International Mathematical Forum, Vol. 6, 2011, no. 5, X. Arul Selvaraj and D. Sivakumar

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1 International Mathematical Forum, Vol. 6, 2011, no. 5, t-norm (λ, μ)-fuzzy Quotient Near-Rings and t-norm (λ, μ)-fuzzy Quasi-Ideals X. Arul Selvaraj and D. Sivakumar Mathematics Wing, D.D.E., Annamalai University Annamalainagar , India xaselvarajmaths@gmail.com sivakumardmaths@yahoo.com Abstract In this paper, we introduce the concept of t-norm (λ, μ)-fuzzy quotient near-ring and establish the isomorphism theorem of t-norm (λ, μ)-fuzzy quotient near-ring, and also we introduce the concept of t-norm (λ, μ)-fuzzy quasi-ideals of near-rings. Further we obtained the characterization of t-norm (λ, μ)-fuzzy quasi-ideal of near-rings and discuss some related properties. Keywords: t-norm (λ, μ)-fuzzy subnear-ring, t-norm (λ, μ) -fuzzy ideal of a near-ring, t-norm (λ, μ)-fuzzy quotiont near-ring,t-norm (λ, μ)-fuzzy quasi-ideal, homomorphism, isomorphism 1 Introduction Throughout this paper N stands for near-ring. For basic terminology and notations for near-ring we refer to Piln [10]. We expand the notion of t- norm (λ, μ)-fuzzy ideals for rings to t-norm (λ, μ)-fuzzy ideals of near-rings from Rajesh Kumar [11]. The concept of fuzzy sets was introduced by Zadeh [15] in Rosenfield [12] introduced the notion of a fuzzy group as easly as The notions of fuzzy subnear-ring and ideals were introduced by S- Abou-Zaid in 1991 [3, 4]. Basicaly this paper was devalaped from Bingxue Yao [5, 2]. Definition 1.1 A triangular norm, [11] t-norm is a function t :[0, 1] [0, 1] [0, 1]satisfying, for each a, b, c, d [0, 1],the following conditions: (i) t(0, 0) = 0, t(a, 1) = a; (ii) t(a, b) t(c, d),whenever a c, b d; (iii) t(a, b) = t(b, a);and t(t(a, b), c)= t(a, t(b, c)) Example 1.2 A funcation t :[0, 1] [0, 1] [0, 1], defined as t(a, b) =ab is a t-norm.

2 204 X. Arul Selvaraj and D. Sivakumar Example 1.3 A funcation t :[0, 1] [0, 1] [0, 1], defined as t(a, b) =a b is a t-norm. Example 1.4 A funcation t :[0, 1] [0, 1] [0, 1], defined as a if b=1 t(a, b) = b if a=1 0 otherwise is a t-norm. 2 t-norm (λ, μ)-fuzzy quotient near-rings Definition 2.1 Let A be a fuzzy subset of N, n N, we define the fuzzy subset n + A of N as follows, (n + A)(x) =t{(a(x n) λ),μ}, x N. Lemma 2.2 To prove t{(a(0) λ),μ} t{(a(x) λ),μ}, x N. Proof. Let x, y N. ThenA(x y) λ t{(a(x) λ),μ}.take y = x, we get A(0) λ t{a(x),μ} (A(0) λ) λ (t{a(x),μ}) λ = t{(a(x) λ), (μ λ)} = t{(a(x) λ),μ}, that is A(0) λ t{(a(x) λ),μ}. Hence t{(a(0) λ),μ} t{(a(x) λ),μ}. Lemma 2.3 Let A be a t-norm (λ, μ)-fuzzy subnear-ring of N and let a, b N. Then t{(a(a b) λ),μ} = t{(a(0) λ),μ} t{(a(b a) λ),μ} = t{(a(0) λ),μ}. Proof. Assume that t{(a(a b) λ),μ} = t{(a(0) λ),μ}, By lemma 2.2. t{(a(0) λ),μ} t{(a(b a) λ),μ} (1). Now consider t{(a(b a) λ),μ} t{t{a(a b),μ} λ),μ} = t{t{(a(a b) λ), (μ λ)},μ} = t{(a(a b) λ),μ} = t{(a(0) λ),μ}, that is t{(a(b a) λ),μ} t{(a(0) λ),μ}. (2). From (1) and(2) we get t{(a(a b) λ),μ} = t{(a(0) λ),μ}. Similarly we can prove the converse part. That ist{(a(b a) λ),μ} = t{(a(0) λ),μ}. Proposition 2.4 Let A be a t-norm (λ, μ)-fuzzy subnear-ring of N and let a, b N. Then a + A = b + A t{(a(a b) λ),μ} = t{(a(0) λ),μ}. Proof. Assume that t{(a(a b) λ),μ} = t{(a(0) λ),μ}. Then, for all x N, we have (b + a)(x) =t{(a(x b) λ),μ} = t{(a(x a)+(a b) λ),μ} = t{(a(x a) +(a b) λ λ),μ} t{(a(x a),a(a b) λ)},μ}, (since A is a fuzzy subnear-ring) = t{t{(a(x a) λ),μ},t{(a(a b) λ),μ}} = t{t{(a(x a) λ),μ},t{(a(0) λ),μ}} = t{t{(a(x a) λ),μ},μ} =(a+a)(x). That is (b + A)(x) (a + A)(x), therefore b + A a + B (1). Again

3 Fuzzy quotient near-rings 205 (a + A)(x) =t{(a(x a) λ),μ} = t{(a(x a)+(b a) λ),μ} = t{(a(x a)+(b a) λ λ),μ} t{(t{t{a(x b),a(b a)},μ} λ),μ} = t{t{(a(x b) λ),μ},t{(a(b a) λ),μ}} = t{t{(a(x b) λ),μ},t{(a(0) λ),μ}} = t{(a(x b) λ),μ} =(b + A)(x). That is (a + A)(x) (b + A)(x), therefore a + A b + A (2). From (1) and (2) we get a + A = b + A. Conversely, assume that a + A = b + A. Then we have, t{(a(a b) λ),μ} = (b + A)(a) = (a + A)(a) = t{(a(a a) λ),μ} = t{(a(0) λ),μ}, Therefore t{(a(a b) λ),μ} = t{(a(0) λ),μ}. Hence proved. Proposition 2.5 Let A be a t-norm (λ, μ)-fuzzy ideal of N, then for all x, y, a, b N. We have x+a = a+a and y+a = b+a (x+y)+a =(a+b)+a and xy + A = ab + A Proof. Let A be a t-norm (λ, μ)-fuzzy ideal of N. If x+a = a+a and y+a = b+a, then from Proposition 2.4 we have t{(a(x a) λ),μ} = t{(a(0) λ),μ} and t{(a(y a) λ),μ} = t{(a(0) λ),μ}. So, t{{(a(x + y) (a + b)) λ},μ} = t{{(a(x a)+(y b)) λ},μ} t{(t{a(x a),a(y b)} λ),μ} = t{t{(a(x a) λ), (A(y b) λ)},μ} = t{t{(a(x a) λ),μ},t{(a(y b) λ),μ}} = t{t{(a(0) λ),μ},t{(a(0) λ),μ}} = t{(a(0) λ),μ}, but t{(a(0) λ),μ} t{{(a(x+y) (a+b)) λ},μ}, Therefore (x+y)+a =(a+b)+a Furthermore we have to prove t{(a(xy ab) λ),μ} = t{(a(0) λ),μ} (1) Given that x+a = a+a and y+a = b+a t{(a(x a) λ),μ} = t{(a(0) λ),μ} (2) and t{(a(y a) λ),μ} = t{(a(0) λ),μ} (3). Let A(0) = t. Then, case(i): If λ t, then always t{(a(xy ab) λ),μ} λ, but t{(a(0) λ),μ} = λ. Therefore t{(a(xy ab) λ),μ} λ t{(a(0) λ),μ}. case(ii): If λ t μ, then from (2), that is t{(a(x a) λ),μ} = t{(a(0) λ),μ}. From (2), A(x a) =t x a A t. From (3), A(y a) =t y a A t. Therefore x a = i 1 and y b = i 2,x= i 1 + a and y = i 2 + b. So, xy =(i 1 + a)(i 2 + b) = i 1 (i 2 + b)+[a(i 2 + b) ab]+ab, xy ab = i 1 (i 2 + b)+[a(i 2 + b) ab] A t,(since A t is an ideal) A(xy ab) t = A(0). But A(0) A(xy ab), therefore A(xy ab) =A(0) t{(a(xy ab) λ),μ} = t{(a(0) λ),μ}, therefore xy + A = ab + A. case (iii): Suppose μ<t, then A(x a) μ and A(y b) μ x a A µ and y b A µ. Since A µ is an ideal, by case (ii),we get A(xy ab) =A(0) t{(a(xy ab) λ),μ} = t{(a(o) λ),μ}, therefore xy + A = ab + A. Proposition 2.6 Let A be a t-norm (λ, μ)-fuzzy ideal of N. Then N/A = {a + A/a N} form a near-ring with the null element 0+A, where (x + A)+ (y + A) =(x + y)+a, (x + A)(y + A) =xy + A. Proof. Proof can be obtain from proposition 2.5.

4 206 X. Arul Selvaraj and D. Sivakumar Definition 2.7 Let A be a t-norm (λ, μ)-fuzzy ideal of N, then N/A is called the t-norm (λ, μ)-fuzzy quotient near-ring of N with respect to A. Proposition 2.8 Let A be a t-norm (λ, μ)-fuzzy ideal of N, then N/A = N/A, where A = t{{x N/(A(x) λ),μ} = t{(a(0) λ),μ}}. Proof. Consider the map f : N N/A defined by f(x) =x + A, x N. Then f is a homomorphism for x, y N, we have f(x + y) =(x + y)+a = (x + A)+(y + A) =f(x)+f(y), and f(xy) =(xy)+a =(x + A)(y + A) = f(x)f(y). Therefore f is a homomorphism for x + A = N/A, we have x/ N such that f(x) = x + A. Hence f is onto homomorpism. Now,Ker(f) = {x N/f(x) =0+A} = {x N/x + A =0+A} = {t{x N/(A(x 0) λ),μ} = t{(a(0) λ),μ}} = {t{x N/(A(x) λ),μ} = t{(a(0) λ),μ}}a. Hence N/A = N/A. 3 t-norm (λ, μ)-fuzzy quasi-ideals of near-ring Definition 3.1 [2] If P and Q are two non-empty subsets of N, we define PQ = {ab a P, b Q} and P Q = {a(b + i) ab a, b P, i Q} Definition 3.2 [2] A near-ring N is called zero-symmetric if x0 = 0 for all x N Definition 3.3 [2] A subgroup Q of (N,+)is called a quasi-ideal of a near-ring NifQN NQ N Q Q. Definition 3.4 [2] A map f from a near-ring N 1 into a near-ring N 2 is called a homomorphism if f(x + y) =f(x) +f(y) and f(xy) =f(x)f(y) for all x, y N 1. Definition 3.5 [7] Let A{ be a fuzzy subset of N. sup x=a(b+i) ab A(i) if x=a(b+i)-ab, for a, b, i N, We define (N A)(x) = 0 otherwise Definition 3.6 [7] A fuzzy subgroup A of N is called a fuzzy quasi-ideal of N if (A N) (N A) (N A) A Definition 3.7 [9] An (, q)-fuzzy subgroup A of N is called an (, q)- fuzzy quasi-ideal of N if for all x N, A(x) min{((a N) (N A) (N A))(x), 0.5}, that is A(x) min{(a N)(x) (N A)(x) (N A)(x), 0.5}. Definition 3.8 [2] A (λ, μ)-fuzzy subgroup A and N is called a (λ, μ)-fuzzy quasi-ideal of N if for all x N, A(x) λ ((A N) (N A) (N A))(x) μ, that is A(x) λ (A N)(x) (N A)(x) (N A)(x) μ.

5 Fuzzy quotient near-rings 207 Definition 3.9 [1] A fuzzy subset A of a group G is said to be an (, q)- fuzzy subgroup of G if for all x, y G, (i)a(xy) min{a(x),a(y), 0.5}, (ii)a(x 1 ) min{a(x), 0.5}. Definition 3.10 [9] A fuzzy subset A of N is said to be an (, q)-fuzzy subnear-ring of N if for all x, y N, (i)a(x + y) min{a(x),a(y), 0.5}, (ii)a( x) min{a(x), 0.5}, (iii)a(xy) min{a(x), A(y), 0.5}. Definition 3.11 [9] A fuzzy subset A of N is said to be an (, q)-fuzzy ideal of N if (i)a is an (, q)-fuzzy subnear-ring of N, (ii)a(y + x y) min{a(x), 0.5} for all x, y N, (iii)a(xy) min{a(x), 0.5} for all x, y N, (iv)a(x(y + i) xy) min{a(i), 0.5} for all x, y, i N. Definition 3.12 A t-norm (λ, μ)-fuzzy subgroup A and N is called a t- norm (λ, μ)-fuzzy quasi-ideal of N if for all x NA(x) λ t[((a N) (N A) (N A))(x),μ], that is A(x) λ t[t[t[(a N)(x), (N A)(x)], (N A)(x)],μ]. Remark 3.13 A fuzzy quasi-ideal is a t-norm (λ, μ)-fuzzy quasi-ideal with λ =0and μ =1, and a (, q)-fuzzy quasi-ideal is a t-norm (λ, μ)-fuzzy quasi-ideal with λ =0and μ =0.5. Theorem 3.14 Every t-norm (λ, μ)-fuzzy quasi-ideal in a zero symmetric near-ring N is a t-norm (λ, μ)-fuzzy subnear-ring of N. Proof. Let A be a t-norm (λ, μ)-fuzzy quasi-ideal in a zero symmetric nearring N. Choose a, b, c, x, y, i in N such that a = bc = x(y +i) xy. Then A(a) λ t[t[t[(a N)(a), (N A)(a)], (N A)(a)],μ]=t[t[t[sup a=bc {min(a(b),n(c))}, sup a=bc {min(n(b),a(c))}], sup a=x(y+i) xy A(i)],μ]=t[t[t[sup a=bc {min(a(b),n (c))}, sup a=bc {min(n(b),a(c))}], sup a=b(0+c) b0 A(c)],μ]=t[t[t[supA(b), supa (c)], supa(c)], μ] (since N is zero symmetric.) t[t[a(b),a(c)],μ]. Therefore, A(bc) λ = A(a) λ t[t[a(b),a(c)],μ]. Hence A is a t-norm (λ, μ)-fuzzy subnear-ring of N. Theorem 3.15 Every t-norm (λ, μ)-fuzzy right ideal of N is a t-norm (λ, μ)- fuzzy quasi-ideal of N. Proof. Let A be a t-norm (λ, μ)-fuzzy right ideal of N. Choose a, b, c, x, y, iin N such that a = bc = x(y + i) xy. Then ((A N) (N A) (N A))(a) = t[t[(a N)(a), (N A)(a)], (N A)(a)] = t[t[sup a=bc {min(a(b),n(c))}, sup a=bc { min(n(b),a(c))}], (N A)(x(y+i) xy)] = t[t[supa(b), supa(c)], (N A)(x(y+ i) xy)]. (since N(z) = 1 for all z N) Nowt[((A N) (N A) (N A))(a),μ]=t[{t[t[supA(b), supa(c)], (N A)(x(y+i) xy)]},μ]=t[t[t{supa(b), μ}, supa(c)], (N A)(x(y +i) xy)] t[t[{a(bc) λ},n(c)],n(x(y +i) xy)]. (since A is a t-norm (λ, μ)-fuzzy right ideal,a(bc) λ t[a(b),μ])= A(bc) λ = A(a) λ. Thus A(a) λ t[((a N) (N A) (N A))(a),μ]. So A is a t-norm (λ, μ)-fuzzy quasi-ideal of N.

6 208 X. Arul Selvaraj and D. Sivakumar Theorem 3.16 Every t-norm (λ, μ)-fuzzy left ideal of N is a t-norm (λ, μ)- fuzzy quasi-ideal of N. Proof. Let A be a t-norm (λ, μ)-fuzzy left ideal of N. Choose a, b, c, x, y, i in N such that a = bc = x(y + i) xy. Then ((A N) (N A) (N A))(a) = t[t[(a N)(a), (N A)(a)], (N A)(a)] = t[t[sup a=bc {min(a(b),n(c))}, sup a=bc { min(n(b),a(c))}], (N A)(x(y+i) xy)] = t[t[supa(b), supa(c)], (N A)(x(y+ i) xy)]. (since N(z) = 1 for all z N) Nowt[((A N) (N A) (N A))(a), μ]=t[t[t[supa(b), supa(c)], supa(i)], μ]=t[t[supa(b), supa(c)], t[sup A(i),μ]] t[t[n(b),n(c)], {A(x(y + i) xy) λ}]. (since A is a t-norm (λ, μ)- fuzzy left ideal, A(x(y+i) xy) λ t[a(i),μ])= A(x(y+i) xy) λ = A(a) λ. Thus A(a) λ t[((a N) (N A) (N A))(a),μ].So A is a t-norm (λ, μ)- fuzzy quasi-ideal of N. Theorem 3.17 Every t-norm (λ, μ)-fuzzy ideal of N is a t-norm (λ, μ)-fuzzy quasi-ideal of N. Proof. The proof is straightforward from Theorem 3.15 and Theorem Theorem 3.18 A non-empty subset Q of N is a quasi-ideal of N if and only if K Q is a t-norm (λ, μ)-fuzzy quasi-ideal of N. Proof. Let Q be a quasi-ideal of N. By Theorem of [1], K Q is a t- norm (λ, μ)-fuzzy quasi-ideal of N. By Remark 3.13, K Q is a t norm(λ, μ)- fuzzy quasi-ideal of N. Conversely, let K Q is a t norm(λ, μ)-fuzzy quasi-ideal of N. Let a be anyelement of QN NQ N Q. Then there exist elements c, x, y of N and elements b, i of Q such that a = bc = x(y + i) xy. Now we have (K Q N)(a) =sup a=pq {t[k Q (p),n(q)]} t[k Q (b),n(c)] = t[1, 1]=1. So (K Q N)(a) =1. Similarly, (N K Q )(a) =1. Moreover, (N K Q )(a) = (N K Q )(x(y+i) xy) K Q (i) =1. Hence, K Q (a) λ t[t[t[(k Q N)(a), (N K Q )(a)], (N K Q )(a)],μ]=μ, and so K Q (a) = 1 which means that a Q. Then QN NQ N Q Q. Hence Q is a (λ, μ)quasi-ideal of N. References [1] S.K. Bhakat and P. Das, (, q)-fuzzy Subgroup, Fuzzy Sets and Systems, 80 (1996), [2] Ezilarasi, Contribution to the algebraic structures in fuzzy theory, Ph.D. Thesis, Annamalai university, India,(2008). [3] S. Abou-Zaid, on Fuzzy subnear-ring and ideals, Fuzzy sets and systems, 44 (1991),

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