Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

241 
https://internationalpubls.com 

A Study of Bipolar Fuzzy Prime Ideals of a Lattice 
 

1Venkata Kalyani U, 2B.V.S.N. Hari Prasad, 3Eswarlal. T, 4Aiyared Iampan 
1Assistant Professor, Department of Mathematics and Statistics, Vignan’s Foundation for Science, Technology and 

Research, Vadlamudi, Guntur-522213, India. Email: u.v.kalyani@gmail.com 

2Professor, Department of Mathematics, Vasireddy Venkatadri Institute of Technology, Nambur -522508, India. 

Email: bvsnhariprasad@gmail.com 

3Associate Professor, Department of Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, 

AP, India. Email: eswarlal@kluniversity.in 

4Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand. 

Email: aiyared.ia@up.ac.th 

 

Article History: 

Received: 15-10-2024 

Revised: 02-12-2024 

Accepted: 10-12-2024 

Abstract:  

This study explores the investigation of Bipolar Fuzzy Prime Ideals (BFPI) in lattices. 

We provide a detailed exploration of their properties, characterizations, and associated 

homomorphisms. 

Introduction: Fuzzy set theory, introduced by Zadeh L.A., is grounded in the concept of 

membership functions where each element in a set is assigned a membership degree 

ranging between 0 and 1. Although this model effectively combines supporting and 

opposing evidence for element membership, it lacks explicit representation of the 

uncertainty or dual nature of these evidence. To address this limitation, Gau and Buehrer 

introduced the concept of vague sets, characterized by two functions: one for membership 

and another for non-membership, where their sum does not exceed one. 

Further contributions to fuzzy set theory came from Atanassov's intuitionistic fuzzy sets 

and Bustince and Burillo's work showing their mathematical equivalence to vague sets. 

The dual-function approach of vague sets has been applied extensively in decision-

making, control systems, and fault diagnosis. Lattice theory has also benefited from these 

advancements, with Ajmal and Thomas pioneering fuzzy sublattice theory, and later 

works exploring intuitionistic fuzzy lattices and vague lattices. Bipolar fuzzy sets (BFS), 

introduced by Lee K.M., extended fuzzy sets by incorporating dual notions of positive 

and negative membership values within a range of [-1, 1]. This extension enables 

interpretations of bipolar information, making BFS a valuable tool in decision-making 

and information processing. 

Objectives: Introduction of Bipolar fuzzy prime ideals of a Lattice, study of their 

characterizations and associated homomorphisms. 

Keywords: Bipolar fuzzy ideal, Bipolar fuzzy prime ideal, Bipolar fuzzy 

homomorphism, Bipolar fuzzy magnified translation. 

 

1. Introduction 

Fuzzy set theory, introduced by Zadeh L.A. [1], is grounded in the concept of membership functions 

where each element in a set is assigned a membership degree ranging between 0 and 1. Although this 

model effectively combines supporting and opposing evidence for element membership, it lacks 

mailto:bvsnhariprasad@gmail.com
mailto:eswarlal@kluniversity.in


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

242 
https://internationalpubls.com 

explicit representation of the uncertainty or dual nature of these evidences. To address this limitation, 

Gau and Buehrer [2] introduced the concept of vague sets, characterized by two functions: one for 

membership and another for non-membership, where their sum does not exceed one. 

Further contributions to fuzzy set theory came from Atanassov's intuitionistic fuzzy sets [5]and 

Bustince and Burillo's [3]work showing their mathematical equivalence to vague sets. The dual-

function approach of vague sets has been applied extensively in decision-making, control systems, and 

fault diagnosis. Lattice theory has also benefited from these advancements, with Ajmal and Thomas 

pioneering fuzzy sublattice theory, and later works exploring intuitionistic fuzzy lattices and vague 

lattices. Bipolar fuzzy sets (BFS), introduced by Lee K.M.[6], extended fuzzy sets by incorporating 

dual notions of positive and negative membership values within a range of [-1, 1]. This extension 

enables interpretations of bipolar information, making BFS a valuable tool in decision-making and 

information processing. 

In particular, Ajmal. N and Thomas.K.V [7] both explored theory of (FL)fuzzy sublattice and 

introduced the idea of fuzzy sets to lattice theory. After then, in 2011, Thomas.K.V and Nair L.S.[4] 

presented idea of intuitionistic fuzzy lattices (IFLs). In 2017, Milles S [13] investigated the 

characterization of IFIs and IFFs based on lattice operations. Rao.R.P [15] later researched rough 

vague lattices in 2019. Nageswara Rao.B., RamaKrishna N and Eswarlal.T [14] introduced vague 

lattices(VL) in 2020. The principal IFI and IFF on a lattice were the subject of Boudaoud.S, Zedam.L 

and Milles S [10] study in 2020. Milles.S.[11] as well as studied the lattice of (IFT)intuitionistic fuzzy 

topologies produced by intuitionistic fuzzy relations in 2020. On residual lattices, Zhang H and 

Qingguo.Li [12] researched the (IFF)intuitionistic fuzzy filter theory. Bipolar vague cosets [9], 

Homomorphism on bipolar vague normal groups [8] were studied by Venkata Kalyani U in 2020. 

Later, Bipolar fuzzy sublattices were introduced in 2023 by Venkata Kalyani U and studied bipolar 

fuzzy ideals of a lattice [16]. Venkata Kalyani U studied the bipolar magnified fuzzy translation of a 

lattice [17] in 2024. Now, in this paper we introduce the theory of bipolar fuzzy prime ideals(BFPI) 

and their characterization using level sets and homomorphism of BFPI of a lattice. 

2. Preliminaries 

Definition 2.1[7]: “A poset (𝔏, ≤ ) is called a lattice if Sup{ 𝑝, 𝑞 } (also denoted by 𝑝 ∨ 𝑞 ) and Inf{𝑝, 𝑞} 

(also denoted by 𝑝 ∧ 𝑞) exists for every pair of elements 𝑝, 𝑞 in 𝔏.” 

Definition 2.2[1]: “Let ϝ be any non-empty set. A mapping 𝜓: ϝ → [0,1] is called a fuzzy subset of ϝ.” 

Definition 2.3[1]: “Let 𝜓: ϝ → [0,1] be any FS. Then the set {𝜓(𝑝)/𝑝 ∈ ϝ} is called the image of 𝜓 

and is denoted by Im(𝜓). For 𝑡 ∈ [0,1], 𝜓𝑡 = {𝑝 ∈ ϝ/𝜓(𝑝) ≥ 𝑡} is called a level subset of 𝜓.” 

Definition 2.4[2]: “A vague set 𝜅 in the universe of discourse ϝ is characterized by two 𝑀ship  functions 

given by 

(i)A truth 𝑀ship function 𝑡𝜅: ϝ → [0,1] and 

(ii) A false 𝑀ship  function 𝑓𝜅: ϝ → [0,1], 

where 𝑡𝜅(𝑝) is a lower bound of the grade of 𝑀ship  of 𝑝 derived from the evidence for 𝑝 and 𝑓𝜅(𝑝) is 

a lower bound on the negation of 𝑝 derived from the evidence against 𝑝, with 𝑡𝜅(𝑝) + 𝑓𝜅(𝑝) ≤ 1 ". 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

243 
https://internationalpubls.com 

We give below a formation of the definition of vague set in the following way, which makes Atanassov, 

K.T.s intuitionistic fuzzy sets and Gau, W.L. and Buehrer, D.J.[2] vague sets in a mathematically 

equivalent form. 

Definition 2.5[2]: “Let 𝜅 be a vague set of a universe ϝ with true 𝑀ship  function 𝑡𝜅 and false 𝑀ship  

function 𝑓𝜅. For 𝛼, Υ ∈ [0,1] with 𝛼 ≤ Υ, the (𝛼, Υ )- cut or vague cut of a vague set 𝜅 is the crisp 

subset of ϝ is given by 𝜅(𝛼,Υ) = {𝑝 ∈ ϝ/𝑉𝜅(𝑝) ≥ [𝛼, Υ]} 

i.e., 𝜅(𝛼,Υ) = {𝑝 ∈ ϝ/𝑡𝜅(𝑝) ≥ 𝛼 and 1 − 𝑓𝜅(𝑝) ≥ Υ}.” 

Definition 2.6[2]: “The 𝛼-cut, 𝜅𝛼 of the vague set 𝜅 is the ( 𝛼, 𝛼 )-cut of 𝜅 and hence given by 𝜅𝛼 =

{𝑝 ∈ ϝ/𝑡𝜅(𝑝) ≥ 𝛼}.” 

Definition 2.7[6]: “Suppose ϝ be a universal set. A (𝐵𝐹𝑆) bipolar fuzzy set 𝔹 in ϝ is an object having 

the form 𝔹 = {< ℏ, 𝔹𝑃(ℏ), 𝔹𝑁(ℏ) >/ℏ ∈ ϝ} where 𝔹𝑃: ϝ → [0,1] and 𝔹𝑁: ϝ → [−1,0] are a positive 

and negative 𝑀ship  functions, respectively.” 

Definition 2.8[6]: “ Let ϝ be a nonempty set, and let 𝔹𝜗 , 𝔹𝜔 ∈ 𝐵𝑃𝐹𝑆(ϝ). 

(i) 𝔹𝜗 is a subset of 𝔹𝜔, denoted by 𝔹𝜗 ⊆ 𝔹𝜔, if for each ℏ ∈ ϝ, 𝔹𝜗
𝑃(ℏ) ≤ 𝔹𝜔

𝑃 (ℏ) and 𝔹𝜗
𝑁(ℏ) ≥ 𝔹𝜔

𝑁 (ℏ). 

(ii) The complement of 𝔹𝜗, denoted by 𝔹𝜗
𝑐 = ((𝔹𝜗

𝑐 )𝑁, (𝔹𝜗
𝑐 )𝑃), is a 𝐵𝐹𝑆 in ϝ defined as: for each ℏ ∈

ϝ, 𝔹𝜗
𝑐 (ℏ) = (−1 − 𝔹𝜗

𝑁(ℏ),1 − 𝔹𝜗
𝑃(ℏ)), i.e., (𝔹𝜗

𝑐 )𝑃(ℏ) = 1 − 𝔹𝜗
𝑁(ℏ), (𝔹𝜗

𝑐 )𝑁(ℏ) = −1 − 𝔹𝜗
𝑁(ℏ). 

(iii) The intersection of 𝔹𝜗 and 𝔹𝜔, denoted by 𝔹𝜗 ∩ 𝔹𝜔, is a 𝐵𝐹𝑆 in ϝ defined as: for each ℏ ∈ ϝ, 

(𝔹𝜗 ∩ 𝔹𝜔)(ℏ) = (𝔹𝜗
𝑁(ℏ) ∨ 𝔹𝜔

𝑁 (ℏ), 𝔹𝜗
𝑃(ℏ) ∧ 𝔹𝜔

𝑃 (ℏ)). 

(iv) The union of 𝔹𝜗 and 𝔹𝜔, denoted by 𝔹𝜗 ∪ 𝔹𝜔, is a 𝐵𝐹𝑆 in ϝ defined as: for each ℏ ∈

ϝ, (𝔹𝜗 ∪ 𝔹𝜔)(ℏ) = (𝔹𝜗
𝑁(ℏ) ∧ 𝔹𝜔

𝑁 (ℏ), 𝔹𝜗
𝑃(ℏ) ∨ 𝔹𝜔

𝑃 (ℏ)). " 

Definition:2.9[17]: “Let 𝔅 =< 𝔅𝑃, 𝔅𝑁 > be a BFS in ϝ and (𝛼, 𝜔) ∈ [0,1], (𝜃, 𝜗) ∈ [∇,0] × [0,△]. 

By a BFMT of 𝐵 =< 𝔅𝑁 , 𝔅𝑃 >, we mean a BFS 𝑀 = {< 𝑟, 𝔅(𝜔,𝜗)
𝑃 (𝑟), 𝔅(𝛼,𝜃)

𝑁 (𝑟) >: 𝑟 ∈ ϝ} or simply 

as 𝑀 = {< 𝑟, 𝔅𝑀
𝑃 (𝑟), 𝔅𝑀

𝑁 (𝑟) >: 𝑟 ∈ ϝ}, where 𝔅𝑀
𝑃 (𝑟) = 𝔅(𝜔,𝜗)

𝑃 : ϝ → [0,1] and 𝔅𝑀
𝑁 = 𝔅(𝛼,𝜃)

𝑁 : ϝ →

[−1,0] 

and defined by 𝔅𝑀
𝑃 (𝑟) = 𝔅(𝜔,𝜗)

𝑃 (𝑟) = 𝜔𝔅𝑃(𝑟) + 𝜗 for all 𝑟 ∈ ϝ and 𝔅𝑀
𝑁 (𝑟) = 𝔅(𝛼,𝜃)

𝑁 (𝑟) =

𝛼𝔅𝑁(𝑟) + 𝜃.” 

Definition:2.10[16]: “Suppose 𝔅 = (𝔅𝑃, 𝔅𝑁) is a BFS in 𝔏 where 𝔅𝑃: ϝ → [0,1] and 𝔅𝑁: ϝ →

[−1,0]Then 𝔅 is known to be a BFL(Bipolar fuzzy sublattice) of 𝔏 when the following conditions are 

fulfilled for all ℏ, 𝑚 ∈ 𝔏, 

(𝑖)𝔅𝑃(ℏ⋁𝑚) ≥ min{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}, 

(ii) 𝔅𝑃(ℏ ∧ 𝑚) ≥ min{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}. 

(iii) 𝔅𝑁(ℏ⋁𝑚) ≤ max{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)}, 

(iv) 𝔅𝑁(ℏ ∧ 𝑚) ≤ max{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)}.” 

Definition:2.11[16]: “Suppose 𝔅 = (𝔅𝑃, 𝔅𝑁) is a BFS in 𝔏 where 𝔅𝑃: ϝ → [0,1] and 𝔅𝑁: ϝ →

[−1,0]Then 𝔅 is known to be a BFIL(Bipolar fuzzy ideal) of 𝔏 when the following conditions are 

fulfilled for all ℏ, 𝑚 ∈ 𝔏, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

244 
https://internationalpubls.com 

(𝑖)𝔅𝑃(ℏ⋁𝑚) ≥ min{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}, 

(ii) 𝔅𝑃(ℏ ∧ 𝑚) ≥ max{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}. 

(iii) 𝔅𝑁(ℏ⋁𝑚) ≤ max{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)}, 

(iv) 𝔅𝑁(ℏ ∧ 𝑚) ≤ min{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)}.” 

3. Bipolar Fuzzy Prime Ideals of a Lattice 

In this section, we explore and study Bipolar Fuzzy prime ideals (BFPI) of 𝔏, their characterizations 

by using level subsets, homomorphism and anti-homomorphism of BFPIs. 

Now, we introduce the following. 

Definition 3.1: Suppose 𝔅 = (𝔅𝑃, 𝔅𝑁) is a BFI in 𝔏. Then 𝔅 is known to be a BFPI of 𝔏 when the 

following conditions are fulfilled for all ℏ, 𝑚 ∈ 𝔏, 

(i) 𝔅𝑃(ℏ ∧ 𝑚) ≤ max{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}, 

(ii) 𝔅𝑁(ℏ ∧ 𝑚) ≥ min{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)}. 

Example 3.2: Consider the lattice 𝔏 of" divisors of 6 ". We get 𝔏 = {1,2,3,6}. Suppose 𝔅 = {< 

1,0.7, −0.3 >, < 2,0.4, −0.3 >, < 3,0.7, −0.1 >, < 6,0.4, −0.1 >}. We can easily prove that 𝔅 is a 

BFPI of 𝔏. 

Theorem 3.3: Suppose 𝔅 = (𝔅𝑃, 𝔅𝑁) be a BFS(𝔏). Then it holds that 𝔅 is a BFPI on 𝔏 iff the 

following four statements hold: 

(i) 𝔅𝑃(ℏ ∨ 𝑚) = min{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}, 

(ii) 𝔅𝑃(ℏ ∧ 𝑚) = max{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}, 

(iii) 𝔅𝑁(ℏ ∨ 𝑚) = max{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)}, 

(iv) 𝔅𝑁(ℏ ∧ 𝑚) = min{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)} for any ℏ, 𝑚 ∈ 𝔏. 

Proof: Proof is clear. 

Theorem 3.4: If 𝜅 and 𝔅 are two BFPIs of a lattice 𝔏, then 𝜅 ∩ 𝔅 is a BFPI of 𝔏. 

Proof: Suppose 𝜅 = (𝜅𝑃, 𝜅𝑁) and 𝔅 = (𝔅𝑃, 𝔅𝑁) be two BFPIs of 𝔏. 

Now, 

(𝜅 ∩ 𝔅)𝑃(ℏ ∧ 𝑚) = min{𝜅𝑃(ℏ ∧ 𝑚), 𝔅𝑃(ℏ ∧ 𝑚)}

 ≤ min{max{𝜅𝑃(ℏ), 𝜅𝑃(𝑚)}, max{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}

 = max{min{𝜅𝑃(ℏ), 𝔅𝑃(ℏ)}, min{𝜅𝑃(𝑚), 𝔅𝑃(𝑚)}

 = max{(𝜅 ∩ 𝔅)𝑃(ℏ), (𝜅 ∩ 𝔅)𝑃(𝑚)}.

 

Thus 

(𝜅 ∩ 𝔅)𝑃(ℏ ∧ 𝑚) ≤ max{𝜅 ∩ 𝔅)𝑃(ℏ), 𝜅 ∩ 𝔅)𝑃(𝑚)} for all ℏ, 𝑚 ∈ 𝔏. 

Now, 

(𝜅 ∩ 𝔅)𝑁(ℏ ∧ 𝑚) = max{𝜅𝑁(ℏ ∧ 𝑚), 𝔅𝑁(ℏ ∧ 𝑚)}

 ≥ max{min{𝜅𝑁(ℏ), 𝜅𝑁(𝑚)}, min{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)}

 = min{max{𝜅𝑁(ℏ), 𝔅𝑁(ℏ)}, max{𝜅𝑁(𝑚), 𝔅𝑁(𝑚)}

 = min{(𝜅 ∩ 𝔅)𝑁(ℏ), (𝜅 ∩ 𝔅)𝑁(𝑚)}.

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

245 
https://internationalpubls.com 

Thus 

(𝜅 ∩ 𝔅)𝑁(ℏ ∧ 𝑚) ≥ min{𝜅 ∩ 𝔅)𝑁(ℏ), 𝜅 ∩ 𝔅)𝑁(𝑚)} for all ℏ, 𝑚 ∈ 𝔏 

Hence, 𝜅 ∩ 𝔅 is a BFPI of 𝔏. 

Remark 3.5: Union of two BFPIs of a lattice 𝔏 need not be a BFPI.  

Since from [ ] union of two BFIs neednot be a BFI, hence, 𝜅 ∪ 𝔅 need not be a BFPI of 𝔏. 

Theorem 3.6: The arbitrary intersection of BFPIs of a complete lattice satisfying infinite meet 

distributive law 𝔏 is also a BFPI of 𝔏. 

Proof: Proof is clear. 

Theorem 3.7: Let B =< BP, BN >∈ BFS(𝔏). Then B is BFPI of 𝔏 if and only if the nonempty 

level subset B (α, ω) is a prime ideal of 𝔏 for each α ∈ [0, 1] and ω ∈ [−1, 0]. 

Proof: 

 Suppose that B =< BP, BN > is BFPI(𝔏).  

Let ℏ, m ∈ B (α, ω). 

⇒ BP (ℏ) ≥ α, BP (m) ≥ α and 

BN (ℏ) ≤ ω, BN (m) ≤ ω. 

To show that B (α, ω) is a prime ideal of 𝔏, 

we show that for each ℏ, m ∈ 𝔏 and ℏ ∧ m ∈ B (α, ω) 

 then either ℏ ∈ B (α, ω) or s ∈ B (α, ω).  

Suppose ℏ, m ∈ 𝔏 and ℏ ∧ s ∈ B (α, ω). 

Then BP (ℏ ∧ m) ≥ α, BN (ℏ ∧ m) ≤ ω 

⇒ max {BP (ℏ), BP (m)} ≥ α, min {BP (ℏ), BP (m)} ≤ ω (since B is a BFPI of 𝔏) 

⇒ BP (ℏ) ≥ α or BP (m) ≥ α, BN (ℏ) ≤ ω or BN (m) ≤ ω 

⇒ BP (ℏ) ≥ α and BN (ℏ) ≤ ω or BP (m) ≥ α and BN (m) ≤ ω 

Thus ℏ ∈ B (α, ω) or s ∈ B (α, ω)  

Hence, B (α, ω) is a prime ideal of 𝔏. 

In converse assume that B (α, ω) is a prime ideal of 𝔏. 

i.e To any ℏ, m ∈ 𝔏 and ℏ ∧ s ∈ B (α, ω) 

 then either ℏ ∈ B (α, ω) or s ∈ B (α, ω).  

We have to show that B is a BFPI of 𝔏. 

Assume that B is not a BFPI of 𝔏. 

Thus BP (ℏ ∧ m) > max {BP (ℏ), BP (m)} and 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

246 
https://internationalpubls.com 

BN (ℏ ∧ m) < min {BN (ℏ), BN (m)} 

⇒ BP (ℏ ∧ m) > BP (ℏ) and BP (ℏ ∧ m) > BP (m),  

BN (ℏ ∧ m) < BN (ℏ) and BN (ℏ ∧ m) < BN (m). 

Suppose that BP (ℏ ∧ m) = α and BN (ℏ ∧ m) = ω 

⇒ BP (ℏ) < α and BN (ℏ) > ω, BP (m) < α and BN (m) > ω. 

Hence ℏ, m ∈/ B (α, ω). 

This is a contradiction to the fact that B (α, ω) is a prime ideal of 𝔏 for any α ∈ [0, 1] and ω ∈ [−1, 0]. 

Hence B is a BFPI of 𝔏.  

Theorem 3.8: Let 𝔏 be a lattice and B ∈ BFS (𝔏). If B, BFPI of 𝔏 then we have Supp(B) forms a crisp 

prime ideal of 𝔏. 

Proof:  

Suppose B = {< ℏ, BP (ℏ), BN (ℏ) > /ℏ ∈ 𝔏 } ∈ BFS (𝔏).  

Given B is a BFPI of 𝔏. 

From [] we have Supp(B) is a crisp ideal in 𝔏. Now we prove that Supp(B) is prime. 

Suppose ℏ, m ∈ 𝔏 so that ℏ ∧ s ∈ Supp(B).  

Thus BP (ℏ ∧ m) ̸= 0 or BN (ℏ ∧ m) ̸= 0. 

But BP (ℏ ∧ m) = max {BP (ℏ), BP (m)} (since B is BFPI of 𝔏). 

⇒ either BP (ℏ) ≠0, or BP (m) ≠ 0 

Likewise, we have either BN (ℏ) ≠ 0, or BN (m) ≠ 0 

⇒ either ℏ ∈ Supp(B) or m ∈ Supp(B). 

Hence Supp(B) is a prime ideal of 𝔏.  

Remark 3.9: Converse part of the above theorem does not hold in general. 

Suppose B = {< 1, 0.5, −0.1 >, < 2, 0.7, −0.2 >, < 3, 0.8, −0.05 >, < 6, 0.4, −0.01 >} be a BF 

subset in 𝔏 = {1, 2, 3, 6} with divisors of 6. 

Supp(B) = {1, 2, 3, 6} is a prime ideal of 𝔏. 

But BP (2 ∧ 3) = BP (1) = 0.5 ≥ max{BP (2), BP (8)} = 0.8 

⇒ B is not a BFPI of 𝔏. 

Theorem 3.10: Let B be a BFS of lattice 𝔏. Then, B is a BFPI of 𝔏 if and only if f the BFMT M of 

B form BF prime ideal of 𝔏. 

Proof: Assume that 𝔅 is a BF primeideal of 𝔏 and 𝑀 be a BFMT of 𝔅. 

Now we have to show that 𝑀 is a BFPI of 𝔏. 

Let ℏ, 𝑚 ∈ 𝔏. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

247 
https://internationalpubls.com 

Now, consider 𝔅(𝜔,𝜗)
𝑃 (ℏ ∧ 𝑚) 

= 𝜔𝔅𝑃(ℏ ∧ 𝑚) + 𝜗. 

≤ 𝜔 max{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)} + 𝜗 。 

= max{𝜔𝔅𝑃(ℏ) + 𝜗, 𝜔𝔅𝑃(𝑚) + 𝜗} 

= max{𝔅(𝜔,𝜗)
𝑃 (ℏ), 𝔅(𝜔,𝜗)

𝑃 (𝑚)} 

Thus 𝔅(𝜔,𝜗)
𝑃 (ℏ ∧ 𝑚) ≤ max{𝔅(𝜔,𝜗)

𝑃 (ℏ), 𝔅(𝜔,𝜗)
𝑃 (𝑚)} 

Now 𝔅(𝛼,𝜃)
𝑁 (ℏ ∧ 𝑚) = 𝛼𝔅𝑁(ℏ ∧ 𝑚) + 𝜃. 

≥ 𝛼min{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)} + 𝜃. 

= min{𝛼𝔅𝑁(ℏ) + 𝜃, 𝛼𝔅𝑁(𝑚) + 𝜃} 

= min{𝔅(𝛼,𝜃)
𝑁 (ℏ), 𝔅(𝛼,𝜃)

𝑁 (𝑚)} 

Thus 𝔅(𝛼,𝜃)
𝑁 (ℏ ∧ 𝑚) ≥ min{𝔅(𝛼,𝜃)

𝑁 (ℏ), 𝔅(𝛼,𝜃)
𝑁 (𝑚)} 

Hence the BFMT 𝑀 of 𝔅 is again a BFPI of 𝔏. 

In converse assume that 𝑀 a BFPI of 𝔏. 

Now, consider 𝔅𝑃(ℏ ∧ 𝑚) 

=
1

𝜔
(𝔅(𝜔,𝜗)

𝑃 (ℏ ∧ 𝑚) − 𝜗) 

≤
1

𝜔
(max{𝔅(𝜔,𝜗)

𝑃 (ℏ), 𝔅(𝜔,𝜗)
𝑃 (𝑚)} − 𝜗) 

= max {
1

𝜔
(𝔅(𝜔,𝜗)

𝑃 (ℏ) − 𝜗),
1

𝜔
(𝔅(𝜔,𝜗)

𝑃 (𝑚) − 𝜗)} 

= max{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}. 

Thus 𝔅𝑃(ℏ ∧ 𝑚) ≤ max{𝔅𝑃(ℏ), 𝔅𝑃(𝑚)}. 

Likewise, we can prove 𝔅𝑁(ℏ ∧ 𝑚) ≥ min{𝔅𝑁(ℏ), 𝔅𝑁(𝑚)}. 

Thus 𝑀 is a BF prime ideal of 𝐿. Hence the proof. 

Theorem 3.11: Let 𝜛: 𝔏 → 𝔏1 be a lattice epimorphism.                                                                        

If 𝔅 is a BFPI of 𝔏, then 𝜛(𝔅) is a BFPI of 𝔏1 . 

Proof: Assume 𝔅 = (𝔅𝑃, 𝔅𝑁) a BFPI of 𝔏. 

Then by known Th we know that 𝜛(𝔅) is a BFI in 𝔏1. 

Now we show that 𝜛(𝔅) is prime. 

Let 𝑠, 𝑧 ∈ 𝔏1. 

Then 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

248 
https://internationalpubls.com 

𝜛(𝔅𝑃)(𝑠 ∧ 𝑧) = sup{𝔅𝑃(ℏ) ∣ ℏ ∈ 𝜛−1(𝑠 ∧ 𝑧)}

 = sup{𝔅𝑃(𝑢 ∧ 𝜉) ∣ 𝑢 ∈ 𝜛−1(𝑚), 𝜉 ∈ 𝜛−1(𝑧)}

 = sup{max{𝔅𝑃(𝑢), 𝔅𝑃(𝜉)} ∣ 𝑢 ∈ 𝜛−1(𝑚), 𝜉 ∈ 𝜛−1(𝑧)}

 = max {sup{𝔅𝑃(𝑢) ∣ 𝑢 ∈ 𝜛−1(𝑚)}, sup{𝔅𝑃(𝜉) ∣ 𝜉 ∈ 𝜛−1(𝑧)}}

 = max{𝜛(𝔅𝑃)(𝑚), 𝜛(𝔅𝑃)(𝑧)}.

𝜛(𝔅𝑁)(𝑠 ∧ 𝑧) = inf{𝔅𝑁(ℏ) ∣ ℏ ∈ 𝜛−1(𝑠 ∧ 𝑧)}

 = inf{𝔅𝑁(𝑢 ∧ 𝜉) ∣ 𝑢 ∈ 𝜛−1(𝑚), 𝜉 ∈ 𝜛−1(𝑧)}

 = inf{min{𝔅𝑁(𝑢), 𝔅𝑁(𝜉)} ∣ 𝑢 ∈ 𝜛−1(𝑚), 𝜉 ∈ 𝜛−1(𝑧)}

 = min {inf{𝔅𝑁(𝑢) ∣ 𝑢 ∈ 𝜛−1(𝑚)}, inf{𝔅𝑁(𝜉) ∣ 𝜉 ∈ 𝜛−1(𝑧)}}

 = min{𝜛(𝔅𝑁)(𝑚), 𝜛(𝔅𝑁)(𝑧)},

 

Hence, 𝜛(𝔅) is a BFPI of 𝔏1. 

Theorem 3.12: Let 𝜛: 𝔏 → 𝔏1 be a lattice homomorphism. If 𝐶 is a BFPI of 𝔏1, then 𝜛−1(𝐶) is a 

BFPI of 𝔏. 

Proof: Suppose 𝐶 = (𝐶𝑃, 𝐶𝑁) be a BFPI of 𝔏1. 

Then by known theorem[], we know that 𝜛−1(𝐶) is a BFI in 𝔏. 

Now we show that 𝜛−1(𝐶) is prime. 

Let ℏ, 𝑚 ∈ 𝔏. 

Then 

𝜛−1(𝐶𝑃)(ℏ ∧ 𝑚) = 𝐶𝑃(𝜛(ℏ ∧ 𝑚))

 = 𝐶𝑃{(𝜛(ℏ) ∧ 𝜛(𝑚)}

 = max{𝐶𝑃(𝜛(ℏ)), 𝐶𝑃(𝜛(𝑚))}

 = max{𝜛−1(𝐶𝑃)(ℏ), 𝜛−1(𝐶𝑃)(𝑚)}

𝜛−1(𝐶𝑁)(ℏ ∧ 𝑚) = 𝐶𝑁(𝜛(ℏ ∧ 𝑚))

 = 𝐶𝑁{(𝜛(ℏ) ∧ 𝜛(𝑚)}

 = min{𝐶𝑁(𝜛(ℏ)), 𝐶𝑁(𝜛(𝑚))}

 = min{𝜛−1(𝐶𝑁)(ℏ), 𝜛−1(𝐶𝑁)(𝑚)}

  

Hence, 𝜛−1(𝐶) is a BFPI of 𝔏. 

4.Conclusion 

This study explores the investigation of Bipolar Fuzzy Prime Ideals (BFPI) in lattices. We provide a 

detailed exploration of their properties, characterizations, and associated homomorphisms. 

References: 

[1]  Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Transactions on Systems, Man, and Cybernetics 23 (1993) 610–614. 

[2]  L.A. Zadeh, Fuzzy Sets, Inform. Control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. 

[3]  Bustince, H., Burillo, P.: Vague sets are intuitionistic fuzzy sets. Fuzzy Sets and Systems 79 (1996) 403–405 

[4]  K.V. Thomas, L.S. Nair, Intuitionistic Fuzzy Sublattices and Ideals, Fuzzy Inform. Eng. 3 (2011), 321-331. https: 

//doi.org/10.1007/s12543-011-0086-5. 

[5]  K.T. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets Syst. 20 (1986), 87-96. https://doi.org/10.1016/ s0165-

0114(86)80034-3. 

https://doi.org/10.1016/s0019-9958(65)
//doi.org/10.1007/s12543-011-0086-5
https://doi.org/10.1016/


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

249 
https://internationalpubls.com 

[6]  K.M. Lee, Bipolar-Valued Fuzzy Sets and Their Operations, in: Proc. Int. Conf. on Intelligent Technologies, Bangkok, 

Thailand, (2000), 307-312. 

[7]  N. Ajmal, K.V. Thomas, Fuzzy Lattices, Inform. Sci. 79 (1994), 271-291. https://doi.org/10.1016/ 0020-0255(94) 

90124-4. 

[8]  U. Venkata Kalyani, T. Eswarlal, Homomorphism on Bipolar Vague Normal Groups, Adv. Math., Sci. J. 9 (2020), 

3315-3324. https://doi.org/10.37418/amsj.9.6.11. 

[9]  U. Venkata Kalyani, T. Eswarlal, Bipolar Vague Cosets, Adv. Math., Sci. J. 9 (2020), 6777-6787. https://doi. 

org/10.37418/amsj.9.9.36. 

[10]  S. Boudaoud, S. Milles, L. Zedam, Principal Intuitionistic Fuzzy Ideals and Filters on a Lattice, Discuss. Math. Gen. 

Algebra Appl. 40 (2020), 75-88. https://doi.org/10.7151/dmgaa.1325. 

[11]  S. Milles, The Lattice of Intuitionistic Fuzzy Topologies Generated by Intuitionistic Fuzzy Relations, Appl. Appl. 

Math. 15 (2020), 942-956. https://digitalcommons.pvamu.edu/aam/vol15/iss2/13. 

[12]  H. Zhang, Q. Li, Intuitionistic Fuzzy Filter Theory on Residuated Lattices, Soft Comput. 23 (2018), 6777-6783. 

https://doi.org/10.1007/s00500-018-3647-2. 

[13]  S. Milles, L. Zedam, E. Rak, Characterizations of Intuitionistic Fuzzy Ideals and Filters Based on Lattice Operations, 

J. Fuzzy Set Valued Anal. 2017 (2017), 143-159. https://doi.org/10.5899/2017/jfsva-00399. 

[14]  B. Nageswararao, N. Ramakrishna, T. Eswarlal, Vague Lattices, Studia Rosenthaliana, 12 (2020), 191-202. 

[15]  R.P. Rao, V.S. Kumar, A.P. Kumar, Rough Vague Lattices, J. Xi'an Univ. Architect. Technol. 9 (2019), 115-124. 

[16]  Uv, Kalyani & Tamma, Eswarlal & Jacob, Kavikumar & Iampan, Aiyared. (2022). Bipolar Fuzzy Sublattices and 

Ideals. International Journal of Analysis and Applications. 20. 45. 10.28924/2291-8639-20-2022-45. 

[17]  Uv, Kalyani & Iampan, Aiyared & Tamma, Eswarlal. (2024). Bipolar Fuzzy Magnified Translation of a Lattice. 

International Journal of Analysis and Applications. 22. 195. 10.28924/2291-8639-22-2024-195. 

https://doi.org/10.1016/
https://doi.org/10.37418/amsj.9.6.11
https://doi/
https://doi.org/10.7151/dmgaa.1325
https://digitalcommons.pvamu.edu/aam/vol15/iss2/13
https://doi.org/10.1007/s00500-018-3647-2
https://doi.org/10.5899/2017/jfsva-00399