An Investigation into Bipolar Fuzzy Hoop Algebras and Their Applications

Abstract

This paper introduces bipolar fuzzy sub-hoops and bipolar fuzzy filters within hoop algebras, extending fuzzy logic to incorporate both positive and negative membership degrees. We define these structures, explore their algebraic properties, and establish their interplay through rigorous theorems. Key results include characterizations of bipolar fuzzy filters via level sets and conditions under which they become implicative filters. These findings enhance the theoretical framework of many-valued logic and offer practical applications in decision-making, image processing, and spatial reasoning under uncertainty. Our work provides a foundation for advanced fuzzy systems handling complex, contradictory information.

1. Introduction

The notion of a hoop was introduced by Bosbach [1,2] as a foundational algebraic structure for examining many-valued logical systems, where the truth values of propositions are represented within a lattice framework. Hoops serve as a valuable mathematical tool for modeling systems in which logical propositions assume a continuum of values, typically within the interval $[0,1]$, rather than being limited to binary truth values. This makes them especially suitable for reasoning in fuzzy logic contexts, where truth is viewed as partial and gradual rather than absolute.

Algebraically, a hoop is a bounded distributive lattice equipped with an additional binary operation, commonly referred to as implication or residuation, that satisfies specific axioms. This structure is particularly significant in fuzzy set theory, where it supports reasoning and decision-making under uncertainty. The implication operation enables the flexible manipulation of fuzzy relations and logical constructs, offering a more expressive framework than classical Boolean logic.

Hoops can be regarded as a generalization of Heyting algebras, extending the semantics of classical logic to accommodate degrees of truth. Their structural properties make hoop algebras indispensable in formalizing the semantics of fuzzy sets and logics and they are widely used in applications such as spatial reasoning, image processing, and decision support systems.

Since Bosbach’s initial formulation, hoop theory has been the subject of extensive research (see [3,4,5,6,7,8,9,10,11]). These studies have deepened our understanding of hoop structures, their properties, and their relevance to many-valued logic systems, particularly within the framework of fuzzy logic.

In 1994, Zhang [12] introduced the concept of bipolar fuzzy sets as a generalization of traditional fuzzy sets [13]. Unlike classical fuzzy sets where membership degrees range from 0 to 1, bipolar fuzzy sets assign degrees within the interval $[-1,1]$. In this setting, a degree of 0 indicates irrelevance to the property in question, values between 0 and 1 suggest partial satisfaction, while values between $-1$ and 0 indicate the partial satisfaction of a counter-property. Although bipolar fuzzy sets share some conceptual similarities with intuitionistic fuzzy sets, they differ fundamentally in structure and interpretation. The distinction between positive and negative information is crucial in many real-world scenarios; positive information reflects what is considered possible, while negative information captures what is regarded as impossible.

To manage bipolar information, both fuzzy and possibilistic frameworks have been proposed. These approaches are particularly relevant in areas such as image processing and spatial reasoning, where bipolarity naturally arises. For instance, in determining an object’s position, positive information may represent plausible locations, while negative information may signify locations that are definitively excluded. In spatial relation modeling, concepts like “left” and “right” are not mere negations of one another but represent symmetric opposites. This symmetry introduces a middle ground, positions that are neither clearly to the left nor the right, emphasizing the inherent indeterminacy and demonstrating that the union of positive and negative information does not exhaust the entire space.

This paper introduces and investigates the concepts of bipolar fuzzy sub-hoops and bipolar fuzzy filters, examining their algebraic properties in depth. It explores the interactions between these two structures, providing a detailed analysis of their relationship. Additionally, it establishes formal characterizations for both bipolar fuzzy sub-hoops and bipolar fuzzy filters, offering new insights into their underlying structure and contributing to their theoretical development within the broader context of fuzzy logic and many-valued reasoning.

The novelty of this work lies in introducing bipolar fuzzy sub-hoops and filters, which generalize fuzzy logic to incorporate both positive and negative information within hoop algebras. Our contributions include defining these structures, proving their algebraic properties, and characterizing their interrelationship, notably that every BFF is a BFSH (Theorem 4). A key technical difficulty was ensuring that bipolar fuzzy sets respect the hoop algebra’s operations while handling dual membership degrees, which we addressed through new conditions and theorems (e.g., Theorems 3 and 6) that align with classical algebraic structures.

The list of acronyms is given in Table 1.

Table 1. List of acronyms.

Acronyms	Representation
BFF	bipolar fuzzy filter
IBFF	implicative bipolar fuzzy filter
BFS	bipolar fuzzy set
BFSH	bipolar fuzzy sub-hoop
$B{F}_S$	bipolar fuzzy structure

2. Preliminaries

Bipolar fuzzy sets, introduced by Zhang [12], extend traditional fuzzy sets by incorporating negative membership degrees, enabling the modeling of both positive and negative information.

Definition 1 

([12]). Let $\mathcal{X}$ be a non-empty set. A bipolar fuzzy set (briefly, BFS) $\mathcal{B}$ in $\mathcal{X}$ is defined as $\mathcal{B}=\left\{\right(\mathfrak{x},{\Omega }^{-}\left(\mathfrak{x}\right),{\Omega }^{+}\left(\mathfrak{x}\right))\mid \mathfrak{x}\in \mathcal{X}\}$, where ${\Omega }^{+}:\mathcal{X}\to [0,1]$ and ${\Omega }^{-}:\mathcal{X}\to [-1,0]$ are two functions. The positive membership degree ${\Omega }^{+}\left(\mathfrak{x}\right)$ represents the degree to which element $\mathfrak{x}$ satisfies the property associated with the BFS $\mathcal{B}$, while the negative membership degree ${\Omega }^{-}\left(\mathfrak{x}\right)$ represents the degree to which $\mathfrak{x}$ satisfies an implicit counter-property related to $\mathcal{B}$. If ${\Omega }^{+}\left(\mathfrak{x}\right)\ne 0$ and ${\Omega }^{-}\left(\mathfrak{x}\right)=0$, this indicates that x fully satisfies the positive property of B. Conversely, if ${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)=0$ and ${\Omega }^{-}\left(\mathfrak{x}\right)\ne 0$, this suggests that x does not meet the property of $\mathcal{B}$ but partially satisfies its counter-property. It is also possible that ${\Omega }^{+}\left(\mathfrak{x}\right)=0$ and ${\Omega }^{-}\left(\mathfrak{x}\right)=0$, which occurs when the membership functions for the property and its counter-property overlap over some subset of $\mathcal{X}$. For simplicity, we denote the BFS $\mathcal{B}=\{(\mathfrak{x},{\Omega }^{-}\left(\mathfrak{x}\right),{\Omega }^{+}\left(\mathfrak{x}\right))\mid \mathfrak{x}\in \mathcal{X}\}$ as $\Omega =({\Omega }^{+},{\Omega }^{-})$.

Hoop algebras, defined by Bosbach [1,2], provide a lattice-based framework for many-valued logic, with operations ⊙ and → supporting fuzzy reasoning. They generalize Heyting algebras and are central to our bipolar fuzzy extensions.

Definition 2 

([4]). A hoop (or hoop algebra) is an algebraic structure denoted as $(\mathcal{H},\odot ,\to ,1)$, where $(\mathcal{H},\odot ,1)$ forms a commutative monoid, and the following properties hold:

• (H1) ${\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_1}=1$,
• (H2) ${\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})={\hslash }_{{\mathfrak{a}}_2}\odot ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_1})$,
• (H3) ${\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})=({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to z$ for all ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H}$.

Next, we define a relation ${\le }_{\mathcal{H}}$ on a hoop $\mathcal{H}$ as follows:

$$\begin{array}{c}\hfill (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left({\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\iff {\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}=1\right).\end{array}$$

(1)

It is straightforward to verify that $(\mathcal{H},{\le }_{\mathcal{H}})$ forms a partially ordered set.

Sub-hoops preserve the algebraic structure of hoops, forming the basis for our bipolar fuzzy sub-hoops, which extend this concept to fuzzy settings.

Definition 3 

([8]). A non-empty subset $\mathcal{S}$ of a hoop algebra $\mathcal{H}$ is called a sub-hoop of $\mathcal{H}$ if it satisfies the following condition:

$$\begin{array}{c}\hfill (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{S})\left({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}\in \mathcal{S}\mathit{and}{\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}\in \mathcal{S}\right).\end{array}$$

(2)

Additionally, every sub-hoop must include the element 1.

Proposition 1 

([8]). Let $(\mathcal{H},\odot ,\to ,1)$ be a hoop algebra. For any ${\hslash }_{\mathfrak{a}},{\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H}$, the following conditions hold:

• (a1) $(\mathcal{H},{\le }_{\mathcal{H}})$ is a meet-semilattice with ${\hslash }_{{\mathfrak{a}}_1}\wedge {\hslash }_{{\mathfrak{a}}_2}={\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})$;
• (a2) ${\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_3}$ if and only if ${\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}$;
• (a3) ${\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}$, and ${\hslash }_{{\mathfrak{a}}_1}^{\mathfrak{n}}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_1}$ for any $\mathfrak{n}\in N$;
• (a4) ${\hslash }_{{\mathfrak{a}}_1}{\le }_{white\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_1}$;
• (a5) $1\to {\hslash }_{\mathfrak{a}}={\hslash }_{\mathfrak{a}}$ and ${\hslash }_{\mathfrak{a}}\to 1=1$;
• (a6) ${\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}){\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_1}\wedge {\hslash }_{{\mathfrak{a}}_2}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}$;
• (a7) ${\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}{\le }_{\mathcal{H}}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})$;
• (a8) ${\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}$ implies ${\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_3}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\odot {\hslash }_{{\mathfrak{a}}_3},{\hslash }_{{\mathfrak{a}}_3}\to {\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_3}\to {\hslash }_{{\mathfrak{a}}_2}$, and ${\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3}$;
• (a9) ${\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})={\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})$.

Filters in hoop algebras capture upward-closed subsets, essential for defining bipolar fuzzy filters that refine fuzzy information.

Definition 4 

([8]). A non-empty subset $\mathcal{F}$ of a hoop algebra $\mathcal{H}$ is referred to as a filter of $\mathcal{H}$ if the following conditions are satisfied:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left({\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{F}\Rightarrow {\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}\in \mathcal{F}\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left({\hslash }_{{\mathfrak{a}}_1}\in \mathcal{F},{\hslash }_{{\mathfrak{a}}_1}\le {\hslash }_{{\mathfrak{a}}_2}\Rightarrow {\hslash }_{{\mathfrak{a}}_2}\in \mathcal{F}\right).\hfill \end{array}$$

(4)

Implicative filters strengthen filter conditions, motivating our definition of implicative bipolar fuzzy filters (IBFFs) for advanced fuzzy logic applications.

Definition 5 

([9]). A non-empty subset $\mathcal{F}$ of a hoop algebra $\mathcal{H}$ is termed an implicative filter of $\mathcal{H}$ if it satisfies the following conditions:

$$\begin{array}{cc}\hfill & 1\in \mathcal{F},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H})\left({\hslash }_{{\mathfrak{a}}_1}\to (({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2})\in \mathcal{F},{\hslash }_{{\mathfrak{a}}_1}\in \mathcal{F}\Rightarrow {\hslash }_{{\mathfrak{a}}_2}\in \mathcal{F}\right).\hfill \end{array}$$

(6)

It is important to note that the conditions in (3) and (4) indicate that the subset $\mathcal{F}$ is closed under the operation ⊙ and upward-closed, respectively. Furthermore, a subset $\mathcal{F}$ of a hoop algebra $\mathcal{H}$ is a filter if and only if it satisfies condition (5) along with the following additional condition:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}\in \mathcal{F},{\hslash }_{{\mathfrak{a}}_1}\in \mathcal{F}\Rightarrow {\hslash }_{{\mathfrak{a}}_2}\in \mathcal{F}\right).\hfill \end{array}$$

(7)

3. Bipolar Fuzzy Hoop Algebras

Definition 6. 

A BFS $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ within a hoop $\mathcal{H}$ is termed a bipolar fuzzy sub-hoop (briefly, BFSH) of $\mathcal{H}$ if the following conditions hold:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\},\hfill \\ {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}\hfill \end{array}\right).\hfill \end{array}$$

(8)

Here, • represents either ⊙ or →.

Example 1. 

Let $\mathcal{H}=\{0,1,2,3,4,5,6\}$ be a set equipped with two binary operations, · and →, whose values are provided in the following Table 2 and Table 3.

Table 2. Cayley table of the binary operation →.

→	0	1	2	3	4	5	6
0	1	1	2	1	1	1	1
1	0	1	2	3	4	5	6
2	1	1	1	1	1	1	1
3	0	1	2	1	1	5	6
4	0	1	2	3	1	5	6
5	0	1	2	1	1	1	6
6	1	2	1	1	1	1	1

Table 3. Cayley table of the binary operation ⊙.

⊙	0	1	2	3	4	5	6
0	0	0	2	0	0	0	0
1	0	1	2	3	4	5	6
2	2	2	2	2	2	2	2
3	0	3	2	3	3	5	6
4	0	4	2	3	4	5	6
5	0	5	2	5	5	5	6
6	0	6	2	6	6	6	0

Then, $(\mathcal{H},\odot ,\to ,1)$ is a hoop algebra. Let $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ be a BFS in $\mathcal{H}$, given by Table 4.

Table 4. Tabular representation of $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$.

$\mathcal{H}$	0	1	2	3	4	5	6
${\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)$	$-0.8$	$-0.8$	$-0.8$	$-0.7$	$-0.7$	$-0.7$	$-0.8$
${\Omega }^{+}\left({\hslash }_{\mathfrak{a}}\right)$	$0.8$	$0.8$	$0.8$	$0.7$	$0.8$	$0.8$	$0.8$

It is straightforward to verify that the BFS $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ within the hoop $\mathcal{H}$ qualifies as a BFSH of $\mathcal{H}$.

Proposition 2. 

For any BFSH $\Omega =({\Omega }^{+},{\Omega }^{-})$ of $\mathcal{H}$, the following holds:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{\mathfrak{a}}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}\left(1\right)\ge {\Omega }^{+}\left({\hslash }_{\mathfrak{a}}\right),\hfill \\ {\Omega }^{-}\left(1\right)\le {\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)\hfill \end{array}\right).\hfill \end{array}$$

(9)

Proof. 

Since ${\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}}=1$ for all ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$, it is straightforward by (8). □

Proposition 3. 

Let $\vartheta :\mathcal{X}\to \mathcal{Y}$ be a homomorphism of a hoop algebra $\mathcal{X}$ into a hoop algebra $\mathcal{Y}$ and $\Omega =(\mathcal{Y},{\Omega }^{-},{\Omega }^{+})$ be a BFSH of $\mathcal{Y}$. Then, the inverse image ${\vartheta }^{-1}(\Omega )$ of Ω is a BFSH of $\mathcal{X}$.

Proof. 

For any $•\in \{\odot ,\to \}$ and ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{X}$. Then,

$$\begin{array}{ccc}{\Omega }_{{\vartheta }^{-1}(\Omega )}^{+}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})& =& {\Omega }^{+}\left(\vartheta ({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\right)\hfill \\ & =& {\Omega }^{+}(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)•\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right))\hfill \\ & \ge & \min \{{\Omega }^{+}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right),{\Omega }^{+}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right)\right)\}\hfill \\ & =& \min \{{\Omega }_{{\vartheta }^{-1}(\Omega )}^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }_{{\vartheta }^{-1}(\Omega )}^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\},\hfill \end{array}$$

and

$$\begin{array}{ccc}{\Omega }_{{\vartheta }^{-1}(\Omega )}^{-}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})& =& {\Omega }^{-}\left(\vartheta ({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\right)\hfill \\ & =& {\Omega }^{-}(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)•\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right))\hfill \\ & \le & \max \{{\Omega }^{-}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right),{\Omega }^{-}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right)\right)\}\hfill \\ & =& \max \{{\Omega }_{{\vartheta }^{-1}(\Omega )}^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }_{{\vartheta }^{-1}(\Omega )}^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}.\hfill \end{array}$$

□

Definition 7. 

A BFS $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ within a hoop algebra $\mathcal{H}$ is said to possess the sup-inf property if for any subset $\mathcal{T}\subset \mathcal{H}$, there exists an element $t_0\in \mathcal{T}$ such that ${\Omega }^{+}\left({\mathfrak{t}}_0\right)=\underset{\mathfrak{t}\in \mathcal{T}}{\sup }{\Omega }^{+}\left(\mathfrak{t}\right)$ and ${\Omega }^{-}\left({\mathfrak{t}}_0\right)=\underset{\mathfrak{t}\in \mathcal{T}}{\inf }{\Omega }^{+}\left(\mathfrak{t}\right)$.

Proposition 4. 

Let $\vartheta :\mathcal{X}\to \mathcal{Y}$ be a homomorphism from a hoop algebra $\mathcal{X}$ to a hoop algebra $\mathcal{Y}$ and let $\Omega =(\mathcal{A},{\Omega }^{-},{\Omega }^{+})$ be a BFSH of $\mathcal{X}$ that satisfies the sup-inf property. Then, the image $\vartheta (\Omega )$ under the homomorphism is a BFSH of $\mathcal{Y}$.

Proof. 

For any $•\in \{\odot ,\to \}$ and $\mathfrak{u},\mathfrak{v}\in \mathcal{Y}$; let ${\hslash }_{{\mathfrak{a}}_1}\in {\alpha }^{-1}\left(\mathfrak{u}\right)$ and ${\hslash }_{{\mathfrak{a}}_2}\in {\alpha }^{-1}\left(\mathfrak{v}\right)$ such that

$${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)=\underset{\mathfrak{t}\in {\vartheta }^{-1}\left(\mathfrak{u}\right)}{\sup }{\Omega }^{+}\left(\mathfrak{t}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)=\underset{\mathfrak{t}\in {\vartheta }^{-1}\left(\mathfrak{v}\right)}{\sup }{\Omega }^{+}\left(\mathfrak{t}\right)$$

and

$${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)=\underset{\mathfrak{t}\in {\vartheta }^{-1}\left(\mathfrak{u}\right)}{\inf }{\Omega }^{-}\left(\mathfrak{t}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)=\underset{\mathfrak{t}\in {\vartheta }^{-1}\left(\mathfrak{v}\right)}{\inf }{\Omega }^{-}\left(\mathfrak{t}\right).$$

Then, by the definition of ${\Omega }_{\vartheta (\Omega )}^{+}$, we have

$$\begin{array}{ccc}{\Omega }_{\vartheta (\Omega )}^{+}(\mathfrak{u}•\mathfrak{v})& =& \underset{\mathfrak{t}\in {\vartheta }^{-1}(\mathfrak{u}•\mathfrak{v})}{\sup }{\Omega }^{+}\left(\mathfrak{t}\right)\hfill \\ & \ge & {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\hfill \\ & \ge & \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}\hfill \\ & =& \min \{\underset{\mathfrak{t}\in {\alpha }^{-1}\left(\mathfrak{u}\right)}{\sup }{\Omega }^{+}\left(\mathfrak{t}\right),\underset{t\in {\alpha }^{-1}\left(\mathfrak{v}\right)}{\sup }{\Omega }^{+}\left(\mathfrak{t}\right)\}\hfill \\ & =& \min \{{\Omega }_{\vartheta (\Omega )}^{+}\left(\mathfrak{u}\right),{\Omega }_{\vartheta (\Omega )}^{+}\left(\mathfrak{v}\right)\}\hfill \end{array}$$

and

$$\begin{array}{ccc}{\Omega }_{\vartheta (\Omega )}^{-}(\mathfrak{u}•\mathfrak{v})& =& \underset{\mathfrak{t}\in {\vartheta }^{-1}(\mathfrak{u}•\mathfrak{v})}{\inf }{\Omega }^{-}\left(\mathfrak{t}\right)\hfill \\ & \le & {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\hfill \\ & \le & \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}\hfill \\ & =& \max \{\underset{t\in {\vartheta }^{-1}\left(\mathfrak{u}\right)}{\inf }{\Omega }^{-}\left(\mathfrak{t}\right),\underset{\mathfrak{t}\in {\vartheta }^{-1}\left(v\right)}{\inf }{\Omega }^{-}\left(t\right)\}\hfill \\ & =& \max \{{\Omega }_{\vartheta (\Omega )}^{-}\left(\mathfrak{u}\right),{\Omega }_{\vartheta (\Omega )}^{-}\left(\mathfrak{v}\right)\}.\hfill \end{array}$$

Hence, $\vartheta (\Omega )$ is a BFSH of $\mathcal{Y}$. □

Definition 8. 

A BFS $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ within a hoop algebra $\mathcal{H}$ is considered to be a bipolar fuzzy filter (briefly, BFF) of $\mathcal{H}$ if for every $•\in \{\odot ,\to \}$, the following conditions hold:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}\hfill \\ {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}\hfill \end{array}\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left(\begin{array}{c}{\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\Rightarrow \hfill \\ \left\{\begin{array}{c}{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)\le {\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\hfill \\ {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)\ge {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\hfill \end{array}\right.\hfill \end{array}\right).\hfill \end{array}$$

(11)

Example 2. 

Let $(\mathcal{H},\odot ,\to ,1)$ be a hoop algebra, given in Example 1. Let $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ be a BFS in $\mathcal{H}$, given by Table 5.

Table 5. Tabular representation of $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$.

$\mathcal{H}$	0	1	2	3	4	5	6
${\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)$	$-0.6$	$-0.8$	$-0.6$	$-0.7$	$-0.8$	$-0.7$	$-0.6$
${\Omega }^{+}\left({\hslash }_{\mathfrak{a}}\right)$	$0.7$	$0.8$	$0.7$	$0.7$	$0.8$	$0.7$	$0.7$

It can be readily verified that the BFS $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ within the hoop algebra $\mathcal{H}$ qualifies as a BFF of $\mathcal{H}$.

The following theorem provides an alternative characterization of bipolar fuzzy filters (BFFs) by replacing the monotonicity condition with a condition involving the implication operation, simplifying their verification in hoop algebras.

Theorem 1. 

A BFS $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ if and only if it satisfies (9) and

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \end{array}\right).\hfill \end{array}$$

(12)

Proof. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be a BFF of $\mathcal{H}$. Since ${\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}1$ for all ${\hslash }_{{\mathfrak{a}}_1}\in \mathcal{H}$, it follows from Equation (11) that Equation (9) holds. For any ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$, we obtain the inequality

$${\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}){\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}.$$

By utilizing Equations (10) and (11), we derive the following bounds:

$${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ge {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}$$

and

$${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\le {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\},$$

which establishes Equation (12).

Conversely, suppose that the BFS $\Omega =({\Omega }^{+},{\Omega }^{-})$ satisfies both Equations (9) and (12). Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$. Since

$${\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}))=({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})=1,$$

it follows from the assumptions (9) and (12) that

$$\begin{array}{ccc}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})& \ge & \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}))\}\hfill \\ & \ge & \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right),\min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})))\}\}\hfill \\ & =& \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right),\min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left(1\right)\}\}\hfill \\ & =& \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)\}\hfill \end{array}$$

and

$$\begin{array}{ccc}{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})& \le & \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}))\}\hfill \\ & \le & \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right),\max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})))\}\}\hfill \\ & =& \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right),\max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left(1\right)\}\}\hfill \\ & =& \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)\}.\hfill \end{array}$$

Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$ be such that ${\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}$. Then, ${\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}=1$, and so

$${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}=\min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left(1\right)\}={\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),$$

$${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}=\max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left(1\right)\}={\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right).$$

Therefore, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

Theorem 2. 

In the context of a hoop algebra $\mathcal{H}$, every BFF is a BFSH.

Proof. 

To prove that every BFF is a BFSH in a hoop algebra $\mathcal{H}$, we show that a BFS $\Omega =({\Omega }^{+},{\Omega }^{-})$ satisfying Definition 13 (BFF) also satisfies Definition 6 (BFSH). By Definition 8, a BFF $\Omega$ satisfies

$${\Omega }^{+}({\hslash }_{a_1}•{\hslash }_{a_2})\ge \min \{{\Omega }^{+}\left({\hslash }_{a_1}\right),{\Omega }^{+}\left({\hslash }_{a_2}\right)\},{\Omega }^{-}({\hslash }_{a_1}•{\hslash }_{a_2})\le \max \{{\Omega }^{-}\left({\hslash }_{a_1}\right),{\Omega }^{-}\left({\hslash }_{a_2}\right)\}$$

for all ${\hslash }_{a_1},{\hslash }_{a_2}\in \mathcal{H}$ and $•\in \{\odot ,\to \}$, and the monotonicity condition:

$$\mathrm{If}{\hslash }_{a_1}{\le }_{\mathcal{H}}{\hslash }_{a_2},\mathrm{then}{\Omega }^{+}\left({\hslash }_{a_1}\right)\le {\Omega }^{+}\left({\hslash }_{a_2}\right),{\Omega }^{-}\left({\hslash }_{a_1}\right)\ge {\Omega }^{-}\left({\hslash }_{a_2}\right).$$

Definition 6 requires only the first condition for $•\in \{\odot ,\to \}$. Since this condition is identical in both definitions, it is immediately satisfied by any BFF. The additional monotonicity condition in Definition 8 is not required for a BFSH, so every BFF automatically satisfies the BFSH conditions. Hence, every BFF is a BFSH. □

The converse of Theorem 2 does not necessarily hold, as demonstrated by the following example.

Example 3. 

Consider the hoop algebra $(\mathcal{H},\odot ,\to ,1)$, as defined in Example 1, and the BFSH $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$, presented in the same example. However, $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ does not qualify as a BFF of $\mathcal{H}$, as evidenced by the following inequalities:

$$\begin{array}{ccc}{\Omega }^{-}\left(3\right)=-0.7\nleqq -0.8& =& \max \{-0.8,-0.8\}\hfill \\ & =& \max \{{\Omega }^{-}\left(1\right),{\Omega }^{-}\left(2\right)\}\hfill \\ & =& \max \{{\Omega }^{-}(2\to 3),{\Omega }^{-}\left(2\right)\}\hfill \end{array}$$

and

$$\begin{array}{ccc}{\Omega }^{+}\left(3\right)=0.7\ngeqq 0.8& =& \min \{0.8,0.8\}\hfill \\ & =& \min \{{\Omega }^{+}\left(1\right),{\Omega }^{+}\left(2\right)\}\hfill \\ & =& \min \{{\Omega }^{+}(2\to 3),{\Omega }^{+}\left(2\right)\}.\hfill \end{array}$$

The following theorem establishes that a bipolar fuzzy set is a BFF if its level sets are filters, bridging fuzzy logic with classical algebraic structures. The proof relies on the definitions of level sets and filter properties in hoop algebras.

Theorem 3. 

Let $\mathcal{H}$ be a hoop algebra. The $B{F}_S$ $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ if and only if for all $(\mathfrak{t},\mathfrak{s})\in [0,1]\times [0,1]$ with $\mathfrak{t}+\mathfrak{s}\le 1$, the non-empty level sets $\mathcal{U}({\Omega }^{+},\mathfrak{t})$ and $\mathcal{L}({\Omega }^{-},\mathfrak{s})$ are filters of $\mathcal{H}$.

Proof. 

Assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. Let $(\mathfrak{t},\mathfrak{s})\in [0,1]\times [0,1]$ be such that $\mathfrak{t}+\mathfrak{s}\le 1$, and suppose that $\mathcal{U}({\Omega }^{+},\mathfrak{t})$ and $\mathcal{L}({\Omega }^{-},\mathfrak{s})$ are non-empty. It is evident that $1\in \mathcal{U}({\Omega }^{+},\mathfrak{t})\cap \mathcal{L}({\Omega }^{-},\mathfrak{s})$.

Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$ be such that ${\hslash }_{{\mathfrak{a}}_1}\in \mathcal{U}({\Omega }^{+},\mathfrak{t})\cap \mathcal{L}({\Omega }^{-},\mathfrak{s})$ and ${\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}\in \mathcal{U}({\Omega }^{+},\mathfrak{t})\cap \mathcal{L}({\Omega }^{-},\mathfrak{s})$. By the properties of $\Omega$, we have ${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)\ge \mathfrak{t}$, ${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)\le \mathfrak{s}$, ${\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\ge \mathfrak{t}$ and ${\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\le s$.

From Equation (12), it follows that

$${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\ge t$$

and

$${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\le s.$$

Hence, ${\hslash }_{{\mathfrak{a}}_2}\in \mathcal{U}({\Omega }^{+},\mathfrak{t})\cap \mathcal{L}({\Omega }^{-},\mathfrak{s})$, which implies that $\mathcal{U}({\Omega }^{+},\mathfrak{t})$ and $\mathcal{L}({\Omega }^{-},\mathfrak{s})$ are filters of $\mathcal{H}$.

Conversely, assume that the non-empty level sets $\mathcal{U}({\Omega }^{+},\mathfrak{t})$ and $\mathcal{L}({\Omega }^{-},\mathfrak{s})$ are filters of $\mathcal{H}$ for all $(\mathfrak{t},\mathfrak{s})\in [0,1]\times [0,1]$ with $\mathfrak{t}+\mathfrak{s}\le 1$. For any ${\hslash }_{{\mathfrak{a}}_1}\in \mathcal{H}$, let ${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)={\mathfrak{t}}_1$ and ${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)={\mathfrak{s}}_1$. Since $1\in \mathcal{U}({\Omega }^{+},{\mathfrak{t}}_1)\cap \mathcal{L}({\Omega }^{-},{\mathfrak{s}}_1)$, we have ${\Omega }^{+}\left(1\right)\ge {\mathfrak{t}}_1={\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)$ and ${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)\le {\mathfrak{s}}_1$.

For any ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$, let ${\mathfrak{t}}_1,{\mathfrak{t}}_2,{\mathfrak{s}}_1,{\mathfrak{s}}_2\in [0,1]$ be such that ${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)={\mathfrak{t}}_1$, ${\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})={\mathfrak{t}}_2$, ${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)={\mathfrak{s}}_1$ and ${\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})={\mathfrak{s}}_2$. Take $\mathfrak{t}=\min \{{\mathfrak{t}}_1,{\mathfrak{t}}_2\}$ and $\mathfrak{s}=\max \{{\mathfrak{s}}_1,{\mathfrak{s}}_2\}$. Then, both ${\hslash }_{{\mathfrak{a}}_1}$ and ${\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}$ belong to $\mathcal{U}({\Omega }^{+},\mathfrak{t})\cap \mathcal{L}({\Omega }^{-},\mathfrak{s})$. Consequently, ${\hslash }_{{\mathfrak{a}}_2}\in \mathcal{U}({\Omega }^{+},\mathfrak{t})\cap L({\Omega }^{-},\mathfrak{s})$ and, thus,

$${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ge t=\min \{{\mathfrak{t}}_1,{\mathfrak{t}}_2\}=\min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}$$

and

$${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\le \mathfrak{s}=\max \{{\mathfrak{s}}_1,{\mathfrak{s}}_2\}=\max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.$$

Therefore, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

The following theorem characterizes BFFs through conditions on their positive and negative membership functions, enhancing their applicability in fuzzy reasoning systems. The proof uses the algebraic properties of hoop operations to ensure consistency.

Theorem 4. 

A $B{F}_S$ $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ if and only if it satisfies Equation (9) and the following condition:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left\{\begin{array}{c}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})=\min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\},\hfill \\ {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})=\max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}.\hfill \end{array}\right.\hfill \end{array}$$

(13)

Proof. 

Assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ and let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$. Since ${\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_1}$ and ${\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}{\le }_{white\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}$, it follows from Equation (11) that

$${\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\le \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\},{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\ge \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}.$$

Since ${\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})$, we also have

$${\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}))\}\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}$$

and

$${\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2}))\}\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}.$$

Thus, by Equations (12) and (11), we prove (13).

Conversely, suppose that $\Omega =({\Omega }^{+},{\Omega }^{-})$ satisfies both Equation (9) and condition (13). Since ${\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}){\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}$ for all ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$, it follows from (9) and (13) that

$${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ge {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))=\min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}$$

and

$${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\le {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))=\max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.$$

Therefore, by Theorem 1, we conclude that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

The following theorem provides a necessary condition for a bipolar fuzzy set to be a BFF, offering a practical test for filter properties. The proof exploits the interplay between positive and negative memberships under hoop operations.

Theorem 5. 

A $B{F}_S$ $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ if and only if it satisfies Equation (9) and the following condition:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H})\left\{\begin{array}{c}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})\ge \min \{{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to y),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\},\hfill \\ {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})\le \max \{{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\}.\hfill \end{array}\right.\hfill \end{array}$$

(14)

Proof. 

Assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ and let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H}$. Since $({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\odot ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}){\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3}$, it follows from Equations (11) and (13) that

$$\begin{array}{cc}\hfill {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})& \ge {\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\odot (y\to {\hslash }_{{\mathfrak{a}}_3}))\hfill \\ \hfill & =\min \{{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})& \le {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\odot ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}))\hfill \\ \hfill & =\max \{{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\}\hfill \end{array}$$

This proves Equation (14).

Conversely, suppose that $\Omega =({\Omega }^{+},{\Omega }^{-})$ satisfies both Equation (9) and condition (14). If we set ${\hslash }_{{\mathfrak{a}}_1}=1$ in (14), we recover Equation (12). Thus, by Theorem 1, we conclude that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

The following theorem refines the characterization of BFFs by introducing additional conditions on membership degrees, strengthening their theoretical foundation. The proof combines hoop algebra axioms with bipolar fuzzy set constraints.

Theorem 6. 

A $B{F}_S$ $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ if and only if it satisfies Equation (9) and the following condition:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H})\left\{\begin{array}{c}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\odot {\hslash }_{{\mathfrak{a}}_3})\ge \min \{{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_3}),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\},\hfill \\ {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\odot {\hslash }_{{\mathfrak{a}}_3})\le \max \{{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_3}),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.\hfill \end{array}\right.\hfill \end{array}$$

(15)

Proof. 

Let us assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ and consider arbitrary elements ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H}$. It is noted that $({\hslash }_{{\mathfrak{a}}_3}\odot {\hslash }_{{\mathfrak{a}}_1})\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})={\hslash }_{{\mathfrak{a}}_3}\odot ({\hslash }_{{\mathfrak{a}}_1}\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})){\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_3}\odot {\hslash }_{{\mathfrak{a}}_2}$. Applying Equations (11) and (13), we obtain the following inequalities:

$$\begin{array}{cc}\hfill {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_3}\odot {\hslash }_{{\mathfrak{a}}_2})& \ge {\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_3}\odot {\hslash }_{{\mathfrak{a}}_1})\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\hfill \\ \hfill & =\min \{{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_3}\odot {\hslash }_{{\mathfrak{a}}_1}),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_3}\odot {\hslash }_{{\mathfrak{a}}_2})& \le {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_3}\odot {\hslash }_{{\mathfrak{a}}_1})\odot ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\hfill \\ \hfill & =\max \{{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_3}\odot {\hslash }_{{\mathfrak{a}}_1}),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.\hfill \end{array}$$

This proves Equation (15).

Conversely, suppose that $\Omega =({\Omega }^{+},{\Omega }^{-})$ satisfies both Equation (9) and condition (15). Setting ${\hslash }_{{\mathfrak{a}}_3}=1$ in (15) leads to Equation (12). Therefore, by Theorem 1, we conclude that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

Theorem 7. 

A $B{F}_S$ $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ if and only if it satisfies the following condition:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H})\left({\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}\Rightarrow \left\{\begin{array}{c}{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_3}\right)\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\},\hfill \\ {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_3}\right)\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}.\hfill \end{array}\right.\right)\hfill \end{array}$$

(16)

Proof. 

Assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ and let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H}$ such that ${\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}$. Since ${\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})=1$, it follows that

$$\begin{array}{cc}\hfill {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})& \ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}))\}\hfill \\ \hfill & =\min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left(1\right)\}\hfill \\ \hfill & ={\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})& \le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}))\}\hfill \\ \hfill & =\max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left(1\right)\}\hfill \\ \hfill & ={\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),\hfill \end{array}$$

which follows from Equation (12). Consequently, we obtain the inequalities

$${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_3}\right)\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\}\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}$$

and

$${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_3}\right)\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\}\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}.$$

Conversely, suppose that $\Omega =({\Omega }^{+},{\Omega }^{-})$ satisfies both Equation (9) and condition (16). Since for all ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$, we have ${\hslash }_{{\mathfrak{a}}_1}{\le }_{\mathcal{H}}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2}$, it follows from (16) that

$${\Omega }^{+}\left(y\right)\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\},{\Omega }^{-}\left(y\right)\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.$$

Thus, by Theorem 1, we conclude that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

Theorem 8. 

A $B{F}_S$ $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$ if and only if it satisfies (9) and the following condition:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})\ge \min \left\{{\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_3}),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\right\},\hfill \\ {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})\le \max \left\{{\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_3}),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\right\}.\hfill \end{array}\right)\hfill \end{array}$$

(17)

Proof. 

Let us assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. By definition, for all ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H}$, the following holds: Since $({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_3}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}$, for any ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H}$, the condition ${\hslash }_{{\mathfrak{a}}_2}\odot (({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_3}){\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_2}\odot ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}){\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_3}{\le }_{\mathcal{H}}{\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3}$ holds. Therefore, we have the following inequalities:

$$\begin{array}{ccc}\hfill {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})& \ge & {\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_3}\right)\hfill \\ \hfill & \ge & {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\odot ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}))\hfill \\ \hfill & =& \min \left\{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\right\}\hfill \\ \hfill & \ge & \min \left\{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_3})\right\}\hfill \end{array}$$

and, similarly for the negative part,

$$\begin{array}{ccc}\hfill {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})& \le & {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_3}\right)\hfill \\ \hfill & \le & {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\odot ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}))\hfill \\ \hfill & =& \max \left\{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\right\}\hfill \\ \hfill & \le & \max \left\{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_3})\right\}.\hfill \end{array}$$

This confirms that the structure $\Omega$ satisfies the condition in (17).

Conversely, assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ satisfies both (9) and (17). By substituting ${\hslash }_{{\mathfrak{a}}_1}=1$ in (17), we obtain the condition in (12). Thus, by applying Theorem 1, we conclude that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is indeed a BFF of $\mathcal{H}$. □

Definition 9. 

A $B{F}_S$ $\Omega =({\Omega }^{+},{\Omega }^{-})$ is called an implicative bipolar fuzzy filter (briefly, IBFF) of a hoop algebra $\mathcal{H}$ if it satisfies the condition given by Equation (9) and

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}(x\to (({\hslash }_{{\mathfrak{a}}_2}\to z)\to y))\}\hfill \\ {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}(x\to (({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}))\}\hfill \end{array}\right).\hfill \end{array}$$

(18)

Example 4. 

Let $\mathcal{H}=\{0,1,2,3,4,5,6\}$ be a set with the binary operations · and → given in the following Table 6 and Table 7.

Table 6. Cayley table of the binary operation →.

→	0	1	2	3	4	5
0	1	1	1	4	4	5
1	0	1	2	3	4	5
2	0	1	1	3	4	5
3	1	1	1	1	1	5
4	0	1	1	0	1	5
5	1	1	1	1	1	1

Table 7. Cayley table of the binary operation ⊙.

⊙	0	1	2	3	4	5
0	0	0	0	3	3	5
1	0	1	2	3	4	5
2	0	2	2	3	4	5
3	3	3	3	3	3	5
4	3	4	4	3	4	5
5	5	5	5	5	5	5

Then, $(\mathcal{H},\odot ,\to ,1)$ is a hoop algebra. Let $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ be a BFS in $\mathcal{H}$ given by Table 8.

Table 8. Tabular representation of $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$.

$\mathcal{H}$	0	1	2	3	4	5
${\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)$	$-0.83$	$-0.83$	$-0.83$	$-0.83$	$-0.83$	$-0.7$
${\Omega }^{+}\left({\hslash }_{\mathfrak{a}}\right)$	$0.7$	$0.7$	$0.7$	$0.7$	$0.7$	$0.6$

Then, it is easy to check that the BFS $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ in $\mathcal{H}$ is an IBFF of $\mathcal{H}$.

The following theorem introduces implicative bipolar fuzzy filters (IBFFs) as a specialized subset of BFFs, enriching the filter hierarchy.

Theorem 9. 

Every IBFF is a BFF.

Proof. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be an IBFF of $\mathcal{H}$. If we take ${\hslash }_{{\mathfrak{a}}_3}=1$ in Equation (18) and use Proposition 1 (a5), then we obtain (12). Therefore, by Theorem 1, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

The converse of Theorem 9 may not be true, as seen in the following example.

Example 5. 

Consider the BFF $\Omega =(\mathcal{H},{\Omega }^{-},{\Omega }^{+})$ of a hoop algebra $(\mathcal{H},\odot ,\to ,1)$ in Example 2. However, it is not an IBFF of $\mathcal{H}$ since

$$\begin{array}{ccc}{\Omega }^{-}\left(5\right)=-0.7\nleqq -0.8& =& \max \{-0.8,-0.8\}\hfill \\ & =& \max \{{\Omega }^{-}\left(1\right),{\Omega }^{-}\left(2\right)\}\hfill \\ & =& \max \{{\Omega }^{-}\left(4\right),{\Omega }^{-}(4\to ((5\to 0)\to 5))\}\hfill \end{array}$$

and

$$\begin{array}{ccc}{\Omega }^{+}\left(5\right)=0.7\ngeqq 0.8& =& \min \{0.8,0.8\}\hfill \\ & =& \max \{{\Omega }^{-}\left(1\right),{\Omega }^{-}\left(2\right)\}\hfill \\ & =& \min \{{\Omega }^{+}\left(1\right),{\Omega }^{+}(1\to ((5\to 6)\to 5))\}.\hfill \end{array}$$

Proposition 5. 

Every IBFF $\Omega =({\Omega }^{+},{\Omega }^{-})$ of $\mathcal{H}$ satisfies the following assertions.

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\le {\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)\hfill \\ {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\ge {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)\hfill \end{array}\right).\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})={\Omega }^{+}\left(1\right)\hfill \\ {\Omega }^{-}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})={\Omega }^{-}\left(1\right)\hfill \end{array}\right).\hfill \end{array}$$

(20)

Proof. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be an IBFF of a hoop algebra H. If we set ${\hslash }_{{\mathfrak{a}}_2}={\hslash }_{{\mathfrak{a}}_1}$, ${\hslash }_{{\mathfrak{a}}_1}=1$ and ${\hslash }_{{\mathfrak{a}}_3}={\hslash }_{{\mathfrak{a}}_2}$ in Equation (18) and use Proposition 1 (a5) and (9), then we obtain (19). Using (19), Definition 2 (H1), Proposition 1 (a5), (a7), (a9), and (11), we obtain

$$\begin{array}{cc}& {\Omega }^{+}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})\hfill \\ & \ge {\Omega }^{+}((((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_2})\to ((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}))\hfill \\ & ={\Omega }^{+}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to ((((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1}))\hfill \\ & \ge {\Omega }^{+}((((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_2})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\hfill \\ & \ge {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to ((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}))\hfill \\ & ={\Omega }^{+}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_1}))\hfill \\ & ={\Omega }^{+}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to 1)\hfill \\ & ={\Omega }^{+}\left(1\right)\hfill \end{array}$$

and

$$\begin{array}{cc}& {\Omega }^{-}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})\hfill \\ & \le {\Omega }^{-}((((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_2})\to ((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}))\hfill \\ & ={\Omega }^{-}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to ((((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1}))\hfill \\ & \le {\Omega }^{-}((((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_2})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\hfill \\ & \le {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to ((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to x))\hfill \\ & ={\Omega }^{-}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_1}))\hfill \\ & ={\Omega }^{-}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to 1)\hfill \\ & ={\Omega }^{-}\left(1\right),\hfill \end{array}$$

for all ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$. It follows from (9) that we have (20). □

Proposition 6. 

Let $\mathcal{H}$ be a bounded hoop algebra. Then, every IBFF $\Omega =({\Omega }^{+},{\Omega }^{-})$ of $\mathcal{H}$ satisfies the following assertions:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})={\Omega }^{+}\left(1\right)\hfill \\ {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1})={\Omega }^{-}\left(1\right)\hfill \end{array}\right)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\ge {\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2})\hfill \\ {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\le {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2})\hfill \end{array}\right)\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H})\left(\begin{array}{c}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})\ge \min \{{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_3}\to {\hslash }_{{\mathfrak{a}}_2}))\}\hfill \\ {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})\le \max \{{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3}),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_3}\to {\hslash }_{{\mathfrak{a}}_2}))\}\hfill \end{array}\right).\hfill \end{array}$$

(23)

Proof. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be an IBFF of a bounded hoop algebra $\mathcal{H}$. Then, by Theorem 9, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$.

If we take ${\hslash }_{{\mathfrak{a}}_2}=0$ in Equation (20), then

$${\Omega }^{+}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})={\Omega }^{+}((({\hslash }_{\mathfrak{a}}\to 0)\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{{\mathfrak{a}}_1})={\Omega }^{+}\left(1\right),$$

$${\Omega }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})={\Omega }^{-}((({\hslash }_{\mathfrak{a}}\to 0)\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{{\mathfrak{a}}_1})={\Omega }^{-}\left(1\right)$$

for all ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$.

Note that ${\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}){\le }_{\mathcal{H}}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})$ for all ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$. It follows from Definition 2 (H3) and Equation (11) that

$${\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2})={\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\le {\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})),$$

$$\begin{array}{cc}\hfill & {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2})={\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\ge {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})).\hfill \end{array}$$

(24)

Combining Proposition 1 (a5) and Equations (9), (18) and (24), we obtain

$$\begin{array}{ccc}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})& \ge & \min \left\{{\Omega }^{+}\left(1\right),{\Omega }^{+}(1\to ((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to 0)\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})))\right\}\hfill \\ & =& {\Omega }^{+}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to 0)\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\hfill \\ & =& {\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\hfill \\ & \ge & {\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2})\hfill \end{array}$$

and

$$\begin{array}{ccc}{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})& \le & \max \left\{{\Omega }^{-}\left(1\right),{\Omega }^{-}(1\to ((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to 0)\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})))\right\}\hfill \\ & =& {\Omega }^{-}((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to 0)\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\hfill \\ & =& {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2}))\hfill \\ & \le & {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2}).\hfill \end{array}$$

This proves Equation (22).

Using (HP3) and Equations (14) and (22), we have

$$\begin{array}{ccc}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})& \ge & {\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_3})\hfill \\ & \ge & \min \left\{{\Omega }^{+}((x\odot z)\to {\hslash }_{{\mathfrak{a}}_2}),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\right\}\hfill \\ & =& \min \left\{{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_3}\to {\hslash }_{{\mathfrak{a}}_2})),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\right\}\hfill \end{array}$$

and

$$\begin{array}{ccc}{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_3})& \le & {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_3})\hfill \\ & \le & \max \left\{{\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\right\}\hfill \\ & =& \max \left\{{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_3}\to {\hslash }_{{\mathfrak{a}}_2})),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\right\},\hfill \end{array}$$

which proves Equation (23). □

Theorem 10. 

Let $\mathcal{H}$ be a hoop algebra. The $B{F}_S$ $\Omega =({\Omega }^{+},{\Omega }^{-})$ is an IBFF of $\mathcal{H}$ if and only if its nonempty bipolar fuzzy level sets $\mathcal{U}({\Omega }^{+},\mathfrak{t})$ and $\mathcal{L}({\Omega }^{-},\mathfrak{s})$ are implicative filters of $\mathcal{H}$ for all $(\mathfrak{t},\mathfrak{s})\in [0,1]\times [0,1]$ with $\mathfrak{t}+\mathfrak{s}\le 1$.

Proof. 

The proof follows similarly to that of Theorem 3. □

We provide conditions under which a BFF is an IBFF.

Theorem 11. 

If a BFF $\Omega =({\Omega }^{+},{\Omega }^{-})$ of $\mathcal{H}$ satisfies condition (20), then it is an IBFF of $\mathcal{H}$.

Proof. 

Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2},{\hslash }_{{\mathfrak{a}}_3}\in \mathcal{H}$. Then, we have

$$\begin{array}{ccc}\hfill {\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)& \ge & \min \left\{{\Omega }^{+}\left((({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{+}\left(({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}\right)\right\}\hfill \\ & =& \min \left\{{\Omega }^{+}\left(1\right),{\Omega }^{+}\left(({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}\right)\right\}\hfill \\ & =& {\Omega }^{+}\left(({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}\right)\hfill \\ & \ge & \min \left\{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\to \left(({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}\right)\right)\right\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)& \le & \max \left\{{\Omega }^{-}\left((({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2}\right),{\Omega }^{-}\left(({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}\right)\right\}\hfill \\ & =& \max \left\{{\Omega }^{-}\left(1\right),{\Omega }^{-}\left(({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}\right)\right\}\hfill \\ & =& {\Omega }^{-}\left(({\hslash }_{{\mathfrak{a}}_2}\to z)\to {\hslash }_{{\mathfrak{a}}_2}\right)\hfill \\ & \le & \max \left\{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\to \left(({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_3})\to {\hslash }_{{\mathfrak{a}}_2}\right)\right)\right\}.\hfill \end{array}$$

This follows from (9), (12) and (20). Therefore, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is an IBFF of $\mathcal{H}$. □

Theorem 12. 

If a BFF $\Omega =({\Omega }^{+},{\Omega }^{-})$ of $\mathcal{H}$ satisfies condition (19), then it is an IBFF of $\mathcal{H}$.

Proof. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be a BFF of $\mathcal{H}$ that satisfies (19). As shown in Proposition 5, (19) implies (20). Thus, by Theorem 11, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is an IBFF of $\mathcal{H}$. □

Theorem 13. 

If a BFF $\Omega =({\Omega }^{+},{\Omega }^{-})$ of $\mathcal{H}$ satisfies condition (21), then it is an IBFF of $\mathcal{H}$.

Proof. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be a BFF of $\mathcal{H}$ that satisfies (21). Note that for all ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$,

$$({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}\le (({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}.$$

By using (9), (11) and (21), we obtain the following:

$$\begin{array}{ccc}\hfill {\Omega }^{+}\left((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}\right)& \le & {\Omega }^{+}\left(1\right)\hfill \\ & =& {\Omega }^{+}\left(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}\right)\hfill \\ & \le & {\Omega }^{+}\left((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\Omega }^{-}\left((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}\right)& \ge & {\Omega }^{-}\left(1\right)\hfill \\ & =& {\Omega }^{-}\left(({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}\right)\hfill \\ & \ge & {\Omega }^{-}\left((({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_1})\to {\hslash }_{{\mathfrak{a}}_1}\right).\hfill \end{array}$$

Thus, (20) holds and, therefore, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is an IBFF of $\mathcal{H}$ by Theorem 11. □

Theorem 14. 

If a BFF $\Omega =({\Omega }^{+},{\Omega }^{-})$ of $\mathcal{H}$ satisfies condition (22), then it is an IBFF of $\mathcal{H}$.

Proof. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be a BFF of $\mathcal{H}$ that satisfies condition (22). For any ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$, we have the following:

$$\begin{array}{ccc}{\Omega }^{+}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})& \le & {\Omega }^{+}\left(1\right)\hfill \\ & =& {\Omega }^{+}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{\mathfrak{a}}))\hfill \\ & =& {\Omega }^{+}(({\hslash }_{\mathfrak{a}}\to x)\to {\hslash }_{\mathfrak{a}})\hfill \\ & \le & {\Omega }^{+}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})\hfill \end{array}$$

and, similarly,

$$\begin{array}{ccc}{\Omega }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})& \ge & {\Omega }^{-}\left(1\right)\hfill \\ & =& {\Omega }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to ({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}}))\hfill \\ & =& {\Omega }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})\hfill \\ & \ge & {\Omega }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}}).\hfill \end{array}$$

These inequalities follow from Definition 2 (H1), (H3), (9), and (22). Therefore, (21) holds and, thus, by Theorem 13, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is an IBFF of $\mathcal{H}$. □

Theorem 15. 

If a BFF $\Omega =({\Omega }^{+},{\Omega }^{-})$ of $\mathcal{H}$ satisfies condition (23), then it is an IBFF of $\mathcal{H}$.

Proof. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be a BFF of $\mathcal{H}$ that satisfies condition (23). Condition (23) implies that

$$\begin{array}{ccc}{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})& \ge & \min \{{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_2}),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_2}))\}\hfill \\ & =& \min \{{\Omega }^{+}\left(1\right),{\Omega }^{+}((x\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ & =& {\Omega }^{+}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2})\hfill \end{array}$$

and, similarly,

$$\begin{array}{ccc}{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})& \le & \max \{{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_2}),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to ({\hslash }_{{\mathfrak{a}}_2}\to {\hslash }_{{\mathfrak{a}}_2}))\}\hfill \\ & =& \max \{{\Omega }^{-}\left(1\right),{\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ & =& {\Omega }^{-}(({\hslash }_{{\mathfrak{a}}_1}\odot {\hslash }_{{\mathfrak{a}}_2})\to {\hslash }_{{\mathfrak{a}}_2}).\hfill \end{array}$$

Thus, (22) is valid and, therefore, by Theorem 14, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is an IBFF of $\mathcal{H}$. □

Theorem 16. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ and $\varrho =({\varrho }^{+},{\varrho }^{-})$ be BFFs of a hoop algebra $\mathcal{H}$ such that

$$\begin{array}{cc}\hfill & {\Omega }^{+}\left(1\right)={\varrho }^{+}\left(1\right),{\Omega }^{-}\left(1\right)={\varrho }^{-}\left(1\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & {\Omega }^{+}\left({\hslash }_{\mathfrak{a}}\right)\le {\varrho }^{+}\left({\hslash }_{\mathfrak{a}}\right),{\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)\ge {\varrho }^{-}\left({\hslash }_{\mathfrak{a}}\right),\hfill \end{array}$$

(26)

for all ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$. If $\Omega =({\Omega }^{+},{\Omega }^{-})$ is an IBFF of $\mathcal{H}$, then $\varrho =({\varrho }^{+},{\varrho }^{-})$ is also an IBFF of $\mathcal{H}$.

Proof. 

Assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is an IBFF of $\mathcal{H}$. By Theorem 9, this implies that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$.

For any ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$, we have

$${\Omega }^{+}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})\ge {\Omega }^{+}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})={\Omega }^{+}\left(1\right)={\varrho }^{+}\left(1\right)$$

and

$${\Omega }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})\le {\Omega }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})={\Omega }^{-}\left(1\right)={\varrho }^{-}\left(1\right),$$

where the equalities follow from (25), (26) and (21).

Since $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$, we know from (9) that

$${\varrho }^{+}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})\le {\varrho }^{+}\left(1\right)\mathrm{and}{\varrho }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})\ge {\varrho }^{-}\left(1\right).$$

Thus, for all ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$, we conclude

$${\varrho }^{+}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})={\varrho }^{+}\left(1\right)\mathrm{and}{\varrho }^{-}(({\hslash }_{\mathfrak{a}}\to {\hslash }_{\mathfrak{a}})\to {\hslash }_{\mathfrak{a}})={\varrho }^{-}\left(1\right).$$

This shows that $\varrho =({\varrho }^{+},{\varrho }^{-})$ satisfies the conditions of an IBFF of $\mathcal{H}$. Therefore, by Theorem 13, $\varrho =({\varrho }^{+},{\varrho }^{-})$ is an IBFF of $\mathcal{H}$. □

Proposition 7. 

If ${\Omega }_{\mathfrak{k}}=({\Omega }_{\mathfrak{k}}^{+},{\Omega }_{\mathfrak{k}}^{-})$ for all $\mathfrak{k}\in \Gamma$ is a family of bipolar fuzzy filters of a hoop algebra $\mathcal{H}$, then

$$\underset{\mathfrak{k}\in \Gamma }{\bigwedge }{\Omega }_{\mathfrak{k}}=(\underset{\mathfrak{k}\in \Gamma }{\bigwedge }{\Omega }_{\mathfrak{k}}^{+},\underset{\mathfrak{k}\in \Gamma }{\bigwedge }{\Omega }_{\mathfrak{k}}^{-})$$

is a BFF of $\mathcal{H}$.

Proof. 

Let ${\Omega }_{\mathfrak{k}}=\{({\Omega }_{\mathfrak{k}}^{+},{\Omega }_{\mathfrak{k}}^{-}):\mathfrak{k}\in \Gamma \}$ be a family of bipolar fuzzy filters of a hoop $\mathcal{H}$.

Let ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$, and we have

$$\left(\underset{\mathfrak{k}\in \Gamma }{\bigwedge }{\Omega }_{\mathfrak{k}}^{+}\right)\left(1\right)=\underset{\mathfrak{k}\in \Gamma }{\inf }\left\{{\Omega }_{\mathfrak{k}}^{+}\left(1\right)\right\}\ge \underset{\mathfrak{k}\in \Gamma }{\inf }\left\{{\Omega }_{\mathfrak{k}}^{+}\left({\hslash }_{\mathfrak{a}}\right)\right\}=\left(\underset{\mathfrak{k}\in \Gamma }{\bigwedge }{\Omega }_{\mathfrak{k}}^{+}\right)\left({\hslash }_{\mathfrak{a}}\right),$$

$$\left(\underset{\mathfrak{k}\in \Gamma }{\bigwedge }{\Omega }_{\mathfrak{k}}^{-}\right)\left(1\right)=\underset{\mathfrak{k}\in \Gamma }{\sup }\left\{{\Omega }_{\mathfrak{k}}^{-}\left(1\right)\right\}\le \underset{\mathfrak{k}\in \Gamma }{\sup }\left\{{\Omega }_{\mathfrak{k}}^{-}\left({\hslash }_{\mathfrak{a}}\right)\right\}=\left(\underset{\mathfrak{k}\in \Gamma }{\bigwedge }{\Omega }_{\mathfrak{k}}^{-}\right)\left({\hslash }_{\mathfrak{a}}\right).$$

Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$. Then, we have

$$\begin{array}{ccc}\left({\displaystyle \underset{\mathfrak{k}\in \Gamma }{\bigwedge }}{\Omega }_{\mathfrak{k}}^{+}\right)\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& \underset{\mathfrak{k}\in \Gamma }{\inf }\left\{{\Omega }_{\mathfrak{k}}^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\right\}\hfill \\ & \ge & \underset{\mathfrak{k}\in \Gamma }{\inf }\left\{\min \{{\Omega }_{\mathfrak{k}}^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }_{\mathfrak{k}}^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\right\}\hfill \\ & =& \min \{\underset{\mathfrak{k}\in \Gamma }{\inf }{\Omega }_{\mathfrak{k}}^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),\underset{\mathfrak{k}\in \Gamma }{\inf }{\Omega }_i^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ & =& \min \{\left({\displaystyle \underset{\mathfrak{k}\in \Gamma }{\bigwedge }}{\Omega }_{\mathfrak{k}}^{+}\right)\left({\hslash }_{{\mathfrak{a}}_1}\right),\left({\displaystyle \underset{\mathfrak{k}\in \Gamma }{\bigwedge }}{\Omega }_{\mathfrak{k}}^{+}\right)({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\left({\displaystyle \underset{\mathfrak{k}\in \Gamma }{\bigwedge }}{\Omega }_{\mathfrak{k}}^{-}\right)\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& \underset{\mathfrak{k}\in \Gamma }{\sup }\left\{{\Omega }_{\mathfrak{k}}^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\right\}\hfill \\ & \le & \underset{\mathfrak{k}\in \Gamma }{\sup }\left\{\max \{{\Omega }_{\mathfrak{k}}^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }_{\mathfrak{k}}^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\right\}\hfill \\ & =& \max \{\underset{\mathfrak{k}\in \Gamma }{\sup }{\Omega }_{\mathfrak{k}}^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),\underset{\mathfrak{k}\in \Gamma }{\sup }{\Omega }_{\mathfrak{k}}^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ & =& \max \{\left({\displaystyle \underset{\mathfrak{k}\in \Gamma }{\bigwedge }}{\Omega }_{\mathfrak{k}}^{-}\right)\left({\hslash }_{{\mathfrak{a}}_1}\right),\left({\displaystyle \underset{\mathfrak{k}\in \Gamma }{\bigwedge }}{\Omega }_{\mathfrak{k}}^{-}\right)({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.\hfill \end{array}$$

Hence, ${\displaystyle \underset{i\in \Gamma }{\bigwedge }}{\Omega }_i$ is a BFF of a hoop $\mathcal{H}$. □

Lemma 1. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be a BFF of a hoop algebra $\mathcal{H}$ if and only if ${\Omega }^{+}$ and $\overline{{\Omega }^{-}}$ are fuzzy filters of $\mathcal{H}$.

Proof. 

Assume that $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. Then, by definition, ${\Omega }^{+}$ is a fuzzy filter of $\mathcal{H}$. For every ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$, we have

$$\overline{{\Omega }^{-}}\left(1\right)=-1-{\Omega }^{-}\left(1\right)\ge -1-{\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)=\overline{{\Omega }^{-}}\left({\hslash }_{\mathfrak{a}}\right).$$

Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$. Then, we have

$$\begin{array}{ccc}\hfill \overline{{\Omega }^{-}}\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& -1-{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\hfill \\ & \ge & -1-\max \{{\Omega }^{-}\left(x\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ & =& \min \{-1-{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),-1-{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ & =& \min \{\overline{{\Omega }^{-}}\left({\hslash }_{{\mathfrak{a}}_1}\right),\overline{{\Omega }^{-}}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.\hfill \end{array}$$

Thus, $\overline{{\Omega }^{-}}$ is a fuzzy filter of $\mathcal{H}$.

Conversely, assume that ${\Omega }^{+}$ and $\overline{{\Omega }^{-}}$ are fuzzy filters of $\mathcal{H}$. Then, for every ${\hslash }_{\mathfrak{a}}\in \mathcal{H}$, we have ${\Omega }^{+}\left(1\right)\ge {\Omega }^{+}\left({\hslash }_{\mathfrak{a}}\right)$ and

$$-1-{\Omega }^{-}\left(1\right)=\overline{{\Omega }^{-}}\left(1\right)\ge \overline{{\Omega }^{-}}\left({\hslash }_{\mathfrak{a}}\right)=-1-{\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right),$$

i.e., ${\Omega }^{-}\left(1\right)\le {\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)$. Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{H}$. Then, we have

$${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.$$

Also,

$$\begin{array}{ccc}\hfill -1-{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& \overline{{\Omega }^{-}}\left({\hslash }_{{\mathfrak{a}}_2}\right)\hfill \\ & \ge & \min \{\overline{{\Omega }^{-}}\left({\hslash }_{{\mathfrak{a}}_1}\right),\overline{{\Omega }^{-}}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ & =& \min \{-1-{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),-1-{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \\ & =& -1-\max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.\hfill \end{array}$$

Thus, ${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}$. Therefore, $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

Theorem 17. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be a BFF of $\mathcal{H}$ if and only if $({\Omega }^{+},\overline{{\Omega }^{+}})$ and $({\Omega }^{-},\overline{{\Omega }^{-}})$ are fuzzy filters of $\mathcal{H}$.

Proof. 

If a BFS $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$, then by Lemma 1, ${\Omega }^{+}$ and ${\Omega }^{-}$ are fuzzy filters of $\mathcal{H}$. Hence, $({\Omega }^{+},\overline{{\Omega }^{+}})$ and $({\Omega }^{-},\overline{{\Omega }^{-}})$ are bipolar fuzzy filters of $\mathcal{H}$.

Conversely, if $({\Omega }^{+},\overline{{\Omega }^{+}})$ and $({\Omega }^{-},\overline{{\Omega }^{-}})$ are fuzzy filters of $\mathcal{H}$, then ${\Omega }^{+}$ and $\overline{{\Omega }^{-}}$ are fuzzy filters of $\mathcal{H}$. Therefore, the BFS $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{H}$. □

Definition 10. 

Let $\Omega =({\Omega }^{+},{\Omega }^{-})$ be a BFS in $\mathcal{H}$. We define the subset ${\Omega }^{-1}(1,1)$ of $\mathcal{H}$ by

$${\Omega }^{-1}(1,1)=\{{\hslash }_{\mathfrak{a}}\in \mathcal{H}\mid {\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)={\Omega }^{-}\left(1\right)\mathit{and}{\Omega }^{+}\left(x\right)={\Omega }^{+}\left(1\right)\}.$$

Theorem 18. 

If $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a bipolar fuzzy sub-hoop of the hoop $\mathcal{H}$, then ${\Omega }^{-1}(1,1)$ is a sub-hoop of $\mathcal{H}$.

Proof. 

Clearly, $1\in {\Omega }^{-1}(1,1)$. Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in {\Omega }^{-1}(1,1)$. Then, ${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)={\Omega }^{-}\left(1\right)$, ${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)={\Omega }^{+}\left(1\right)$, ${\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)={\Omega }^{-}\left(1\right)$, and ${\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)={\Omega }^{+}\left(1\right)$. Thus, for $•\in \{\odot ,\to \}$, we have

$${\Omega }^{-}\left(1\right)\le {\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\le \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}=\max \{{\Omega }^{-}\left(1\right),{\Omega }^{-}\left(1\right)\}={\Omega }^{-}\left(1\right)$$

and

$${\Omega }^{+}\left(1\right)\ge {\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})\ge \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)\}=\min \{{\Omega }^{+}\left(1\right),{\Omega }^{+}\left(1\right)\}={\Omega }^{+}\left(1\right).$$

Thus, ${\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})={\Omega }^{-}\left(1\right)$ and ${\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})={\Omega }^{+}\left(1\right)$, which means that ${\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2}\in {\Omega }^{-1}(1,1)$. Therefore, ${\Omega }^{-1}(1,1)$ is a sub-hoop of $\mathcal{H}$. □

A mapping $\alpha :\mathcal{X}\to \mathcal{Y}$ of hoops is called a homomorphism if

$$\vartheta ({\hslash }_{{\mathfrak{a}}_1}•{\hslash }_{{\mathfrak{a}}_2})=\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)•\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right)$$

for all ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{X}$ and $•\in \{\odot ,\to \}$. Note that if $\vartheta :\mathcal{X}\to \mathcal{Y}$ is a homomorphism of hoops, then $\vartheta \left(1\right)=1$.

Let $\vartheta :\mathcal{X}\to \mathcal{Y}$ be a homomorphism of hoops. For any BFS $\Omega =({\Omega }^{+},{\Omega }^{-})$ in $\mathcal{X}$, we define a new BFS $\vartheta (\Omega )=({\vartheta }_{\sup }\left({\Omega }^{+}\right),{\vartheta }_{\inf }\left({\Omega }^{-}\right))$ in $\mathcal{Y}$, where

$${\vartheta }_{\sup }\left({\Omega }^{+}\right)\left({\hslash }_{{\mathfrak{a}}_2}\right)=\left\{\begin{array}{cc}\underset{{\hslash }_{{\mathfrak{a}}_1}\in {\vartheta }^{-1}\left({\hslash }_{{\mathfrak{a}}_2}\right)}{\sup }{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)\hfill & \mathrm{if}{\vartheta }^{-1}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ne \varnothing ,\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

and

$${\vartheta }_{\inf }\left({\Omega }^{-}\right)\left({\hslash }_{{\mathfrak{a}}_2}\right)=\left\{\begin{array}{cc}\underset{{\hslash }_{{\mathfrak{a}}_1}\in {\vartheta }^{-1}\left({\hslash }_{{\mathfrak{a}}_2}\right)}{\inf }{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)\hfill & \mathrm{if}{\vartheta }^{-1}\left({\hslash }_{{\mathfrak{a}}_2}\right)\ne \varnothing ,\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

for each ${\hslash }_{{\mathfrak{a}}_2}\in \mathcal{Y}$.

For any BFS $\Omega =({\Omega }^{+},{\Omega }^{-})$ in $\mathcal{Y}$, we define a new BFS ${\vartheta }^{-1}(\Omega )=\left\{\right({\hslash }_{\mathfrak{a}},{\vartheta }^{-1}\left({\Omega }^{+}\right)\left({\hslash }_{\mathfrak{a}}\right)$, ${\vartheta }^{-1}\left({\Omega }^{-}\right)\left({\hslash }_{\mathfrak{a}}\right))\mid {\hslash }_{\mathfrak{a}}\in \mathcal{X}\}$.

The following theorem demonstrates that BFFs are preserved under hoop homomorphisms, ensuring their applicability in transformed algebraic systems.

Theorem 19. 

Let $\vartheta :\mathcal{X}\to \mathcal{Y}$ be a homomorphism of hoops. If a BFS $\Omega =(\mathcal{X},{\Omega }^{-},{\Omega }^{+})$ is a BFF of $\mathcal{X}$, then $\vartheta (\Omega )=\{({\hslash }_{\mathfrak{a}},{\vartheta }_{\sup }\left({\Omega }^{+}\right)\left({\hslash }_{\mathfrak{a}}\right),{\alpha }_{\inf }\left({\Omega }^{-}\right)\left({\hslash }_{\mathfrak{a}}\right))\mid {\hslash }_{\mathfrak{a}}\in \mathcal{Y}\}$ is a BFF of $\mathcal{Y}$.

Proof. 

For all ${\hslash }_{\mathfrak{a}}\in white\mathcal{X}$, we have

$$\vartheta \left({\Omega }^{+}\right)\left({\hslash }_{\mathfrak{a}}\right)={\Omega }^{+}\left(\vartheta \left({\hslash }_{\mathfrak{a}}\right)\right)\le {\Omega }^{+}\left(1\right)={\Omega }^{+}\left(\vartheta \left(1\right)\right)=\vartheta \left({\Omega }^{+}\right)\left(1\right),$$

$$\vartheta \left({\Omega }^{-}\right)\left({\hslash }_{\mathfrak{a}}\right)={\Omega }^{-}\left(\vartheta \left({\hslash }_{\mathfrak{a}}\right)\right)\ge {\Omega }^{-}\left(1\right)={\Omega }^{-}\left(\vartheta \left(1\right)\right)=\vartheta \left({\Omega }^{-}\right)\left(1\right).$$

Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{X}$. Then,

$$\begin{array}{ccc}\vartheta \left({\Omega }^{+}\right)\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& {\Omega }^{+}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right)\right)\hfill \\ & \ge & \min \{{\Omega }^{+}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right),{\Omega }^{+}\left(\vartheta ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\right)\}\hfill \\ & =& \min \{\vartheta \left({\Omega }^{+}\right)\left({\hslash }_{{\mathfrak{a}}_1}\right),\vartheta \left({\Omega }^{+}\right)({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\},\hfill \end{array}$$

$$\begin{array}{ccc}\vartheta \left({\Omega }^{-}\right)\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& {\Omega }^{-}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right)\right)\hfill \\ & \le & \max \{{\Omega }^{-}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right),{\Omega }^{-}\left(\vartheta ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\right)\}\hfill \\ & =& \max \{\vartheta \left({\Omega }^{-}\right)\left({\hslash }_{{\mathfrak{a}}_1}\right),\vartheta \left({\Omega }^{-}\right)({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.\hfill \end{array}$$

Hence, $\vartheta (\Omega )=(\vartheta \left({\Omega }^{+}\right),\vartheta \left({\Omega }^{-}\right))$ is a BFF of $\mathcal{Y}$. □

The following theorem extends the preservation of BFSH properties under homomorphisms, reinforcing their algebraic stability.

Theorem 20. 

Let $\vartheta :\mathcal{X}\to \mathcal{Y}$ be an epimorphism of hoops and let $\Omega =({\Omega }^{-},{\Omega }^{+})$ be a BFS in $\mathcal{X}$. If $\vartheta (\Omega )=(\vartheta \left({\Omega }^{+}\right),\vartheta \left({\Omega }^{-}\right))$ is a BFF of $\mathcal{Y}$, then $\Omega =({\Omega }^{-},{\Omega }^{+})$ is a BFF of $\mathcal{X}$.

Proof. 

For any ${\hslash }_{\mathfrak{a}}\in \mathcal{Y}$, there exist $\mathfrak{p}\in \mathcal{X}$ such that $\vartheta \left(\mathfrak{p}\right)={\hslash }_{\mathfrak{a}}$. Then,

$${\Omega }^{+}\left({\hslash }_{\mathfrak{a}}\right)={\Omega }^{+}\left(\vartheta \left(\mathfrak{p}\right)\right)=\alpha \left({\Omega }^{+}\right)\left(\mathfrak{p}\right)\le \alpha \left({\Omega }^{+}\right)\left(1\right)={\Omega }^{+}\left(\vartheta \left(1\right)\right)={\Omega }^{+}\left(1\right),$$

$${\Omega }^{-}\left({\hslash }_{\mathfrak{a}}\right)={\Omega }^{-}\left(\vartheta \left(\mathfrak{p}\right)\right)=\vartheta \left({\Omega }^{-}\right)\left(\mathfrak{p}\right)\ge \alpha \left({\Omega }^{-}\right)\left(1\right)={\Omega }^{-}\left(\vartheta \left(1\right)\right)={\Omega }^{-}\left(1\right).$$

Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{Y}$. Then, there exist $\mathfrak{p},\mathfrak{q}\in \mathcal{X}$ such that $\vartheta \left(\mathfrak{p}\right)={\hslash }_{{\mathfrak{a}}_1}$, $\vartheta \left(\mathfrak{q}\right)={\hslash }_{{\mathfrak{a}}_2}$. It follows that

$$\begin{array}{ccc}{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& {\Omega }^{+}\left(\vartheta \left(\mathfrak{q}\right)\right)\hfill \\ & \ge & \min \{{\Omega }^{+}\left(\vartheta \left(\mathfrak{p}\right)\right),{\Omega }^{+}(\vartheta \left(\mathfrak{p}\right)\to \vartheta \left(\mathfrak{q}\right))\}\hfill \\ & =& \min \{{\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}\hfill \end{array}$$

and

$$\begin{array}{ccc}{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& {\Omega }^{-}\left(\vartheta \left(\mathfrak{q}\right)\right)\hfill \\ & \le & \max \{{\Omega }^{-}\left(\vartheta \left(\mathfrak{p}\right)\right),{\Omega }^{-}(\vartheta \left(\mathfrak{p}\right)\to \alpha \left(\mathfrak{q}\right))\}\hfill \\ & =& \max \{{\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.\hfill \end{array}$$

Hence, $\Omega =({\Omega }^{-},{\Omega }^{+})$ is a BFF of $\mathcal{X}$. □

Theorem 21. 

Let $\vartheta :\mathcal{X}\to \mathcal{Y}$ be a homomorphism of hoops and $\Omega =({\Omega }^{-},{\Omega }^{+})$ be a BFS in $\mathcal{Y}$. If $\Omega =({\Omega }^{+},{\Omega }^{-})$ is a BFF of $\mathcal{Y}$, then ${\vartheta }^{-1}(\Omega )=({\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{+},{\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{-})$ is a BFF of $\mathcal{X}$.

Proof. 

Since $\vartheta$ is a homomorphism of $\mathcal{X}$ into $\mathcal{Y}$, then $\vartheta \left(1\right)=1\in \mathcal{Y}$ and, by the assumption, ${\Omega }^{+}\left(\vartheta \left(1\right)\right)={\Omega }^{+}\left(1\right)\ge {\Omega }^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)$ for every ${\hslash }_{{\mathfrak{a}}_2}\in \mathcal{Y}$. In particular, ${\Omega }_{\mathcal{Y}}^{+}\left(\vartheta \left(1\right)\right)\ge {\Omega }_{\mathcal{Y}}^{+}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right)$ for ${\hslash }_{{\mathfrak{a}}_1}\in \mathcal{X}$. Hence, ${\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{+}\left(1\right)\ge {\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right)$. Also, ${\Omega }^{-}\left(\vartheta \left(1\right)\right)={\Omega }^{-}\left(1\right)\le {\Omega }^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)$ for every ${\hslash }_{{\mathfrak{a}}_2}\in \mathcal{Y}$. In particular, ${\Omega }_{\mathcal{Y}}^{-}\left(\alpha \left(1\right)\right)\le {\Omega }_{\mathcal{Y}}^{-}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right)$ for ${\hslash }_{{\mathfrak{a}}_1}\in \mathcal{X}$. Hence, ${\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{-}\left(1\right)\le {\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right)$. Let ${\hslash }_{{\mathfrak{a}}_1},{\hslash }_{{\mathfrak{a}}_2}\in \mathcal{X}$. Then, by assumption,

$$\begin{array}{ccc}{\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{+}\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& {\Omega }^{+}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right)\right)\hfill \\ & \ge & \min \{{\Omega }^{+}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right),{\Omega }^{+}(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\to \vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right))\}\hfill \\ & =& \min \{{\Omega }^{+}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right),{\Omega }^{+}\left(\vartheta ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\right)\}\hfill \\ & =& \min \{{\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{+}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{+}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\},\hfill \end{array}$$

$$\begin{array}{ccc}{\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{-}\left({\hslash }_{{\mathfrak{a}}_2}\right)& =& {\Omega }^{-}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right)\right)\hfill \\ & \le & \max \{{\Omega }^{-}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right),{\Omega }^{-}(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\to \vartheta \left({\hslash }_{{\mathfrak{a}}_2}\right))\}\hfill \\ & =& \max \{{\Omega }^{-}\left(\vartheta \left({\hslash }_{{\mathfrak{a}}_1}\right)\right),{\Omega }^{-}\left(\vartheta ({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\right)\}\hfill \\ & =& \max \{{\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{-}\left({\hslash }_{{\mathfrak{a}}_1}\right),{\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{-}({\hslash }_{{\mathfrak{a}}_1}\to {\hslash }_{{\mathfrak{a}}_2})\}.\hfill \end{array}$$

Hence, ${\vartheta }^{-1}(\Omega )=({\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{+},{\Omega }_{{\vartheta }^{-1}\left(\mathcal{Y}\right)}^{-})$ is a BFF of $\mathcal{X}$. □

4. Conclusions

This study advances fuzzy logic by introducing bipolar fuzzy sub-hoops (BFSHs) and bipolar fuzzy filters (BFFs) within hoop algebras, enabling the modeling of both positive and negative information in many-valued logic systems. Below, we summarize the key results and achievements of this study:

• Introduced and defined bipolar fuzzy sub-hoops (BFSHs) and bipolar fuzzy filters (BFFs) in hoop algebras, extending fuzzy logic to handle positive and negative memberships.
• Established the interplay between BFSHs and BFFs, proving that every BFF is a BFSH (Theorem 2).
• Characterized BFFs through level sets (Theorem 3) and alternative conditions involving implication and monotonicity (Theorems 1 and 4).
• Defined and characterized implicative bipolar fuzzy filters (IBFFs) as a specialized subset of BFFs (Theorems 10–15).
• Proved the preservation of BFSH and BFF properties under hoop homomorphisms, ensuring robustness in transformed systems (Theorems 19–21).
• Demonstrated that the intersection of BFFs is a BFF (Proposition 7), supporting their algebraic stability.
• Showed that level sets of BFSHs form sub-hoops, connecting fuzzy and classical structures (Theorem 18).

These results find applications in image processing, spatial reasoning, and decision support systems, as detailed in the subsection below. Future research will explore additional algebraic structures, such as bipolar fuzzy rings, and develop computational algorithms for practical applications in image processing, decision-making, and control systems. This work lays a robust foundation for advanced fuzzy logic systems handling complex, contradictory information.

Future work could explore topological indices for fuzzy graphs, such as the fuzzy misbalance prodeg index, which has shown promise in multi-criteria decision-making [new reference]. Applying bipolar fuzzy sub-hoops and filters to such indices could enhance their ability to handle contradictory data in graph-based fuzzy systems.

In image processing, bipolar fuzzy filters enhance edge detection by modeling pixel intensities with positive and negative membership degrees, capturing both desired features (edges) and undesired features (noise). Using the hoop algebra framework, a bipolar fuzzy filter (BFF), as defined in Definition 8, assigns a positive degree ${\Omega }^{+}\left(x\right)\in [0,1]$ to pixels with high gradient magnitudes, indicating likely edge presence, and a negative degree ${\Omega }^{-}\left(x\right)\in [-1,0]$ to pixels with erratic intensity changes, indicating likely noise. For instance, a pixel at an object boundary may have ${\Omega }^{+}\left(x\right)=0.9$ and ${\Omega }^{-}\left(x\right)=-0.1$, while a noisy pixel in a uniform region may have ${\Omega }^{+}\left(x\right)=0.2$ and ${\Omega }^{-}\left(x\right)=-0.8$. The BFF processes these memberships using hoop operations ⊙ and →, retaining pixels with high ${\Omega }^{+}$ and low ${\Omega }^{-}$ while suppressing noise, as supported by Theorems 1 and 4 on BFF properties. This dual-membership approach improves segmentation accuracy over traditional fuzzy filters, enabling precise edge detection in applications like medical imaging, where clear tissue boundaries are critical [12].