Ratio Mathematica Volume 44, 2022 Properties of Intuitionistic Multi-Anti Fuzzy Normal Ring Dr. R. Muthuraj1 S. Yamuna2 Abstract In this paper, we discuss the properties of an intuitionistic multi-anti fuzzy normal ring of a ring is defined and discussed its properties. some results based on cartesian product, homomorphism and anti homomorphism of an intuitionistic multi-anti fuzzy normal ring of a ring are also discussed. Keywords –R - Intuitionistic multi-anti fuzzy ring, H - Intuitionistic multi-anti fuzzy normal subring, R1, R2- rings. Mathematics Subject Classification:03E72, 47S40,08A05,08A72,16Y30,08A20N253. 1PG and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai - 622001, TamilNadu, India. E-mail: rmr1973@yahoo.co.i; rmr1973@gmail.com 2 Part time Research Scholar, PG and Research, Department Mathematics, H.H. The Rajah’s College, Pudukkottai - 622001, TamilNadu, India. E - mail: bassyam1@gmail.com (Affiliated to Bharathidasan University, Tiruchirappalli - 24) 3Received on June 29th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.910. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 222 mailto:rmr1973@gmail.com R. Muthuraj and S. Yamuna 1. Introduction The idea of fuzzy sets introduced by L. A. Zadeh 1965 [19] is an approach to mathematical representation of vagueness in everyday curriculum, The idea of fuzzy set is welcome because it handles uncertainty and vagueness which ordinary set could not address. In fuzzy set theory membership function of an element is single value between 0 and 1. Therefore, a generalization of fuzzy set was introduced by Attanassov [1], 1983 called intuitionistic fuzzy set (IFS) which deals with the degree of non-membership function and the degree of hesitation. After several year, Sabu Sebastian [13] introduced the theory of multi-fuzzy sets in terms of multi-dimensional membership function. R. Muthuraj and S. Balamurugan [15] introduced the concept of multi-anti fuzzy subgroup and discussed some of its properties. R. Muthuraj and S. Balamurugan [17] introduced the concept of multi-anti fuzzy ideal of a ring under homomorphism. In this paper, we discuss the properties of an intuitionistic multi-anti fuzzy normal ring of a ring is defined and discussed its properties. some results based on cartesian product, homomorphism and anti homomorphism of an intuitionistic multi-anti fuzzy normal ring of a ring. 2. Preliminaries 2.1 Definition [1, 2, 4] A fuzzy subset A of a ring R is called a fuzzy subring of R if for all x, y R i. A (x– y) ≥ min {A(x), A (y)} and ii. A (xy) ≥ min {A (x), A (y)}. 2.2 Definition [2, 7] A fuzzy subset A of a ring R is called an anti-fuzzy subring of R if for all x, y R i. A (x– y) ≤ max {A(x), A (y)} and ii. A (xy) ≤ max {A (x), A (y)}. 2.3 Proposition [7] Let R1 and R2be rings and let f be a homomorphism from R1 onto R2. If A is a anti fuzzy ideal of R2 then f -1(A) is a anti fuzzy ideal of R1. 2.4 Definition [2] Let R be a ring. Let G ={  x, A(x), B(x)  / xR} be an intuitionistic fuzzy set defined on a ring R, where A: R→[0,1], B: R→[0,1] such that 0  A(x) + B(x)  1. An intuitionistic fuzzy subset G of R is called an intuitionistic fuzzy ring on R if the following conditions are satisfied. For all x, y R, i. A ( x – y) ≥ min {A (x), A(y)} , ii. A (xy) ≥ min {A(x), A(y)}, iii. B (x− y)  max {B (x), B (y)}, iv. B (xy)  max {B (x), B (y)}. 223 Properties Of Intuitionistic Multi- Anti Fuzzy Normal Ring 2.5 Definition [7,17] Let R be a ring. Let G = {x, A(x), B(x) / xR} be an intuitionistic fuzzy set defined on a ring R, where A: R→[0,1], B: R→[0,1] such that 0  A(x) + B(x)  1. An intuitionistic fuzzy subset G of R is called an intuitionistic anti- fuzzy ring on R if the following conditions are satisfied. For all x, y R, i. A (x – y)  max {A (x), A (y)}, ii. A (xy)  max {A(x), A (y)}, iii. B (x− y) ≥ min {B (x) , B (y)}, iv. B (xy) ≥ min {B (x) , B (y)}. 2.6 Definition [18] An intuitionistic multi-anti fuzzy ring G = {x, A(x), B(x) / xR} on a ring R is said to be an intuitionistic multi-anti fuzzy normal ring on R if for every x, y  R, A (xy) = A (yx) and B(xy) = B(yx). 2.7 Example [18] Consider the intuitionistic fuzzy sets, G = {x, A(x), B(x) / xR} of dimension 2 on Z is defined as, A1(x) = 0.2 if x = 0; A1(x) = 0.7 if x 0 and A2(x) = 0.3 if x = 0; A2(x) = 0.9 if x 0. B1(x) = 0.7 if x = 0; B1(x) = 0.2 if x 0 and B2(x) = 0.6 if x = 0; B2(x) = 0.1 if x 0. Then the intuitionistic multi-fuzzy set G = (A, B) of dimension 2 on Z is defined as,      = 0 x if(0.7,0.9) 0 x if(0.2,0.3) = (x)) 2 A (x), 1 (A = A(x) .B(x)= (B1(x), B2(x)) = { (0.7,0.6)if x = 0 (0.2,0.1)if x ≠ 0 Clearly, G is an intuitionistic multi-anti fuzzy normal ring on Z. 3. Properties of Intuitionistic multi-anti fuzzy normal ring In this section, the properties of an intuitionistic multi-anti fuzzy normal ring is discussed. 3.1 Theorem Let G = {x, A(x), B(x) / xR} and H = {x, C(x), D(x) / xR} be any two intuitionistic multi-anti fuzzy normal subrings of rings R1 and R2 respectively. Then their anti cartesian product G  H is an intuitionistic multi-anti fuzzy normal subring of R1  R2. Proof Let G and H be any two intuitionistic multi-anti fuzzy normal subrings of rings R1 and R2 respectively. Then, by Theorem 2.2.5, G  H is an intuitionistic multi-anti fuzzy subring of R1 R2. Let (p, q), (r, s)  R1R2.For each i = 1, 2, ..., k, Now, (A  C) ((p, q)(r, s)) = (( Ai  Ci )(pr, qs )) 224 R. Muthuraj and S. Yamuna = (max {Ai(pr), Ci(qs)}) = (max {Ai(rp), Ci(sq)}) = ((Ai Ci )( rp, sq )) = (A  C) ((r, s) (p, q)) Therefore, (A  C)((p, q)(r, s)) = (A  C) ((r, s) (p, q)) and (B D) ((p, q)(r, s)) = (( Bi Di )(pr, qs )) = (min {Bi(pr), Di(qs)}) = (min {Bi(rp), Di(sq)}) = (( Bi Di )( rp, sq )) = (B  D) ((r, s) (p, q)) (B D) ((p, q)(r, s))=(B  D) ((r, s) (p, q)) Hence, (G  H) ((p, q)(r, s) = (G  H) ((r, s) (p, q)). Hence, the anti cartesian product GH is an intuitionistic multi-anti fuzzy normal subring of R1  R2. 3.2 Theorem Let G = {x, A(x), B(x) / xR}and H = {x, C(x), D(x) / xR }be intuitionistic multi-fuzzy subsets of R1 and R2 respectively, such that C(02)  A(x) and D(02)  B(x) for all x in R1, where 02 is the additive identity element of R2. The anti cartesian product G  H is an intuitionistic multi-anti fuzzy normal subring of R1  R2, and then G is an intuitionistic multi-anti fuzzy normal subring of R1. Proof Let p, r  R1 and 02R2.Let G  H be an intuitionistic multi-anti fuzzy normal subring of R1 R2. Then, by Theorem 2.2.7, G is an intuitionistic multi-anti fuzzy subring of R1. For each i = 1, 2,..., k, A (pr) = (A1(pr ), A2(pr ), ... , Ak(pr )) = (max {A1(pr ), C1(0202)} , ... , max{Ak(pr ), Ck(0202)}) A (pr) = (max (Ai(pr ), Ci(0202))) and B (pr) = (B1(pr ), B2(pr ), ... , Bk(pr )) = (min {B1(pr ), D1(0202)} , ... , min{Bk(pr ), Dk(0202)}) B(pr) = (min (Bi(pr ), Di(0202))) That is, A (pr) = (max (Ai (pr), Ci (0202))) and B(pr) = (min (Bi(pr ), Di(0202))). Hence, (G  H )( pr, 0202 ) = (G  H)(pr, 0202) = (G  H)((p, 02)  (r, 02)) = (G  H)((r, 02)  (p, 02)) (G  H )( pr, 0202) = (G  H)(pr, 0202). That is, A(pr) = A(rp) and B(pr) = B(rp). 225 Properties Of Intuitionistic Multi- Anti Fuzzy Normal Ring 3.3 Theorem. Let G = {x, A(x), B(x) / xR}and H = {x, C(x), D(x) / xR} be intuitionistic multi-fuzzy subsets of R1 and R2 respectively, such that A (01)  C(y) and B (01)  D(y)for all y in R2, where 01 is the additive identity element of R1. The anti cartesian product G  An intuitionistic multi-anti fuzzy normal subring of R1  R2, then His a multi-anti fuzzy normal subring of R2. Proof Let q, s  R2 and 01R1.Let G  H is an intuitionistic multi-anti fuzzy normal subring of R1  R2. Then, by Theorem 2.2.8, An intuitionistic multi-anti fuzzy subring of R1. For each i = 1, 2, k, C (qs) = (C1(qs), C2(qs), ..., Ck(qs)) = (max {C1(qs), A1(0101)}, ..., max {Ck (qs), Ak (0101)}) C (qs) = (max (Ci(qs), Ai (0101))) and D (qs) = (D1(qs), D2(qs), ..., Dk(qs)) = (min {D1(qs), B1(0101)}, ..., min {Dk (qs), Bk (0101)}) D(qs) = (min (Bi (0101), Di (qs))) That is, C (qs) = (max (Ai (0101), Ci (qs))) and D(qs) = (min (Bi(0101), Di(qs))). Hence, (G  H )( 0101,qs ) = (G  H )(0101, qs) = (G  H )(01 ,q)  (01 , s) ) = (G  H )((01 , s)  ( 01 ,p) ) (G H) (01 01, qs) = (G  H) (01 01, sq). That is, C(qs) = C(sq) and D(qs) = D(sq). Hence, H is an intuitionistic multi-anti fuzzy normal subring of R1. 3.4 Remark Let G = {x, A(x), B(x) / xR}and H = {x, C(x), D(x) / xR}be intuitionistic multi-fuzzy subsets of rings R1 and R2 respectively. The anti cartesian product G  H is an intuitionistic multi-anti fuzzy normal subring of R1  R2, then it is not necessarily that both G and H are intuitionistic multi-anti fuzzy normal subrings of R1 andR2 respectively. 226 R. Muthuraj and S. Yamuna 4. Properties of intuitionistic multi-anti fuzzy normal subring of a ring under homomorphism and anti homomorphism In this section, the properties of intuitionistic multi-anti fuzzy normal subring of a ring under homomorphism and anti homomorphism are discussed. 4.1 Theorem Let R1 and R2 be any two rings. Let f: R1 → R2 be an onto homomorphism. If G = {x, A(x), B(x) / xR1} is an intuitionistic multi-anti fuzzy normal subring of R1, then f(G) is an intuitionistic multi-anti fuzzy normal subring of R2, if G has inf property and G is f-invariant. Proof Let G be an intuitionistic multi-anti fuzzy normal subring of R1. Then, by Theorem 2.3.2, f(G) is an intuitionistic multi-anti fuzzy subring of R2. Then if x, yR1, then f(x), f(y)R2. Now, f(A)(f(x)f(y)) = f(A)(f(xy)) = A(xy) = A(yx) = f(A)(f(yx)) = f(A)(f(y)f(x)) There fore, f(A)(f(x)f(y)) = f(A)(f(y)f(x)) and f(B)(f(x)f(y)) = f(B)(f(xy)) = B(xy) = B(yx) = f(B)(f(yx)) = f(B)(f(y)f(x)) There fore, f(B)(f(x)f(y)) = f(B)(f(y)f(x)). Hence, G(f(x)f(y) = G(f(y)f(x)). Hence, f(G) is an intuitionistic multi-anti fuzzy normal subring of R2. 4.2 Theorem Let R1 and R2 be any two rings. Let f: R1 → R2 be a homomorphism. If H = { x, C(x), D(x) / xR1} is an intuitionistic multi-anti fuzzy normal subring of R2, then f–1(H) is an intuitionistic multi-anti fuzzy normal subring of R1. Proof Let H be an intuitionistic multi-anti fuzzy normal subring of R2. Then, by Theorem 2.3.3, f–1(H) is an intuitionistic multi-anti fuzzy subring of R1. For any x, yR1, Now, f–1(C)(xy) = C(f(xy)) = C(f(x)f(y)) = C(f(y)f(x)) 227 Properties Of Intuitionistic Multi- Anti Fuzzy Normal Ring = C(f(yx)) = f–1(C)(yx) Therefore, f–1(C)(xy) = f–1(C)(yx) and f–1(D)(xy) = D(f(xy)) = D(f(x)f(y)) = D(f(y)f(x)) = D(f(yx)) = f–1(D)(yx) Therefore, f–1(D)(xy) = f–1(D)(yx). Hence, f–1(H)(xy) = f–1(H)(yx). Hence, f–1(H) is an intuitionistic multi-anti fuzzy normal subring of R1. 4.3 Theorem Let R1 and R2 be any two rings. Let f: R1 → R2 be an onto anti homomorphism. If G = { x, A(x), B(x) / xR1} is an intuitionistic multi-anti fuzzy normal subring of R1, then f(G) is an intuitionistic multi-anti fuzzy normal subring of R2, if G has inf property and G is f-invariant. Proof Let G be an intuitionistic multi-anti fuzzy normal subring of R1. Then, by Theorem 2.3.4, f(G) is an intuitionistic multi-anti fuzzy subring of R2. Then if x, yR1, then f(x), f(y)R2. Now, f(A)(f(x)f(y)) = f(A)(f(yx)) = A(yx) = A(xy) = f(A)(f(xy)) = f(A)(f(y)f(x)) There fore, f(A)(f(x)f(y)) = f(A)(f(y)f(x)) and f(B)(f(x)f(y)) = f(B)(f(yx)) = B(yx) = B(xy) = f(B)(f(xy)) = f(B)(f(y)f(x)) There fore, f(B)(f(x)f(y) = f(B)(f(y)f(x)). Hence, G(f(x)f(y)) = G(f(y)f(x)). Hence, f(G) is an intuitionistic multi-anti fuzzy normal subring of R2. 4.4 Theorem Let R1 and R2 be any two rings. Let f: R1 → R2 be an anti homomorphism. If H = {x, A(x), B(x) / xR1} is an intuitionistic multi-anti fuzzy normal subring of R2, then f –1(H) is an intuitionistic multi-anti fuzzy normal subring of R1. Proof Let B be an intuitionistic multi-anti fuzzy normal subring of R2. Then, by Theorem 2.3.5, f–1(H) is an intuitionistic multi-anti fuzzy subring of R1. 228 R. Muthuraj and S. Yamuna For any x, yR1, Now, f–1(C)(xy) = C(f(xy)) = C(f(y)f(x)) = C(f(x)f(y)) = C(f(yx)) = f–1(C)(yx) Therefore, f–1(C)(xy) = f–1(C)(yx) and f–1(D)(xy) = D(f(xy)) = D(f(y)f(x)) = D(f(x)f(y)) = D(f(yx)) = f–1(D)(yx) Therefore, f–1(D)(xy) = f–1(D)(yx). f–1(H)(xy) = f–1(H)(yx). Hence, f–1(H) is an intuitionistic multi-anti fuzzy normal subring of R1. 5. Conclusion In this paper, we discuss the properties of an intuitionistic multi-anti fuzzy normal ring of a ring is defined and discussed its properties. Homomorphism and anti homomorphism of an intuitionistic multi-anti fuzzy normal ring of a ring. References [1] Attanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 20,87-89,1986. [2] Azan. F. A, A. A. Mamun, and F. Nasrin, Anti fuzzy ideals of a ring, Annals of Fuzzy Mathematics and Informatics, Volume 5, No.2, pp 349 – 390, March 2013. [3] Biswas. R., fuzzy subgroups and Anti – fuzzy subgroups, Fuzzy sets and Systems, 5, 121 – 124, 1990. [4] Bingxue. 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