Generalized Fuzzy Interior Ideals in Abel Grassmann's Groupoids

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 838392, 14 pages doi:10.1155/2010/838392 Research Article Generalized Fuzzy Interior Ideals in Abel Grassmann’s Groupoids Asghar Khan,1 Young Bae Jun,2 and Tahir Mahmood3 1 Department of Mathematics, COMSATS Institute of Information Technology, 22060 Abbottabad, Pakistan Department of Mathematics Education and RINS, Gyeongsang National University, Chinju 660-701, South Korea 3 Department of Applied Mathematics, International Islamic University, 45320 Islamabad, Pakistan 2 Correspondence should be addressed to Asghar Khan, Received 27 August 2009; Accepted 24 February 2010 Academic Editor: Peter Basarab-Horwath Copyright q 2010 Asghar Khan et al. 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. Using the notion of a fuzzy point and its belongness to and quasicoincidence with a fuzzy subset, some new concepts of a fuzzy interior ideal in Abel Grassmann’s groupoids S are introduced and their interrelations and related properties are invesitigated. We also introduce the notion of a strongly belongness and strongly quasicoincidence of a fuzzy point with a fuzzy subset and characterize fuzzy interior ideals of S in terms of these relations. 1. Introduction The idea of a quasicoincidence of a fuzzy point with a fuzzy set, which is mentioned in 1, 2, played a vital role to generate some different types of fuzzy subgroups. It is worth pointing out that Bhakat and Das 2 gave the concepts of α, β-fuzzy subgroups by using the “belongs to” relation ∈ and “quasicoincident with” relation q between a fuzzy point and a fuzzy subgroup, and they introduced the concept of an ∈, ∈ ∨q-fuzzy subgroup. In particular, ∈, ∈ ∨q-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems of other algebraic structures. With this objective in view, Davvaz 3, 4 introduced the concept of ∈, ∈ ∨q-fuzzy sub-near-rings R-subgroups, ideals of a near-ring and investigated some of their interesting properties. Jun and Song 5 discussed general forms of fuzzy interior ideals in semigroups. Kazanci and Yamak introduced the concept of a generalized fuzzy bi-ideal 6. Also Davvaz and many others used this concept in several other algebraic structures see 7–16. Jun 13, 17, gave the concept of α, βfuzzy subalgebra of a BCK/BCI-algebras. In 18, Luo introduced the concept of a strong neighborhood. According to him, a fuzzy point xλ 0 < λ < 1 is said to be strongly belong to 2 International Journal of Mathematics and Mathematical Sciences a fuzzy subset F, denoted by xλ ∈F, if and only if Fx > λ. λ-strong cut set Fλ of F is given by Fλ  {x ∈ X | Fx > λ}, where X is a nonempty set. The idea of Q-neighborhood in fuzzy topology was introduced by Pu and Liu in 19. According to them, a fuzzy point xλ is said to be strongly quasicoincident with F, denoted by xλ qF, if and only if λ  Fx > 1. An Abel Grassmann’s groupoid, abbreviated as AG-groupoid, is a groupoid S whose elements satisfy the left invertive law: abc  cba for all a, b, c ∈ S. An AG-groupoid is the midway structure between a commutative semigroup and a groupoid. It is a useful nonassociative structure with wide applications in theory of flocks. In an AG-groupoid the medial law, abcd  acbd for all a, b, c ∈ S see 20. If there exists an element e in an AGgroupoid S such that ex  x for all x ∈ S then S is called an AG-groupoid with left identity e. If an AG-groupoid S has the right identity then S is a commutative monoid. If an AG-groupoid S contains left identity then abcd  dcba holds for all a, b, c ∈ S. Also abc  bac holds for all a, b, c ∈ S. In this paper, we define α, β-fuzzy interior ideals of an AG-groupoid and give some interesting characterizations of an AG -groupoids in terms of α, β-fuzzy interior ideals. We also introduce the notion of α, β-fuzzy interior ideals of an AG-groupoid. 2. Preliminaries For subsets A, B of an AG-groupoid S, we denote AB  {ab ∈ S | a ∈ A, b ∈ B}. A nonempty subset A of an AG-groupoid S is called an AG-subgroupoid of S if A2 ⊆ A. A is called an interior ideal of S if SAS ⊆ A. Let S be an AG-groupoid. By a fuzzy subset F of S, we mean a mapping, F : S → 0, 1. For fuzzy subsets F1 and F2 of S, define F1 ◦ F2 : S −→ 0, 1, a −→ F1 ◦ F2 a  ⎧   ⎪ min F1 y , F2 z , ⎨  if a  yz ∀a, x, y ∈ S , 2.1 ayz ⎪ ⎩0, if a /  yz. We denote by FS the set of all fuzzy subsets of S. One can easily see that FS, ◦ becomes an AG-groupoid as shown in 21. The order relation “⊆” on FS is defined as follows: F1 ⊆ F2 iff F1 x ≤ F2 x ∀x ∈ S, ∀F1 , F2 ∈ FS. 2.2 For a nonempty family of fuzzy subsets {Fi }i∈I , of an AG-groupoid S, the fuzzy subsets i∈I Fi and i∈I Fi of S are defined as follows:   Fi : G −→ 0, 1, a −→ i∈I  Fi : G −→ 0, 1, a −→ i∈I Fi a : sup{Fi a}, i∈I i∈I   Fi a : inf{Fi a}.  i∈I i∈I 2.3 International Journal of Mathematics and Mathematical Sciences 3 If I is a finite set, say I  {1, 2, . . . , n}, then clearly Fi a  max{F1 a, F2 a, . . . , Fn a}, i∈I  Fi a  min{F1 a, F2 a, . . . , Fn a}. 2.4 i∈I Definition 2.1 cf. 21. Let S be an AG-groupoid and F a fuzzy subset of S. Then F is called a fuzzy interior ideal of S, if it satisfies the following conditions. B1 for all x, y ∈ S Fxy ≥ min{Fx, Fy}. B2 for all x, y, a ∈ S Fxay ≥ Fa. Let F be a fuzzy subset of S and ∅  / A ⊆ S, then the characteristic function χA of A is defined as χA : S −→ 0, 1, a −→ χA a : ⎧ ⎨1, if a ∈ A, ⎩0, if a / ∈ A. 2.5 Lemma 2.2 cf. 21. Let S be an AG-groupoid and F a fuzzy subset of S. Then F is a fuzzy interior ideal of S if and only if χA is a fuzzy interior ideal of S. Let S be an AG-groupoid and F a fuzzy subset of S. Then for every λ ∈ 0, 1 the set UF; λ := {x | x ∈ S, Fx ≥ λ} 2.6 is called a level set of F. The proof of the following lemma is easy and we omit it. Lemma 2.3. Let S be an AG-groupoid and F a fuzzy subset of S. Then F is a fuzzy interior ideal of S if and only if UF; λ  / ∅ is an interior ideal of S for every λ ∈ 0, 1. 3. α, β-Fuzzy Interior Ideal In what follows let S denote an AG-groupoid and let α, β denote any one of ∈, q, ∈ ∨q, ∈ ∧q. Let S be an AG-groupoid and F a fuzzy subset of S, then the set of the form  F y : ⎧ ⎨λ /  0, if y  x, ⎩0, if y / x 3.1 is called a fuzzy point with support x and value λ and is denoted by xλ . A fuzzy point xλ is said to belong to resp., quasicoincident with a fuzzy set F, written as xλ ∈ F resp., xλ qF if Fx ≥ λ resp., (...truncated)