Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2615 https://internationalpubls.com Various Types of Translation in Bipolar Valued Multi I-Fuzzy Normal Subrings of a Ring 1K.Vairamuthu & 2S. Loganathan 1 Department Of Mathematics, Sethupathy Government Arts College(Affiliated To Alagappa University, Karaikudi), Ramanathapuram -623 502, Tamilnadu, India. Email: Vairammathi83@Gmail.Com 2Department Of Mathematics, Sethupathy Government Arts College(Affiliated To Alagappa University, Karaikudi), Ramanathapuram -623 502, Tamilnadu, India. Email: Logaamaths2010@Gmail.Com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this paper, various types of translation in B_V MIFNSR of a ring are studied and dealt. Some theorems are given and,they are proved. Keywords: Interval valued fuzzy subset,bipolar valued fuzzy subset,〖 B〗_V MIFS, 〖 B〗_V MIFSR,〖 B〗_V MIFNSR and translations. INTRODUCTION. Zadeh [14]had introduced the fuzzy subset in 1965. It is one of the generalizations of crisp set. 𝐺𝑟𝑜𝑢𝑝 𝑤𝑎𝑠 𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑧𝑒𝑑 𝑎𝑠 𝑓𝑢𝑧𝑧𝑦 𝑔𝑟𝑜𝑢𝑝 𝑏𝑦 Azriel Rosenfeld [3]. 𝐴𝑓𝑡𝑒𝑟, 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑡𝑦𝑝𝑒𝑠 𝑜𝑓 𝑓𝑢𝑧𝑧𝑦 𝑤𝑒𝑟𝑒 𝑖𝑛𝑡𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑏𝑦 𝑣𝑎𝑟𝑖𝑜𝑢𝑠 𝑎𝑢𝑡ℎ𝑜𝑟𝑠. In 1994, bipolar valued fuzzy subset was introduced by W.R.Zhang[15]. Bipolar valued multi I-fuzzy subring has been introduced by K.Vairamuthu, S.Loganathan[11]. The following papers [1], [2], [4], [5], [6], [7], [8], [9], [10], [12] and [13] were useful to write the this paper. In this paper, 𝑣𝑎𝑟𝑖𝑜𝑢𝑠 𝑡𝑦𝑝𝑒𝑠 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 𝑜𝑓 𝑎 𝑟𝑖𝑛𝑔 𝑎𝑟𝑒 𝑖𝑛𝑡𝑟𝑜𝑑𝑢𝑐𝑒𝑑 and established some results. 1.PRELIMINARIES. Definition 1.1 [14] 𝐴 𝑚𝑎𝑝 ℜ: 𝕄 → 𝔻[0,1] 𝑖𝑠 𝑠𝑎𝑖𝑑 𝑡𝑜 𝑏𝑒 𝑎𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑣𝑎𝑙𝑢𝑒𝑑 𝑓𝑢𝑧𝑧𝑦 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝕄, 𝑤ℎ𝑒𝑟𝑒 𝔻[0,1] 𝑚𝑒𝑎𝑛𝑠 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎𝑙𝑙 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑢𝑏𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑜𝑓 [0, 1]. Definition 1.2 [15] 𝑇ℎ𝑒 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝔗 = {(𝔷, 𝔗+(𝔷), 𝔗−(𝔷)): 𝔷 ∈ 𝕎} 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 a bipolar 𝑣𝑎𝑙𝑢𝑒𝑑 𝑓𝑢𝑧𝑧𝑦 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝕨, 𝑤ℎ𝑒𝑟𝑒 𝔗+: 𝕨 → [0,1] 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝 map and 𝔗−: 𝕨 → [−1,0] is a negative membership map. Definition 1.3 [11] 𝑇ℎ𝑒 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝔗 = {( 𝔷, 𝔗1 +(𝔷), 𝔗2 +(𝔷), … , 𝔗𝑛 +(𝔷), 𝔗1 −(𝔷), 𝔗2 −(𝔷), …,𝔗𝑛 −(𝔷)): 𝔷 ∈ 𝕎} 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑎 𝑏𝑖𝑝𝑜𝑙𝑎𝑟 𝑣𝑎𝑙𝑢𝑒𝑑 𝑚𝑢𝑙𝑡𝑖 𝐼 − 𝑓𝑢𝑧𝑧𝑦 𝑠𝑢𝑏𝑠𝑒𝑡 (𝐵𝑉𝑀𝐼𝐹𝑆) of 𝕨, where 𝔗𝑖 +: 𝕨 → 𝔻[0,1] 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝 map and 𝔗𝑖 −: 𝕨 → 𝔻[−1,0] is a negative membership map. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2616 https://internationalpubls.com Definition 1.4 [11] 𝐴 𝐵𝑉𝑀𝐼𝐹𝑆 Ħ =  Ħ1 +, Ħ2 +,…,Ħ𝑛 +, Ħ1 −, Ħ2 −,…,Ħ𝑛 − 𝑜𝑓 𝑎 𝑟𝑖𝑛𝑔 Ṏ 𝑖𝑠 𝑠𝑎𝑖𝑑 to 𝑏𝑒 𝑎 bipolar valued multi I − fuzzy subring of Ṏ (𝐵𝑉𝑀𝐼𝐹𝑆𝑅) 𝑖𝑓 Ħ ℎ𝑎𝑠, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, (i) Ħ𝑖 +(ჲ − ծ) ≥ 𝑟𝑚𝑖𝑛{Ħ𝑖 +(ჲ), Ħ𝑖 +(ծ)}, (ii) Ħ𝑖 +(ჲծ) ≥ 𝑟𝑚𝑖𝑛{Ħ𝑖 +(ჲ), Ħ𝑖 +(ծ)}, (iii) Ħ𝑖 −(ჲ − ծ) ≤ 𝑟𝑚𝑎𝑥{Ħ𝑖 −(ჲ), Ħ𝑖 −(ծ)}, (iv) Ħ𝑖 −(ჲծ) ≤ 𝑟𝑚𝑎𝑥{Ħ𝑖 −(ჲ), Ħ𝑖 −(ծ)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 ჲ, ծ ∈ Ṏ, where 𝑟𝑚𝑖𝑛{[𝔯, ą], [ъ, ծ]} = [min{𝔯, ъ} , min{ą, ծ}] and 𝑟𝑚𝑎𝑥{[𝔯, ą], [ъ, ծ]} = [max{𝔯, ъ} , max{ą, ծ}]. Example 1.5 Let 𝑅 = 𝕫3 = {0, 1, 2} 𝑏𝑒 𝑎 𝑟𝑖𝑛𝑔 𝑤𝑖𝑡ℎ ⨁3 𝑎𝑛𝑑 ⨂3. 𝑇ℎ𝑒𝑛 ℭ = {(0, [0.72, 0.81], [0.91, 1], [0.52, 0.63], [− 0.91,−0.81], [− 1,−0.91], [− 0.81, −0.72] ), (1, [0.51, 0.61], [0.71, 0.81], [0.31, 0.41],[− 0.71,−0.61], [− 0.61,−0.51], [− 0.51, −0.41]), (2, [0.51, 0.61], [0.71, 0.81], [0.31, 0.41], [−0.71, −0.61], [−0.61,−0.51], [−0.51, −0.41])} is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of R. Definition 1.6 𝐴 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 ɮ =  ɮ1 +, ɮ2 +,…,ɮ𝑛 +, ɮ1 −, ɮ2 −,…,ɮ𝑛 − 𝑜𝑓 𝑎 𝑟𝑖𝑛𝑔 ₢ 𝑖𝑠 𝑠𝑎𝑖𝑑 to 𝑏𝑒 𝑎 bipolar valued multi I − fuzzy normal subring of R (𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅) 𝑖𝑓 ɮ ℎ𝑎𝑠, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖, (i) ɮ𝑖 +(𝔶𝔴) = ɮ𝑖 +(𝔴𝔶), (ii) ɮ𝑖 −(𝔶𝔴) = ɮ𝑖 −(𝔴𝔶), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝔶, 𝔴 ∈ ₢. Definition 1.7 [11] Let Œ =  Œ1 +, Œ2 +,…,Œ𝑛 +, Œ1 −, Œ2 −,…,Œ𝑛 − and Ϥ =  Ϥ1 +, Ϥ2 +,…,Ϥ𝑛 +, Ϥ1 −, Ϥ2 −,…,Ϥ𝑛 − be 𝑡𝑤𝑜 𝐵𝑉𝑀𝐼𝐹𝑆𝑠 𝑤𝑖𝑡ℎ 𝑑𝑒𝑔𝑟𝑒𝑒 𝑛 𝑜𝑓 𝑎 𝑠𝑒𝑡 𝒲. 𝑇ℎ𝑒𝑛 (i) Œ ⊂ Ϥ 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓  𝑖, Œ𝑖 +(ҁ) ≤ Ϥ𝑖 +(ҁ) and Œ𝑖 −(ҁ) ≥ Ϥ𝑖 −(ҁ), ҁ ∈ 𝒲. (ii) Œ ∩ Ϥ = {ҁ, rmin(Œ1 +(ҁ), Ϥ1 +(ҁ)), rmin(Œ2 +(ҁ), Ϥ2 +(ҁ)), …, rmin(Œ𝑛 +(ҁ), Ϥ𝑛 +(ҁ)), rmax(Œ1 −(ҁ), Ϥ1 −(ҁ)), rmax(Œ2 −(ҁ), Ϥ2 −(ҁ)),…, rmax(Œ𝑛 −(ҁ), Ϥ𝑛 −(ҁ)) / ҁ ∈ 𝒲}. Definition 1.8. 𝐿𝑒𝑡 Ж =  Ж1 +, Ж2 +,…,Ж𝑛 +, Ж1 −, Ж2 −,…,Ж𝑛 − be 𝐵𝑉𝑀𝐼𝐹𝑆 of the set 𝒲. The transformations are defined as,  i = 1, 2, …, n, (i) ≬ (Ж) =  ≬(Ж1 +), ≬(Ж2 +), …, ≬(Ж𝑛 +), ≬(Ж1 −), ≬(Ж2 −), …,≬(Ж𝑛 −) , where ≬(Ж𝑖 +)(𝜚) = 𝑟𝑚𝑖𝑛 {[½, ½], Ж𝑖 +(𝜚)} and ≬(Ж𝑖 −)(𝜚) = 𝑟𝑚𝑎𝑥 {[−½, −½], Ж𝑖 −(𝜚)},  𝜚 ∈ 𝒲. (ii) ⋈ (Ж) =  ⋈(Ж1 +), ⋈(Ж2 +), …, ⋈(Ж𝑛 +), ⋈(Ж1 −), ⋈(Ж2 −), …,⋈(Ж𝑛 −) , where ⋈(Ж𝑖 +)(𝜚) = 𝑟𝑚𝑎𝑥 {[½, ½], Ж𝑖 +(𝜚)} and ⋈(Ж𝑖 −)(𝜚) = 𝑟𝑚𝑖𝑛 {[−½, −½], Ж𝑖 −(𝜚)},  𝜚 ∈ 𝒲. (iii) 𝔔(𝜛,𝜍)(Ж) =  𝔔(𝜛,𝜍)(Ж1 +), 𝔔(𝜛,𝜍)(Ж2 +), …, 𝔔(𝜛,𝜍)(Ж𝑛 +), 𝔔(𝜛,𝜍)(Ж1 −), 𝔔(𝜛,𝜍)(Ж2 −), …, 𝔔(𝜛,𝜍)(Ж𝑛 −) , where 𝔔(𝜛,𝜍)(Ж𝑖 +)(𝜚) = 𝑟𝑚𝑖𝑛 {𝜛𝑖 , Ж𝑖 +(𝜚)} and 𝔔(𝜛,𝜍)(Ж𝑖 −)(𝜚) = 𝑟𝑚𝑎𝑥 {𝜍𝑖 , Ж𝑖 −(𝜚)},  𝜚 ∈ 𝒲, 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ 𝐷[0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ 𝐷[−1, 0]. (iv) ℜ(𝜛,𝜍)(Ж) =  ℜ(𝜛,𝜍)(Ж1 +), ℜ(𝜛,𝜍)(Ж2 +), …, ℜ(𝜛,𝜍)(Ж𝑛 +), ℜ(𝜛,𝜍)(Ж1 −), ℜ(𝜛,𝜍)(Ж2 −), …, ℜ(𝜛,𝜍)(Ж𝑛 −) , where ℜ(𝜛,𝜍)(Ж𝑖 +)(𝜚) = 𝑟𝑚𝑎𝑥 {𝜛𝑖 , Ж𝑖 +(𝜚)} and ℜ(𝜛,𝜍)(Ж𝑖 −)(𝜚) = 𝑟𝑚𝑖𝑛 {𝜍𝑖 , Ж𝑖 −(𝜚)}, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2617 https://internationalpubls.com  𝜚 ∈ 𝒲, 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ 𝐷[0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ 𝐷[−1, 0]. (v) 𝔖(𝜛,𝜍)(Ж) =  𝔖(𝜛,𝜍)(Ж1 +), 𝔖(𝜛,𝜍)(Ж2 +), …, 𝔖(𝜛,𝜍)(Ж𝑛 +), 𝔖(𝜛,𝜍)(Ж1 −), 𝔖(𝜛,𝜍)(Ж2 −), …, 𝔖(𝜛,𝜍)(Ж𝑛 −) , where 𝔖(𝜛,𝜍)(Ж𝑖 +)(𝜚) = 𝜛𝑖Ж𝑖 +(𝜚) and 𝔖(𝜛,𝜍)(Ж𝑖 −)(𝜚) = −𝜍𝑖Ж𝑖 −(𝜚),  𝜚 ∈ 𝒲, 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ [0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ [−1, 0]. 2 – THEOREMS. Theorem 2.1. 𝐼𝑓 Њ =  Њ1 +, Њ2 +, … , Њ𝑛 +, Њ1 −, Њ2 −, … , Њ𝑛 − and 𝔉 =  𝔉1 +, 𝔉2 +, … , 𝔉𝑛 +, 𝔉1 −, 𝔉2 −, … , 𝔉𝑛 − 𝑎𝑟𝑒 𝑡𝑤𝑜 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 of a ring Ĕ1, then their intersection Њ ∩ 𝔉 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of Ĕ1. Theorem 2.2. 𝐼𝑓 Ѝ =  Ѝ1 +, Ѝ2 +, … , Ѝ𝑛 +, Ѝ1 −, Ѝ2 −, … , Ѝ𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of a ring ß1, then ≬ (Ѝ)𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of ß1. Theorem 2.3. 𝐼𝑓 ℭ =  ℭ1 +, ℭ2 +, … , ℭ𝑛 +, ℭ1 −, ℭ2 −, … , ℭ𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of a ring ℧1, then ≬ (ℭ)𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1. Proof. Let 𝜁, 𝜐 be in ℧1. For all i = 1, 2, …, n, by Theorem 2.2, ≬(ℭ) is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of ℧1, ≬(ℭ𝑖 +)(𝜁𝜐) = rmin{[½,½], ℭ𝑖 +(𝜁𝜐)}= rmin{[½,½], ℭ𝑖 +(𝜐𝜁)}= ≬(ℭ𝑖 +)(𝜐𝜁), for all 𝜁, 𝜐 in ℧1. Also ≬(ℭ𝑖 −)(𝜁𝜐) = rmax{[−½,−½], ℭ𝑖 −(𝜁𝜐)} = rmax{[−½,−½], ℭ𝑖 −(𝜐𝜁)}= ≬(ℭ𝑖 −)(𝜐𝜁), for all 𝜁, 𝜐 in ℧1. Hence ≬(ℭ) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Corollary 2.4. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔 ℧1, then ≬(𝔓 ∩ 𝔚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Corollary 2.5. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1and ℧2, then ≬ 𝔓 ∩≬ 𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1 ∩ ℧2. Proof. From the above Theorems, it is trivial. Corollary 2.6. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1, then ≬ 𝔓 ∩≬ 𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Theorem 2.7. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then ≬ (𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.8. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then ≬ 𝔓1 ∩≬ 𝔓2 ∩ … ∩≬ 𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.9. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then ≬ (𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 𝑜𝑓 the ring ℧1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2618 https://internationalpubls.com Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.10. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then ≬ 𝔓1 ∩≬ 𝔓2 ∩ … ∩≬ 𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Theorem 2.11. 𝐼𝑓 Њ =  Њ1 +, Њ2 +, … , Њ𝑛 +, Њ1 −, Њ2 −, … , Њ𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of a ring Ѝ1, then ⋈ (Њ)𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of Ѝ1. Theorem 2.12. 𝐼𝑓 ℭ =  ℭ1 +, ℭ2 +, … , ℭ𝑛 +, ℭ1 −, ℭ2 −, … , ℭ𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of a ring ℧1, then ⋈ (ℭ)𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1. Proof. Let 𝜁, 𝜐 be in ℧1. For all i = 1, 2, …, n, by Theorem 2.11, ⋈(ℭ) is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of ℧1, ⋈(ℭ𝑖 +)(𝜁𝜐) = rmax{[½,½], ℭ𝑖 +(𝜁𝜐)}= rmax{[½,½], ℭ𝑖 +(𝜐𝜁)}= ⋈(ℭ𝑖 +)(𝜐𝜁), for all 𝜁, 𝜐 in ℧1. Also ⋈(ℭ𝑖 −)(𝜁𝜐) = rmin{[−½,−½], ℭ𝑖 −(𝜁𝜐)} = rmin{[−½,−½], ℭ𝑖 −(𝜐𝜁)}= ⋈(ℭ𝑖 −)(𝜐𝜁), for all 𝜁, 𝜐 in ℧1. Hence ⋈(ℭ) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Corollary 2.13. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔 ℧1, then ⋈(𝔓 ∩ 𝔚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Corollary 2.14. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1and ℧2, then ⋈ 𝔓 ∩⋈ 𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1 ∩ ℧2. Proof. From the above Theorems, it is trivial. Corollary 2.15. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1, then ⋈ 𝔓 ∩⋈ 𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Theorem 2.16. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then ⋈ (𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.17. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then ⋈ 𝔓1 ∩⋈ 𝔓2 ∩ … ∩⋈ 𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.18. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then ⋈ (𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 𝑜𝑓 the ring ℧1. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.19. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then ⋈ 𝔓1 ∩⋈ 𝔓2 ∩ … ∩ ⋈ 𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2619 https://internationalpubls.com Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Theorem 2.20. 𝐼𝑓 Ⅎ =  Ⅎ1 +, Ⅎ2 +, … , Ⅎ𝑛 +, Ⅎ1 −, Ⅎ2 −, … , Ⅎ𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of a ring ℵ1, then 𝔔(𝜛,𝜍)(Ⅎ) is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of ℵ1, where 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ 𝐷[0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ 𝐷[−1, 0]. Theorem 2.21. 𝐼𝑓 𝔓 =  𝔓1 +, 𝔓2 +, … , 𝔓𝑛 +, 𝔓1 −, 𝔓2 −, … , 𝔓𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of a ring ℜ1, then 𝔔(𝜛,𝜍)(𝔓) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℜ1, where 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ 𝐷[0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ 𝐷[−1, 0]. Proof. Let 𝜉, ℏ be in ℜ1, 𝜛𝑖 ∈ 𝐷[0, 1]𝑎𝑛𝑑 𝜍𝑖 ∈ 𝐷[−1, 0]. For all i = 1, 2, …, n, by Theorem 2.20, 𝔔(𝜛,𝜍)(𝔓) is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of ℜ1, 𝔔(𝜛,𝜍)(𝔓𝑖 +)(𝜉ℏ) = rmin{𝜛𝑖, 𝔓𝑖 +(𝜉ℏ)} = rmin{𝜛𝑖, 𝔓𝑖 +(ℏ𝜉)}= 𝔔(𝜛,𝜍)(𝔓𝑖 +)(ℏ𝜉), for all 𝜉, ℏ in ℜ1. And 𝔔(𝜛,𝜍)(𝔓𝑖 −)(𝜉ℏ) = rmax{𝜍𝑖, 𝔓𝑖 −(𝜉ℏ)} = rmax{𝜍𝑖, 𝔓𝑖 −(ℏ𝜉)}= 𝔔(𝜛,𝜍)(𝔓𝑖 −)(ℏ𝜉), for all 𝜉, ℏ in ℜ1. Hence 𝔔(𝜛,𝜍)(𝔓) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℜ1. Corollary 2.22. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔 ℧1, then 𝔔(𝜛,𝜍)(𝔓 ∩ 𝔚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Corollary 2.23. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1and ℧2, then 𝔔(𝜛,𝜍)𝔓 ∩ 𝔔(𝜛,𝜍)𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1 ∩ ℧2. Proof. From the above Theorems, it is trivial. Corollary 2.24. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1, then 𝔔(𝜛,𝜍)𝔓 ∩ 𝔔(𝜛,𝜍)𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Theorem 2.25. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then 𝔔(𝜛,𝜍)(𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.26. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then 𝔔(𝜛,𝜍)𝔓1 ∩ 𝔔(𝜛,𝜍)𝔓2 ∩ … ∩ 𝔔(𝜛,𝜍)𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.27. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then 𝔔(𝜛,𝜍)(𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 𝑜𝑓 ℧1. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.28. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then 𝔔(𝜛,𝜍)𝔓1 ∩ 𝔔(𝜛,𝜍)𝔓2 ∩ … ∩ 𝔔(𝜛,𝜍)𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2620 https://internationalpubls.com Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Theorem 2.29. 𝐼𝑓 ℔ =  ℔1 +, ℔2 +, … , ℔𝑛 +, ℔1 −, ℔2 −, … , ℔𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of a ring Ṏ1, then ℜ(𝜛,𝜍)(℔) is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of Ṏ1, where 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ 𝐷[0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ 𝐷[−1, 0]. Theorem 2.30. 𝐼𝑓 ℭ =  ℭ1 +, ℭ2 +, … , ℭ𝑛 +, ℭ1 −, ℭ2 −, … , ℭ𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of a ring ℝ1, then ℜ(𝜛,𝜍)(ℭ) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℝ1, where 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ 𝐷[0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ 𝐷[−1, 0]. Proof. Let 𝜉, ℏ be in ℝ1, 𝜛𝑖 ∈ 𝐷[0, 1]𝑎𝑛𝑑 𝜍𝑖 ∈ 𝐷[−1, 0]. For all i = 1, 2, …, n, by Theorem 2.29, ℜ(𝜛,𝜍)(ℭ) is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of ℝ1, ℜ(𝜛,𝜍)(ℭ𝑖 +)(𝜉ℏ) = rmax{𝜛𝑖, ℭ𝑖 +(𝜉ℏ)} = rmax{𝜛𝑖, ℭ𝑖 +(ℏ𝜉)}= ℜ(𝜛,𝜍)(ℭ𝑖 +)(ℏ𝜉), for all 𝜉, ℏ in ℝ1. And ℜ(𝜛,𝜍)(ℭ𝑖 −)(𝜉ℏ) = rmin{𝜍𝑖, ℭ𝑖 −(𝜉ℏ)} = rmin{𝜍𝑖, ℭ𝑖 −(ℏ𝜉)}= ℜ(𝜛,𝜍)(ℭ𝑖 −)(ℏ𝜉), for all 𝜉, ℏ in ℝ1. Hence ℜ(𝜛,𝜍)(ℭ) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℝ1. Corollary 2.31. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔 ℧1, then ℜ(𝜛,𝜍)(𝔓 ∩ 𝔚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Corollary 2.32. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1and ℧2, then ℜ(𝜛,𝜍)𝔓 ∩ ℜ(𝜛,𝜍)𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1 ∩ ℧2. Proof. From the above Theorems, it is trivial. Corollary 2.33. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1, then ℜ(𝜛,𝜍)𝔓 ∩ ℜ(𝜛,𝜍)𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Theorem 2.34. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then ℜ(𝜛,𝜍)(𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.35. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then ℜ(𝜛,𝜍)𝔓1 ∩ ℜ(𝜛,𝜍)𝔓2 ∩ … ∩ ℜ(𝜛,𝜍)𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.36. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then ℜ(𝜛,𝜍)(𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 𝑜𝑓 ℧1. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.37. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then ℜ(𝜛,𝜍)𝔓1 ∩ ℜ(𝜛,𝜍)𝔓2 ∩ … ∩ ℜ(𝜛,𝜍)𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2621 https://internationalpubls.com Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Theorem 2.38. 𝐼𝑓 Ж =  Ж1 +, Ж2 +, … , Ж𝑛 +, Ж1 −, Ж2 −, … , Ж𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of a ring ɮ1, then 𝔖(𝜛,𝜍)(Ж) is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of ɮ1, where 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ [0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ [−1, 0]. Theorem 2.39. 𝐼𝑓 ℬ =  ℬ1 +, ℬ2 +, … , ℬ𝑛 +, ℬ1 −, ℬ2 −, … , ℬ𝑛 − is 𝑎 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of a ring ℧1, then 𝔖(𝜛,𝜍)(ℬ) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1, where 𝜛 = (𝜛1, 𝜛2, … , 𝜛𝑛) and 𝜍 = (𝜍1, 𝜍2, … , 𝜍𝑛), 𝜛𝑖 ∈ [0, 1] 𝑎𝑛𝑑 𝜍𝑖 ∈ [−1, 0]. Proof. Let ℴ, 𝜐 be in ℧1, 𝜛𝑖 ∈ [0, 1]𝑎𝑛𝑑 𝜍𝑖 ∈ [−1, 0]. For all i = 1, 2, …, n, by Theorem 2.38, 𝔖(𝜛,𝜍)(ℬ) is a 𝐵𝑉𝑀𝐼𝐹𝑆𝑅 of ℧1, 𝔖(𝜛,𝜍)(ℬ𝑖 +)(ℴ𝜐) = 𝜛𝑖 ℬ𝑖 +(ℴ𝜐) = 𝜛𝑖 ℬ𝑖 +(𝜐ℴ) = 𝔖(𝜛,𝜍)(ℬ𝑖 +)(𝜐ℴ), for all ℴ, 𝜐 in ℧1. And 𝔖(𝜛,𝜍)(ℬ𝑖 −)(ℴ𝜐) = (−𝜍𝑖) ℬ𝑖 −(ℴ𝜐) = (−𝜍𝑖) ℬ𝑖 −(𝜐ℴ) = 𝔖(𝜛,𝜍)(ℬ𝑖 −)(𝜐ℴ), for all ℴ, 𝜐 in ℧1. Hence 𝔖(𝜛,𝜍)(ℬ) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Corollary 2.40. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔 ℧1, then 𝔖(𝜛,𝜍)(𝔓 ∩ 𝔚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Corollary 2.41. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1and ℧2, then 𝔖(𝜛,𝜍)𝔓 ∩ 𝔖(𝜛,𝜍)𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1 ∩ ℧2. Proof. From the above Theorems, it is trivial. Corollary 2.42. If 𝔓 and 𝔚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑖𝑛𝑔𝑠 ℧1, then 𝔖(𝜛,𝜍)𝔓 ∩ 𝔖(𝜛,𝜍)𝔚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of ℧1. Proof. From the above Theorems, it is trivial. Theorem 2.43. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then 𝔖(𝜛,𝜍)(𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.44. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 𝑎𝑟𝑒 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the rings ℧1, ℧2, … , ℧m respectively, then 𝔖(𝜛,𝜍)𝔓1 ∩ 𝔖(𝜛,𝜍)𝔓2 ∩ … ∩ 𝔖(𝜛,𝜍)𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1 ∩ ℧2 ∩ … ∩ ℧m. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.45. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then 𝔖(𝜛,𝜍)(𝔓1 ∩ 𝔓2 ∩ … ∩ 𝔓𝑚) is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 𝑜𝑓 the ring ℧1. Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Corollary 2.46. 𝐼𝑓 𝔓1, 𝔓2, … , 𝔓𝑚 are 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s of the ring ℧1, then 𝔖(𝜛,𝜍)𝔓1 ∩ 𝔖(𝜛,𝜍)𝔓2 ∩ … ∩ 𝔖(𝜛,𝜍)𝔓𝑚 is a 𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅 of the ring ℧1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2622 https://internationalpubls.com Proof. From the above Theorems, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. CONCLUSION In this paper, 𝑣𝑎𝑟𝑖𝑜𝑢𝑠 𝑡𝑦𝑝𝑒𝑠 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 bipolar valued multi I- fuzzy normal subring of a ring have been introduced. Some useful theorems have been found and using these theorem we can find more results. It can be extended into different types of algebra. REFERENCES [1]. Anitha.M.S., Muruganantha Prasad & K.Arjunan, “Notes on Bipolar-valued fuzzy subgroups of a group”, Bulletin of Society for Mathematical Services and Standards,Vol. 2 No. 3 (2013), pp. 52 − 59. [2]. Arsham Borumand Saeid, “Bipolar-valued fuzzy BCK/BCI-algebras”, World Applied Sciences Journal, 7 (11) (2009), 1404 − 1411. [3]. Azriel Rosenfeld, “Fuzzy groups”, Journal of mathematical analysis and applications, 35(1971), 512 − 517. [4]. Balasubramanian.A, K.L.Muruganantha Prasad & K.Arjunan, “Properties of Bipolar interval valued fuzzy subgroups of a group”, International Journal of Scientific Research, Vol. 4, Iss. 4 (2015), 262 - 268. [5]. Grattan-Guiness, “Fuzzy membership mapped onto interval and many valued quantities”, Z.Math.Logik. Grundladen Math. 22 (1975), 149 − 160. [6]. Kyoung Ja Lee, “Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI- algebras”, Bull. Malays.Math. Sci. Soc., (2) 32(3) (2009), 361 – 373. [7]. K.M.Lee, “Bipolar-valued fuzzy sets and their operations”. Proc. Int. Conf. on Intelligent Technologies, Bangkok, Thailand, (2000), 307 − 312. [8]. K.M.Lee, “Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets and bipolarvalued fuzzy sets”. J. fuzzy Logic Intelligent Systems, 14 (2) (2004), 125 −129. [9]. Murugalingam.K and K.Arjunan, “A study on interval valued fuzzy subsemirings of a semiring”, International Journal of Applied Mathematics and Modeling, Vol. 1, No. 5 (2013), 1 − 6. [10]. Sabu Sebastian, T.V.Ramakrishnan, “Multi fuzzy sets”, International Mathematical Forum, 5, no.50 (2010), 2471 −2476. [11]. Vairamuthu.K,S.Loganathan, “Product in bipolar valued multi I-fuzzy subrings of a ring”, Gradiva Review Journal, Vol.8, Issue 11(2022). [12]. Vairamuthu.K,S.Loganathan, “Bipolar valued multi I-fuzzy subrings of a ring”, Journal for Basic Sciences, Vol.23, Issue 1(2023). [13]. Yasodara.S, KE. Sathappan, “Bipolar-valued multi fuzzy subsemirings of a semiring”, International Journal of Mathematical Archive, 6(9) (2015), 75 −80. [14]. L.A.Zadeh, fuzzy sets, Inform. And Control, 8(1965), 338 −353. [15]. W.R.Zhang, Bipolar Fuzzy sets and Relations, a computational Frame work for cognitive modeling and multiple decision Analysis, proceedings of Fuzzy IEEE conferences, (1994), 305− 309.