Ratio Mathematica Volume 45, 2023 Translations of Bipolar Valued Multi Fuzzy Subnearring of a Nearring S. Muthukumaran1 B. Anandh2 Abstract In this paper, some translations of bipolar valued multi fuzzy subnearring of a nearing are introduced and using these translations, some theorems are stated and proved. Key Words. Bipolar valued fuzzy subset, bipolar valued multi fuzzy subset, bipolar valued multi fuzzy subnearring, translations, intersection. Subject Classification. 97H40, 03B52, 03E723. 1Research Scholar, P. G. and Research Department of Mathematics, H. H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Tiruchirappalli, Tamilnadu, India. Email: muthumaths28@gmail.com. 2Assistant Professor, P.G. and Research Department of Mathematics, H. H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Tiruchirappalli, Tamilnadu, India. Email: drbaalaanandh@gmail.com. 3Received on July 10, 2022. Accepted on October 15, 2022. Published on January 30, 2023.doi: 10.23755/rm. v45i0.999. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY license agreement. 127 S. Muthukumaran and B. Anandh 1. Introduction In 1965, Zadeh [9] introduced the notion of a fuzzy subset of a set, fuzzy sets are a kind of useful mathematical structure to represent a collection of objects whose boundary is vague. Since then, it has become a vigorous area of research in different domains, there have been a number of generalizations of this fundamental concept such as intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets, soft sets etc. W. R. Zhang [10, 11] introduced an extension of fuzzy sets named bipolar valued fuzzy sets in 1994 and bipolar valued fuzzy set was developed by Lee [2, 3]. Bipolar valued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to [−1, 1]. In a bipolar valued fuzzy set, the membership degree 0 means that elements are irrelevant to the corresponding property, the membership degree (0, 1] indicates that elements somewhat satisfy the property and the membership degree [−1, 0) indicates that elements somewhat satisfy the implicit counter property. Bipolar valued fuzzy sets and intuitionistic fuzzy sets look similar each other. However, they are different each other [3]. Vasantha kandasamy. W. B [7] introduced the basic idea about the fuzzy group and fuzzy bigroup. M.S. Anithat et.al [1] introduced the bipolar valued fuzzy subgroup. Sheena. K. P and K. Uma Devi [6] have introduced the bipolar valued fuzzy subbigroup of a bigroup. Shanthi. V.K and G. Shyamala [5] have introduced the bipolar valued multi fuzzy subgroups of a group. Yasodara. S, KE. Sathappan [8] defined the bipolar valued multi fuzzy subsemirings of a semiring. Bipolar valued multi fuzzy subnearring of a nearing has been introduced by S. Muthukumaran and B. Anandh [4]. In this paper, the concept of translations of bipolar valued multi fuzzy subnearring of a nearing is introduced and established some results. Definition 1.1. ([11])A bipolar valued fuzzy set (BVFS) B in X is defined as an object of the form B = {< x, B+ (u), B −(u) >/ xX}, where B+: X→ [0, 1] and B −: X→ [−1, 0]. The positive membership degree B+(u) denotes the satisfaction degree of an element x to the property corresponding to a bipolar valued fuzzy set B and the negative membership degree B−(u) denotes the satisfaction degree of an element x to some implicit counter-property corresponding to a bipolar valued fuzzy set B. Definition 1.2. ([8]) A bipolar valued multi fuzzy set (BVMFS) A in X is defined as an object of the form B = {  x, B1 +(u), B2 +(u), …, Bn +(u), B1 −(u), B2 −(u), …, Bn −(u)  / xX}, where Bi + : X→ [0, 1] and Bi −: X→ [−1, 0], for all i. The positive membership degrees Bi +(u) denote the satisfaction degree of an element x to the property corresponding to a bipolar valued multi fuzzy set B and the negative membership degrees Bi −(u) denote the satisfaction degree of an element x to some implicit counter- property corresponding to a bipolar valued multi fuzzy set B. 128 Translations of Bipolar Valued Multi Fuzzy Subnearring of a Nearring Definition 1.3. ([4])Let (N, +, ) be a nearring. A BVMFS B of N is said to be a bipolar valued multi fuzzy subnearring of N (BVMFSNR) if the following conditions are satisfied, for all i, (i) Bi + (u−v)  min {Bi+ (u), Bi + (v)} (ii) Bi + (uv)  min {Bi + (u), Bi + (v)} (iii) Bi −(u−v)  max {Bi −(u), Bi −(v)} (iv) Bi −(uv)  max{Bi −(u), Bi −(v)},  u, vN. Definition 1.4. ([8])Let A =  A1 +, A2 +, …, An +, A1 −, A2 −, …, An − and B =  B1 +, B2 +, …, Bn +, B1 −, B2 −, …, Bn − be two bipolar valued multi fuzzy subsets with degree n of a set X. We define the following relations and operations: (i) A  B if and only if for all i, Ai +(u) ≤ Bi +(u) and Ai −(u) ≥ Bi −(u),  uX. (ii) AB = {  u, min(A1 +(u), B1 +(u)), min(A2 +(u), B2 +(u)), …, min(An +(u), Bn +(u)), max (A1 −(u), B1 −(u) ), max (A2 −(u), B2 −(u) ), …, max (An −(u), Bn −(u) )  / uX }. Definition 1.5. Let C =  C1 +, C2 +, …, Cn +, C1 −, C2 −, …, Cn − be a bipolar valued multi fuzzy subnearring of a nearring R and sR. Then the pseudo bipolar valued multi fuzzy coset (sC)p =  (sC1 +)p1 +, (sC2 +)p2 +, …, (sCn +)pn +, (sC1 −)p1 −, (sC2 −)p2 −, …, (sCn −)pn − is defined by (sCi +)pi +(a) = pi +(s) Ci +(a) and (sCi −)pi −(a) = − pi −(s) Ci −(a), for all i and every aR and pP, where P is a collection of bipolar valued multi fuzzy subsets of R. Definition 1.6. [8] Let A =  A1 +, A2 +, …, An +, A1 −, A2 −, …, An − be a bipolar valued multi fuzzy subset of X. Then the height H(A) =  H(A1 +), H(A2 +), …, H(An +), H(A1 −), H(A2 −), …, H(An −)  is defined for all i as H(Ai +) = sup Ai +(x) for all xX and H(Ai −) = inf Ai −(x) for all xX. Definition 1.7. [6]Let A =  A1 +, A2 +, …, Ai +, A1 −, A2 −, …, Ai − be a bipolar valued multi fuzzy subset of X. Then 0A = 0A1 +, 0A2 +,…, 0An +, 0A1 −, 0A2 −,…, 0An −is defined for all i as 0Ai +(x) = Ai +(x) H (Ai +) for all xX and 0Ai −(x) = −Ai −(x)H(Ai −) for all xX. Definition 1.8. [6] Let A =  A1 +, A2 +, …, An +, A1 −, A2 −, …, An − be a bipolar valued multi fuzzy subset of X. Then A = A1 +, A2 +, …, An +, A1 −, A2 −, …, An −is defined for all i as Ai +(x) = Ai +(x) / H(Ai +) for all xX and Ai −(x) = −Ai −(x) / H(Ai −) for all xX. Definition 1.9. [6] Let A =  A1 +, A2 +, …, An +, A1 −, A2 −, …, An − be a bipolar valued multi fuzzy subset of X. Then A = A1 +, A2 +, …, An +, A1 −, A2 −, …, An −is defined for all i as Ai +(x) = Ai +(x) + 1− H(Ai +) for all xX and Ai −(x) = Ai −(x) −1−H(Ai −) for all xX. 129 S. Muthukumaran and B. Anandh Definition 1.10. [6] Let A =  A1 +, A2 +, …, An +, A1 −, A2 −, …, An − be a bipolar valued multi fuzzy subset of X. Then A is called bipolar valued normal multi fuzzy subset of X if H(Ai +) = 1 and H(Ai −) = −1 for all I. 2. Properties Theorem 2.1.([4]) If B =  B1 +, B2 +, …, Bn +, B1 −, B2 −, …, Bn −and C =  C1 +, C2 +, …, Cn +, C1 −, C2 −, …, Cn − are two bipolar valued multi fuzzy subnearrings with degree n of a nearring R, then their intersection BC is a bipolar valued multi fuzzy Subnearring of R. Theorem 2.2.Let K =  K1 +, K2 +… Kn +, K1 −, K2 −… Kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring R. Then the pseudo bipolar valued multi fuzzy coset (a1K) m is a bipolar valued multi fuzzy subnearring of the nearring R, for every a1 in R and m in M, where M is a collection of bipolar valued multi fuzzy subset of R. Proof. Let b1, c1 in R and a1R. For each i, then ( ) )()()( 111111 cbKamcbKa ii m i i −=− +++ + ≥ mi +(a1) min{Ki +(b1), Ki +(c1)} = min{mi +(a1) Ki +(b1), mi +(a1)Ki +(c1)}= min{ ( ) )( 11 bKa im i + + , ( ) )( 11 cKa im i + + }. Therefore ( ) )( 111 cbKa im i − + + ≥ min { ( ) )( 11 bKa im i + + , ( ) )( 11 cKa im i + + }, for b1, c1R. And for each i, then ( ) )()()( 111111 cbKamcbKa ii m i i +++ = + ≥ mi +(a1) min{Ki +(b1), Ki +(c1)}= min{mi +(a1)Ki +(b1), mi +(a1)Ki +(c1)}= min{ ( ) )( 11 bKa im i + + , ( ) )( 11 cKa im i + + }. Therefore ( ) )( 111 cbKa im i + + ≥ min { ( ) )( 11 bKa im i + + , ( ) )( 11 cKa im i + + }, for all b1, c1R. For each i, ( ) )()()( 111111 cbKamcbKa ii m i i −=− −−− −  mi −(a1) max {Ki −(b1), Ki −(c1)} = max{mi −(a1)Ki −(b1), mi −(a1)Ki −(c1)}= max{ ( ) )( 11 bKa im i − − , ( ) )( 11 cKa im i − − }. Therefore ( ) )( 111 cbKa im i − − −  max { ( ) )( 11 bKa im i − − , ( ) )( 11 cKa im i − − }, for b1, c1R. Also for each i, then ( ) )()()( 111111 cbKamcbKa ii m i i −−− = −  mi −(a1) max{Ki −(b1), Ki −(c1)} = max{mi −(a1)Ki −(b1), mi −(a1)Ki −(c1)}= max{ ( ) )( 11 bKa im i − − , ( ) )( 11 cKa im i − − }. Therefore ( ) )( 111 cbKa im i − − max { ( ) )( 11 bKa im i − − , ( ) )( 11 cKa im i − − }, for all b1, c1R. Hence (a1K) m is a bipolar valued multi fuzzy subnearring of the nearring R. Theorem 2.3. If K =  K1 +, K2 +, …, Kn +, K1 −, K2 −, …, Kn − is a bipolar valued multi fuzzy subnearring with degree n of a nearring R, thenK = K1 +, K2 +, …, Kn +, K1 −, K2 −, …, Kn −is a bipolar valued multi fuzzy subnearring of R. 130 Translations of Bipolar Valued Multi Fuzzy Subnearring of a Nearring Proof. Let a1, b1 in R. For each i, Ki +(a1− b1) = Ki +(a1− b1)+1–H(Ki +) min{Ki +(a1), Ki +(b1)}+1–H(Ki +)= min{Ki +(a1)+1–H(Ki +), Ki +(b1)+1–H(Ki +)}= min{Ki +(a1), Ki +(b1)}implies Ki +(a1− b1)  min{ Ki +(a1), Ki +(b1) } for all a1, b1R. And for all i, Ki +( a1b1) = Ki +(a1b1)+1–H(Ki +) min{Ki +(a1), Ki +(b1) }+1–H(Ki +) = min{Ki +(a1)+1– H(Ki +), Ki +(b1)+1–H(Ki +)}= min{Ki +(a1), Ki +(b1)}which implies Ki +(a1b1)  min{Ki +(a1), Ki +(b1) } for all a1, b1R. Also for all i, Ki −(a1− b1) = Ki −(a1− b1)−1– H(Ki −) max{Ki −(a1), Ki −(b1) }−1–H(Ki −) = max{Ki −(a1)−1–H(Ki −), Ki −(b1)−1–H(Ki −)}= max{Ki −(a1), Ki −(b1)}implies Ki −(a1− b1)  max{ Ki −(a1), Ki −(b1) } for all a1, b1R. And for all i, Ki −(a1b1) = Ki −(a1b1)−1–H(Ki −) max{Ki −(a1), Ki −(b1)}−1–H(Ki −)= max{Ki −(a1)−1–H(Ki −), Ki −(b1)−1–H(Ki −)}= max{Ki −(a1), Ki −(b1)}implies Ki −(a1b1)  max{Ki −(a1), Ki −(b1) } for all a1, b1R. Hence K is abipolar valued multi fuzzy subnearring of R. Corollary 2.4. Let K =  K1 +, K2 +, …, Kn +, K1 −, K2 −, …, Kn − is a bipolar valued multi fuzzy subnearring with degree n of a nearring R. (i) If eR, then for each i, Ki +(e) = 1 and Ki −(e) = −1, where e is an Identity element of R; (ii)For each i, there exists eR such that Ki +(e) = 1 and Ki −(e) = −1 if and only if Ki +(a1) = Ki +(a1) and Ki −(a1) = Ki −(a1) for all a1R; (iii)For each i, there exists a1R such that Ki +(a1) = Ki +(e) and Ki −(a1) = Ki −(e) if and only if Ki +(a1) = 1 and Ki −(a1) = −1, for somea1R; (iv)For each i, if there exists a1R such that Ki +(a1) = 1 and Ki −(a1) = −1, then Ki +(a1) = 1 and Ki −(a1) = −1; (v)For each i, if Ki +(e) = 1, Ki −(e) = −1, Ki +(a1) = 0 and Ki −(a1) = 0, then Ki +(a1) = 0, Ki −(a1) = 0; (vi) (K)= K, (vii)K is a bipolar valued normal multi fuzzy subnearring of R containing K; (viii) K is a bipolar valued normal multi fuzzy subnearring of R if and only if K = K; (ix)If there exists a bipolar valued multi fuzzy subnearring P of R satisfying P  K; then K is a bipolar valued normal fuzzy subnearring of R; (x)If there exists a bipolar valued multi fuzzy subnearring P of R satisfying P  K, then K = K. Proof. (i), (ii), (iii), (iv), (v) and (x) are trivial.(vi) Let a1, b1R. For each i, then (Ki +) +(a1) = Ki +(a1)+1– Ki +(e)= {Ki+(e)+1–Ki +(e)}+1–{Ki +(e)+1–Ki +(e)}= Ki +(a1)+1–Ki +(e) = Ki +(a1). Also for each i, (Ki −)−(a1) = Ki −(a1) –1– Ki −(e)= {Ki −(a1)–1–Ki −(e)}–1– {Ki −(e)–1–Ki −(e)} = Ki −(a1)–1–Ki −(e) = Ki −(a1).Hence  (K)= K.(vii) Let eR. Clearly Ki +(e) = 1 and Ki −(e) = −1. Thus K is a bipolar valued normal multi fuzzy subnearring of R and K K.(viii) If K = K, then it is obvious that K is a bipolar valued normal multi fuzzy subnearring of R. Assume that K is a bipolar valued normal multi fuzzy subnearring of R. Let a1R. Then Ki +(a1) = Ki +(a1)+1–Ki +(e) = Ki +(a1) and 131 S. Muthukumaran and B. Anandh Ki −(a1) = Ki −(a1)−1–Ki −(e) = Ki −(a1). Hence K = K. (ix) Suppose there exists a bipolar valued multi fuzzy subnearring P of H such that P  K. Then 1 = Pi + (e) ≤ Ki + (e) and −1 = Pi −(e) ≥ Ki −(e). Hence Ki +(e) =1 and Ki −(e) = −1. Theorem 2.5. If K =  K1 +, K2 +, …, Kn +, K1 −, K2 −, …, Kn − is a bipolar valued multi fuzzy subnearring with degree n of a nearring R, then0K = 0K1 +, 0K2 +, …, 0Kn +, 0K1 −, 0K2 −, …, 0Kn −is a bipolar valued multi fuzzy subnearring of R. Proof. Let a1, b1 in R. For each i, 0Ki +(a1− b1) = Ki +(a1−b1)H(Ki +) min{Ki +(a1), Ki +(b1)}H(Ki +) = min{Ki +(a1)H(Ki +), Ki +(b1)H(Ki +)}= min{0Ki +(a1), 0Ki +(b1)}implies 0Ki +(a1−b1)  min{ 0Ki +(a1), 0Ki +(b1)} for all a1, b1R. And for all i, 0Ki +(a1b1) = Ki +(a1b1)H(Ki +)  min{Ki +(a1), Ki +(b1)} H(Ki +)= min{Ki +(a1)H(Ki +), Ki +(b1)H(Ki +)}= min{0Ki +(a1), 0Ki +(b1)}. Thus 0Ki +(a1b1) min{ 0Ki +(a1), 0Ki +(b1)} for all a1, b1R. Also for all i, 0Ki −(a1−b1) = −K −(a1− b1)H(Ki −) −max{Ki −(a1), Ki −(b1)}H(Ki −)= max{−Ki −(a1)H(Ki −), −Ki −(b1)H(Ki −)} = max{0Ki −(a1), 0Ki −(b1)}implies 0Ki −(a1− b1)  max{0Ki −(a1), 0Ki −(b1)} for all a1, b1R. And for all i, 0Ki −(a1b1) = −Ki −(a1b1)H(Ki −)−max{Ki −(a1), Ki −(b1)} H(Ki −) = max{−Ki −(a1)H(Ki −), −Ki −(b1)H(Ki −)} = max{0Ki −(a1), 0Ki −(b1)}.Therefore 0Ki −(a1b1) max{ 0Ki −(a1), 0Ki −(b1)} for all a1, b1R. Hence 0Kis a bipolar valued multi fuzzy subnearring of R. Theorem 2.6. If K =  K1 +, K2 +, …, Kn +, K1 −, K2 −, …, Kn − is a bipolar valued multi fuzzy subnearring with degree n of a nearring R, thenK = K1 +, K2 +, …, Kn +, K1 −, K2 −, …, Kn −is a bipolar valued multi fuzzy subnearring of R. Proof. Let a1, b1 in R. For each i, then Ki +(a1− b1) = Ki +(a1− b1) / H(Ki +) min{Ki +(a1), Ki +(b1)} / H(Ki +) = min{Ki +(a1) / H(Ki +), Ki +(b1) / H(Ki +)}= min{Ki +(a1), Ki +(b1)}implies Ki +(a1− b1)  min{ Ki +(a1), Ki +(b1)} for all a1, b1R. And for all i, Ki +(a1b1) = Ki +(a1b1) / H(Ki +)  min{Ki +(a1), Ki +(b1)} / H(Ki +)= min{Ki +(a1) / H(Ki +), Ki +(b1) / H(Ki +)}= min{Ki +(a1), Ki +(b1)}. Therefore Ki +(a1b1) min{ Ki +(a1), Ki +(b1)} for all a1, b1R. Also for all i, Ki −(a1− b1) = −Ki −(a1− b1) / H(Ki −) − max{Ki −(a1), Ki −(b1)} / H(Ki −)= max{−Ki −(a1) / H(Ki −), −Ki −(a1) / H(Ki −)} = max{Ki −(a1), Ki −(b1)}implies Ki −(a1− b1)  max{ Ki −(a1), Ki −(b1)} for all a1, b1R. And for all i, Ki −(a1b1) = −Ki −(a1b1) / H(Ki −)−max{Ki −(a1), Ki −(b1)} / H(Ki −) = max{−Ki −(a1) / H(Ki −), −Ki −(b1) / H(Ki −)}= max{Ki −(a1), Ki −(b1)}. Therefore Ki −(a1b1) max{ Ki −(a1), Ki −(b1)}, for all a1, b1R. Hence K is a bipolar valued multi fuzzy subnearring of R. Corollary 2.7. Let K =  K1 +, K2 +, …, Kn +, K1 −, K2 −, …, Kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring R. (i) If for each i, H(Ki +) < 1, then 0Ki +< Ki +; (ii) If for each i, H(Ki −) −1, then 0Ki − Ki −, (iii) If for each i, H(Ki +) < 1 and H(Ki −) −1, then 0K< K; 132 Translations of Bipolar Valued Multi Fuzzy Subnearring of a Nearring (iv) If for each i, H(Ki +) < 1, then Ki + Ki +; (v) If for each i, H(Ki −) −1, then Ki − Ki −; (vi) If for each i, H(Ki +) < 1 and H(Ki −) −1, thenK K; (vii) If for each i, H(Ki +) < 1 and H(Ki −) −1, thenK is a bipolar valued normal multi fuzzy subnearring of R. Proof. (i), (ii), (iii), (iv), (v), (vi) and (vii) are trivial. Corollary 2.8. If K is a bipolar valued normal multi fuzzy subnearring of a nearring R, then (i) 0K = K, (ii) K = K. Proof. The proof follows from Definitions 1.8, 1.9 and 1.11. Theorem 2.9.Let K =  K1 +, K2 +… Kn +, K1 −, K2 −… Kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring R. If (a1K) m and (b1K) m are two pseudo bipolar valued multi fuzzy coset of K, then their intersection (a1K) m ∩(b1K) m is also a bipolar valued multi fuzzy subnearring of the nearring R, for every a1, b1R and m in M, where M is a collection of bipolar valued multi fuzzy subset of R. Proof. The Proof follows from the Theorem 2.1 and 2.2. Theorem 2.10.Let K =  K1 +, K2 +… Kn +, K1 −, K2 −… Kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring R. If (a1K) m and (b1K) m are two pseudos bipolar valued multi fuzzy coset of K and m (a1) m(b1) or m(a1)  m(b1), then their union (a1K) m(b1K) m is also a bipolar valued multi fuzzy subnearring of the nearring R, for every a1, b1R and m in M, where M is a collection of bipolar valued multi fuzzy subset of R. Proof. The proof follows from the Theorem 2.2. Theorem 2.11. Let K =  K1 +, K2 +, …, Kn +, K1 −, K2 −, …, Kn − be a bipolar valued multi fuzzy subnearring with degree n of a nearring R. Then K is a bipolar valued multi fuzzy subnearring of R if and only if each (Ki +, Ki −) is a bipolar valued fuzzy subnearring of R. Proof. Let a1, b1 in R. Suppose K is a bipolar valued multi fuzzy subnearring of R, for each i, Ki +(a1−b1) min {Ki +(a1), Ki +(b1)}, Ki +(a1b1) min {Ki +(a1), Ki +(b1)},Ki −(a1− b1) ≤ max{Ki −(a1), Ki −(b1)} and Ki −(a1b1) ≤ max{Ki −(a1), Ki −(b1)}. Hence each (Ki +, Ki −) is bipolar valued fuzzy subnearring of R. Conversely, assume that each (Ki +, Ki −) is bipolar valued fuzzy subnearring of R. As per the definition of bipolar valued multi fuzzy subnearring of R, K is a bipolar valued multi fuzzy subnearring of R. 133 S. Muthukumaran and B. Anandh References [1] Anitha. M.S., Muruganantha Prasad &Arjunan. K, “Notes on Bipolar valued fuzzy subgroups of a group”, Bulletin of Society for Mathematical Services and Standards, Vol. 2 No. 3 (2013), 52 −59. [2] Lee K.M., “Bipolar valued fuzzy sets and their operations”, Proc. Int. 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