Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

507 https://internationalpubls.com 

Characterization of Bipolar Valued Vague Ideals of a Semiring 
 

K. Anitha1, M. Muthusamy2, K. Arjunan3 
1Research Scholar, Department of Mathematics, Dr. Zakir Husain College, Ilayangudi-630702, Tamilnadu, India. 

Email:anitha.kesav13@gmail.com 

2Department of Mathematics, Dr. Zakir Husain College, Ilayangudi-630702, Tamilnadu, India. 

Email: msamy0207@yahoo.com 
3Department of Mathematics, Alagappa Government Arts college, Karaikudi – 630003, Tamilnadu, India. Email: 

arjunan.karmegam@gmail.com 

 

Article History: 

Received: 05-08-2024 

Revised: 13-09-2024 

Accepted: 21-09-2024 

Abstract:  

Bipolar valued vague ideal of a semiring (BVVI) is described and examined in the present 

paper. Some characterization theorems are introduced in this paper and intersection, product 

and strongest bipolar valued vague relation of bipolar valued vague ideals (BVVI) of a 

semirings are introduced. 

Keywords: Fuzzy subset, vague subset, bipolar valued fuzzy subset, bipolar valued vague 

subset, bipolar valued vague ideals. 

 

1   Introduction 

Zadeh's [17] research from 1965 was the first to create the idea of a fuzzy subset of a set, and the 

mathematical construct known as a fuzzy set is useful for expressing a group of things with ambiguous 

borders. There have been many generalisations of this core idea since it has developed into a lively 

area of research in other fields, including intuitionistic fuzzy sets, interval valued fuzzy sets, 

vague sets, soft sets, etc. Grattan-Guiness Fuzzy membership mapped onto interval and multiple 

valued quantities was discussed in [9].  A fuzzy set extension known as a vague set is a special instance 

of a fuzzy set that depends on the context. D.J.Buehrer and W.L. Gau [8] introduced the vague set. Lee 

presented the idea of bipolar valued fuzzy sets in his article from [8]. Fuzzy sets with membership 

degree ranges that vary from  [0, 1] to [-1, 1] are considered extensions of fuzzy sets. In a bipolar 

valued fuzzy set, elements with a membership degree of 0 are unrelated to the associated property, 

those with a membership degree of (0, 1] are somewhat in agreement with the property, and those with 

a membership degree of [-1, 0) are somewhat in agreement with the implicit counter property. Both 

intuitionistic and bipolar valued fuzzy sets have a similar appearance. They differ from one another, 

but, [10,11].  Azriel Rosenfeld[5] introduced the fuzzy subgroup. [5]. The vague groups were 

introduced by Ranjit Biswas ([13]). A new class of generalised bipolar ambiguous sets has been 

presented by S. Cicily Flora and I. Arockiarani in [7]. Described as bipolar valued fuzzy subgroups of 

a group by Anitha.M.S., et. al.[1] and  The bipolar interval valued fuzzy subgroups of a group were 

defined by A. Balasubramanian. [6] . K. Murugalingam and K. Arjunan talked about the interval-

valued fuzzy subsemiring of a semiring in their discussion in [12]. and  Yasodara.B and 

KE.Sathappan [14] developed bipolar valued multi fuzzy semiring subsemirings. Bipolar valued 

vague  subsemirings of a semiring were defined by Anitha.K., et al [2,3,4].This article makes use of 

the idea of bipolar valued vague ideals (BVVI) of a semiring. 



Communications on Applied Nonlinear Analysis 

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2   Preliminaries 

In this step, we recollect a few key standards and definitions that are likely to be significant for this 

work. 

Definition 2.1 [12] A mapping 𝔜: 𝔊 ⟶ [0, 1] 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑓𝑢𝑧𝑧𝑦 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙  𝑠𝑒𝑡 𝔊.  

Definition 2.2 [5] 𝑇ℎ𝑒 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝔑 = {(𝔶, 𝒱𝔑(𝔶)): 𝔶 ∈ 𝕋} 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 a vague set  

 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑡 𝕋, 𝑤ℎ𝑒𝑟𝑒 𝒱𝔑(𝔶) = [ 𝒯𝔑(𝔶), 1 −  ℱ𝔑(𝔶)], 𝒯𝔑: 𝕋 → [0,1] 𝑖𝑠 𝑎 𝑡𝑟𝑢𝑡ℎ  𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝 map 

and ℱ𝔑: 𝕋 → [0,1]  is a false membership map. 

Example 2.3 𝔑 = { (𝔶, [0.4, 0.7]), (𝔳, [0.5, 0.8]), (𝓀, [0.6, 0.9])} is a vague subset of the Universal 

set 𝕋 = {𝔶, 𝔳, 𝓀}. 

Definition 2.4 [9] 𝑇ℎ𝑒 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝔗 = {(𝔳, 𝔗+(𝔳), 𝔗−(𝔳)): 𝔳 ∈ 𝕋} 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 a bipolar 

𝑣𝑎𝑙𝑢𝑒𝑑 𝑓𝑢𝑧𝑧𝑦 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝕋, 𝑤ℎ𝑒𝑟𝑒 𝔗+: 𝕋 → [0,1] 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝 map         and 

𝔗−: 𝕋 → [−1,0] is a negative membership map. 

Example 2.5 𝔑 = { (𝔶, 0.05, −0.003), (𝔳, 0.04, −0.6), (𝓀, 0.004, −0.07)} is a bipolar valued fuzzy 

subset of the set 𝕋 = { 𝔶, 𝔳, 𝓀 }. 

Definition 2.6 [7] 𝑇ℎ𝑒 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝔈 = {(𝔳,  𝒱𝔈
+(𝔳),  𝒱𝔈

−(𝔳)): 𝔳 ∈ 𝕋} 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 a bipolar 

𝑣𝑎𝑙𝑢𝑒𝑑 𝑣𝑎𝑔𝑢𝑒 𝑠𝑢𝑏𝑠𝑒𝑡 (𝔹𝕍𝕍𝕊𝕊) 𝑜𝑓 𝕋, 𝑤ℎ𝑒𝑟𝑒  𝒱𝔈
+(𝔳) = [  𝒯𝔈

+(𝔳), 1 −  ℱ𝔈
+(𝔳)] and  

 𝒱𝔈
−(𝔳) = [ −1 −  ℱ𝔈

−(𝔳),  𝒯𝔈
−(𝔳)],  𝒯𝔈

+: 𝕋 → [0, 1],  ℱ𝔈
+: 𝕋 → [0, 1],  𝒯𝔈

−: 𝕋 → [−1, 0] and  

 ℱ𝔈
−: 𝕋 → [−1, 0] 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡  𝒯𝔈

+(𝔳) +  ℱ𝔈
+(𝔳)  ≤ 1 𝑎𝑛𝑑 − 1 ≤  ℱ𝔈

−(𝔳) +  𝒯𝔈
−(𝔳).                      

Example 2.7 𝔈 = { (𝔶, [0.03, 0,6], [−0.6, −0.02]), (𝔳, [0.002, 0.04], [−0.05, −0.004]),                           

(𝓀, [0.02, 0.7], [ −0.05, −0.005])} is a 𝔹𝕍𝕍𝕊𝕊 of 𝕋 = {𝔶, 𝔳, 𝓀}. 

Definition 2.8. [4] Let A = 
+

AV , 
−

AV  and B = 
+

BV , 
−

BV  be two 𝔹𝕍𝕍𝕊𝕊s of a set X. We define the 

following relations and operations: 

(i)  [A]  [B] if and only if 
+

AV (u) ≤ 
+

BV (u) and 
−

AV (u) ≥ 
−

BV (u),  uX. 

(ii)  [A] = [B] if and only if 
+

AV (u) = 
+

BV (u) and 
−

AV (u) = 
−

BV (u),  uX. 

(iii) [A][B] = {  u, rmin (
+

AV (u), 
+

BV (u) ), rmax (
−

AV (u), 
−

BV (u) )  / uX }. 

(iv) [A][B] = {  u, rmax(
+

AV (u), 
+

BV (u)), rmin (
−

AV (u), 
−

BV (u)) / uX}. Here rmin (
+

AV (u), 
+

BV (u) 

) = [ min { ),(xt A

+ )(xtB

+
 }, min {1− )(xf A

+
, 1− )(xf B

+
} ], rmax (

+

AV (u), 
+

BV (u) ) = [max { ),(xt A

+ )(xtB

+
 

}, max {1− )(xf A

+
, 1− )(xf B

+
 } ], rmin (

−

AV (u), 
−

BV (u) ) = [min {−1− )(xf A

−
, −1− )(xf B

−
}, min {

),(xt A

− )(xtB

−
} ], rmax (

−

AV (u), 
−

BV (u) ) = [max {−1− )(xf A

−
, −1− )(xf B

−
 }, max { ),(xt A

− )(xtB

−
} ]. 

Definition 2.9. A 𝔹𝕍𝕍𝕊𝕊 𝔈 = 〈 𝒱𝔈
+,  𝒱𝔈

−〉 𝑜𝑓 𝑎 𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔 ℝ is said to be a bipolar valued vague 

subsemiring (𝔹𝕍𝕍𝕊𝕊ℝ) 𝑜𝑓 ℝ if  

(i)  𝒱𝔈
+(𝔳 + 𝔥)  rmin{ 𝒱𝔈

+(𝔳),  𝒱𝔈
+(𝔥)}, 



Communications on Applied Nonlinear Analysis 

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(ii)  𝒱𝔈
+(𝔳𝔥)  rmin{ 𝒱𝔈

+(𝔳),  𝒱𝔈
+(𝔥)}, 

(iii)  𝒱𝔈
−(𝔳 + 𝔥)  rmax{ 𝒱𝔈

−(𝔳),  𝒱𝔈
−(𝔥)}, 

(iv)  𝒱𝔈
−(𝔳𝔥) rmax{ 𝒱𝔈

−(𝔳),  𝒱𝔈
−(𝔥)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝔳, 𝔥 ∈ ℝ. 

Example 2.10. Let R = Z3 = {0, 1, 2} be a semiring in terms of standard addition and multiplication. 

Then A = { <0, [0.5, 0.7], [− 0.8, − 0.5]>, <1, [0.4, 0.6], [−0.7, −0.4]>, <2, [0.4, 0.6], [−0.7, −0.4] >} 

is a BVVSSR of R. 

3  BIPOLAR VALUED VAGUE IDEALS: 

This section introduced bipolar valued vague ideals (BVVIs) and looked at their characteristics. 

Definition 3.1. A 𝔹𝕍𝕍𝕊𝕊 𝔈 = 〈 𝒱𝔈
+,  𝒱𝔈

−〉 𝑜𝑓 𝑎 𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔 ℝ is said to be a bipolar valued vague ideal 

(𝔹𝕍𝕍𝕀) 𝑜𝑓 ℝ if  

(i)  𝒱𝔈
+(𝔳 + 𝔥)  rmin{ 𝒱𝔈

+(𝔳),  𝒱𝔈
+(𝔥)}, 

(ii)  𝒱𝔈
+(𝔳𝔥)  rmax{ 𝒱𝔈

+(𝔳),  𝒱𝔈
+(𝔥)}, 

(iii)  𝒱𝔈
−(𝔳 + 𝔥)  rmax{ 𝒱𝔈

−(𝔳),  𝒱𝔈
−(𝔥)}, 

(iv)  𝒱𝔈
−(𝔳𝔥) rmin{ 𝒱𝔈

−(𝔳),  𝒱𝔈
−(𝔥)}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝔳, 𝔥 ∈ ℝ. 

Theorem 3.2. Let 𝔈 = 〈 𝒱𝔈
+,  𝒱𝔈

−〉 be a 𝔹𝕍𝕍𝕀 of a semiring ℝ.  

(i) If  𝒱𝔈
+(𝔳 + 𝔥) = [0] then either   𝒱𝔈

+(𝔳)= [0] or  𝒱𝔈
+(𝔥) = [0] for 𝔳, 𝔥 ∈ ℝ  

(ii) If  𝒱𝔈
+(𝔳𝔥) = [0] then either  𝒱𝔈

+(𝔳) = [0] or  𝒱𝔈
+(𝔥)= [0] for 𝔳, 𝔥 ∈ ℝ  

(iii) If  𝒱𝔈
−(𝔳 + 𝔥)= [0] then either  𝒱𝔈

−(𝔳) = [0] or  𝒱𝔈
−(𝔥) = [0] for  𝔳, 𝔥 ∈ ℝ 

(iv) If  𝒱𝔈
−(𝔳𝔥) = [0] then either   𝒱𝔈

−(𝔳) = [0] or  𝒱𝔈
−(𝔥) = [0] for  𝔳, 𝔥 ∈ ℝ 

Proof. Let 𝔳, 𝔥 ∈ ℝ. (i) By the definition 𝒱𝔈
+(𝔳 + 𝔥)  rmin {  𝒱𝔈

+(𝔳),  𝒱𝔈
+(𝔥) } which implies that [0] 

 rmin {𝒱𝔈
+(𝔳),  𝒱𝔈

+(𝔥)}. Therefore either 𝒱𝔈
+(𝔳) = [0] or  𝒱𝔈

+(𝔥) = [0]. (ii) By the definition   𝒱𝔈
+(𝔳𝔥)  

rmax { 𝒱𝔈
+(𝔳),  𝒱𝔈

+(𝔥)} which implies that [0]  rmax { 𝒱𝔈
+(𝔳),  𝒱𝔈

+(𝔥)}. Therefore either  𝒱𝔈
+(𝔳) = 

[0] or  𝒱𝔈
+(𝔥)= [0]. (iii) By the definition 𝒱𝔈

−(𝔳 + 𝔥)≤ rmax                              {𝒱𝔈
−(𝔳),  𝒱𝔈

−(𝔥)} which 

implies that [0] ≤ rmax  {𝒱𝔈
−(𝔳),  𝒱𝔈

−(𝔥)}. Therefore either  𝒱𝔈
−(𝔳) = [0] or  𝒱𝔈

−(𝔥) = [0] . (iv) By the 

definition 𝒱𝔈
−(𝔳𝔥) ≤ rmin  {𝒱𝔈

−(𝔳),  𝒱𝔈
−(𝔥)} which implies that [0] ≤ rmin {𝒱𝔈

−(𝔳),  𝒱𝔈
−(𝔥)}. Therefore 

either  𝒱𝔈
−(𝔳) = [0] or  𝒱𝔈

−(𝔥) = [0]. 

Theorem 3.3. 𝐼𝑓 𝔈 = 〈 𝒱𝔈
+,  𝒱𝔈

−〉 𝑖𝑠 𝑎 𝔹𝕍𝕍𝕀 𝑜𝑓 𝑎 𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔 ℜ, 𝑡ℎ𝑒𝑛 ℋ = {𝔬ℜ / 𝒱𝔈
+(𝔬) = [1],  𝒱𝔈

−(𝔬) 

= [−1] } is either a subideal or empty of ℜ. 

Proof. There is no 𝔬ℜ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝒱𝔈
+(𝔬) = [1] and 𝒱𝔈

−(𝔬) = [−1], then ℋ is empty. If 𝔬 and 𝔰 in ℋ, 

then 𝒱𝔈
+(𝔬 +  𝔰)  rmin{ 𝒱𝔈

+(𝔬), 𝒱𝔈
+(𝔰)} = rmin{[1], [1]} = [1]. Thus  𝒱𝔈

+(𝔬+ 𝔰) =[1]. And 𝒱𝔈
+(𝔬𝔰)  

rmax{ 𝒱𝔈
+(𝔬), 𝒱𝔈

+(𝔰)} = rmax{[1], [1]}= [1]. So,  𝒱𝔈
+(𝔬𝔰) =[1]. Also 𝒱𝔈

−(𝔬 + 𝔰)  rmax{ 𝒱𝔈
−(𝔬), 𝒱𝔈

−(𝔰)}= 

rmax{[−1],[−1]} = [−1]. That is,  𝒱𝔈
−(𝔬+ 𝔰) = [−1]. And  𝒱𝔈

−(𝔬𝔰)  rmin{ 𝒱𝔈
−(𝔬),  𝒱𝔈

−(𝔰)} = 

rmin{[−1], [−1]} = [−1]. So,  𝒱𝔈
−(𝔬𝔰) = [−1]. That is 𝔬+ 𝔰, 𝔬𝔰 ℋ. Hence ℋ 𝑖𝑠 𝑎 𝑠𝑢𝑖𝑑𝑒𝑎𝑙 𝑜𝑓 ℜ.  



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 2 (2025) 

 

510 https://internationalpubls.com 

Theorem 3.4. If  𝕰 = 〈 𝓥𝕰
+,  𝓥𝕰

−〉 and 𝕳 = 〈 𝓥𝕳
+,  𝓥𝕳

−〉 are two 𝔹𝕍𝕍𝕀s of a ring R, then their intersection 

𝕰  𝕳 is a 𝔹𝕍𝕍𝕀 of ℝ . 

Proof. Let Ҫ = 𝔈  ℌ and let 𝔳, 𝔥 ∈ ℝ. Now  𝒱Ҫ
+(𝔳 + 𝔥) = rmin { 𝒱𝔈

+(𝔳 + 𝔥), 𝒱ℌ
+(𝔳 + 𝔥) }  

rmin{rmin{ 𝒱𝔈
+(𝔳),  𝒱𝔈

+(𝔥)}, rmin{ 𝒱ℌ
+(𝔳),  𝒱ℌ

+(𝔥) }}  rmin{rmin{ 𝒱𝔈
+(𝔳),  𝒱ℌ

+(𝔳)}, rmin{ 𝒱𝔈
+(𝔥), 

 𝒱ℌ
+(𝔥)}} = rmin{ 𝒱Ҫ

+(𝔳),  𝒱Ҫ
+(𝔥)}. Therefore  𝒱Ҫ

+(𝔳 + 𝔥)  rmin{ 𝒱Ҫ
+(𝔳),  𝒱Ҫ

+(𝔥), for all 𝔳, 𝔥 ∈ ℝ. 

And  𝒱Ҫ
+(𝔳𝔥)= rmin{ 𝒱𝔈

+(𝔳𝔥), 𝒱ℌ
+(𝔳𝔥)}  rmin{rmax{ 𝒱𝔈

+(𝔳), 𝒱𝔈
+(𝔥)}, rmax{ 𝒱ℌ

+(𝔳), 𝒱ℌ
+(𝔥)}}  

rmax{rmin{ 𝒱ℌ
+(𝔳),  𝒱ℌ

+(𝔥)}, rmin{ 𝒱ℌ
+(𝔳),  𝒱ℌ

+(𝔥)}} = rmax{ 𝒱Ҫ
+(𝔳),  𝒱Ҫ

+(𝔥)}. Therefore   𝒱Ҫ
+(𝔳𝔥)  

 rmax{ 𝒱Ҫ
+(𝔳),  𝒱Ҫ

+(𝔥)}, for all   𝒱Ҫ
+(𝔳 + 𝔥). Also  𝒱Ҫ

−(𝔳 + 𝔥) = rmax{ 𝒱𝔈
−(𝔳 + 𝔥), 𝒱ℌ

−(𝔳 + 𝔥)}  

rmax{rmax{ 𝒱𝔈
−(𝔳),  𝒱𝔈

−(𝔥)}, rmax{ 𝒱ℌ
−(𝔳),  𝒱ℌ

−(𝔥)}} rmax{rmax{  𝒱𝔈
−(𝔳),  𝒱ℌ

−(𝔳) , rmax{ 𝒱𝔈
−(𝔥), 

 𝒱ℌ
−(𝔥)}} = rmax{ 𝒱Ҫ

−(𝔳),  𝒱Ҫ
−(𝔥)}. Therefore  𝒱Ҫ

−(𝔳 + 𝔥)  rmax{ 𝒱Ҫ
−(𝔳),  𝒱Ҫ

−(𝔥)}, for all𝔳, 𝔥 ∈ ℝ. 

And  𝒱Ҫ
−(𝔳𝔥)  = rmax{ 𝒱𝔈

−(𝔳𝔥), 𝒱ℌ
−(𝔳𝔥)}  rmax{rmin{ 𝒱𝔈

−(𝔳),  𝒱𝔈
−(𝔥)}, rmin { 𝒱ℌ

−(𝔳),  𝒱ℌ
−(𝔥)}}  

rmin{rmax{ 𝒱𝔈
−(𝔳),  𝒱ℌ

−(𝔳) }, rmax{  𝒱𝔈
−(𝔥),  𝒱ℌ

−(𝔥)}}= rmin{  𝒱Ҫ
−(𝔳),  𝒱Ҫ

−(𝔥)}. Therefore 
 
 𝒱Ҫ

−(𝔳𝔥) 

 rmin{ 𝒱Ҫ
−(𝔳),  𝒱Ҫ

−(𝔥)}, for all  𝔳, 𝔥 ∈ ℝ. Hence   𝔈  ℌis a 𝔹𝕍𝕍𝕀  of Ҫ = 𝔈  ℌ. 

Theorem 3.5. The intersection of a family of 𝔹𝕍𝕍𝕀s of a semiring ℝ is a 𝔹𝕍𝕍𝕀 of ℝ. 

Proof. The proof follows from the Theorem 3.4. 

Theorem 3.6. If   𝕰 = 〈 𝓥𝕰
+,  𝓥𝕰

−〉 and 𝕳 = 〈 𝓥𝕳
+,  𝓥𝕳

−〉 are two 𝔹𝕍𝕍𝕀s of a semiring ℝ, then their union 

𝕰  𝕳 need not be a 𝔹𝕍𝕍𝕀 of ℝ. 

Proof. It can be easily proved. 

Remark 3.7. If one is contained other, then the union is a 𝔹𝕍𝕍𝕀s of a semiring ℝ. 

Definition 3.8. Let 𝔈 = 〈 𝒱𝔈
+,  𝒱𝔈

−〉 and 𝔓 = 〈 𝒱𝔓
+,  𝒱𝔓

−〉 be any two 𝔹𝕍𝕍𝕊𝕊𝑠 of sets ℜ1 and ℜ2. The 

product of 𝔈 𝑎𝑛𝑑 𝔓, denoted by 𝔈 × 𝔓, is defined as 𝔈 × 𝔓 = {(𝔳, 𝔥),  𝒱𝔈×𝔓
+ (𝔳, 𝔥),  𝒱𝔈×𝔓

− (𝔳, 𝔥) / for 

all 𝔳 ∈ ℜ1 and 𝔥 ∈ ℜ2 }, where 𝒱𝔈×𝔓
+ (𝔳, 𝔥) = rmin{ 𝒱𝔈

+(𝔳),  𝒱𝔓
+(𝔥)} and  𝒱𝔈×𝔓

− (𝔳, 𝔥) = 

rmax{ 𝒱𝔈
−(𝔳),  𝒱𝔓

−(𝔥)} for all 𝔳 ∈ ℜ1 and 𝔥 ∈ ℜ2. 

Theorem 3.9. If A = 
+

AV , 
−

AV  and B = 
+

BV , 
−

BV  are any two 𝔹𝕍𝕍𝕀s of the semirings R1 and R2 

respectively, then A×B =  
+

BAV , 
−

BAV   is a 𝔹𝕍𝕍𝕀 of R1×R2.  

Proof. Let x1, x2 be in R1, y1 and y2 be in R2. Then ( x1, y1 ) and ( x2, y2 ) are in R1×R2. Now, 
+

BAV [(x1, 

y1)+(x2, y2)] = 
+

BAV (x1+x2, y1+y2) = rmin{
+

AV (x1+x2), 
+

BV (y1+y2)}  rmin{rmin          {
+

AV (x1), 
+

AV

(x2)}, rmin{
+

BV (y1), 
+

BV (y2)}} = rmin{rmin{
+

AV (x1), 
+

BV (y1)}, rmin{
+

AV (x2), 
+

BV (y2)}} = rmin{
+

BAV

(x1, y1), 
+

BAV (x2, y2)}. Therefore 
+

BAV [ (x1, y1)+(x2, y2)]  rmin                      {
+

BAV (x1, y1), 
+

BAV (x2, 

y2)}. And 
+

BAV [(x1, y1)(x2, y2)] = 
+

BAV (x1x2, y1y2) = rmin{
+

AV (x1x2), 
+

BV (y1y2)}  rmin{rmax{
+

AV (x1), 

+

AV (x2)}, rmax{
+

BV (y1), 
+

BV (y2)}} = rmax{rmin{
+

AV (x1), 
+

BV (y1)}, rmin{
+

AV (x2), 
+

BV (y2)}} = rmax{

+

BAV (x1, y1), 
+

BAV (x2, y2)}.Therefore 
+

BAV [(x1, y1) (x2, y2)]  rmax{
+

BAV (x1, y1), 
+

BAV (x2, y2)}. Also 

−

BAV [(x1, y1)+(x2, y2)] = 
−

BAV (x1+x2,                   y1+y2) = rmax{
−

AV (x1+x2), 
−

BV (y1+y2)} ≤ rmax{rmax{



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 2 (2025) 

 

511 https://internationalpubls.com 

−

AV (x1), 
−

AV (x2)}, rmax{ −

BV (y1),             (y2)}} = rmax{rmax{ −

AV (x1), 
−

BV (y1)}, rmax{ −

AV (x2), 
−

BV

(y2)}} = rmax{
−

BAV (x1, y1), 
−

BAV (x2, y2)}. Therefore 
−

BAV [(x1, y1)+(x2, y2)] ≤ rmax{
−

BAV (x1, y1), 
−

BAV

(x2, y2)}. And            
−

BAV [(x1, y1)(x2, y2)] = 
−

BAV (x1x2, y1y2) = rmax{
−

AV (x1x2), 
−

BV (y1y2)} ≤ 

rmax{rmin{
−

AV (x1), 
−

AV (x2)}, rmin{
−

BV (y1), 
−

BV (y2)}} = rmin{rmax{
−

AV (x1), 
−

BV (y1)}, rmax{
−

AV (x2), 

−

BV (y2)}}   = rmin{
−

BAV (x1, y1), 
−

BAV (x2, y2)}. Therefore 
−

BAV [(x1, y1)(x2, y2)] ≤ rmin{
−

BAV (x1, y1),               

−

BAV (x2, y2) }. Hence A×B is a 𝔹𝕍𝕍𝕀 of R1×R2. 

Theorem 3.10. A product of 𝔹𝕍𝕍𝕀s of the semirings is also a 𝔹𝕍𝕍𝕀 in . 

Proof. From the Theorem 3.9, the proof follows. 

Definition 3.11. Let A = 
+

AV , 
−

AV  be a 𝔹𝕍𝕍𝕊𝕊 in a set S, the strongest 𝔹𝕍𝕍 relation on S, that is a 

𝔹𝕍𝕍 relation on A is V = {(x, y), +

VV (x, y), −

VV (x, y) / x, yS} given by +

VV (x, y) = rmin{ 
+

AV (x), 

+

AV (y) } and −

VV (x, y) = rmax{
−

AV (x), 
−

AV (y)},  x, yS. 

Theorem 3.12. Let A = 
+

AV , 
−

AV  be a 𝔹𝕍𝕍𝕊𝕊 of a semiring R and V =  +

VV , −

VV  be the strongest 

𝔹𝕍𝕍 relation of R. Then A is a 𝔹𝕍𝕍𝕀 of R  V is a 𝔹𝕍𝕍𝕀 of R×R. 

Proof. Suppose that A is a 𝔹𝕍𝕍𝕀 of R. Then for any x = (x1, x2), y = (y1, y2) are in R×R. Now +

VV

(x+y) = +

VV [(x1, x2)+(y1, y2)] = +

VV (x1+y1, x2+y2) = rmin{
+

AV (x1+y1), 
+

AV (x2+y2)}  rmin{rmin{
+

AV

(x1), 
+

AV (y1)}, rmin{
+

AV (x2), 
+

AV (y2)}} = rmin{rmin{
+

AV (x1), 
+

AV (x2)}, rmin               {
+

AV (y1), 
+

AV

(y2)}} = rmin{ +

VV (x1, x2), 
+

VV (y1, y2)} = rmin{ +

VV (x), +

VV (y)}. Therefore              +

VV (x+y)  rmin{

+

VV (x), +

VV (y)},  x, yR×R. And +

VV (xy) = +

VV [(x1, x2)(y1, y2)] =                 +

VV (x1y1, x2y2) = rmin{

+

AV (x1y1), 
+

AV (x2y2)}  rmin{rmax{
+

AV (x1), 
+

AV (y1)}, rmax{
+

AV (x2), 
+

AV (y2)}}  rmax{rmin{
+

AV (x1), 

+

AV (x2)}, rmin{
+

AV (y1), 
+

AV (y2)}} = rmax{ +

VV (x1, x2),                   
+

VV (y1, y2)}= rmax{ +

VV (x), +

VV (y)}. 

Therefore +

VV (xy)  rmax{ +

VV (x), +

VV (y)},  x, yR×R. Also we have −

VV (x+y) = −

VV [(x1, x2)+(y1, 

y2)] = −

VV (x1+y1, x2+y2) = rmax{
−

AV (x1+y1),              
−

AV (x2+y2)}  rmax{rmax{
−

AV (x1), 
−

AV (y1)}, 

rmax{
−

AV (x2), 
−

AV (y2)}} = rmax{rmax {
−

AV (x1), 
−

AV (x2)}, rmax{
−

AV (y1), 
−

AV (y2)}}= rmax{ −

VV (x1, 

x2), 
−

VV (y1, y2)} = rmax{ −

VV (x), −

VV (y)}. Therefore −

VV (x+y)  rmax{ −

VV (x), −

VV (y)},  x, yR×R. 

And −

VV (xy) = −

VV [(x1, x2)(y1, y2)] = −

VV (x1y1, x2y2) = rmax{
−

AV (x1y1), 
−

AV (x2y2)}  rmax{rmin{
−

AV

(x1), 
−

AV (y1)}, rmin{
−

AV (x2), 
−

AV (y2)}} = rmin{rmax{
−

AV (x1), 
−

AV (x2)}, rmax{
−

AV (y1), 
−

AV (y2)}}= 

rmin{ −

VV (x1, x2),                 
−

VV (y1, y2) } = rmin{ −

VV (x), −

VV (y)}. Therefore −

VV (xy)  rmin{ −

VV (x), 

−

VV (y)},  x, yR×R. This proves that V is a 𝔹𝕍𝕍𝕀 of R×R. Conversely assume that V is a 𝔹𝕍𝕍𝕀 of 

R×R, then for any x = (x1, x2) and y = (y1, y2) are in R×R, we have rmin{
+

AV (x1+y1), 
+

AV (x2+y2)} = 

+

VV (x1+y1, x2+y2) = +

VV [(x1, x2)+(y1, y2)] = +

VV (x+y)  rmin{ +

VV (x), +

VV (y)} = rmin                        {

+

VV (x1, x2), 
+

VV (y1, y2)} = rmin{rmin{
+

AV (x1), 
+

AV (x2)}, rmin{
+

AV (y1), 
+

AV (y2)}}. If                 
+

AV



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 2 (2025) 

 

512 https://internationalpubls.com 

(x1+y1)  +

AV (x2+y2), we get, +

AV (x1+y1)  rmin{ +

AV (x1), 
+

AV (y1)},  x1, y1R. And rmin                   {

+

AV (x1y1), 
+

AV (x2y2)} = +

VV (x1y1, x2y2) = +

VV [(x1, x2)(y1, y2)] = +

VV (xy)  rmax{ +

VV (x),             +

VV

(y)} = rmax{ +

VV (x1, x2), 
+

VV (y1, y2)} = rmax{rmin{
+

AV (x1), 
+

AV (x2)}, rmin{
+

AV (y1),              
+

AV (y2)}}. 

If +

AV (x1y1)  +

AV (x2y2), we get +

AV (x1y1)  rmax{ +

AV (x1), 
+

AV (y1)},  x1, y1R. Also we have rmax{

−

AV (x1+y1), 
−

AV (x2+y2)} = −

VV (x1+y1, x2+y2) = −

VV [(x1, x2)+(y1, y2)]                 = −

VV (x+y)  rmax{ −

VV

(x), −

VV (y)} = rmax{ −

VV (x1, x2), 
−

VV (y1, y2)}= rmax{rmax{
−

AV (x1), 
−

AV (x2)}, rmax{
−

AV (y1), 
−

AV (y2)}}. 

If 
+

AV (x1+y1)  
+

AV (x2+y2), we get 
−

AV (x1+y1)  rmax               {
−

AV (x1), 
−

AV (y1)},  x1, y1R. And 

rmax{ −

AV (x1y1), 
−

AV (x2y2) } = −

VV (x1y1, x2y2) = −

VV [(x1, x2)(y1, y2)] = −

VV (xy)  rmin{ −

VV (x), −

VV (y)} 

= rmin{ −

VV (x1, x2), 
−

VV (y1, y2) }= rmin{rmax            {
−

AV (x1), 
−

AV (x2)}, rmax{
−

AV (y1), 
−

AV (y2)}}. If 

+

AV (x1y1)  
+

AV (x2y2), we get 
−

AV (x1y1)  rmin{
−

AV (x1), 
−

AV (y1)}, x1, y1R. Hence A is a 𝔹𝕍𝕍𝕀 of 

R. 

4  CONCLUSION 

 The concept of characterization of bipolar valued vague ideal a semiring is discussed in this section  

and  Bipolar valued vague ideal a semiring properties have been introduced. These ideas are applied 

to further research in the creation of bipolar valued vague semiring subsemirings. As a result, our 

upcoming research will examine some of the qualities based on the idea of bipolar valued vague ideals 

with translations. 

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Vol 32 No. 2 (2025) 

 

513 https://internationalpubls.com 

[11] K.M.Lee, Comparison of interval valued fuzzy sets, intuitionistic fuzzy sets and bipolar valued fuzzy sets. J. fuzzy 

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