Q-cubic ideals of near-rings

Inter national Journal of Pure and Applied Mathematics Volume 113 No. 10 2017, 56 64 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Q-cubic ideals of near-rings V. Chinnadurai 1 and K. Bharathivelan 2 1 Department of Mathematics, Annamalai University, Chidambaram, India. kv.chinnadurai@yahoo.com 2 Department of Mathematics, Annamalai University, Chidambaram, India. bharathivelan81@gmail.com February 10, 2017 Abstract In this paper, we introduce the notion of Q-cubic ideals of near-rings, which is a combination of an interval-valued Q-fuzzy set and a Q-fuzzy set. Interval-valued Q-fuzzy set is another generalization of Q-fuzzy set. Some of its characterizations with examples is given. AMS Subject Classification: 08A72, 20N25, 20M12, 97H40. Key Words and Phrases: Near-rings, Q-fuzzy ideals, interval-valued Q-fuzzy ideals, Q-cubic ideals, Homomorphism of Q-cubic ideals. 1 Introduction The theory of fuzzy sets was introduced by Zadeh [12]. Zadeh [13] introduced the notion of interval-valued fuzzy set, where the values of the membership function are closed subintervals of [0,1] instead of a single value from it. Rosenfeld [10] first introduced the fuzzification of the algebraic structures and defined fuzzy subgroups. Abou-zaid [1] introduced the notion of a fuzzy sub near-ring and studied fuzzy ideals of a near ring. The notion of intutionistic fuzzy ijpam.eu 56 2017

sets was introduced by Atanasov [2] as a generalization of the notion of fuzzy sets. Biswas [3] applied the concept of intuitionistic fuzzy sets to the theory of groups and studied intutionistic fuzzy subgroups of a group. The notion of an intutionistic fuzzy ideal of a near-ring was given by Jun et al.[7]. Cho et al.[5] introduced the notion of intutionistic fuzzy R-subgroups in a near-ring and investigated related properties. Kazanci et al.[9] introduced intutionistic Q-fuzzy R-subgroups of near-rings. Janardan D. Yadav et al.[6] discussed intutionistic Q-fuzzy ideals of near-ring. Thillaigovindan et al.[11] introduced the notion of interval valued fuzzy ideals of near-rings. Jun et al.[8] introduced the concept of cubic sets. Chinnadurai et al.[4] discussed cubic ideals of Γ-near rings. In this paper we introduced the notion of Q-cubic ideals of near-rings and investigated some related properties. 2 Preliminaries In this section, we first review some elementary aspects that are necessary for this paper. Definition 1. [1] A near-ring is an algebraic system (R, +, ) consisting of a non-empty set R together with two binary operations called + and such that (R, +) is a group not necessarily abelian and (R, ) is a semigroup connected by the following distributive law: x (y + z) = x y + x z valid for all x, y, z R. We use the word near-ring to means left near-ring. We denote xy insted of x y. Definition 2. [1] An ideal I of a near-ring R is a subset of R such that 1. (I, +) is a normal subgroup of (R, +), 2. RI I, 3. (r + i)s rs I for all i I and r, s R. Definition 3. [11] Let R and S be near-rings. A map θ : R S is called a (near-ring) homomorphism if θ(x + y) = θ(x) + θ(y) and θ(xy) = θ(x) θ(y) for all x, y R. Definition 4. µ : X [0, 1]. Definition 5. [1] A fuzzy subset µ of a set X is a function [9] A function µ : X Q [0, 1] is called a ijpam.eu 57 2017

Q-fuzzy set. Definition 6. [9] A Q-fuzzy set µ in a near-ring R is called Q-fuzzy subnear-ring of R if 1. µ(x y, q) min{µ(x, q), µ(y, q)} 2. µ(xy, q) min{µ(x, q), µ(y, q)} for all x, y R and q Q. Definition 7. [9] A Q-fuzzy set µ in a near-ring R is called Q- fuzzy ideal of R if 1.µ(x y, q) min{µ(x, q), µ(y, q)}, 2.µ(rx, q) µ(x, q), 3.µ((x + i)y xy, q) µ(i, q) for all x, y R. Definition 8. [11] Let X be a non-empty set. A mapping µ : X D[0, 1] is called an interval-valued (in short i-v) fuzzy subset of X, if for all x X, µ(x) = [µ (x), µ + (x)], where µ and µ + are fuzzy subsets of X such that µ (x) µ + (x). Thus µ(x) is an interval (a closed subset of [0,1]) and not a number from the interval [0,1] as in the case of fuzzy set. Definition 9. [8] Let X be a non-empty set. A cubic set A in X is a structure of the form A = { x, µ A (x), λ(x) : x X} and denoted by A = µ A, λ where µ A = [µ 1 A, µ+ A ] is an interval-valued fuzzy set in X and λ is a fuzzy set in X. 3 Q-Cubic ideals of near-rings In this section, we introduced the notion of Q-cubic ideals of nearrings and dicuss some of its properties. Throughout this paper, R denotes near-ring unless otherwise specified. Definition 10. Let X and Q be non-empty sets. A Q- cubic set A with respect to Q is an object of the form A = { (x, q), µ(x, q), ω(x, q) x X, q Q} denoted by A = µ, ω where µ : X Q D[0, 1] is an interval-valued Q fuzzy set over Q and ω : X Q [0, 1] is a Q fuzzy set over Q. Definition 11. A Q-cubic set A = µ, ω of R is called a Q-cubic subnear-ring of R, if 1. µ(x y, q) min{µ(x, q), µ(y, q)} and ω(x y, q) max{ω(x, q), ω(y, q)}, 2. µ(xy, q) min{µ(x, q), µ(y, q)} and ω(xy, q) max{ω(x, q),ω(y, q)}, x, y R and q Q. ijpam.eu 58 2017

Definition 12. A Q-cubic set A = µ, ω of R is called Q- cubic ideal of R, if 1. µ(x y, q) min{µ(x, q), µ(y, q)} and ω(x y, q) max{ω(x, q), ω(y, q)} 2. µ(y + x y, q) µ(x, q) and ω(y + x y, q) ω(x, q) 3. µ(nx, q) µ(x, q) and ω(nx, q) ω(x, q) 4. µ(n(x + y) nx, q) µ(y, q) and ω(n(x + y) nx, q) ω(y, q) forall x, y, n R and q Q. Example 13. Let Q = {q} and R = {k, l, m, n} with two binary operations + and are defined as follows: + k l m n k k l m n l l k n m m m n l k n n m k l k l m n k k k k k l k k k k m k k k k n k l m n Then (R, +, ) is a near-ring. Define a Q-cubic set A = µ, ω in R as follows µ(k, q) = [1, 1], µ(l, q) = [ 1, 1 ], µ(m, q) = [0, 0] = µ(n, q) 3 2 and ω(k, q) = 0, ω(l, q) = 1, ω(m, q) = 1 = ω(n, q) for all q Q. 3 Hence A = µ, ω is a Q-cubic ideal of near-ring R. Example 14. Consider the near-ring R = Z 0, the set of all negative integers with usual addition and multiplication and let Q = {q 1, q 2 }. Let the Q-cubic subset A = µ, ω in R be defined by { { µ(x, q 1 )= [0.6, 0.7] ifx ( 3) [0.1, 0.2] otherwise and ω(x, q 0.2 ifx ( 3) 1)= 0.8 otherwise µ(x, q 2 )= { [0.8, 0.9] ifx ( 3) [0.2, 0.3] otherwise and ω(x, q 2)= Then A = µ, ω is a Q-cubic ideal of R. { 0.5 ifx ( 3) 0.9 otherwise Definition 15. Let R, R 1 be near-rings and Q be non-empty set. A mapping f : R Q R 1 Q is said to be homomorphism if f(x + y, q) = f(x, q) + f(y, q) and f(xy, q) = f(x, q) f(y, q) for all x, y R, and q Q. Definition 16. Let f be a mapping from a set X Q to a set Y Q and A = (µ, ω) be a Q-cubic set of X then the image of ijpam.eu 59 2017

X, f(a ) = (f(µ), f(ω)) is a Q-cubic set of Y is defined by sup µ(y, q) if f 1 (x) f(µ)(x, q) = y f 1 (x) f(a )(x, q) = [0, 0] otherwise inf ω(y, q) if f 1 (x) f(ω)(x, q) = y f 1 (x) 1 otherwise Let f be a mapping from a set X Q to Y Q and A = (µ, ω) be a Q-cubic set of Y then the pre image of Y f 1 (A ) = (f 1 (µ), f 1 (ω)) is a Q-cubic set of X is defined by { f 1 f 1 (µ)(x, q) = µ(f(x), q) (A ) = f 1 (ω)(x, q) = ω(f(x), q) Theorem 17. A Q-cubic set A = µ, ω in R is a Q-cubic ideal of R if and only if µ, µ + and ω are Q-cubic ideals of R. Proof. Let A = µ, ω be a Q-cubic ideal of R. For any x, y, n R we have [µ (x y, q), µ + (x y, q)] = µ(x y, q) min{ µ(x, q), µ(y, q)} = min{[µ (x, q), µ + (x, q)], [µ (y, q), µ + (y, q)]}=[min{µ (x, q), µ (y, q)}, min{µ + (x, q), µ + (y, q)}]. It follows that µ (x y, q) min{µ (x, q), µ (y, q)} and µ + (x y, q) min{µ + (x, q), µ + (y, q)}. We know that, ω(x y, q) max{ω(x, q), ω(y, q)},[µ (y + x y, q), µ (y+x y, q)] = µ(y+x y, q) µ(x, q) = [µ (x, q), µ + (x, q)]. It follows that µ (y + x y, q) µ (x, q) and µ + (y + x y, q) µ + (x, q). Clearly ω(y + x y, q) ω(x, q). Similarly, µ (nx, q) µ (x, q), µ + (nx, q) µ + (x, q), ω(nx, q) ω(x, q) and µ (n(x + y) nx, q) µ (y, q), µ + (n(x + y) nx, q) µ + (y, q), ω(n(x + y) nx, q) ω(y, q). Hence µ, µ + and ω are Q-fuzzy ideals of R. Conversely, suppose that µ, µ + and ω are Q-fuzzy ideals of R. Let x, y, n R and q Q. µ(x y, q) = [µ (x y, q), µ + (x y, q)] [min{µ (x, q), µ (y, q)}, min{µ + (y, q), µ + (y, q)}] min{ µ(x, q), µ(y, q)}. Clearly, ω(x y, q) max{ω(x, q), ω(y, q)}. µ(y + x y, q) = [µ (y + x y, q), µ (y + x y, q)] [µ (x, q), µ (x, q)] = µ(x, q). Clearly, ω(x y, q) ω(x, q). Similarly, µ(nx, q) µ(nx, q), ω(nx, q) ω(x, q) and µ(n(x + y) nx, q) µ(y, q), ω(n(x + y) nx, q) ω(y, q). Thus A = µ, ω is Q-cubic ideal of R. ijpam.eu 60 2017

Theorem 18. Let a Q-cubic set A = µ, ω be Q-cubic ideal of R. Then R A = {A (x, q) = A (0, q) x R, q Q} is ideal of R. Proof. Let x, y, n R A and q Q. Then µ(x, q) = µ(y, q) = µ(0, q) and ω(x, q) = ω(y, q) = ω(0, q). Since A = µ, ω is a Q- cubic ideal of R. µ(x y, q) min{ µ(x, q), µ(y, q)} = µ(0, q) and ω(x y, q) max{ω(x, q), ω(y, q)} = ω(0, q). Thus x y R A. Similarly, we can prove that for y R, x R A and q Q. Thus y + x y R A. For n R, x R A and q Q. Thus nx R A. Similarly, For n, x R, y R A and q Q. Thus n(x + y) nx R A. Hence R A is an ideal of R. Lemma 19. Let A be Q-cubic ideal of R. If A (x) A (y) that is µ(x, q) < µ(y, q) and ω(x, q) > ω(y, q) then µ(x y, q) = µ(x, q) = µ(y x, q) and ω(x y, q) = ω(x, q) = ω(y x, q) Proof. Let A be Q-cubic ideal of R. Let x, y R and q Q. µ(x y, q) min{ µ(x, q), µ(y, q)} = µ(x, q). Now µ(x, q) = µ(x y + y, q) = µ((x y) ( y), q) min{ µ(x y, q), µ( y, q)} = min{ µ(x y, q), µ(y, q)} = µ(x y, q). Also µ(y x, q) min{ µ(y, q), µ(x, q)} = µ(x, q), µ(x, q) = µ(y + x y, q) = µ(y (y x), q) min{ µ(y, q), µ(y x, q)} = µ(y x, q). Thus µ(x y, q) = µ(x, q) = µ(y x, q). Similarly, we can prove the other result. Lemma 20. Let A = µ, ω be Q-cubic ideal of R. 1. µ(0, q) µ(x, q) and ω(0, q) ω(x, q) 2. µ( x, q) µ(x, q) and ω( x, q) ω(x, q) for all x R and q Q. Proof. Since A = µ, ω is a Q-cubic ideal of R. 1. µ(0, q) = µ(x x, q) min{ µ(x, q), µ(x, q)} µ(x, q). Similarly, ω(0, q) ω(x, q). 2. µ( x, q) = µ(0 x, q) min{ µ(0, q), µ(x, q)} µ(x, q). Similarly, ω( x, q) ω(x, q). Lemma 21. Let A = µ, ω be Q-cubic ideal of R. 1. µ(x y, q) µ(0, q) implies µ(x, q) = µ(y, q) 2. ω(x y, q) ω(0, q) implies ω(x, q) = ω(y, q) for all x, y R and q Q. ijpam.eu 61 2017

Proof. It is clear by using Lemma 19. Theorem 22. Let R, R 1 be near-rings, Q be any non-empty set and φ : R Q R 1 Q be a homomorphism. If A = µ, ω is a Q-cubic ideal of R 1 then the preimage φ 1 (A ) = φ 1 (µ), φ 1 (ω) is a Q-cubic ideal of R. Proof. Let A = µ, ω is a Q-cubic ideal of R 1, let x, y, n R and q Q. Then φ 1 ( µ)(x y, q) = µ(φ(x y, q)) = µ(φ(x, q) φ(y, q)) min{ µ(φ(x, q)), (φ(y, q))} φ 1 (ω)(x y, q) = ω(φ(x y, q)) = min{φ 1 ( µ)(x, q), φ 1 ( µ)(y, q)} = ω(φ(x, q) φ(y, q)) max{ω(φ(x, q)), (φ(y, q))} = min{φ 1 (ω)(x, q), φ 1 (ω)(y, q)}. φ 1 ( µ)(y + x y, q) = µ(φ(y + x y, q)) = µ(φ(y, q) + µ(φ(x, q) φ(y, q)) µ(φ(x, q)) = φ 1 ( µ)(x, q). φ 1 (ω)(y + x y, q) = ω(φ(y + x y, q)) = ω(φ(y, q) + ω(φ(x, q) φ(y, q)) ω(φ(x, q)) = φ 1 (ω)(x, q). Similarly we can prove φ 1 ( µ)(nx, q) φ 1 ( µ)(x, q), and φ 1 (ω)(nx, q) φ 1 (ω)(x, q) φ 1 ( µ)(n(x + y) nx, q) φ 1 ( µ)(x, q), and φ 1 (ω)(n(x + y) nx, q) φ 1 (ω)(x, q). Thus φ 1 (A ) = φ 1 (µ), φ 1 (ω) is a Q-cubic ideal of R. Theorem 23. Let R, R 1 be near-rings, Q be any non-empty set and φ : R Q R 1 Q be a homomorphism. If A = µ, ω is a Q-cubic ideal of R then the image φ(a ) = φ( µ), φ(ω) is a Q-cubic ideal of R 1. Proof. The proof is straight forward. ijpam.eu 62 2017

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