼[𝑎]. Therefore 𝐵[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎] ⊆ 𝐵[𝑎]𝐿[𝑎]𝐼[𝑎]. Hence by Theorem 24, 𝑆 is right weakly regular semigroup. Theorem 26. For a semigroup 𝑆, the following conditions are equivalent: (i) 𝑆 is right weakly regular; (ii) 𝑄 ∩ 𝐿 ∩ 𝐼 ⊆ 𝑄𝐿𝐼 for every quasi-ideal 𝑄, left ideal 𝐿, and interior ideal 𝐼 of 𝑆, respectively; (iii) 𝑄[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎] ⊆ 𝑄[𝑎]𝐿[𝑎]𝐼[𝑎].

6


Proof. (i) ⇒ (ii): Let 𝑆 be right weakly regular semigroup, and let 𝑄, 𝐿, and 𝐼 be quasi-ideal, left ideal, and interior ideal of 𝑆, respectively. Let 𝑎 ∈ 𝑄 ∩ 𝐿 ∩ 𝐼 then 𝑎 ∈ 𝑄, 𝑎 ∈ 𝐿, and 𝑎 ∈ 𝐼. Since 𝑆 is right weakly regular semigroup so for 𝑎 there exist 𝑥, 𝑦 ∈ 𝑆 such that 𝑎 = 𝑎𝑥𝑎𝑦 = (𝑎𝑥𝑎𝑦) (𝑥𝑎𝑦) = 𝑎 (𝑥𝑎) (𝑦𝑥𝑎𝑦) ∈ 𝑄 (𝑆𝐿) (𝑆𝐼𝑆) ⊆ 𝑄𝐿𝐼.

≥ 𝑓 (𝑎) ∧ 𝑔 (𝑥𝑎) ∧ ℎ (𝑦𝑥𝑎𝑦) ∧

(𝑎 ∪ (𝑎𝑆 ∩ 𝑆𝑎)) ∩ (𝑎 ∪ 𝑆𝑎) ∩ (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆) ⊆ (𝑎 ∪ (𝑎𝑆 ∩ 𝑆𝑎)) (𝑎 ∪ 𝑆𝑎) (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆) ⊆ (𝑎 ∪ 𝑎𝑆) (𝑎 ∪ 𝑆𝑎) (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆) (20)

1−𝑘 2

≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎) ∧ ℎ (𝑎) ∧

1−𝑘 . 2 (22)

Therefore 𝑓∧𝑘 𝑔∧𝑘 ℎ ≤ 𝑓∘𝑘 𝑔∘𝑘 ℎ. (iii) ⇒ (ii) is obvious. (ii) ⇒ (i): Let 𝑄[𝑎], 𝐿[𝑎], and 𝐼[𝑎] be quasi-ideal, left ideal and interior ideal of 𝑆 generated by 𝑎, respectively. Then (𝐶𝑄[𝑎] )𝑘 , (𝐶𝐿[𝑎] )𝑘 , and (𝐶𝐼[𝑎] )𝑘 are (∈, ∈ ∨𝑞𝑘 )-fuzzy quasi-ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy left ideal, and (∈, ∈ ∨𝑞𝑘 )-fuzzy interior ideal of semigroup 𝑆. Let 𝑎 ∈ 𝑆 and let 𝑏 ∈ 𝑄[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎]. Then 𝑏 ∈ 𝑄[𝑎], 𝑏 ∈ 𝐿[𝑎], and 𝑏 ∈ 𝐼[𝑎]. Now 1−𝑘 ≤ (𝐶𝑄[𝑎]∩𝐿[𝑎]∩𝐼[𝑎] )𝑘 (𝑏) 2 = ((𝐶𝑄[𝑎] )𝑘 ∧𝑘 (𝐶𝐿[𝑎] )𝑘 ∧𝑘 (𝐶𝐼[𝑎] )𝑘 ) (𝑏)

= 𝑎3 ∪ 𝑎4 ∪ 𝑎2 𝑆𝑎𝑆 ∪ 𝑎𝑆𝑎2 ∪ 𝑎𝑆𝑎3 ∪ 𝑎𝑆𝑎𝑆𝑎𝑆

= (𝐶𝑄[𝑎]𝐿[𝑎]𝐼[𝑎] )𝑘 (𝑏) .

⊆ 𝑎3 ∪ 𝑎4 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑎𝑆. Thus 𝑎 = 𝑎4 = 𝑎𝑎𝑎𝑎 or 𝑎 = 𝑎3 = 𝑎𝑎𝑎 = 𝑎𝑎𝑎3 = 𝑎𝑎𝑎2 𝑎 or 𝑎 = 𝑎𝑥𝑎 = 𝑎𝑥𝑎𝑥𝑎 or 𝑎 = 𝑎𝑢𝑎V, for some 𝑥, 𝑢, V in 𝑆. Hence 𝑆 is right weakly regular semigroup. Theorem 27. For a semigroup 𝑆, the following conditions are equivalent: (i) 𝑆 is right weakly regular; (ii) 𝑓 ∧ 𝑔 ∧ ℎ ≤ 𝑓 ∘ 𝑔 ∘ ℎ for every fuzzy quasi-ideal, fuzzy left ideal, and fuzzy interior ideal of 𝑆, respectively. Proof. (i) ⇒ (iii): Let 𝑓, 𝑔, and ℎ be any (∈, ∈ ∨𝑞𝑘 )-fuzzy quasi-ideal, (∈, ∈ ∨𝑞𝑘 )-fuzzy left ideal, and (∈, ∈ ∨𝑞𝑘 )-fuzzy interior ideal of 𝑆. Since 𝑆 is right weakly regular therefore for each 𝑎 ∈ 𝑆 there exist 𝑥, 𝑦 ∈ 𝑆 such that 𝑎 = 𝑎𝑥𝑎𝑦 = (𝑎𝑥𝑎𝑦) (𝑥𝑎𝑦) = 𝑎 (𝑥𝑎) (𝑦𝑥𝑎𝑦) . Then (𝑓∘𝑘 𝑔∘𝑘 ℎ) (𝑎) 1−𝑘 2

= (𝑎=𝑝𝑞 {𝑓 (𝑝) ∧ (𝑔 ∘ ℎ) (𝑞)}) ∧

(23)

≤ ((𝐶𝑄[𝑎] )𝑘 ∘𝑘 (𝐶𝐿[𝑎] )𝑘 ∘𝑘 (𝐶𝐼[𝑎] )𝑘 ) (𝑏)

∪ 𝑎𝑆𝑆𝑎2 ∪ 𝑎𝑆𝑆𝑎3 ∪ 𝑎𝑆𝑆𝑎𝑆𝑎𝑆

= (𝑓 ∘ 𝑔 ∘ ℎ) (𝑎) ∧

1−𝑘 2

1−𝑘 2

≥ 𝑓 (𝑎) ∧ 𝑔 (𝑎) ∧ ℎ (𝑎) ∧ (19)

Therefore 𝑎 ∈ 𝑄𝐿𝐼. So 𝑄 ∩ 𝐿 ∩ 𝐼 ⊆ 𝑄𝐿𝐼. (ii) ⇒ (iii) is obvious. (𝑖𝑖𝑖) ⇒ (𝑖): As 𝑎 ∪ (𝑎𝑆 ∩ 𝑆𝑎), 𝑎 ∪ 𝑆𝑎, and 𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆 are quasi-ideal, left ideal, and interior ideal of 𝑆 generated by an element 𝑎 of 𝑆, respectively, thus by assumption we have

= (𝑎2 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑎 ∪ 𝑎𝑆𝑆𝑎) (𝑎 ∪ 𝑎2 ∪ 𝑆𝑎𝑆)

≥ 𝑓 (𝑎) ∧ ((𝑥𝑎)(𝑦𝑥𝑎𝑦)=𝑏𝑐 {𝑔 (𝑏) ∧ ℎ (𝑐)}) ∧

1−𝑘 2

≥ 𝑓 (𝑎) ∧ (𝑔 ∘ ℎ) ((𝑥𝑎) (𝑦𝑥𝑎𝑦)) ∧

1−𝑘 2

(21)

Thus 𝑏 ∈ 𝑄[𝑎]𝐿[𝑎]𝐼[𝑎]. Therefore 𝑄[𝑎] ∩ 𝐿[𝑎] ∩ 𝐼[𝑎] ⊆ 𝑄[𝑎]𝐿[𝑎]𝐼[𝑎]. Hence by Theorem 26, 𝑆 is right weakly regular semigroup.

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Journal of Mathematics [13] O. Kazanci and S. Yamak, “Generalized fuzzy bi-ideals of semigroup,” Soft Computing, vol. 12, pp. 1119–1124, 2008. [14] Y. B. Jun, “Generalizations of (∈, ∈, 𝑉𝑞)-fuzzy subalgebras in BCK/BCI-algebras,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1383–1390, 2009. [15] M. Shabir, Y. B. Jun, and Y. Nawaz, “Semigroups characterized by (∈, ∈, 𝑉𝑞𝑘)-fuzzy ideals,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1473–1493, 2010.

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