Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 231 https://internationalpubls.com Some Product in Bipolar Valued Multi I-Fuzzy Subrings of a Ring 1K.Vairamuthu, 2 S. Loganathan 1 Department of Mathematics, Sethupathy Government Arts College, Ramanathapuram -623 502, Affiliated to Alagappa University, Tamilnadu, India. Email: vairammathi83@gmail.com 2Department of Mathematics, Sethupathy Government Arts College, Ramanathapuram -623 502, Affiliated to Alagappa University, Tamilnadu, India. Email: logaamaths2010@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: The cited sources help to construct this paper. Here, theorems on products in bipolar valued multi-I-fuzzy subrings of rings are presented together with their attributes, which are stated and demonstrated. Keywords: Interval-valued fuzzy subset, bipolar valued fuzzy subset, bipolar valued multi fuzzy subset, bipolar valued multi I-fuzzy subset, bipolar valued multi fuzzy subring, bipolar valued multi I-fuzzy subring, product, and strongest relation. Introduction The concept of a fuzzy subset of a set was first suggested by Zadeh [17] in 1965. Fuzzy sets are a helpful mathematical structure that can be used to describe a group of objects whose boundaries are not clearly defined. Since then, there have been many generalizations of this basic idea, including intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets, soft sets, etc. It has also become a burgeoning field of study in other disciplines. 1, 1] are called bipolar-valued fuzzy sets. Intuitionistic fuzzy sets and bipolar-valued fuzzy sets have a similar appearance. They differ from one another, nevertheless [9, 10]. Azriel Rosenfeld introduced the fuzzy group [4]. Following that, Anthony J. M. and H. Sherwood proposed fuzzy groups redefined[2], and Chitra V. and K. Arjunan extended Q-fuzzy principles to nearring[6]. T.V. Ramakrishnan and Sabu Sebastian introduced multi fuzzy sets[12]. The concept of bipolar-valued fuzzy sets was suggested by Lee [9]. Fuzzy sets that have their membership degree range expanded from [0, 1] to [−1, 1]. Following that, Anitha M.S et al. [1] introduced bipolar- valued fuzzy subgroups of a group, while Arsham Borum and Saeid [3] introduced bipolar-valued fuzzy BCK/BCI-algebras. Balasubramanian introduced properties of Bipolar interval-valued fuzzy subgroups of a group and associates [5]. Kyoung Ja Lee introduced bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras[8]. Murugalingam.K and K. Arjunan[11] presented a study on interval-valued fuzzy subsemirings of a semiring, while Shanmugapriya.M.M & K. Arjunan[13] presented the (Q, L)-Fuzzy subnearrings of a nearing. Somasundra Moorthy's work, "A study on interval valued fuzzy, anti-fuzzy, intuitionistic fuzzy subrings of a ring, [14], writing this work benefited from the thesis. The idea of product in the bipolar valued multi I-fuzzy subring of an is explored in this article. 1. Prelirrminaries. Definition 1.1. [17] An interval-valued fuzzy subset Ƒ of the set  is a function Ƒ:  →D[0, 1]. Here D[0, 1] denotes the family of all closed subintervals of [0, 1]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 232 https://internationalpubls.com Definition 1.2. [9] 𝑇ℎ𝑒 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝔗 = {(𝔷, 𝔗+(𝔷), 𝔗−(𝔷)): 𝔷 ∈ 𝕎} 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 a bipolar valued 𝑓𝑢𝑧𝑧𝑦 𝑠𝑢𝑏𝑠𝑒𝑡(𝔹𝕍𝔽𝕊) 𝑜𝑓 𝕨, 𝑤ℎ𝑒𝑟𝑒 𝔗+: 𝕨 → [0,1] 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 membership map and 𝔗−: 𝕨 → [−1,0] is a negative membership map. Example 1.3. Let  = {, , } be a set. Then 𝜑 = {, 0.4, −0.7, , 0.9, −0.3, , 0.8, −0.03} is a bipolar valued fuzzy subset of . Definition 1.4. [16] 𝑇ℎ𝑒 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 ℘ = {(𝔷, ℘1 +(𝔷), ℘2 +(𝔷), … , ℘𝑛 +(𝔷), ℘1 −(𝔷), ℘2 −(𝔷), … , ℘𝑛 −(𝔷)) ∶ 𝔷 ∈ ℳ} 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 a bipolar-valued multi-fuzzy subset (𝔹𝕍𝕄𝔽𝕊)𝑜𝑓 ℳ𝑤𝑖𝑡ℎ 𝑜𝑟𝑑𝑒𝑟 𝑛, 𝑤ℎ𝑒𝑟𝑒 ℘𝑖 +: ℳ → [0,1] 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 membership maps and ℘𝑖 −: ℳ → [−1,0] are negative membership maps, where i = 1, 2, …, n. Example 1.5. Let  = {, , } be a set. Then 𝜑 = {, 0.4, 0.5, 0.2, −0.7, −0.4, −0.1, , 0.9, 0.5, 0.8, −0.3, −0.2, −0.8, , 0.8, 0.1, 0.4, −0.4, −0.3, −0.6} is a bipolar valued multi fuzzy subset of  with order 3. Definition 1.6. [15] 𝑇ℎ𝑒 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 ℘ = {(𝔷, ℘1 +(𝔷), ℘2 +(𝔷), … , ℘𝑛 +(𝔷), ℘1 −(𝔷), ℘2 −(𝔷), … , ℘𝑛 −(𝔷)) ∶ 𝔷 ∈ ℳ} 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 a bipolar-valued multi-I-fuzzy subset (𝔹𝕍𝕄𝕀𝔽𝕊)𝑜𝑓 ℳ𝑤𝑖𝑡ℎ 𝑜𝑟𝑑𝑒𝑟 𝑛, 𝑤ℎ𝑒𝑟𝑒 ℘𝑖 +: ℳ → 𝐷[0,1] 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 membership maps and ℘𝑖 −: ℳ → 𝐷[−1,0] are negative membership maps, where i = 1, 2, …, n. Here D[0, 1] denotes the family of all closed subintervals of [0, 1] and 𝐷[−1,0] denotes the family of all closed subintervals of [−1, 0]. Note that [0] = [0, 0], [1] = [1, 1] and [−1] = [−1, −1]. Example 1.7. Let  = {, , } be a set. Then 𝜑 = {, [0.4, 0.6], [0.5, 0.7], [0.2, 0.6], [−0.7, −0.4], [−0.4, −0.1], [−0.3, −0.1], , [0.5, 0.9], [0.5, 0.7], [0.8, 0.9], [−0.3, −0.2], [−0.2, −0.1], [−0.8, −0.5], , [0.8, 0.9], [0.1, 0.6], [0.4, 0.7], [−0.4, −0.2], [−0.3, −0.1], [−0.6, −0.2]} is a bipolar valued multi I- fuzzy subset of  with order 3. Definition 1.8. [15] 𝐴 𝔹𝕍𝕄𝕀𝔽𝕊 ℘ =  ℘1 +, ℘2 +, …, ℘𝑛 +, ℘1 −, ℘2 −, …, ℘𝑛 −  of a ring 𝔜 𝑖𝑠 𝑠𝑎𝑖𝑑 to be a bipolar valued multi I − fuzzy subring of 𝔜 (𝔹𝕍𝕄𝕀𝔽𝕊ℝ) 𝑖𝑓 ℘ ℎ𝑎𝑠 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 , (i) ℘𝑖 +(𝔶 − 𝔴) ≥ 𝑟𝑚𝑖𝑛{℘𝑖 +(𝔶), ℘𝑖 +(𝔴)}, (ii) ℘𝑖 +(𝔶𝔴) ≥ 𝑟𝑚𝑖𝑛{℘𝑖 +(𝔶), ℘𝑖 +(𝔴)}, (iii) ℘𝑖 −(𝔶 − 𝔴) ≤ 𝑟𝑚𝑎𝑥{℘𝑖 −(𝔶), ℘𝑖 −(𝔴)}, (iv) ℘𝑖 −(𝔶𝔴) ≤ 𝑟𝑚𝑎𝑥{℘𝑖 −(𝔶), ℘𝑖 −(𝔴)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝔶, 𝔴 ∈ 𝔜. Example 1.9. Let 𝕫3 = {0, 1, 2} 𝑏𝑒 𝑎 𝑟𝑖𝑛𝑔 𝑤𝑖𝑡ℎ ⊕3 𝑎𝑛𝑑 ⊗3. Then ℘ is defined as ℘ = {(0, [0.7, 0.8], [0.8, 0.9], [0.9, 1.0], [− 0.9, −0.8], [− 0.8, −0.7], [− 0.7, −0.6]), (1, [0.5, 0.6], [0.6, 0.7], [0.7, 0.8], [− 0.6, −0.5], [− 0.5, −0.4], [− 0.4, −0.3]), (2, [0.5, 0.6], [0.6, 0.7], [0.7, 0.8], [−0.6, −0.5], [− 0.5, −0.4], [− 0.4, −0.3])}, is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of 𝕫3. Definition 1.10. 𝐿𝑒𝑡 𝔎 =  𝔎1 +, 𝔎2 +, …, 𝔎𝑛 +, 𝔎1 −, 𝔎2 −, … , 𝔎𝑛 − 𝑏𝑒 𝔹𝕍𝕄𝕀𝔽𝕊 of the set 𝔏1, the strongest 𝔹𝕍𝕄𝕀𝔽 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑛 𝔏1, that is a 𝔹𝕍𝕄𝕀𝔽 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 on 𝔎 𝑖𝑠 ℘ = {(𝜚, 𝜁), ℘1 +(𝜚, 𝜁), ℘2 +(𝜚, 𝜁),…, ℘𝑛 +(𝜚, 𝜁), ℘1 −(𝜚, 𝜁), ℘2 −(𝜚, 𝜁), …, ℘𝑛 −(𝜚, 𝜁) / for all 𝜚, 𝜁𝔏1}, where ℘𝑖 +(𝜚, 𝜁) = rmin{𝔎i +(𝜚), 𝔎i +(𝜁)} and ℘𝑖 −(𝜚, 𝜁) = rmax{𝔎i −(𝜚), 𝔎i −(𝜁)}, for all 𝜚, 𝜁𝔏1, i = 1, 2, …, n. Definition 1.11. 𝐿𝑒𝑡 𝔎 =  𝔎1 +, 𝔎2 +, …, 𝔎𝑛 +, 𝔎1 −, 𝔎2 −, … , 𝔎𝑛 − and ℘ =  ℘1 +, ℘2 +, …, ℘𝑛 +, ℘1 −, ℘2 −, … , ℘𝑛 − 𝑏𝑒 𝔹𝕍𝕄𝕀𝔽𝕊s of the sets 𝔏1 and 𝔏2 respectively. The product of 𝔎 and ℘, denoted by 𝔎 × ℘, is defined as 𝔎 × ℘ = {(𝜚, 𝜁), (𝔎1×℘1) +(𝜚, 𝜁), (𝔎2×℘2) +(𝜚, 𝜁), …, (𝔎n×℘n) +(𝜚, 𝜁), (𝔎1×℘1)−(𝜚, 𝜁), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 233 https://internationalpubls.com (𝔎2×℘2)−(𝜚, 𝜁), …, (𝔎n×℘n)−(𝜚, 𝜁) / for all (𝜚, 𝜁)𝔏1 × 𝔏2}, where (𝔎i×℘i) +(𝜚, 𝜁) = rmin{𝔎i +(𝜚), ℘i +(𝜁)} and (𝔎i×℘i)−(𝜚, 𝜁) = rmax{𝔎i −(𝜚), ℘i −(𝜁)} , i = 1, 2, …, n. 2. Some Theorems. Theorem 2.1. 𝐼𝑓 𝔎 =  𝔎1 +, 𝔎2 +, … , 𝔎𝑛 +, 𝔎1 −, 𝔎2 −, … , 𝔎𝑛 − and ℘ =  ℘1 +, ℘2 +, … , ℘𝑛 +, ℘1 −, ℘2 −, … , ℘𝑛 − 𝑎𝑟𝑒 𝔹𝕍𝕄𝕀𝔽𝕊ℝs of the rings 𝔏1 and 𝔏2 respectively, then 𝔎 × ℘ is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of the ring 𝔏1 × 𝔏2. Proof. Let 𝜚, 𝜐 be in 𝔏1 and 𝜁, 𝜉 be in 𝔏2. Then (𝜚, 𝜁) and (𝜐, 𝜉) are in 𝔏1×𝔏2. For all i, i = 1, 2, …, n, (𝔎i×℘i) +[(𝜚, 𝜁)−(𝜐, 𝜉)] = (𝔎i×℘i) +(𝜚−𝜐, 𝜁− 𝜉) = rmin{𝔎i +(𝜚−𝜐), ℘i +(𝜁− 𝜉)}  rmin {rmin{𝔎i +(𝜚), 𝔎i +(𝜐)}, rmin{℘i +( 𝜁), ℘i +(𝜉)}} = rmin{rmin{𝔎i +(𝜚), ℘i +(𝜁)}, rmin{𝔎i +(𝜐), ℘i +( 𝜉)}} = rmin{(𝔎i×℘i) +(𝜚, 𝜁), (𝔎i×℘i) +(𝜐, 𝜉)}, for all (𝜚, 𝜁), (𝜐, 𝜉) in 𝔏1×𝔏2. And (𝔎i×℘i) +[(𝜚, 𝜁)(𝜐, 𝜉)] = (𝔎i×℘i) +(𝜚𝜐, 𝜁 𝜉) = rmin{𝔎i +(𝜚𝜐), ℘i +(𝜁 𝜉)}  rmin {rmin{𝔎i +(𝜚), 𝔎i +(𝜐)}, rmin{℘i +( 𝜁), ℘i +( 𝜉)}} = rmin{rmin{𝔎i +(𝜚), ℘i +(𝜁)}, rmin{𝔎i +(𝜐), ℘i +(𝜉)}} = rmin {(𝔎i×℘i) +(𝜚, 𝜁), (𝔎i×℘i) +(𝜐, 𝜉)}, for all (𝜚, 𝜁), (𝜐, 𝜉) in 𝔏1×𝔏2. Also (𝔎i×℘i)−[(𝜚, 𝜁)−(𝜐, 𝜉)] = (𝔎i×℘i)−(𝜚−𝜐, 𝜁− 𝜉) = rmax{𝔎i −(𝜚−𝜐), ℘i −(𝜁− 𝜉)}  rmax {rmax{𝔎i −(𝜚), 𝔎i −(𝜐)}, rmax{℘i −(𝜁), ℘i −( 𝜉)}} = rmax{rmax{𝔎i −(𝜚), ℘i −(𝜁)}, rmax{𝔎i −(𝜐), ℘i −( 𝜉)}} = rmax{ (𝔎i×℘i)−(𝜚, 𝜁), (𝔎i×℘i)−(𝜐, 𝜉)}, for all (𝜚, 𝜁), (𝜐, 𝜉) in 𝔏1×𝔏2. And (𝔎i×℘i)−[(𝜚, 𝜁)(𝜐, 𝜉)] = (𝔎i×℘i)−(𝜚𝜐, 𝜁𝜉) = rmax{𝔎i −(𝜚𝜐), ℘i −(𝜁𝜉)}  rmax {rmax{𝔎i −(𝜚), 𝔎i −(𝜐)}, rmax{℘i −(𝜁), ℘i −( 𝜉)}} = rmax{max{𝔎i −(𝜚), ℘i −(𝜁)}, rmax{𝔎i −(𝜐), ℘i −( 𝜉)}} = rmax{(𝔎i×℘i)−(𝜚, 𝜁), (𝔎i×℘i)−(𝜐, 𝜉)}, for all (𝜚, 𝜁), (𝜐, 𝜉) in 𝔏1×𝔏2. Hence 𝔎×℘ is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of 𝔏1×𝔏2. Theorem 2.2. 𝐼𝑓 ℘1, ℘2, … , ℘𝑚 𝑎𝑟𝑒 𝔹𝕍𝕄𝕀𝔽𝕊ℝs of the rings 𝔏1, 𝔏2, … , 𝔏m respectively, then ℘1 × ℘2 × … × ℘𝑚 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of the ring 𝔏1 × 𝔏2 × … × 𝔏m. Proof. 𝐹𝑟𝑜𝑚 𝑡ℎ𝑒 𝑡ℎ𝑒𝑜𝑟𝑒𝑚 2.1, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. Theorem 2.3. 𝐼𝑓 𝔎×℘ is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ with degree n of a ring 𝔏1×𝔏2, then for all i, i = 1, 2, …, n, (𝔎i×℘i) +(−𝜐, 𝜉−1) = (𝔎i×℘i) +(𝜐, 𝜉 ), (𝔎i×℘i)−(−𝜐, 𝜉−1) = (𝔎i×℘i)−(𝜐, 𝜉 ), (𝔎i×℘i) +(𝜐, 𝜉 )  (𝔎i×℘i) +(0, 1) and (𝔎i×℘i)−(𝜐, 𝜉 ) ≥ (𝔎i×℘i)−(0, 1), for all (𝜐, 𝜉 ) in 𝔏1×𝔏2, where (0, 1) is the identity element of 𝔏1×𝔏2. Proof. Let (𝜐, 𝜉) be in 𝔏1×𝔏2 and (0, 1) be the identity element of 𝔏1×𝔏2. For all i, i = 1, 2, …, n, (𝔎i×℘i) +(𝜐, 𝜉) = (𝔎i×℘i) +(−(−𝜐), (𝜉−1)−1) ≥ (𝔎i×℘i) +(−𝜐, 𝜉−1) ≥ (𝔎i×℘i) +(𝜐, 𝜉). Thus (𝔎i×℘i) +(−𝜐, 𝜉−1) = (𝔎i×℘i) +(𝜐, 𝜉 ), for all (𝜐, 𝜉 ) in 𝔏1×𝔏2. And (𝔎i×℘i)−(𝜐, 𝜉) = (𝔎i×℘i)−(−(−𝜐), (𝜉−1)−1)  (𝔎i×℘i)−(−𝜐, 𝜉−1)  (𝔎i×℘i)−(𝜐, 𝜉). Thus (𝔎i×℘i)−(−𝜐, 𝜉−1) = (𝔎i×℘i)−(𝜐, 𝜉 ), for all (𝜐, 𝜉 ) in 𝔏1×𝔏2. Also (𝔎i×℘i) +(0, 1) = (𝔎i×℘i) +(𝜐 − 𝜐), 𝜉𝜉−1) = rmin {𝔎i +(𝜐 − 𝜐), ℘i +(𝜉𝜉−1)}  rmin { rmin{ 𝔎i +(𝜐), 𝔎i +(𝜐)}, rmin { ℘i +(𝜉), ℘i +(𝜉)}} = rmin { 𝔎i +(𝜐), ℘i +(𝜉)} = (𝔎i×℘i) +(𝜐, 𝜉 ). Thus (𝔎i×℘i) +(𝜐, 𝜉 )  (𝔎i×℘i) +(0, 1), for all (𝜐, 𝜉 ) in 𝔏1×𝔏2. And (𝔎i×℘i)−(0, 1) = (𝔎i×℘i)−(𝜐 − 𝜐), 𝜉𝜉−1) = rmax{𝔎i −(𝜐 − 𝜐), ℘i −(𝜉𝜉−1)}  rmax { rmax{ 𝔎i −(𝜐), 𝔎i −(𝜐)}, rmax{℘i −(𝜉), ℘i −(𝜉)}} = rmax{𝔎i −(𝜐), ℘i −(𝜉)} = (𝔎i×℘i)−(𝜐, 𝜉 ). Thus (𝔎i×℘i)−(𝜐, 𝜉 ) ≥ (𝔎i×℘i)−(0, 1), for all (𝜐, 𝜉 ) in 𝔏1×𝔏2. Theorem 2.4. 𝐿𝑒𝑡 𝔜 and 𝔚 𝑏𝑒 𝑎𝑛𝑦 𝑡𝑤𝑜 𝔹𝕍𝕄𝕀𝔽𝕊s of the rings ℌ1 and ℌ2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 234 https://internationalpubls.com respectively. If 𝔜 × 𝔚 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of the ring ℌ1 × ℌ2, then at least one of the following two statements must hold; (i) For all i = 1, 2, …, n, 𝔚𝑖 +(𝔬) ≥ 𝔜𝑖 +(𝜚), 𝔚𝑖 −(𝔬) ≤ 𝔜𝑖 −(𝜚), for all 𝜚ℌ1, (ii) 𝔚𝑖 +(𝜁) ≤ 𝔜𝑖 +(𝔢), 𝔚𝑖 −(𝜁) ≥ 𝔜𝑖 −(𝔢), for all 𝜁ℌ2, where 𝔢, 𝔬 𝑎𝑟𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 ℌ1and ℌ2. Proof. By contraposition, suppose that none of the statements (i) and (ii) holds. For 𝜚 ∈ ℌ1 and 𝜁 ∈ ℌ2 such that 𝔚𝑖 +(𝔬) < 𝔜𝑖 +(𝜚), 𝔚𝑖 −(𝔬) > 𝔜𝑖 −(𝜚) and 𝔚𝑖 +(𝜁) > 𝔜𝑖 +(𝔢), 𝔚𝑖 −(𝜁) < 𝔜𝑖 −(𝔢). For all i = 1, 2, …, n, (𝔜i×𝔚i) +(𝜚, 𝜁) = rmin{𝔜i +(𝜚), 𝔚i +(𝜁)}> rmin{ 𝔜i +(𝔢), 𝔚i +(𝔬)} = (𝔜i×𝔚i) +(𝔢, 𝔬). Also (𝔜i×𝔚i) +(𝜚, 𝜁) = rmax{𝔜i +(𝜚), 𝔚i +(𝜁)}< rmax{ 𝔜i +(𝔢), 𝔚i +(𝔬)}= (𝔜i×𝔚i) +(𝔢, 𝔬). Thus 𝔜 × 𝔚 is not a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of the ring ℌ1 × ℌ2. Hence either 𝔚𝑖 +(𝔬) ≥ 𝔜𝑖 +(𝜚), 𝔚𝑖 −(𝔬) ≤ 𝔜𝑖 −(𝜚), for all 𝜚ℌ1 or 𝔚𝑖 +(𝜁) ≤ 𝔜𝑖 +(𝔢), 𝔚𝑖 −(𝜁) ≥ 𝔜𝑖 −(𝔢), for all 𝜁ℌ2. Theorem 2.5. 𝐿𝑒𝑡 𝔓 and 𝔚 𝑏𝑒 𝑎𝑛𝑦 𝑡𝑤𝑜 𝔹𝕍𝕄𝕀𝔽𝕊s of the rings 𝔒1 and 𝔒2 respectively and 𝔓 × 𝔚 be a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of the ring 𝔒1 × 𝔒2. Then the following are true; (i) For all i = 1, 2, …, n, if 𝔚𝑖 +(𝔬) ≥ 𝔓𝑖 +(𝜚), 𝔚𝑖 −(𝔬) ≤ 𝔓𝑖 −(𝜚), for all 𝜚𝔒1, then 𝔓 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of 𝔒1; (ii) if 𝔚𝑖 +(𝜁) ≤ 𝔓𝑖 +(𝔢), 𝔚𝑖 −(𝜁) ≥ 𝔓𝑖 −(𝔢), for all 𝜁𝔒2, then 𝔚 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of 𝔒2; where 𝔢, 𝔬 𝑎𝑟𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝔒1and 𝔒2. Proof. Let 𝜚, 𝜐 be in 𝔒1. Then (𝜚, 𝔬) and (𝜐, 𝔬) are in 𝔒1×𝔒2. For all i, i = 1, 2, …, n, (i) 𝔓i +(𝜚−𝜐) = rmin{𝔓i +(𝜚−𝜐), 𝔚i +(𝔬− 𝔬)} = (𝔓i×𝔚i) +(𝜚−𝜐, 𝔬− 𝔬) = (𝔓i×𝔚i) +[(𝜚, 𝔬)−(𝜐, 𝔬)]  rmin{(𝔓i×𝔚i) +(𝜚, 𝔬), (𝔓i×𝔚i) +(𝜐, 𝔬)} = rmin{rmin{𝔓i +(𝜚), 𝔚i +(𝔬)}, rmin{𝔓i +(𝜐), 𝔚i +(𝔬)}} = rmin{ 𝔓i +(𝜚), 𝔓i +(𝜐)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜚, 𝜐 in 𝔒1. And 𝔓i +(𝜚𝜐) = rmin{𝔓i +(𝜚𝜐), 𝔚i +(𝔬𝔬)}= (𝔓i×𝔚i) +(𝜚𝜐, 𝔬𝔬) = (𝔓i×𝔚i) +[(𝜚, 𝔬)(𝜐, 𝔬)]  rmin{(𝔓i×𝔚i) +(𝜚, 𝔬), (𝔓i×𝔚i) +(𝜐, 𝔬)} = rmin{rmin{𝔓i +(𝜚), 𝔚i +(𝔬)}, rmin{𝔓i +(𝜐), 𝔚i +(𝔬)}} = rmin{ 𝔓i +(𝜚), 𝔓i +(𝜐)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜚, 𝜐 in 𝔒1. Also 𝔓𝑖 −(𝜚−𝜐) = rmax{𝔓𝑖 −(𝜚−𝜐), 𝔚𝑖 −(𝔬− 𝔬)}= (𝔓i×𝔚i)−(𝜚−𝜐, 𝔬− 𝔬) = (𝔓i×𝔚i)−[(𝜚, 𝔬)−(𝜐, 𝔬)]  rmax{(𝔓i×𝔚i)−(𝜚, 𝔬), (𝔓i×𝔚i)−(𝜐, 𝔬)} = rmax{rmax{𝔓𝑖 −(𝜚), 𝔚𝑖 −(𝔬)}, rmax{𝔓𝑖 −(𝜐), 𝔚𝑖 −(𝔬)}} = rmax{ 𝔓𝑖 −(𝜚), 𝔓𝑖 −(𝜐)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜚, 𝜐 in 𝔒1. And 𝔓𝑖 −(𝜚𝜐) = rmax{𝔓𝑖 −(𝜚𝜐), 𝔚𝑖 −(𝔬𝔬)} = (𝔓i×𝔚i)−(𝜚𝜐, 𝔬𝔬) = (𝔓i×𝔚i)−[(𝜚, 𝔬)(𝜐, 𝔬)]  rmax{(𝔓i×𝔚i)−(𝜚, 𝔬), (𝔓i×𝔚i)−(𝜐, 𝔬)} = rmax{rmax{𝔓𝑖 −(𝜚), 𝔚𝑖 −(𝔬)}, rmax{𝔓𝑖 −(𝜐), 𝔚𝑖 −(𝔬)}} = rmax{ 𝔓𝑖 −(𝜚), 𝔓𝑖 −(𝜐)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜚, 𝜐 in 𝔒1. Hence 𝔓 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of 𝔒1. (ii) Let 𝜚, 𝜐 be in 𝔒2. Then (𝔢, 𝜚) and (𝔢, 𝜐) are in 𝔒1×𝔒2. For all i, i = 1, 2, …, n, 𝔚𝑖 +(𝜚−𝜐) = rmin{𝔓i +(𝔢 − 𝔢), 𝔚𝑖 +(𝜚−𝜐)}= (𝔓i×𝔚i) +(𝔢 − 𝔢, 𝜚−𝜐) = (𝔓i×𝔚i) +[(𝔢, 𝜚)− (𝔢, 𝜐)]  rmin{(𝔓i×𝔚i) +(𝔢, 𝜚), (𝔓i×𝔚i) +(𝔢, 𝜐) } = rmin{rmin{𝔓i +(𝔢), 𝔚𝑖 +(𝜚)}, rmin{𝔓i +(𝔢), 𝔚𝑖 +(𝜐)}} = rmin{𝔚𝑖 +(𝜚), 𝔚𝑖 +(𝜐)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜚, 𝜐 in 𝔒2. And 𝔚𝑖 +(𝜚𝜐) = rmin{𝔓i +(𝔢𝔢), 𝔚𝑖 +(𝜚𝜐)}= (𝔓i×𝔚i) +(𝔢𝔢, 𝜚𝜐) = (𝔓i×𝔚i) +[(𝔢, 𝜚)(𝔢, 𝜐)]  rmin{(𝔓i×𝔚i) +(𝔢, 𝜚), (𝔓i×𝔚i) +(𝔢, 𝜐) } = rmin{rmin{𝔓i +(𝔢), 𝔚𝑖 +(𝜚)}, rmin{𝔓i +(𝔢), 𝔚𝑖 +(𝜐)}} = rmin{𝔚𝑖 +(𝜚), 𝔚𝑖 +(𝜐)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜚, 𝜐 in 𝔒2. Also 𝔚𝑖 −(𝜚−𝜐) = rmax{𝔓𝑖 −(𝔢 − 𝔢), 𝔚𝑖 −(𝜚−𝜐)}= (𝔓i×𝔚i)−(𝔢 − 𝔢, 𝜚−𝜐) = (𝔓i×𝔚i)−[(𝔢, 𝜚)− (𝔢, 𝜐)]  rmax{(𝔓i×𝔚i)−(𝔢, 𝜚), (𝔓i×𝔚i)−(𝔢, 𝜐) } = rmax{rmax{𝔓𝑖 −(𝔢), 𝔚𝑖 −(𝜚)}, rmax{𝔓𝑖 −(𝔢), 𝔚𝑖 −(𝜐)}} = rmax{𝔚𝑖 −(𝜚), 𝔚𝑖 −(𝜐)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜚, 𝜐 in 𝔒2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 235 https://internationalpubls.com And 𝔚𝑖 −(𝜚𝜐) = rmax{𝔓𝑖 −(𝔢𝔢), 𝔚𝑖 −(𝜚𝜐)}= (𝔓i×𝔚i)−(𝔢𝔢, 𝜚𝜐) = (𝔓i×𝔚i)−[(𝔢, 𝜚)(𝔢, 𝜐)]  rmax{(𝔓i×𝔚i)−(𝔢, 𝜚), (𝔓i×𝔚i)−(𝔢, 𝜐) } = rmax{rmax{𝔓𝑖 −(𝔢), 𝔚𝑖 −(𝜚)}, rmax{𝔓𝑖 −(𝔢), 𝔚𝑖 −(𝜐)}} = rmax{𝔚𝑖 −(𝜚), 𝔚𝑖 −(𝜐)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜚, 𝜐 in 𝔒2. Hence 𝔚 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of 𝔒2. Theorem 2.6. 𝐿𝑒𝑡 𝔓1, 𝔓2, …, 𝔓n 𝑏𝑒 𝑡ℎ𝑒 𝔹𝕍𝕄𝕀𝔽𝕊s of the rings 𝔒1, 𝔒2, …, 𝔒n respectively and 𝔓1 × 𝔓2 × … × 𝔓n be a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of the ring 𝔒1 × 𝔒2 × … × 𝔒n. Then the following are true; For all i, j, k = 1, 2, …, n, if 𝔓𝑘𝑗 + (𝔬) ≥ 𝔓𝑖𝑗 + (𝜚), 𝔓𝑘𝑗 − (𝔬) ≤ 𝔓𝑖𝑗 − (𝜚), for all 𝜚𝔒i, then 𝔓i is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of 𝔒i. where 𝔢, 𝔬 𝑎𝑟𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝔒iand 𝔒k. Proof. The proof follows from the theorem 2.5. Theorem 2.7. 𝐼𝑓 𝔓 × 𝔚 𝑖𝑠 𝑎 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of the ring 𝔎1 × 𝔎2, then 𝔉 = {(𝔥, 𝔷) ∈ 𝔎1 × 𝔎2: (𝔓𝑖 + × 𝔚𝑖 +)(𝔥, 𝔷) = (𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡) 𝑎𝑛𝑑 (𝔓𝑖 − × 𝔚𝑖 −)(𝔥, 𝔷) = (𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡), for all i = 1, 2, … , n} is either empty or 𝑎 subring 𝔎1 × 𝔎2, where 𝔬, 𝔡 are first operation identity elements of 𝔎1 and 𝔎2. Proof. 𝐼𝑓 𝑎𝑛𝑦 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑛𝑜𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠 𝑡ℎ𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛, 𝑡ℎ𝑒𝑛 𝔉 𝑖𝑠 𝑒𝑚𝑝𝑡𝑦. Let (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝔉. 𝐹or all i = 1, 2, … , n, then (𝔓𝑖 + × 𝔚𝑖 +)[(𝔥1, 𝔷1) − (𝔥2, 𝔷2)]  rmin{(𝔓𝑖 + × 𝔚𝑖 +)(𝔥1, 𝔷1), (𝔓𝑖 + × 𝔚𝑖 +)(𝔥2, 𝔷2)} = 𝑟𝑚𝑖𝑛 {(𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡), (𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡)} = (𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡). Thus (𝔓𝑖 + × 𝔚𝑖 +)[(𝔥1, 𝔷1) − (𝔥2, 𝔷2)] = (𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡), 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝔉. And (𝔓𝑖 − × 𝔚𝑖 −)[(𝔥1, 𝔷1) − (𝔥2, 𝔷2)]  rmax{(𝔓𝑖 − × 𝔚𝑖 −)(𝔥1, 𝔷1), (𝔓𝑖 − × 𝔚𝑖 −)(𝔥2, 𝔷2)} = 𝑟𝑚𝑎𝑥 {(𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡), (𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡)} = (𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡). Thus (𝔓𝑖 − × 𝔚𝑖 −)[(𝔥1, 𝔷1) − (𝔥2, 𝔷2)] = (𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡), 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝔉. Therefore (𝔥1, 𝔷1) − (𝔥2, 𝔷2) ∈ 𝔉. 𝐴𝑙𝑠𝑜 (𝔓𝑖 + × 𝔚𝑖 +)[(𝔥1, 𝔷1)(𝔥2, 𝔷2)]  rmin {(𝔓𝑖 + × 𝔚𝑖 +)(𝔥1, 𝔷1), (𝔓𝑖 + × 𝔚𝑖 +)(𝔥2, 𝔷2)} = 𝑟𝑚𝑖𝑛 {(𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡), (𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡)} = (𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡). Thus (𝔓𝑖 + × 𝔚𝑖 +)[(𝔥1, 𝔷1)(𝔥2, 𝔷2)] = (𝔓𝑖 + × 𝔚𝑖 +)(𝔬, 𝔡), 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝔉. And (𝔓𝑖 − × 𝔚𝑖 −)[(𝔥1, 𝔷1)(𝔥2, 𝔷2)]  rmax{(𝔓𝑖 − × 𝔚𝑖 −)(𝔥1, 𝔷1), (𝔓𝑖 − × 𝔚𝑖 −)(𝔥2, 𝔷2)} = 𝑟𝑚𝑎𝑥 {(𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡), (𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡)} = (𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡). Thus (𝔓𝑖 − × 𝔚𝑖 −) [(𝔥1, 𝔷1)(𝔥2, 𝔷2)] = (𝔓𝑖 − × 𝔚𝑖 −)(𝔬, 𝔡), 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝔉. Therefore (𝔥1, 𝔷1)(𝔥2, 𝔷2) ∈ 𝔉. 𝐻𝑒𝑛𝑐𝑒 𝔉 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑟𝑖𝑛𝑔 𝑜𝑓 𝔎1 × 𝔎2. Theorem 2.8. 𝐼𝑓 𝔓 × 𝔚 𝑖𝑠 𝑎 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of the ring 𝔊1 × 𝔊2, then 𝕐 = {(𝔥, 𝔷) ∈ 𝔊1 × 𝔊2: (𝔓𝑖 + × 𝔚𝑖 +)(𝔥, 𝔷) = [1] 𝑎𝑛𝑑 (𝔓𝑖 − × 𝔚𝑖 −)(𝔥, 𝔷) = [−1], for all i = 1, 2, … , n} is either empty or 𝑎 subring 𝔊1 × 𝔊2. Proof. 𝐼𝑓 𝑎𝑛𝑦 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑛𝑜𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠 𝑡ℎ𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛, 𝑡ℎ𝑒𝑛 𝕐 𝑖𝑠 𝑒𝑚𝑝𝑡𝑦. Let (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝕐. 𝐹or all i = 1, 2, … , n, then (𝔓𝑖 + × 𝔚𝑖 +)[(𝔥1, 𝔷1) − (𝔥2, 𝔷2)]  rmin{(𝔓𝑖 + × 𝔚𝑖 +)(𝔥1, 𝔷1), (𝔓𝑖 + × 𝔚𝑖 +)(𝔥2, 𝔷2)} = 𝑟𝑚𝑖𝑛 {[1], [1]} = [1]. Thus (𝔓𝑖 + × 𝔚𝑖 +)[(𝔥1, 𝔷1) − (𝔥2, 𝔷2)] = [1], 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝕐. And (𝔓𝑖 − × 𝔚𝑖 −)[(𝔥1, 𝔷1) − (𝔥2, 𝔷2)]  rmax{(𝔓𝑖 − × 𝔚𝑖 −)(𝔥1, 𝔷1), (𝔓𝑖 − × 𝔚𝑖 −)(𝔥2, 𝔷2)} = 𝑟𝑚𝑎𝑥 {[−1], [−1]} = [−1]. Thus (𝔓𝑖 − × 𝔚𝑖 −)[(𝔥1, 𝔷1) − (𝔥2, 𝔷2)] = [−1], 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝕐. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 236 https://internationalpubls.com Therefore (𝔥1, 𝔷1) − (𝔥2, 𝔷2) ∈ 𝕐. 𝐴𝑙𝑠𝑜 (𝔓𝑖 + × 𝔚𝑖 +)[(𝔥1, 𝔷1)(𝔥2, 𝔷2)]  rmin {(𝔓𝑖 + × 𝔚𝑖 +)(𝔥1, 𝔷1), (𝔓𝑖 + × 𝔚𝑖 +)(𝔥2, 𝔷2)} = 𝑟𝑚𝑖𝑛 {[1], [1]} = [1]. Thus (𝔓𝑖 + × 𝔚𝑖 +)[(𝔥1, 𝔷1)(𝔥2, 𝔷2)] = [1], 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝕐. And (𝔓𝑖 − × 𝔚𝑖 −)[(𝔥1, 𝔷1)(𝔥2, 𝔷2)]  rmax{(𝔓𝑖 − × 𝔚𝑖 −)(𝔥1, 𝔷1), (𝔓𝑖 − × 𝔚𝑖 −)(𝔥2, 𝔷2)} = 𝑟𝑚𝑎𝑥 {[−1], [−1]} = [−1]. Thus (𝔓𝑖 − × 𝔚𝑖 −) [(𝔥1, 𝔷1)(𝔥2, 𝔷2)] = [−1], 𝑓𝑜𝑟 𝑎𝑙𝑙 (𝔥1, 𝔷1), (𝔥2, 𝔷2) ∈ 𝕐. Therefore (𝔥1, 𝔷1)(𝔥2, 𝔷2) ∈ 𝕐. 𝐻𝑒𝑛𝑐𝑒 𝕐 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑟𝑖𝑛𝑔 𝑜𝑓 𝔊1 × 𝔊2. Theorem 2.9. 𝐿𝑒𝑡 𝔓 𝑏𝑒 𝑎 𝔹𝕍𝕄𝕀𝔽𝕊 of a ring ℨ and 𝔐 be the stronget 𝔹𝕍𝕄𝕀𝔽 relation of ℨ. Then 𝔓 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of ℨ if and only if 𝔐 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of ℨ×ℨ. Proof. Let 𝜚, 𝜐 be in ℨ and 𝜁, 𝜉 be in ℨ. Then (𝜚, 𝜁) and (𝜐, 𝜉) are in ℨ×ℨ. For all i, i = 1, 2, …, n, if 𝔓 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of ℨ, then 𝔚𝑖 +[(𝜚, 𝜁)−(𝜐, 𝜉)] = 𝔚𝑖 +(𝜚−𝜐, 𝜁− 𝜉) = rmin{𝔓i +(𝜚−𝜐), 𝔓i +(𝜁− 𝜉)}  rmin{rmin{𝔓i +(𝜚), 𝔓i +(𝜐)}, rmin{𝔓i +( 𝜁), 𝔓i +(𝜉)}} = rmin{rmin{𝔓i +(𝜚), 𝔓i +(𝜁)}, rmin{𝔓i +(𝜐), 𝔓i +( 𝜉)}} = rmin{𝔚𝑖 +(𝜚, 𝜁), 𝔚𝑖 +(𝜐, 𝜉)}, for all (𝜚, 𝜁), (𝜐, 𝜉) in ℨ×ℨ. And 𝔚𝑖 +[(𝜚, 𝜁)(𝜐, 𝜉)] = 𝔚𝑖 +(𝜚𝜐, 𝜁 𝜉) = rmin{𝔓i +(𝜚𝜐), 𝔓i +(𝜁 𝜉)} rmin{rmin{𝔓i +(𝜚), 𝔓i +(𝜐)}, rmin{𝔓i +( 𝜁), 𝔓i +( 𝜉)}} = rmin{rmin{𝔓i +(𝜚), 𝔓i +(𝜁)}, rmin{𝔓i +(𝜐), 𝔓i +(𝜉)}} = rmin {𝔚𝑖 +(𝜚, 𝜁), 𝔚𝑖 +(𝜐, 𝜉)}, for all (𝜚, 𝜁), (𝜐, 𝜉) in ℨ×ℨ. Also 𝔚𝑖 −[(𝜚, 𝜁)−(𝜐, 𝜉)] = 𝔚𝑖 −(𝜚−𝜐, 𝜁− 𝜉) = rmax{𝔓i −(𝜚−𝜐), 𝔓i −(𝜁− 𝜉)}  rmax{rmax{𝔓i −(𝜚), 𝔓i −(𝜐)}, rmax{𝔓i −(𝜁), 𝔓i −( 𝜉)}} = rmax{rmax{𝔓i −(𝜚), 𝔓i −(𝜁)}, rmax{𝔓i −(𝜐), 𝔓i −( 𝜉)}} = rmax{ 𝔚𝑖 −(𝜚, 𝜁), 𝔚𝑖 −(𝜐, 𝜉)}, for all (𝜚, 𝜁), (𝜐, 𝜉) in ℨ×ℨ. And 𝔚𝑖 −[(𝜚, 𝜁)(𝜐, 𝜉)] = 𝔚𝑖 −(𝜚𝜐, 𝜁𝜉) = rmax{𝔓i −(𝜚𝜐), 𝔓i −(𝜁𝜉)}  rmax{rmax{𝔓i −(𝜚), 𝔓i −(𝜐)}, rmax{𝔓i −(𝜁), 𝔓i −(𝜉)}} = rmax{rmax{𝔓i −(𝜚), 𝔓i −(𝜁)}, rmax{𝔓i −(𝜐), 𝔓i −(𝜉)}} = max{𝔚𝑖 −(𝜚, 𝜁), 𝔚𝑖 −(𝜐, 𝜉)}, for all (𝜚, 𝜁), (𝜐, 𝜉) in ℨ×ℨ. Hence 𝔚 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of ℨ×ℨ. Conversely, assume that 𝔚 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of ℨ×ℨ. For all i = 1, 2, …, n, rmin{𝔓i +(𝜚−𝜐), 𝔓i +(𝜁− 𝜉)} = 𝔚𝑖 +(𝜚−𝜐, 𝜁− 𝜉) = 𝔚𝑖 +[(𝜚, 𝜁)−(𝜐, 𝜉)]  rmin{𝔚𝑖 +(𝜚, 𝜁), 𝔚𝑖 +(𝜐, 𝜉)} = rmin{rmin{𝔓i +(𝜚), 𝔓i +(𝜁)}, rmin{𝔓i +(𝜐), 𝔓i +(𝜉)}}, put 𝜁 = 𝔬 and 𝜉 = 𝔬, where 𝔬 is an first operation identity element of ℨ, then 𝔓i +(𝜚−𝜐)  rmin{𝔓i +(𝜚), 𝔓i +(𝜐)}, for all 𝜚, 𝜐 in ℨ. And rmin{𝔓i +(𝜚𝜐), 𝔓i +(𝜁𝜉)} = 𝔚𝑖 +(𝜚𝜐, 𝜁𝜉) = 𝔚𝑖 +[(𝜚, 𝜁)(𝜐, 𝜉)]  rmin{𝔚𝑖 +(𝜚, 𝜁), 𝔚𝑖 +(𝜐, 𝜉)} = rmin{rmin{𝔓i +(𝜚), 𝔓i +(𝜁)}, rmin{𝔓i +(𝜐), 𝔓i +(𝜉)}}, put 𝜁 = 𝔬 and 𝜉 = 𝔬, where 𝔬 is an first operation identity element of ℨ, then 𝔓i +(𝜚𝜐)  rmin{𝔓i +(𝜚), 𝔓i +(𝜐)}, for all 𝜚, 𝜐 in ℨ. Also rmax{𝔓i −(𝜚−𝜐), 𝔓i −(𝜁− 𝜉)} = 𝔚𝑖 −(𝜚−𝜐, 𝜁− 𝜉) = 𝔚𝑖 −[(𝜚, 𝜁)−(𝜐, 𝜉)]  rmax{𝔚𝑖 −(𝜚, 𝜁), 𝔚𝑖 −(𝜐, 𝜉)} = rmax{rmax{𝔓i −(𝜚), 𝔓i −(𝜁)}, rmax{𝔓i −(𝜐), 𝔓i −(𝜉)}}, put 𝜁 = 𝔬 and 𝜉 = 𝔬, where 𝔬 is an first operation identity element of ℨ, then 𝔓i −(𝜚−𝜐)  rmax{𝔓i −(𝜚), 𝔓i −(𝜐)}, for all 𝜚, 𝜐 in ℨ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 237 https://internationalpubls.com And rmax{𝔓i −(𝜚𝜐), 𝔓i −(𝜁𝜉)} = 𝔚𝑖 −(𝜚𝜐, 𝜁𝜉) = 𝔚𝑖 −[(𝜚, 𝜁)(𝜐, 𝜉)]  rmax{𝔚𝑖 −(𝜚, 𝜁), 𝔚𝑖 −(𝜐, 𝜉)} = rmax{rmax{𝔓i −(𝜚), 𝔓i −(𝜁)}, rmax{𝔓i −(𝜐), 𝔓i −(𝜉)}}, put 𝜁 = 𝔬 and 𝜉 = 𝔬, where 𝔬 is an first operation identity element of ℨ, then 𝔓i −(𝜚𝜐)  rmax{𝔓i −(𝜚), 𝔓i −(𝜐)}, for all 𝜚, 𝜐 in ℨ. Hence 𝔓 is a 𝔹𝕍𝕄𝕀𝔽𝕊ℝ of ℨ. Theorem 2.10. 𝐿𝑒𝑡 𝔓1, 𝔓2, … , 𝔓𝑚 be 𝔹𝕍𝕄𝕀𝔽𝕊𝑠 of a ring ℨ and 𝔐 be the strongest 𝔹𝕍𝕄𝕀𝔽 n- dimensional relation of ℨ. Then 𝔓1, 𝔓2, …, 𝔓𝑚 are 𝔹𝕍𝕄𝕀𝔽𝕊ℝ 𝑜𝑓 ℨ 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝔐 𝑖𝑠 𝑎 𝔹𝕍𝕄𝕀𝔽𝕊ℝ 𝑜𝑓 ℨ×ℨ…×ℨ (m times). Proof. 𝐹𝑟𝑜𝑚 𝑡ℎ𝑒 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 2.9, 𝑡ℎ𝑒 𝑝𝑟𝑜𝑜𝑓 𝑖𝑠 𝑡𝑟𝑖𝑣𝑖𝑎𝑙. 3. Conclusion Properties of transformations of 𝔹𝕍𝕄𝕀𝔽𝕊ℝof a ring have been discussed. The above concepts can be extended into bipolar valued multi I-fuzzy subfield of a field, bipolar interval valued multi fuzzy subspace of a linear space and any other algebraic system. 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