Characterization of regular LA-semigroups by interval

Characterization of regular LA-semigroups by interval-valued $$(\overline{\alpha }, \overline{\beta })$$ -fuzzy ideals Muhammad Aslam, Saleem Abdullah & Samreen Aslam

Afrika Matematika ISSN 1012-9405 Volume 25 Number 3 Afr. Mat. (2014) 25:501-518 DOI 10.1007/s13370-012-0130-6

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Author's personal copy Afr. Mat. (2014) 25:501–518 DOI 10.1007/s13370-012-0130-6

Characterization of regular LA-semigroups by interval-valued (α, β)-fuzzy ideals Muhammad Aslam · Saleem Abdullah · Samreen Aslam

Received: 27 September 2012 / Accepted: 14 December 2012 / Published online: 11 January 2013 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2013

Abstract The concept of interval-valued (α, β)-fuzzy ideals, interval-valued (α, β)-fuzzy generalized bi-ideals are introduced in LA-semigroups, using the ideas of belonging and quasi-coincidence of an interval-valued fuzzy point with an interval-valued fuzzy set and some related properties are investigated. We define the lower and upper parts of intervalvalued fuzzy subsets of an LA-semigroup. Also regular LA-semigroups are characterized by the properties of the upper part of interval-valued (∈, ∈ ∨ q)-fuzzy left ideals, interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideals and interval-valued (∈, ∈ ∨ q)-fuzzy generalized bi-ideals. Keywords Interval-valued (α, β)-fuzzy sub LA-semigroups · Interval-valued (α, β)-fuzzy ideals · Interval-valued (α, β)-fuzzy bi-ideals · Interval-valued (α, β)-fuzzy quasi-ideals Mathematics Subject Classification (2000)

16D25, 20N99, 03E72

1 Introduction The fundamental concept of a fuzzy set, introduced by Zadeh in his classic paper [28] of 1965, provides a natural framework for generalizing some of the basic notions of algebra. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. The notion of fuzzy subgroups was defined by Rosenfeld [21] and its structure was investigated. In [6] Bhakat and Das generalized Rosenfeld’s fuzzy subgroups and introduced the (∈, ∈ ∨q)fuzzy subgroup, by using the notions of “belongingness” and “quasi-coincidence” of fuzzy point and fuzzy set, which was introduced by Pu and Liu [20]. Some authors investigated

M. Aslam · S. Abdullah (B) · S. Aslam Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Paksitan e-mail: [email protected] M. Aslam e-mail: [email protected] S. Aslam e-mail: [email protected]

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similar type of generalizations of other algebraic structures, for example Davvaz [8] applied this theory to nearrings. On the other hand Kuroki [16] laid the foundations of a theory of fuzzy semigroups and Jun and Song [10] initiated the study of (α, β)-fuzzy interior ideals of a semigroup. In [11] Kazanci and Yamak study (∈, ∈ ∨q)-fuzzy bi-ideals of a semigroup. In [23], Shabir et al. characterized regular semigroups by the properties of (∈, ∈ ∨q)-fuzzy ideals, bi-ideals and quasi-ideals. In [11], Kazanci and Yamak defined (∈, ∈ ∨ q)-fuzzy bi-ideals of a semigroup. In [24], Shabir et al. characterized regular semigroups by the properties of (∈, ∈ ∨ q)-fuzzy ideals, generalized bi-ideals and quasi-ideals of a semigroup. Khan et al., studied ideals theory of LA-semigroup by another aspect of fuzzy set see [13–15]. For more detail study on Fuzzy ideals theory of LA-semigroups see [1–4,25–27]. Interval-valued fuzzy subsets were proposed about 30 years ago as a natural extension of fuzzy sets by Zadeh [29]. Interval-valued fuzzy subsets have many applications in several areas such as the method of approximate inference. In [19] Narayanan and Manikantan introduced the notions of interval-valued fuzzy ideals generated by an interval-valued fuzzy subset in semigroups. In this paper we introduced the concepts of interval-valued (∈, ∈ ∨ q)fuzzy ideals, interval-valued (∈, ∈ ∨ q)-fuzzy bi-ideals and interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideals of an LA-semigroup. We characterize the different class of la-semigroups by the properties of interval valued (∈, ∈ ∨ q)-fuzzy ideals.

2 Preliminaries We first recall some basic concepts. A groupoid (S, ∗) is called a left almost semigroup, abbreviated as an LA-semigroup, if it satisfies left invertive law: (a ∗ b) ∗ c = (c ∗ b) ∗ a ∀a, b, c ∈ S A non-empty subset A of S is called a sub LA-semigroup of S if A A ⊆ A and is called left (resp. right) ideal of S if S A ⊆ A(AS ⊆ A). By a two-sided ideal or simply an ideal we mean a non-empty subset of S which is both a left and a right ideal of S. A non-empty subset B of S is called a generalized bi-ideal of S if (B S)B ⊆ B. A sub LA-semigroup B of S is called a bi-ideal of S if (B S)B ⊆ B. A non-empty subset Q of S is called a quasi-ideal of S if Q S ∩ S Q ⊆ Q. Obviously every one-sided ideal of an LA-semigroup S is a quasi-ideal, every quasi-ideal is a bi-ideal and every bi-ideal is a generalized bi-ideal. An LA-semigroup S is called regular if for each element a of S, there exists an element x in S such that a = (ax)a. An LA-semigroup S is called left regular if for each element a of S, there exists an element x in S such that a = (aa)x. We now review some concepts of fuzzy subsets. A fuzzy subset λ of an LA-semigroup S is a mapping λ : S → [0, 1]. Throughout this paper S denotes an LA-semigroup. Definition 2.1 [30] A fuzzy subset λ of S is called a fuzzy sub LA-semigroup of S if for all x, y ∈ S λ(x y) ≥ λ(x) ∧ λ(y). A fuzzy subset λ of S is called a fuzzy left (resp. right) ideal of S if for all x, y ∈ S λ(x y) ≥ λ(y). (resp. λ(x y) ≥ λ(x)) A fuzzy subset λ of S is called a fuzzy two-sided ideal or simply fuzzy ideal of S if it is both a fuzzy left and a fuzzy right ideal of S.

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Definition 2.2 [30] A fuzzy sub LA-semigroup λ of S is called a fuzzy bi-ideal of S if for all x, y, z ∈ S λ((x y)z) ≥ λ(x) ∧ λ(z). We will describe some results of an interval number. By an interval number on [0, 1], we mean [22], say a , is a closed subinterval of [0, 1], that is, a = [a − , a + ], where 0 ≤ a − ≤ a + ≤ 1. Let D[0, 1] denote the family of all closed subintervals of [0, 1], 0 = [0, 0] and 1 = [1, 1]. Now we define ≤, =, ∧, ∨ in case of two elements in D[0, 1]. Consider two elements a = [a − , a + ] and b = [b− , b+ ] in D[0, 1]. Then, a ≤ b if and only if a − ≤ b− and a + ≤ b+ . a = b if and only if a − = b− and a + = b+ . min{ a, b} = a ∧ b = [min{a − , b− }, min{a + , b+ }]. max{ a, b} = a ∨ b = [max{a − , b− }, max{a + , b+ }]. Let X be a set. A mapping λ : X → D[0, 1] is called an interval-valued fuzzy subset (briefly, an i-v fuzzy subset) of X, where λ(x) = [λ− (x), λ+ (x)] for all x ∈ X , where λ− and λ+ are − fuzzy subsets in X such that λ (x) ≤ λ+ (x) for all x ∈ X . Let λ and μ be two interval-valued fuzzy subsets of X. Define the relation ⊆ between λ and μ as follows: λ⊆ μ if and only if λ(x) ≤ μ(x) for all x ∈ X , that is, λ− (x) ≤ μ− (x) and λ+ (x) ≤ μ+ (x). An interval-valued fuzzy subset λ in a universe X of the form t( = [0, 0]) if y = x λ(y) = 0 if y = x for all y ∈ X is said to be an interval-valued fuzzy point with support x and interval value t, and is denoted by xt . Zhan et al. [31] gave meaning to the symbol xt α λ, where α ∈ {∈, q, ∈ ∨q, ∈ ∧q}. An interval-valued fuzzy point xt is said to belongs to (resp. quasi-coincident with) an intervalvalued fuzzy set λ written xt ∈ λ (resp. xt q λ) if λ(x) ≥ t (resp. λ(x) + t > 1), and in this case xt ∈ ∨q λ (resp. xt ∈ ∧q λ) means that xt ∈ λ or xt q λ (resp. xt ∈ λ and xt q λ). To say that xt α λ means that xt α λ does not hold. Let λ and μ be two interval-valued fuzzy subsets of an LA-semigroup S. The product λ◦ μ is defined by ∨ { λ(y) ∧ μ(z)} if there exist y, z ∈ S such that x = yz (λ ◦ μ)(x) = x=yz 0 otherwise

3 Interval-valued fuzzy ideals of LA-semigroups Definition 3.1 [26] An interval-valued fuzzy subset λ of an LA-semigroup S is called an interval-valued fuzzy sub LA-semigroup of S if λ(x y) ≥ λ(x) ∧ λ(y). ∀ x, y ∈ S An interval-valued fuzzy subset λ of an LA-semigroup S is called an interval-valued fuzzy left (resp. right) ideal of S if λ(x y) ≥ λ(y) (resp. λ(x y) ≥ λ(x)) ∀ x, y ∈ S

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An interval-valued fuzzy subset λ of an LA-semigroup S is called an interval-valued fuzzy generalized bi-ideal of S if λ((x y)z) ≥ λ(x) ∧ λ(z). ∀ x, y, z ∈ S An interval-valued fuzzy sub LA-semigroup λ of S is called an interval-valued fuzzy bi-ideal of S if λ((x y)z) ≥ λ(x) ∧ λ(z). ∀x, y, z ∈ S An interval-valued fuzzy subset λ of an LA-semigroup S is called an interval-valued fuzzy quasi-ideal of S if ( λ ◦ δ ) ∧ ( δ ◦ λ) ≤ λ, where δ : S → [1, 1]. Theorem 3.2 [30] Let S be an LA-semigroup with left identity e such that (xe)S = x S for all x ∈ S. Then, the following are equivalent (1) S is regular. (2) R ∩ L = R L for every right ideal R and left ideal L of S. (3) A = (AS)A for every quasi-ideal A of S. 4 Interval-valued (α, β)-fuzzy ideals Let S be an LA-semigroup and α and β denote any one of ∈, q, ∈ ∨q, ∈ ∧q unless otherwise specified. Definition 4.1 An interval-valued fuzzy subset λ of an LA-semigroup S is called an intervalvalued (α, β)-fuzzy sub LA-semigroup of S, where α = ∈ ∨q , if the following condition holds If (x y)min{ t1 ,t2 } α λ implies that x t1 β λ or yt2 β λ. for all x ∈ S. Let Let λ be an interval-valued fuzzy subset of S such that λ(x) ≥ 0.5 x ∈ S and t ∈ D[0, 1] be such that xt ∈ ∨q λ. Then λ(x) < t and λ(x) + t ≤ 1 . It This means that follows that 2 λ(x) = λ(x) + λ(x) < λ(x) + t ≤ 1 so that λ(x) < 0.5. {xt : xt ∈ ∨q λ} = φ. Therefore the case α = ∈ ∨q , in the above definition is omitted. Definition 4.2 An interval-valued fuzzy subset λ of an LA-semigroup S is called an intervalvalued (α, β)-fuzzy left (resp. right) ideal of S, where α = ∈ ∨q, if it satisfies, (x y)t α λ implies that yt β λ (resp. xt β λ) for all x, y ∈ S. An interval-valued fuzzy subset λ of an LA-semigroup S is called an interval-valued (α, β)-fuzzy ideal of S, if it is both an interval-valued (α, β)-fuzzy left ideal of S and an interval-valued (α, β)-fuzzy right ideal of S. Definition 4.3 An interval-valued fuzzy subset λ of an LA-semigroup S is called an intervalvalued (α, β)-fuzzy generalized bi-ideal of S, where α = ∈ ∨q, if it satisfies, For all x, y, z ∈ S and for all t3 , t4 ∈ D[0, 1], where t3 , t4 = 0, ((x y)z)min{ t3 ,t4 } α λ implies that x t3 β λ or z t4 β λ. Definition 4.4 An interval-valued fuzzy subset λ of an LA-semigroup S is called an intervalvalued (α, β)-fuzzy bi-ideal of S, where α = ∈ ∨q, if it satisfies the following two conditions.

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(i) For all x, y ∈ S and for all t1 , t2 ∈ D[0, 1], where t1 , t2 = 0, (x y)min{ t1 ,t2 } α λ implies λ or yt2 β λ. that xt1 β (ii) For all x, y, z ∈ S and for all t3 , t4 ∈ D[0, 1], where t3 , t4 = 0, ((x y)z)min{ t3 ,t4 } α λ implies that x t3 β λ or z t4 β λ. Lemma 4.5 An interval-valued fuzzy subset λ of an LA-semigroup S is an interval-valued fuzzy sub LA-semigroup of S if and only if it satisfies, For all x, y ∈ S and for all t1 , t2 ∈ D[0, 1], where t1 , t2 = 0, (x y)min{ t1 ,t2 } ∈λ implies thatxt1 ∈λ or yt2 ∈λ. Proof Suppose that λ is an interval-valued fuzzy sub LA-semigroup of an LA-semigroup S. Let x, y ∈ S and t1 , t2 ∈ D[0, 1], where t1 , t2 = 0, such that (x y)min{ t1 ,t2 } ∈λ. Then λ(x y) < min{ t1 , t2 }. Since λ is an interval-valued fuzzy sub LA-semigroup of S. So min{ t1 , t2 } > λ(x y) ≥ min{ λ(x), λ(y)} which implies that min{ t1 , t2 } < t1 and min{ t1 , t2 } < t2 . So if min{ λ(x), λ(y)} = λ(x), then λ(x) < t1 , that is, x t1 ∈λ. If min{λ(x), λ(y)} = λ(y), then λ(y) < t2 , that is, yt2 λ. Hence, x λ. ∈ ∈ t1 ∈λ or yt2 Conversely, assume that λ satisfies the given condition. We show that λ(x y) ≥ min{λ(x), λ(y)}. On contrary assume that there exist x, y ∈ S such that λ(x y) < min{ λ(x), λ(y)}. Let t ∈ D[0, 1] be such that λ(x y) < t < min{ λ(x), λ(y)}. Then (x y)t ∈ λ, but xt ∈ λ and yt ∈ λ. This contradicts our hypothesis. Thus λ(x y) ≥ min{ λ(x), λ(y)}. Lemma 4.6 An interval-valued fuzzy subset λ of an LA-semigroup S is an interval-valued fuzzy left (resp. right) ideal of S if and only if it satisfies, For all x, y ∈ S and for all t ∈ D[0, 1], where t = 0, (x y)t ∈ λ implies that yt ∈ λ (resp. xt ∈ λ). Proof The proof is similar to Lemma 4.5.

Remark 4.7 The above lemma shows that every fuzzy left (right) ideal of S is an (∈, ∈)-fuzzy left (right) ideal of S. Lemma 4.8 An interval-valued fuzzy subset λ of an LA-semigroup S is an interval-valued fuzzy generalized bi-ideal of S if and only if it satisfies, For all x, y, z ∈ S and for all t3 , t4 ∈ D[0, 1], where t3 , t4 = 0, ((x y)z)min{ t3 ,t4 } ∈λ implies that x ∈ λ or z ∈ λ. t3 t4 Proof The proof is similar to Lemma 4.5.

Lemma 4.9 An interval-valued fuzzy subset λ of an LA-semigroup S is an interval-valued fuzzy bi-ideal of S if and only if it satisfies, (1) For all x, y ∈ S and for all t1 , t2 ∈ D[0, 1], where t1 , t2 = 0, (x y)min{ t1 ,t2 } ∈λ implies that xt1 ∈ λ or yt2 ∈ λ. (2) For all x, y, z ∈ S and for all t3 , t4 ∈ D[0, 1], where t3 , t4 , ((x y)z)min{ t3 ,t4 } ∈λ implies that x t3 ∈λ or z t4 ∈λ. Proof The proof is similar to Lemma 4.5. So we omit its proof. Theorem 4.10 Let λ be a nonzero interval-valued (α, β)-fuzzy sub LA-semigroup of S. Then the set λ◦ = {x ∈ S : λ(x) > 0} is a sub LA-semigroup of S. λ(x y) = 0. Let λ(x) > 0 and λ(y) > 0. If α ∈ {∈, q, ∈ Proof Let x y ∈ / λ◦ . Then ∨q, ∈ ∧q}, then (x y)min{λ(x),λ(x)} α λ but xλ(x) β λ and yλ(x) β λ for every β ∈ {∈, ∈ ∨q}. Which is a contradiction. Hence, λ(x) = 0 or λ(y) = 0, that is, x or y ∈ / λ◦ . Also for α ∈ {∈, q, ∈ ∨q, ∈ ∧q}, (x y)1 α λ but x1 q λ and x1 q λ. Hence, λ(x) = 0 or λ(y) = 0, that is, x or y ∈ / λ◦ . Thus λ◦ is sub LA-semigroup of S.

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Theorem 4.11 Let λ be a nonzero (α, β)-fuzzy left (resp. right) ideal of S. Then the set λ◦ = {x ∈ S : λ(x) > 0} is a left (resp. right) ideal of S. Proof The proof is similar to Theorem 4.10.

Theorem 4.12 Let λ be a nonzero (α, β)-fuzzy generalized bi-ideal of S. Then the set λ◦ = {x ∈ S : λ(x) > 0} is a generalized bi-ideal of S. Proof The proof is similar to the Theorem 4.10.

Theorem 4.13 Let λ be a nonzero (α, β)-fuzzy bi-deal of S. Then the set λ◦ = {x ∈ S : λ(x) > 0} is a bi-ideal of S. Proof The proof is similar to Theorem 4.10. So we omit its proof.

5 Interval-valued (∈, ∈ ∨ q)-fuzzy ideals Theorem 5.1 Let A be a sub LA-semigroup of S and let λ be an interval-valued fuzzy subset in S such that 1 if x ∈ A λ(x) = if x ∈ S\A ≤ 0.5 Then (1) λ is an interval-valued (q, ∈ ∨ q)-fuzzy sub LA-semigroup of S. (2) λ is an interval-valued (∈, ∈ ∨ q)-fuzzy sub LA-semigroup of S. Proof (1) Let x, y ∈ S and t1 , t2 ∈ D[0, 1], where t1 , t2 = 0, such that (x y)min{t1 ,t2 } q λ. Then λ(x y) + min{ t1 , t2 } ≤ 1. So x y ∈ / L. Therefore x ∈ / L or y ∈ / L. If t1 , t2 > 0.5, ≤ ≤ then then λ(x) ≤ 0.5 t1 so xt1 ∈ λ, and λ(y) ≤ 0.5 t1 so yt2 ∈ λ. If t1 , t2 ≤ 0.5, + 0.5 = + 0.5 = λ(x) + t1 ≤ 0.5 1 so xt1 q λ, and λ(y) + t2 ≤ 0.5 1. Hence yt2 q λ. Therefore xt1 ∈ ∨ q λ or yt2 ∈ ∨ q λ. Thus λ is an interval-valued (∈, ∈ ∨ q)-fuzzy sub LA-semigroup of S. (2) Let x, y ∈ S and t1 , t2 ∈ D[0, 1], where t1 , t2 = 0, such that (x y)min{t1 ,t2 } ∈ λ. Then then λ(x y) < min{t1 , t2 }. So x y ∈ / L. Therefore x ∈ / L or y ∈ / L. Thus, if t1 , t2 > 0.5, ≤ ≤ then λ(x) ≤ 0.5 t1 and so xt1 ∈ λ and λ(y) ≤ 0.5 t1 so yt2 ∈ λ. If t1 , t2 ≤ 0.5, λ(x) + t1 ≤ + 0.5 = + 0.5 = 0.5 1 so xt1 q λ, and λ(y) + t2 ≤ 0.5 1. Hence yt2 q λ. Therefore xt1 ∈ ∨ q λ or yt2 ∈ ∨ q λ. Thus λ is an interval-valued (∈, ∈ ∨ q)-fuzzy sub LA-semigroup of S. Theorem 5.2 Let L be a left (resp. right) ideal of S and let λ be an interval-valued fuzzy subset in S such that 1 if x ∈ L λ(x) = if x ∈ S\L ≤ 0.5 Then, (1) λ is an interval-valued (q, ∈ ∨ q)-fuzzy left (resp. right) ideal of S. (2) λ is an interval-valued (∈, ∈ ∨ q)-fuzzy left (resp. right) ideal of S. Proof The proof is similar to the proof of Theorem 5.1.

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Theorem 5.3 Let B be a generalized bi-ideal of S and let λ be an interval-valued fuzzy subset in S such that 1 if x ∈ B λ(x) = if x ∈ S\B ≤ 0.5 Then (1) λ is an interval-valued (q, ∈ ∨ q)-fuzzy generalized bi-ideal of S. (2) λ is an interval-valued (∈, ∈ ∨ q)-fuzzy generalized bi-ideal of S. Proof The proof is similar to the proof of Theorem 5.1. Theorem 5.4 Let B be a bi-ideal of S and let λ be an interval-valued fuzzy subset in S such that 1 if x ∈ B λ(x) = if x ∈ S\B ≤ 0.5 Then (1) λ is an interval-valued (q, ∈ ∨ q)-fuzzy bi-ideal of S. (2) λ is an interval-valued (∈, ∈ ∨ q)-fuzzy bi-ideal of S. Proof Follows from Theorem 5.1. So we omit its proof. Lemma 5.5 Let λ be an interval-valued fuzzy subset of an LA-semigroup S. Then λ is an interval-valued (∈, ∈ ∨ q)-fuzzy left (resp. right) ideal of S if and only if ≥ ≥ max{ λ(x y), 0.5} λ(y) (resp. max{ λ(x y), 0.5} λ(x)) Proof Let λ be an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of S. On contrary assume that 0.5, λ(y) < t. So yt ∈ λ. Hence yt ∈ ∨ q λ . Thus λ be an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of S. Lemma 5.6 Let λ be an interval-valued fuzzy subset of an LA-semigroup S. Then λ is an interval-valued (∈, ∈ ∨ q)-fuzzy two-sided ideal of S if and only if ≥ ≥ max{ λ(x y), 0.5} λ(y) and max{ λ(x y), 0.5} λ(x). Lemma 5.7 The intersection of an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of an LAsemigroup S is an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of S. Proof Let { λi }i∈I be a family of an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of S. Let x, y ∈ S. Then λi )(x y) = ∧i∈I ( λi (x y)). (∧i∈I ≥ (Since each λi is an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of S, so λi (x y) ∨ 0.5 λi (y) for all i ∈ I ). Thus, = ∧i∈I ( (∧i∈I λi )(x y) ∨ 0.5 λi (x y)) ∨ 0.5 = ∧i∈I ( λi (x y) ∨ 0.5) ≥ ∧i∈I ( λi (y)) = (∧i∈I λi )(y). Hence, ∧i∈I λi is an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of S.

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Similarly, we can prove that the intersection of interval-valued (∈, ∈ ∨ q)-fuzzy right ideals of an LA-semigroup S is an interval-valued (∈, ∈ ∨ q)-fuzzy right ideal of S. Thus the intersection of an interval-valued (∈, ∈ ∨ q)-fuzzy two-sided ideal of an LA-semigroup S is an interval-valued (∈, ∈ ∨ q)-fuzzy two-sided ideal of S. Now we show that if λ and μ are (∈, ∈ ∨ q)-fuzzy ideals of an LA-semigroup S, then λ◦ μ ≮ λ∧ μ. Example 5.8 Consider the LA-semigroup S = {1, 2, 3, 4} · 1 2 3 4

1 3 2 4 4

2 2 2 2 2

3 3 2 4 4

4 4 2 4 4

Let λ and μ be fuzzy subsets of S such that λ(1) = [0.3, 0.4], λ(2) = [0.8, 0.9], λ(3) = [0.1, 0.2], λ(4) = [0.8, 0.9] μ(1) = [0, 0], μ(2) = [0.7, 0.8], μ(3) = [0.6, 0.65] = μ(4). Then λ and μ are (∈, ∈ ∨ q)-fuzzy ideals of S. Now λ(x) ∧ μ(y)] ( λ◦ μ)(3) = ∨3=x y [ = ∨{[0, 0], [0.3, 0.4]} = [0.3, 0.4] ≮ ( λ∧ μ)(3) = [0.1, 0.2]. λ∧ μ) in general. Hence ( λ◦ μ) ≮ ( Theorem 5.9 An interval-valued fuzzy subset λ of an LA-semigroup S is an interval-valued (∈, ∈ ∨ q)-fuzzy sub LA-semigroup of S if and only if ≥ λ(x y) ∨ 0.5 λ(x) ∧ λ(y) ∀ x, y ∈ S. Proof Suppose that λ is an interval-valued (∈, ∈ ∨ q)-fuzzy sub LA-semigroup of S. On 0.5, λ(x) < t1 so xt1 ∈ λ. If t1 ≤ 0.5, which implies that + 0.5 = λ(x) < 0.5 λ(x) + t1 < 0.5 1. So xt1 q λ. Hence xt1 ∈ ∨ q λ. And . Now there are two possibilities either if λ(x) ∧ λ(y) = λ(y), then λ(y) < t2 ∧ 0.5 t2 > 0.5 or t2 ≤ 0.5. If t2 > 0.5, then λ(y) < t2 . So yt2 ∈λ. If t2 ≤ 0.5, then λ(y) < 0.5 which implies + 0.5 = that λ(y) + t2 < 0.5 1. So yt2 q λ. Hence yt2 ∈ ∨ q λ. Thus λ is an interval-valued (∈, ∈ ∨ q)-fuzzy sub LA-semigroup of S. Theorem 5.10 An interval-valued fuzzy subset λ of an LA-semigroup S is an interval-valued (∈, ∈ ∨ q)-fuzzy generalized bi-ideal of S if and only if ≥ λ((x y)z) ∨ 0.5 λ(x) ∧ λ(z) ∀ x, y, z ∈ S.

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Proof The proof is similar to the proof of Theorem 5.10.

Theorem 5.11 An interval-valued fuzzy subset λ of an LA-semigroup S is an interval-valued (∈, ∈ ∨ q)-fuzzy bi-ideal of S if and only if it satisfies the following conditions ≥ (1) λ(x y) ∨ 0.5 λ(x) ∧ λ(y) ∀ x, y ∈ S. (2) λ((x y)z) ∨ 0.5 ≥ λ(x) ∧ λ(z) ∀ x, y, z ∈ S. Proof Follows from Theorems 5.9 and 5.10.

Lemma 5.12 Every interval-valued (∈, ∈ ∨ q)-fuzzy generalized bi-ideal of a left regular LA-semigroup S is an interval-valued (∈, ∈ ∨ q)-fuzzy bi-ideal of S. Proof Let λ be any interval-valued (∈, ∈ ∨ q)-fuzzy generalized bi-ideal of a left regular LA-semigroup S, with left identity e. Let a, b ∈ S be any elements of S. Then there exists an element x ∈ S such that a = (aa)x. Thus we have ab = ((aa)x)b = ((aa)(ex))b = ((ae)(ax))b = (a((ae)x))b. Thus we have λ(ab) = λ(a((ae)x)b) ≥ min{ λ(a), λ(b), 0.5} This shows that λ is an interval-valued (∈, ∈ ∨ q)-fuzzy LA-semigroup of S and so λ is an interval-valued (∈, ∈ ∨ q)-fuzzy bi-ideal of S. Definition 5.13 An interval-valued fuzzy subset λ of an LA-semigroup S is called an intervalvalued (∈, ∈ ∨ q)-fuzzy quasi-ideal of S, if it satisfies ≥ min{( λ(x) ∨ 0.5 λ ◦ δ )(x), ( δ ◦ λ)(x)}. where δ is an interval-valued fuzzy subset of S mapping every element of S on 1, that is, δ : S → 1. Theorem 5.14 Let λ be an interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal of an LA is a quasi-ideal of S. semigroup S. Then, the set λ0.5 λ(x) > 0.5} = {x ∈ S : λ0.5 λ0.5 λ0.5 Proof In order to show that λ0.5 is a quasi-ideal of S, we have to show that S ∩ S ⊆ . Let a ∈ S λ0.5 ∩ λ S. This implies that a ∈ S λ and a ∈ λ S. So a = sx and a = yr for 0.5 0.5 0.5 and Now some s, r ∈ S and x, y ∈ λ0.5 λ(x) > 0.5 λ(y) > 0.5. . Thus ≥ ( λ(a) ∨ 0.5 λ ◦ δ )(a) ∧ ( δ ◦ λ)(a). Since ( δ ◦ λ)(a) = ∨a= pq { δ ( p) ∧ λ(q)} ≥ δ (s) ∧ λ(x) because a = sx = λ(x). Similarly, ( λ ◦ δ )(a) = ∨a= pq { λ( p) ∧ δ (q)} ≥ λ(y) ∧ δ (r ) because a = yr = λ(y).

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Thus, ≥ ( λ(a) ∨ 0.5 λ ◦ δ )(a) ∧ ( δ ◦ λ)(a) ≥ (λ ◦ δ )(a) ∧ (δ ◦ λ)(a) ∧ 0.5 ≥ λ(x) ∧ λ(y) ∧ 0.5 because and > 0.5 λ(x) > 0.5 λ(y) > 0.5. Therefore, a ∈ λ0.5 λ0,5 λ0.5 λ0.5 λ0.5 this imply S ∩ S ⊆ . Thus, is a quasi-ideal of S.

Remark 5.15 Every interval-valued fuzzy quasi-ideal of S is an interval-valued (∈, ∈ ∨ q)fuzzy quasi-ideal of S. Lemma 5.16 A non-empty subset Q of an LA-semigroup S is a quasi-ideal of S if and only if Q is an interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal interval-valued characteristic function C of S. Q be an interval-valued characteristic Proof Suppose that Q is a quasi-ideal of S. Let C function of Q and let x ∈ S. If x ∈ / Q, then x ∈ / S Q or x ∈ / Q S. If x ∈ / S Q, then Q )(x) = Q ◦ Q )(x) = Q (x). If x ∈ Q, then C Q (x) ∨ 0.5. ( δ◦C 0. So (C δ )(x) ∨ ( δ◦C 0=C Q is an interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal of S. Hence, C Q is an interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal of S. Let Conversely, assume that C a ∈ Q S ∩ S Q. Then there exist b, c ∈ S and x, y ∈ Q such that a = xb and a = cy. Then Q ◦ Q ( p) ∧ Q (x) ∧ Q ◦ (C δ )(a) = ∨a= pq {C δ (q)} ≥ C δ (b) = 1 ∧ 1 = 1. So, (C δ )(a) = 1. Similarly (δ ◦ C Q )(a) = ∨a= pq {δ ( p) ∧ C Q (q)} ≥ δ (c) ∧ C Q (y) = 1 ∧ 1} = 1. So = 0.5. Thus C Q )(a) = Q (a) ≥ (C Q ◦ Q )(x) ∧ 0.5 Q (a) = ( δ◦C 1. Hence C δ )(x) ∧ ( δ◦C 1. Which implies that a ∈ Q. Hence S Q ∩ Q S ⊆ Q, that is, Q is a quasi-ideal of S. The following is the corollary of Theorem 5.2. So we omit its proof. L is an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal Lemma 5.17 The characteristic function C of S if and only if L is a left ideal of S. R is an interval-valued (∈, ∈ ∨ q)-fuzzy right ideal Similarly the characteristic function C I is of S if and only if R is a right ideal of S. Hence it follows that characteristic function C an interval-valued (∈, ∈ ∨ q)-fuzzy two-sided ideal of S if and only if I is a two-sided ideal of S. Theorem 5.18 Every interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of S is an interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal of S. Proof Let x, y ∈ S. Then δ (y) ∧ λ(z)} ( δ ◦ λ)(x) = ∨x=yz { = ∨x=yz λ(z). This implies that λ(z) ( δ ◦ λ)(x) = ∨x=yz λ(yz) ∨ 0.5} ≤ ∨x=yz {

because λ is an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal ofS.

= λ(x) ∨ 0.5. Thus, ( δ ◦ λ)(x) ≤ max{ λ(x), 0.5}.

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≥ ( Hence λ(x) ∨ 0.5 δ ◦ λ)(x) ≥ ( λ ◦ δ )(x) ∧ ( δ ◦ λ)(x). Thus λ is an interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal of S. Similarly, we can show that every interval-valued (∈, ∈ ∨ q)-fuzzy right ideal of S is an interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal of S. Lemma 5.19 Let S be an LA-semigroup with left identity e, such that (xe)S = x S. Then, every interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal of S is an interval-valued (∈, ∈ ∨ q)-fuzzy bi-ideal of S. Proof Suppose that λ is an interval-valued (∈, ∈ ∨ q)-fuzzy quasi-ideal of an LA-semigroup S. Now let x, y, z ∈ S. Then, we have ≥ ( λ(x y) ∨ 0.5 λ ◦ δ )(x y) ∧ ( δ ◦ λ)(x y) = [∨x y=ab { λ(a) ∧ δ (b)}] ∧ [∨x y= pq { δ ( p) ∧ λ(q)}] ≥ [ λ(x) ∧ δ (y)] ∧ [ δ (x) ∧ λ(y)] ≥ [ λ(x) ∧ 1] ∧ [ 1 ∧ λ(y)] = λ(x) ∧ λ(y). So ≥ λ(x y) ∨ 0.5 λ(x) ∧ λ(y). Also ≥ {( λ((x y)z) ∨ 0.5 λ ◦ δ )((x y)z) ∧ ( δ ◦ λ)((x y)z)}. Now δ ( p) ∧ λ(q)} ( δ ◦ λ)((x y)z) = ∨((x y)z)= pq { = ∨((x y)z)= pq { 1 ∧ λ(q)} ≥ λ(z).

Then, ( δ ◦ λ)((x y)z) ≥ λ(z). Also, λ(a) ∧ δ (b)} ( λ ◦ δ )((x y)z) = ∨((x y)z)=ab { = ∨((x y)z)=ab {{ λ(a) ∧ 1}. Since (x y)z = (x y)(ez) = (xe)(yz) ∈ (xe)S = x S.So,(x y)z = xr for some r ∈ S. Then, λ(a) ∧ 1} ≥ λ(x). ( λ ◦ δ )((x y)z) = ∨xr =ab { Thus ≥ λ((x y)z) ∨ 0.5 λ(x) ∧ λ(z). Hence λ is an interval-valued (∈, ∈ ∨ q)-fuzzy bi-ideal of S.

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6 Lower and upper parts of interval-valued (∈, ∈ ∨ q)-fuzzy ideals Lemma 6.1 Let A and B be non-empty subsets of an LA-semigroup S. Then the following properties hold (i) (ii) (iii)

B )+ = C + A ∧ C (C A∩B + + (C A ∨ C B ) = C A∪B A ◦ C B )+ = C + (C AB

Proof Proof is obvious.

+ is an intervalLemma 6.2 The upper part of an interval-valued characteristic function C L valued (∈, ∈ ∨ q)-fuzzy left ideal of S if and only if L is a left ideal of S. + is an interval-valued (∈, ∈ ∨ q)Proof Let L be a left ideal of S. Then by Theorem 5.2, C L fuzzy left ideal of S. + is an interval-valued (∈, ∈ ∨ q)-fuzzy left ideal of S. Let Conversely assume that C L +