EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 1, 2020, 113-129 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Anti fuzzy interior ideals on Ordered AG-groupoids Nasreen Kausar1,∗, Meshari Alesemi2, Salahuddin2 1 Department of Mathematics, University of Agriculture FSD Pakistan 2 Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia Abstract. The purpose of this paper is to investigate, the characterizations of different classes of non-associative ordered semigroups by using anti fuzzy left (resp. right, interior) ideals. 2020 Mathematics Subject Classifications: 13Cxx, 94D05, 13Axx, 18B40 Key Words and Phrases: Fuzzy sets, anti fuzzy AG-subgroupoids, anti fuzzy left (resp. right, interior) ideals, left (resp. right, weakly, intra-, (2, 2)-) regular ordered AG-groupoids. 1. Introduction In 1972, a generalization of commutative semigroup has been established by Naseerud- din et al. [14]. In ternary commutative law, abc = cba, they introduced the braces on the left side of this law and explored a new pseudo associative law, that is (ab)c = (cb)a. This they called the left invertive law. A groupoid S is a left almost semigroup (abbreviated as LA-semigroup), if it satisfies the left invertive law: (ab)c = (cb)a. This structure is also known as Abel-Grassmann’s groupoid (abbreviated as AG-groupoid) by Protic et al. [27]. In fact an AG-groupoid is non-commutative and non-associative semigroup. Ideals in AG-groupoids have been investigated in [26]. In [6] (resp. [3]) , a groupoid S is said to be medial (resp. paramedial) if (ab)(cd) = (ac)(bd) (resp. (ab)(cd) = (db)(ca)). In [14], an AG-groupoid is medial, but in general an AG-groupoid needs not to be paramedial. However by Protic et al. [27], every AG- groupoid with left identity is paramedial and also satisfies a(bc) = b(ac), (ab)(cd) = (dc)(ba). In [15], if (S, ·,≤) is an ordered semigroup and A ⊆ S, we define (A] = {s ∈ S : s ≤ a for some a ∈ A}. A non-empty subset A of S is an ordered subsemigroup of S if A2 ⊆ A. The notions of ideals play a crucial role in the study of (ring, semiring, near-ring, semigroup, ordered semigroup) theory etc. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i1.3576 Email addresses: kausar.nasreen57@gmail.com (K. Nasreen) malesemi@jazanu.edu.sa (M. Alesemi), drsalah12@hotmail.com (Salahuddin) http://www.ejpam.com 113 c© 2020 EJPAM All rights reserved. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 114 A non-empty subset A of S is a left (resp. right) ideal of S, if following hold (1) SA ⊆ A (resp. AS ⊆ A). (2) If a ∈ A and b ∈ S such that b ≤ a implies b ∈ A. Equivalent definition: A is a left ( resp. right) ideal of S if (A] ⊆ A and SA ⊆ A (resp. AS ⊆ A). A non-empty subset A of S is an interior ideal of S if (1) SAS ⊆ A. (2) If a ∈ A and b ∈ S such that b ≤ a implies b ∈ A. In [17, 18], an ordered semigroup S is said to be regular, if for every a ∈ S, there exists x ∈ S such that a ≤ axa. Equivalent definitions are as follows: (1) A ⊆ (ASA] for every A ⊆ S. (2) a ∈ (aSa] for every a ∈ S. An ordered semigroup S is said to be (2, 2)-regular, if for every a ∈ S, there exists x ∈ S such that a ≤ a2xa2. Equivalent definitions are as follows: (1) A ⊆ (A2SA2] for every A ⊆ S. (2) a ∈ (a2Sa2] for every a ∈ S. An ordered semigroup S is said to be weakly regular, if for every a ∈ S, there exist x, y ∈ S such that a ≤ axay. Equivalent definitions are as follows: (1) A ⊆ ((AS)2] for every A ⊆ S. (2) a ∈ ((aS)2] for every a ∈ S. In [16, 18], an ordered semigroup S is an intra-regular if for every a ∈ S there exist x, y ∈ S such that a ≤ xa2y. Equivalent definitions are as follows: (1) A ⊆ (SA2S] for every A ⊆ S. (2) a ∈ (Sa2S] for every a ∈ S. We define anti fuzzy left (resp. right, interior) ideals in an ordered AG-groupoids, basically an ordered AG-groupoid is non-commutative and non-associative ordered semi- group. In this present paper, we characterize regular (resp. right regular, left regular, (2, 2)- regular, weakly regular and intra-regular) ordered AG-groupoids in terms of anti fuzzy left (resp. right, interior) ideals. In this regard, we prove that in (regular, right regular, weakly regular ) ordered AG-groupoids, the concept of anti fuzzy ( interior, two-sided) ideals coincide. The concept of anti fuzzy (interior, two-sided) ideals coincide in ((2, 2) , left, intra-) regular ordered AG-groupoids with left identity. 2. Preliminaries In [31], an ordered AG-groupoid S, is a partially ordered set, at the same time an AG- groupoid such that a ≤ b, implies ac ≤ bc and ca ≤ cb for all a, b, c ∈ S. Two conditions are equivalent to the one condition (ca)d ≤ (cb)d, for all a, b, c, d ∈ S. An ordered AG-groupoid is also called a po-AG-groupoid for short. Example 1. Consider a set S = {e, f, a, b, c} with the following multiplication “·” and order relation “≤”: · e f a b c e e f a b c f f f f b c a a f c b c b c c c f b c b b b c f ≤ : = {(e, e), (e, a), (e, b), (e, c), (f, f), (f, b), (f, c), (a, a), (a, c), (b, b), (b, c), (c, c)}. Then (S, ·,≤) is an ordered AG-groupoid with left identity e. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 115 Let S be an ordered AG-groupoid and A ⊆ S, we define a subset (A] = {s ∈ S : s ≤ a for some a ∈ A} of S and obviously A ⊆ (A]. If A = {a}, then we write (a] instead of ({a}]. For A,B ⊆ S, then AB = {ab | a ∈ A, b ∈ B}, ((A]] = (A], (A](B] ⊆ (AB], ((A](B]] = (AB], if A ⊆ B then (A] ⊆ (B], (A ∩B] 6= (A] ∩ (B] in general. For ∅ 6= A ⊆ S. A is an ordered AG-subgroupoid of S if A2 ⊆ A. A is left (resp. right) ideal of S if (1) SA ⊆ A (resp. AS ⊆ A) . (2) if a ∈ A and b ∈ S such that b ≤ a implies b ∈ A. Equivalent definition: A is left (resp. right ) ideal of S if (A] ⊆ A and SA ⊆ A (resp. AS ⊆ A) . A is an ideal of S if A is both a left and a right ideal of S. If A,B are ideals of S, then A ∪B and A ∩B are also ideals of S. A non-empty subset A of an ordered AG-groupoid S is an interior ideal of S if (1) (SA)S ⊆ A. (2) If a ∈ A and b ∈ S such that b ≤ a implies b ∈ A (or (A] ⊆ A) . An ordered AG-groupoid S is left ( resp. right) regular, if for every a ∈ S, there exists x ∈ S such that a ≤ xa2 ( resp. a ≤ a2x ) . Equivalent definitions are as follows: (1) A ⊆ (SA2] (resp. A ⊆ (A2S]) for every A ⊆ S. (2) a ∈ (Sa2] (resp. a ∈ (a2S]) for every a ∈ S. An ordered AG-groupoid S is regular, if for every a ∈ S, there exists x ∈ S such that a ≤ (ax)a. Equivalent definitions: (1) A ⊆ ((AS)A] for every A ⊆ S. (2) a ∈ ((aS)a] for every a ∈ S. An ordered AG-groupoid S is completely regular, if it is regular, left regular, right regular. An ordered AG-groupoid S is strongly regular, if for every a ∈ S, there exists x ∈ S such that a ≤ (ax)a and ax = xa. Every strongly regular ordered AG-groupoid is right regular ordered AG-groupoid. An ordered AG-groupoid S is said to be weakly regular, if for every a ∈ S, there exist x, y ∈ S such that a ≤ (ax)(ay). Equivalent definitions are as follows: (1) A ⊆ ((AS)2] for every A ⊆ S. (2) a ∈ ((aS)2] for every a ∈ S. An ordered AG-groupoid S is an intra-regular, if for every a ∈ S, there exist x, y ∈ S such that a ≤ (xa2)y. Equivalent definitions are as follows: (1) A ⊆ ((SA2)S] for every A ⊆ S. (2) a ∈ ((Sa2)S] for every a ∈ S. We denote by L(a), R(a), I(a) the left ideal, the right ideal and the ideal of S, re- spectively generated by a. We have L(a) = {s ∈ S : s ≤ a or s ≤ xa for some x ∈ S} = (a ∪ Sa], R(a) = (a ∪ aS], I(a) = (a ∪ Sa ∪ aS ∪ (Sa)S]. Example 2. Let S = {a, b, c, d, e}. Define multiplication “·” in S as follows : · a b c d e a a a a a a b a a a a a c a a e c d d a a d e c e a a c d e and ≤ : = {(a, a), (b, b), (c, c), (d, d), (e, e)}. Then S is an ordered AG-groupoid. A = {c, d, e} is an AG-subgroupoid of S and I = {a, c, d, e} is an ideal of S. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 116 Remark 1. Every ideal (whether right, left or two-sided) is an AG-subgroupoid but the converse is not true in general. An ordered AG-groupoid S is to be locally associative, if (a.a).a = a.(a.a) for every a ∈ S. Example 3. Let S = {a, b, c}. Define multiplication “·” in S as follows : · a b c a c c b b b b b c b b b and ≤ : = {(a, a), (b, b), (c, c)}. Then (S, ·,≤) is a locally associative ordered AG-groupoid. In a locally associative ordered AG-groupoids S, we define powers of an element as follow: a1 = a, an+1 = ana. If S has a left identity e, we define a0 = e, as left identity is unique in an ordered AG-groupoid.A locally associative ordered AG-groupoid S with left identity e has associative powers. 3. Anti fuzzy interior ideals on ordered AG-groupoids A fuzzy set µ on a given set X is described as an arbitrary function µ : X → [0, 1], where [0, 1] is the unit closed interval of real numbers. The fundamental concept of a fuzzy set, introduced by Zadeh in his classic paper [33] 1965, which gives a natural frame work for the generalizations of some basic notions of algebra, for example set (resp. semigroup, group, ring, near-ring, semiring) theory, groupoids, real analysis, topology, differential equations and so forth. Rosenfeld [29], introduced the concept of fuzzy set in groups. The study of fuzzy set in semigroups investigated by Kuroki [21–23]. He studied fuzzy (interior, bi-, quasi-, semiprime quasi- ) ideals in semigroups. Dib and Galham in [4], examined the definition of fuzzy groupoid (resp. semigroup). They studied fuzzy ideals and fuzzy bi-ideals of fuzzy semigroups. A systematic exposition of fuzzy semigroups by Mordeson et al. appeared in [24], where one can find theoretical results on fuzzy semigroups and their use in fuzzy finite state machines and fuzzy languages. Fuzzy sets in ordered semigroups/ordered groupoids established by Kehayopulu and Tsingelis [19]. They also studied fuzzy bi-ideals and fuzzy quasi-ideals in ordered semigroups [19, 20]. In [2], Biswas introduced the concept of anti fuzzy subgroups of groups and studied the basic properties of groups in terms of anti fuzzy subgroups. Hong and Jun [5] modified the Biswas idea and applied it into BCK-algebra. Akram and Dar defined anti fuzzy left h-ideals of hemiring and discussed the basic properties of hemiring [1]. By a fuzzy set µ of an ordered AG-groupoid S, we mean a function µ : S → [0, 1] and the complement of µ is denoted by µ′, is a fuzzy set in S given by µ′(x) = 1− µ(x) for all x ∈ S. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 117 A fuzzy set µ of an ordered AG-groupoid S is an anti fuzzy AG-subgroupoid of S if µ(xy) ≤ max{µ(x), µ(y)} for all x, y ∈ S. µ is an anti fuzzy left (resp. right) ideal of S, if (1) µ(xy) ≤ µ(y) (resp. µ(xy) ≤ µ(x)). (2) x ≤ y, implies µ(x) ≤ µ(y) for all x, y ∈ S. µ is an anti fuzzy ideal of S, if µ is both an anti fuzzy left ideal and an anti fuzzy right ideal of S. Equivalently, µ is an anti fuzzy ideal of S if (1) µ(xy) ≤ max{µ(x), µ(y)}. (2) x ≤ y, implies µ(x) ≤ µ(y) for all x, y ∈ S. Every anti fuzzy ideal (whether left, right, two-sided) is an anti fuzzy AG-subgroupoid but the converse is not true in general. A fuzzy set µ of S is an anti fuzzy interior ideal of S, if (1) µ((xa)y) ≤ µ(a). (2) x ≤ y, implies µ(x) ≤ µ(y) for all x, a, y ∈ S. We denote by F (S), the set of all fuzzy subsets of S. We define an order relation ”⊆” on F (S) such that f ⊆ g if and only if f(x) ≤ g(x) for all x ∈ S. Then (F (S), ◦,⊆) is an ordered AG-groupoid. For f ∧ g and f ∨ g, we define (f ∧ g)(x) = min{f(x), g(x)} and (f ∨ g)(x) = max{f(x), g(x)}. For a ∈ S, we define Aa = {(y, z) ∈ S × S | a ≤ yz}. Let f and g be fuzzy subsets of S, the product f ◦ g of f and g is defined by: (f ◦ g)(a) = { ∧(y,z)∈Aa max{f(y), g(z)} if Aa 6= ∅ 0 if Aa = ∅ For a non-empty family of fuzzy subsets {fi}i∈I , of S, the fuzzy subsets ∨i∈Ifi and ∧i∈Ifi of S are defined as follows: (∨i∈Ifi)(a) : = sup i∈I {fi(a)} and (∧i∈Ifi)(a) : = inf i∈I {fi(a)}. If I is a finite set, say I = {1, 2, ...n}, then clearly, ∨i∈Ifi(a) = max{f1(a), f2(a), ..., fn(a)} and ∧i∈I fi(a) = min{f1(a), f2(a), ..., fn(a)}. For S, the fuzzy subsets “0 ”and “1” are defined as 0(x) : = 0 and 1(x) := 1. 0 : S → [0, 1], x 7→ 0(x) : = 0. 1 : S → [0, 1], x 7→ 1(x) : = 1. Clearly, the fuzzy subset “0 ”(resp.“1 ”) of S is the least ( resp. the greatest) element of the ordered set (F (S),≤). The fuzzy subset “0” is the zero element of (F (S), ◦,≤) (that is, f ◦ 0 = 0 ◦ f = 0 and 0 ≤ f for every f ∈ F (S)). For ∅ 6= A ⊆ S, the anti characteristic function of A is denoted by χC A and defined as χC A(a) = { 0 if a ∈ A 1 if a /∈ A K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 118 An ordered AG-groupoid S can be considered a fuzzy subset of itself and we write S = χC S , i.e., S(x) = χC S (x) = 0 for all x ∈ S. This implies that S(x) = 0 for all x ∈ S. For A,B ⊆ S, then A ⊆ B if and only if χC A ≥ χC B, χ C A ∩ χC B = χC A∩B and χC A ◦ χC B = χC (AB]. Let µ be a fuzzy subset of S, then for all t ∈ (0, 1], we define a set L(µ; t) = {x ∈ S | µ(x) ≤ t}, which is called lower t-level set of µ and can be used for the characterization of µ. Example 4. Let S = {a, b, c, d}. Define multiplication “·” in S as follows : · a b c d a c d a b b b c d a c a b c d d d a b c and ≤ : = {(a, a), (b, b), (c, c), (d, d)}. Then S is an ordered AG-groupoid. Let µ be a fuzzy subset of S. We define µ(a) = µ(c) = 0.7, µ(b) = µ(d) = 0. Hence µ is an anti fuzzy AG-subgroupoid of S. Example 5. Let S = {a, b, c, d}. Define multiplication “·” in S as follows : · a b c d a a a a a b a a a a c a a d a d a a c d and ≤ : = {(a, a), (b, b), (c, c), (d, d)}. Then S is an ordered AG-groupoid. Let µ be a fuzzy subset of S. We define µ(a) = µ(c) = µ(d) = 0, µ(b) = 0.7. Hence µ is an anti fuzzy right ideal of S. Remark 2. Example 4 and Example 5 show that, every anti fuzzy ideal (whether right, left, two-sided) is an anti fuzzy AG-subgroupoid, but the converse is not true. Lemma 1. Let S be an ordered AG-groupoid and ∅ 6= A ⊆ S. Then the anti characteristic function χC (A] of (A] is a fuzzy subset of S satisfying the condition x ≤ y ⇒ χC (A](x) ≤ χC (A](y) for all x, y ∈ S. Proof. By the definition, χC (A] is a mapping of S into {0, 1} ⊆ [0, 1]. Let x ≤ y, x, y ∈ S. If y /∈ (A], by definition χC (A](y) = 1, thus χC (A](x) ≤ χC (A](y). If y ∈ (A], by definition χC (A](y) = 0. Since y ∈ (A], so there exists z ∈ A such that y ≤ z. Thus x ≤ z, i.e., x ∈ (A] and χC (A](x) = 0. Hence χC (A](x) ≤ χC (A](y). K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 119 Proposition 1. Let S be an ordered AG-groupoid and ∅ 6= A ⊆ S. Then A = (A] if and only if fuzzy subset χC A of S has the property x ≤ y ⇒ χC A(x) ≤ χC A(y) for all x, y ∈ S. Proof. Suppose A = (A], then the anti characteristic function χC A of A is a fuzzy subset of S satisfying the condition x ≤ y ⇒ χC A(x) ≤ χC A(y), by the Lemma 1. Conversely, let x ∈ (A], this imply that there exists y ∈ A such that x ≤ y. By the given condition, we have χC A(x) ≤ χC A(y). Since y ∈ A, we have χC A(y) = 0. Thus χC A(x) = 0, i.e., x ∈ A. Hence A = (A]. Lemma 2. Let S be an ordered AG-groupoid and ∅ 6= A ⊆ S. Then A is an AG- subgroupoid of S if and only if the anti characteristic function χC A of A is an anti fuzzy AG-subgroupoid of S. Proof. Suppose A is an AG-subgroupoid of S and x, y ∈ S. If x, y /∈ A, by definition χC A(x) = 1 = χC A(y). Thus χC A(xy) ≤ χC A(x) ∨ χC A(y). If x, y ∈ A, by definition χC A(x) = 0 = χC A(y). xy ∈ A, A being an AG-subgroupoid of S, this imply that χC A(xy) = 0. Thus χC A(xy) ≤ χC A(x) ∨ χC A(y). Hence the anti characteristic function χC A of A is an anti fuzzy AG-subgroupoid of S. Conversely, let xy ∈ A2, x, y ∈ A. By definition of anti characteristic function χC A(x) = 0 = χC A(y). χC A(xy) ≤ χC A(x) ∨ χC A(y) = 0, χC A being an anti fuzzy AG-subgroupoid of S. This imply that χC A(xy) = 0, i.e., xy ∈ A. Hence A is an AG-subgroupoid of S. Lemma 3. Let S be an ordered AG-groupoid and ∅ 6= A ⊆ S. Then A is a left (resp. right) ideal of S if and only if the anti characteristic function χC A of A is an anti fuzzy left (resp. right) ideal of S. Proof. Suppose A is a left ideal of S and x, y ∈ S such that x ≤ y. This imply that A = (A], A being a left ideal of S. Then χC A(x) ≤ χC A(y), by the Proposition 1. If y /∈ A, by definition χC A(y) = 1. Thus χC A(xy) ≤ χC A(y). If y ∈ A, by definition χC A(y) = 0. xy ∈ A, A being a left ideal, so χC A(xy) = 0. Thus χC A(xy) ≤ χC A(y). Hence the anti characteristic function χC A of A is an anti fuzzy left ideal of S. Conversely, let y ∈ A and x ∈ S such that x ≤ y. This imply that χC A(x) ≤ χC A(y), χC A being an anti fuzzy left ideal of S. Then A = (A], by the Proposition 1. Let xy ∈ SA, where y ∈ A, x ∈ S. By definition of anti characteristic function χC A(y) = 0. χC A(xy) ≤ χC A(y) = 0, χC A being an anti fuzzy left ideal of S. Thus χC A(xy) = 0, i.e., xy ∈ A. Hence A is a left ideal of S. Proposition 2. Let S be an ordered AG-groupoid and ∅ 6= A ⊆ S. Then A is an interior ideal of S if and only if the anti characteristic function χC A of A is an anti fuzzy interior ideal of S. Proof. Suppose A is an interior ideal of S and a, x, y ∈ S such that x ≤ y. This imply that A = (A], A being an interior-ideal. Then χC A(x) ≤ χC A(y), by the Proposition 1. If a /∈ A, by definition χC A(a) = 1. Thus χC A((xa)y) ≤ χC A(a). If a ∈ A, by definition K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 120 χC A(a) = 0. (xa)y ∈ A, A being an interior ideal, this imply that χC A((xa)y) = 0. Thus χC A((xa)y) ≤ χC A(a). Hence the anti characteristic function χC A of A is an anti fuzzy interior ideal of S. Conversely, let y ∈ A and x ∈ S such that x ≤ y. This imply that χC A(x) ≤ χC A(y), χC A being an anti fuzzy interior ideal of S. Then A = (A], by the Proposition 1. Let t ∈ (SA)S, implies t = (xa)y, where a ∈ A and x, y ∈ S. By definition of anti characteristic function χC A(a) = 0. χC A((xa)y) ≤ χC A(a) = 0, χC A being an anti fuzzy interior ideal of S. Thus χC A((xa)y) = 0, i.e., (xa)y ∈ A. Hence A is an interior ideal of S. Lemma 4. Let µ be a fuzzy subset of an ordered AG-groupoid S. Then µ is an anti fuzzy AG-subgroupoid of S if and only if lower t-level L(µ; t) of µ is an AG-subgroupoid of S for all t ∈ (0, 1]. Proof. Suppose µ is an anti fuzzy AG-subgroupoid of S and x, y ∈ L(µ; t), this imply that µ(x), µ(y) ≤ t. µ(xy) ≤ µ(x) ∨ µ(y) ≤ t, µ being an anti fuzzy AG-subgroupoid, i.e., xy ∈ L(µ; t). Hence L(µ; t) is an AG-subgroupoid of S. Conversely, we have to show that µ(xy) ≤ µ(x) ∨ µ(y), x, y ∈ S. We suppose a contradiction µ(xy) > µ(x)∧µ(y). Assume µ(x) = t = µ(y), this imply that µ(x), µ(y) ≤ t, i.e., x, y ∈ L(µ; t). But µ(xy) > t, i.e., xy /∈ U(µ; t), which is a contradiction. Hence µ(xy) ≤ µ(x) ∨ µ(y). Lemma 5. Let µ be a fuzzy subset of an ordered AG-groupoid S. Then µ is an anti fuzzy left (resp. right) ideal of S if and only if lower t-level L(µ; t) of µ is a left (resp. right) ideal of S for all t ∈ (0, 1]. Proof. Suppose µ is an anti fuzzy left ideal of S. Let y ∈ L(µ; t) and x ∈ S such that x ≤ y, this imply that µ(y) ≤ t. µ(x) ≤ µ(y) ≤ t and µ(xy) ≤ µ(y) ≤ t, µ being an anti fuzzy left ideal of S. Thus x, xy ∈ L(µ; t). Hence L(µ; t) is a left ideal of S. Conversely, suppose L(µ; t) is a left ideal of S and x, y ∈ S such that x ≤ y. We have to show that µ(x) ≤ µ(y) and µ(xy) ≤ µ(y). We suppose a contradiction µ(x) > µ(y) and µ(xy) > µ(y). Let µ(y) = t, this imply that µ(y) ≤ t, i.e., y ∈ L(µ; t). But µ(x) > t and µ(xy) > t, i.e., x, xy /∈ L(µ; t), which is a contradiction. Hence µ(x) ≤ µ(y) and µ(xy) ≤ µ(y). Proposition 3. Let µ be a fuzzy subset of an ordered AG-groupoid S. Then µ is an anti fuzzy interior ideal of S if and only if the lower t-level L(µ; t) of µ is an interior ideal of S for all t ∈ (0, 1]. Proof. Suppose µ is an anti fuzzy interior ideal of S. Let y ∈ L(µ; t) and x ∈ S such that x ≤ y, this imply that µ(y) ≤ t. µ(x) ≤ µ(y) ≤ t, µ being an anti fuzzy interior ideal of S. Thus µ(x) ≤ t, i.e., x ∈ L(µ; t). Let a ∈ L(µ; t) and x, y ∈ S, by definition µ(a) ≤ t. µ((xa)y) ≤ µ(a) ≤ t, µ being an anti fuzzy interior ideal of S. Thus µ((xa)y) ≤ t, i.e., (xa)y ∈ L(µ; t). Hence L(µ; t) is an interior ideal of S. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 121 Conversely, suppose L(µ; t) is an interior ideal of S and x, y, a ∈ S such that x ≤ y. We have to show that µ(x) ≤ µ(y), we suppose a contradiction µ(x) > µ(y). Let µ(y) = t, this imply that µ(y) ≤ t, i.e., y ∈ L(µ; t). But µ(x) > t, i.e., x /∈ L(µ; t), which is a contradiction. Hence µ(x) ≤ µ(y). We have to show that µ((xa)y) ≤ µ(a), we suppose a contradiction µ((xa)y) > µ(a). Let µ(a) = t, this imply that µ(a) ≤ t, i.e., a ∈ L(µ; t). But µ((xa)y) > t, i.e., (xa)y /∈ L(µ; t), which is a contradiction. Hence µ((xa)y) ≤ µ(a). Lemma 6. Every anti fuzzy right ideal of an ordered AG-groupoid S with left identity e, is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy right ideal of S and x, y ∈ S. Now µ(xy) = µ((ex)y) = µ((yx)e) ≤ µ(yx) ≤ µ(y). Hence µ is an anti fuzzy ideal of S. Remark 3. The concept of anti fuzzy (right, two-sided) ideals coincide in ordered AG-groupoids S with left identity. Lemma 7. Every anti fuzzy ideal of an ordered AG-groupoid S is an anti fuzzy interior ideal of S. Proof. Let µ be an anti fuzzy two-sided ideal of S and x, a, y ∈ S. Now µ((xa)y) ≤ µ(xa) ≤ µ(a). Hence µ is an anti fuzzy interior ideal of S. Proposition 4. Let S be an ordered AG-groupoid with left identity e. Then µ is an anti fuzzy interior ideal if and only if µ is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy interior ideal of S and x, y ∈ S. Now µ(xy) = µ((ex)y) ≤ µ(x). Thus µ is an anti fuzzy right ideal of S. Hence µ is an anti fuzzy ideal of S by Lemma 6. Converse is true by Lemma 7. Lemma 8. Every anti fuzzy right ideal of a regular ordered AG-groupoid S, is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy right ideal of S and x, y ∈ S, this imply that there exists a ∈ S such that x ≤ (xa)x. Now µ(xy) ≤ µ(((xa)x)y) = µ((yx)(xa)) ≤ µ(yx) ≤ µ(y). Hence µ is an anti fuzzy ideal of S. Remark 4. The concept of anti fuzzy (right, two-sided) ideals coincide in regular ordered AG-groupoids S. Proposition 5. Let S be a regular ordered AG-groupoid. Then µ is an anti fuzzy interior ideal if and only if µ is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy interior ideal of S and x, y ∈ S, this imply that there exists a ∈ S such that x ≤ (xa)x. Now µ(xy) ≤ µ(((xa)x)y) = µ((yx)(xa)) ≤ µ(x). Thus µ is an anti fuzzy right ideal of S. Hence µ is an anti fuzzy ideal of S by Lemma 8. Converse is true by Lemma 7. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 122 Lemma 9. Every anti fuzzy right (resp. left) ideal of (2, 2)-regular ordered AG-groupoid S, is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy right ideal of S and x, y ∈ S, this imply that there exists a ∈ S such that x ≤ (x2a)x2. Now µ(xy) ≤ µ(((x2a)x2)y) = µ((yx2)(x2a)) ≤ µ(yx2) ≤ µ(y). Hence µ is an anti fuzzy ideal of S. Let µ be an anti fuzzy left ideal of S. Now µ(xy) ≤ µ(((x2a)x2)y) = µ((yx2)(x2a) ≤ µ((xx)a) = µ((ax)x) ≤ µ(x). Hence µ is an anti fuzzy ideal of S. Remark 5. The concept of anti fuzzy (right, left, two-sided) ideals coincide in (2, 2)- regular ordered AG-groupoids S. Proposition 6. Let S be a (2, 2)-regular ordered AG-groupoid with left identity e. Then µ is an anti fuzzy interior ideal if and only if µ is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy interior ideal of S and x, y ∈ S, this imply that there exists a ∈ S such that x ≤ (x2a)x2. Now µ(xy) ≤ µ(((x2a)x2)y) = µ((yx2)(x2a)) ≤ µ(x2) = µ(xx) = µ((ex)x) ≤ µ(x). Thus µ is an anti fuzzy right ideal of S. Hence µ is an anti fuzzy ideal of S by Lemma 9. Converse is true by Lemma 7. Lemma 10. Let S be a right regular ordered AG-groupoid. Then every anti fuzzy right (resp. left) ideal of S is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy right ideal of S and x, y ∈ S, this imply that there exists a ∈ S such that x ≤ x2a. Now µ(xy) ≤ µ((x2a)y) = µ(((xx)a)y) = µ(((ax)x)y) = µ((yx)(ax)) ≤ µ(yx) ≤ µ(y). Hence µ is an anti fuzzy ideal of S. Let µ be an anti fuzzy left ideal of S. Now µ(xy) ≤ µ((x2a)y) = µ(((xx)a)y) = µ(((ax)x)y) = µ((yx)(ax)) ≤ µ(ax) ≤ µ(x). Hence µ is an anti fuzzy ideal of S. Remark 6. The concept of anti fuzzy (right, left, two-sided) ideals coincide in right regular ordered AG-groupoids S. Proposition 7. Let S be a right regular ordered AG-groupoid. Then µ is an anti fuzzy interior ideal if and only if µ is an anti fuzzy ideal of S. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 123 Proof. Let µ be an anti fuzzy interior ideal of S and x, y ∈ S, this imply that there exists a ∈ S such that x ≤ x2a. Now µ(xy) ≤ µ((x2a)y) = µ(((xx)a)y) = µ(((ax)x)y) ≤ µ(x). Thus µ is an anti fuzzy right ideal of S. Hence µ is an anti fuzzy ideal of S by Lemma 10. Converse is true by Lemma 7. Lemma 11. Let S be a left regular ordered AG-groupoid with left identity e. Then every anti fuzzy right (resp. left) ideal of S is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy right ideal of S and x, y ∈ S, this imply that there exists a ∈ S such that x ≤ ax2. Now µ(xy) ≤ µ((ax2)y) = µ((a(xx))y) = µ((x(ax))y) = µ((y(ax))x) ≤ µ(y(ax)) ≤ µ(y). Hence µ is an anti fuzzy ideal of S. Let µ be an anti fuzzy left ideal of S. Now µ(xy) ≤ µ((ax2)y) = µ((a(xx))y) = µ((x(ax))y) = µ((y(ax))x) ≤ µ((ax)x) ≤ µ(x). Hence µ is an anti fuzzy ideal of S. Remark 7. The concept of anti fuzzy (right, left, two-sided) ideals coincide in left regular ordered AG-groupoids S with left identity. Proposition 8. Let S be a left regular ordered AG-groupoid with left identity e. Then µ is an anti fuzzy interior ideal if and only if µ is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy interior ideal of S and x, y ∈ S, this imply that there exists a ∈ S such that x ≤ ax2. Now µ(xy) ≤ µ((ax2)y) = µ((a(xx))y) = µ((x(ax))y) = µ(((ex)(ax))y) = µ(((xx)(ae))y) = µ((((ae)x)x)y) ≤ µ(x). Thus µ is an anti fuzzy right ideal of S. Hence µ is an anti fuzzy ideal of S by Lemma 11. Converse is true by Lemma 7. Theorem 1. Let S be a right regular locally associative ordered AG-groupoid with left identity e. Then for every anti fuzzy interior ideal µ of S, µ(an) = µ(a2n), where n is any positive integer, for all a ∈ S. Proof. For n = 1. Let a ∈ S, this imply that there exists x ∈ S such that a ≤ a2x. Thus µ(a) ≤ µ(a2x) = µ((ea2)x) ≤ µ(a2) ≤ max{µ (a) , µ (a)} = µ (a) , (µ is an anti fuzzy ideal of S by Proposition 7). Hence µ (a) = µ ( a2 ) . Now a2 = aa ≤ (a2x)(a2x) = a4x2, then K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 124 the result is true for n = 2. Suppose that result is true for n = k, i.e., µ(ak) = µ(a2k). Now ak+1 = aka ≤ (a2kxk)(a2x) = a2(k+1)x(k+1). Thus µ(ak+1) ≤ µ(a2(k+1)x(k+1)) = µ((ea2(k+1))x(k+1)) ≤ µ(a2(k+1)) = µ(a2k+2) = µ(ak+1ak+1) ≤ max{µ ( ak+1 ) , µ ( ak+1 ) } = µ ( ak+1 ) . Therefore µ(ak+1) = µ(a2(k+1)). Hence by induction method, the result is true for all positive integers. Lemma 12. Let S be a right regular locally associative ordered AG-groupoid with left identity e. Then for every anti fuzzy interior ideal µ of S, µ(ab) = µ(ba) for all a, b ∈ S. Proof. Let a, b ∈ S. By using Theorem (for n = 1). Now µ(ab) = µ((ab)2) = µ((ab)(ab)) = µ((ba)(ba)) = µ((ba)2) = µ(ba). Theorem 2. Let S be a regular and right regular locally associative ordered AG-groupoid with left identity e. Then for every anti fuzzy interior ideal µ of S, µ(an) = µ(a3n), where n is any positive integer, for all a ∈ S. Proof. For n = 1. Let a ∈ S, this imply that there exists x ∈ S such that a ≤ (ax)a and a ≤ a2x. Now a ≤ (ax)a ≤ (ax)(a2x) = a3x2. Thus µ(a) ≤ µ(a3x2) = µ((ea3)x2) ≤ µ(a3) = µ(aa2) ≤ max{µ (a) , µ ( a2 ) } ≤ max{µ (a) , µ (a) , µ (a)} = µ (a) . Hence µ (a) = µ ( a3 ) . Now a2 = aa ≤ (a3x2)(a3x2) = a6x4, then the result is true for n = 2. Suppose that result is true for n = k, i.e., µ(ak) = µ(a3k). Now ak+1 = aka ≤ (a3kx2k)(a3x2) = a3(k+1)x2(k+1). Thus µ(ak+1) ≤ µ(a3(k+1)x2(k+1)) = µ((ea3(k+1))x2(k+1)) ≤ µ(a3(k+1)) = µ(a3k+3) = µ(ak+1a2k+2) ≤ maxµ ( ak+1 ) , µ ( a2k+2 ) } ≤ max{µ ( ak+1 ) , µ ( ak+1 ) , µ ( ak+1 ) } = µ ( ak+1 ) . Therefore µ(ak+1) = µ(a3(k+1)). Hence by induction method, the result is true for all positive integers. Lemma 13. Let S be a weakly regular ordered AG-groupoid. Then every anti fuzzy right (resp. left) ideal is an anti fuzzy ideal of S. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 125 Proof. Let µ be an anti fuzzy right ideal of S and x, y ∈ S, this imply that there exist a, b ∈ S such that x ≤ (xa)(xb). Now µ(xy) ≤ µ(((xa)(xb))y) = µ((((xb)a)x)y) = µ((((ab)x)x)y) = µ((yx)((ab)x)) = µ((yx)(nx)) say ab = n ≤ µ(yx) ≤ µ(y). Hence µ is an anti fuzzy ideal of S. Let µ be an anti fuzzy left ideal of S. Now µ(xy) ≤ µ(((xa)(xb))y) = µ((((xb)a)x)y) = µ((((ab)x)x)y) = µ((yx)((ab)x)) = µ((yx)(nx)) say ab = n ≤ µ(nx) ≤ µ(x). Hence µ is an anti fuzzy ideal of S. Remark 8. The concept of anti fuzzy (right, left, two-sided) ideals coincide in weakly regular ordered AG-groupoids S. Proposition 9. Let S be a weakly regular ordered AG-groupoid. Then µ is an anti fuzzy interior ideal if and only if µ is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy interior ideal of S and x, y ∈ S, this imply that there exist a, b ∈ S such that x ≤ (xa)(xb). Now µ(xy) ≤ µ(((xa)(xb))y) = µ((((xb)a)x)y) ≤ µ(x). Thus µ is an anti fuzzy right ideal of S. Hence µ is an anti fuzzy ideal of S by Lemma 13. Converse is true by Lemma 7. Theorem 3. Let S be an ordered AG-groupoid with left identity e. Then S is a weakly regular if and only if S is completely regular. Proof. Suppose S is a weakly regular ordered AG-groupoid. Let a ∈ S, then there exist x, y ∈ S such that a ≤ (ax)(ay). Now a ≤ (ax)(ay) = (aa)(xy) = a2t, for some t ∈ S, this imply that a ≤ a2t. Thus S is a right regular ordered AG-groupoid. Now a ≤ (ax)(ay) = (yx)(aa) = ta2, for some t ∈ S, this imply that a ≤ ta2. Thus S is a left regular ordered AG-groupoid. Now a ≤ (ax)(ay) = (aa)(xy) = a2t = (aa)t = (ta)a ≤ (t(ta2))a = (t(t(aa)))a = (t(a(ta)))a = (a(t(ta)))a = (as)a, say t(ta) = s This imply that a ≤ (as)a, for some s ∈ S. Thus S is a regular ordered AG-groupoid. Hence S is a completely regular ordered AG-groupoid. K. Nasreen, M. Alesemi, Salahuddin / Eur. J. Pure Appl. Math, 13 (1) (2020), 113-129 126 Conversely, let S be a completely regular ordered AG-groupoid. Let a ∈ S, then there exists x ∈ S such that a ≤ (ax)a, a ≤ a2x and a ≤ xa2. Now a ≤ (ax)a ≤ (ax)(xa2) = (ax)(x(aa)) = (ax)(a(xa)) = (ax)(ay), say xa = y This imply that a ≤ (ax)(ay), for some x, y ∈ S. Hence S is weakly regular ordered AG-groupoid. Lemma 14. Every anti fuzzy right ideal of an intra-regular ordered AG-groupoid S is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy right ideal of S and x, y ∈ S, this imply that there exist a, b ∈ S such that x ≤ (ax2)b. Now µ(xy) ≤ µ(((ax2)b)y) = µ((yb)(ax2)) ≤ µ(yb) ≤ µ(y). Hence µ is an anti fuzzy ideal of S. Remark 9. The concept of anti fuzzy (right, two-sided) ideals coincide in intra- regular ordered AG-groupoids S. Proposition 10. Let S be an intra-regular ordered AG-groupoid with left identity e. Then µ is an anti fuzzy interior ideal if and only if µ is an anti fuzzy ideal of S. Proof. Let µ be an anti fuzzy interior ideal of S and x, y ∈ S, this imply that there exist a, b ∈ S such that x ≤ (ax2)b. Now xy ≤ ((ax2)b)y = (yb)(ax2) = n(a(xx)) = n(x(ax)), say yb = n = (en)(x(ax)) = (ex)(n(ax)) = (ex)m, say n(ax) = m Thus µ(xy) ≤ µ((ex)m) ≤ µ(x). Hence µ is an anti fuzzy ideal of S. Converse is true by Lemma 7. Theorem 4. Let S be an intra-regular locally associative ordered AG-groupoid. Then for every anti fuzzy interior ideal µ of S, µ(an) = µ(a2n), where n is any positive integer, for all a ∈ S. Proof. For n = 1. Let a ∈ S, this imply that there exist x, y ∈ S such that a ≤ (xa2)y. Thus µ (a) ≤ µ((xa2)y) ≤ µ(a2) = µ(aa) ≤ max{µ (a) , µ (a)} = µ (a) , (µ is an anti fuzzy ideal of S by Proposition 10). Hence µ(a) = µ(a2). Now a2 = aa ≤ ((xa2)y)((xa2)y) = ((xa2)(xa2))y2 = (x2a4)y2, then the result is true for n = 2. Suppose that the result is true for n = k, i.e., µ(ak) = µ(a2k). Now ak+1 = aka ≤ ((xka2k)yk)((xa2)y) = (xk+1a2(k+1))yk+1. Thus µ ( ak+1 ) ≤ µ((xk+1a2(k+1))yk+1) ≤ µ(a2(k+1)) = µ(a(k+1)a(k+1)) ≤ max{µ ( a(k+1) ) , µ ( a(k+1) ) } = µ ( a(k+1) ) . Therefore µ(ak+1) = µ(a2(k+1)). Hence by induction method, the result is true for all positive integers. REFERENCES 127 Lemma 15. Let S be an intra-regular locally associative ordered AG-groupoid with left identity e. Then for every anti fuzzy interior ideal µ of S, µ(ab) = µ(ba) for all a, b ∈ S. Proof. 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