Bipolar Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras and Its Applications in Decision-Making

Abstract

In this paper, we merge the concepts of soft set theory and a bipolar intuitionistic fuzzy set. A bipolar intuitionistic fuzzy soft ideal in a BCK-algebra is defined as a soft set over the set of elements in the BCK-algebra, with each element associated with an intuitionistic fuzzy set. This relationship captures degrees of uncertainty, hesitancy, and non-membership degrees within the context of BCK-algebras. We investigate basic operations on bipolar intuitionistic fuzzy soft ideals such as union, intersection, AND, and OR. The intersection, union, AND, and OR of two bipolar intuitionistic fuzzy soft ideals is a bipolar intuitionistic fuzzy soft ideal. We also demonstrate how to use a bipolar intuitionistic fuzzy soft set to solve a problem involving decision making. Finally, we provide a general approach for handling decision-making problems using a bipolar intuitionistic fuzzy soft set.

1. Introduction

Imai and Iséki [1,2] acquainted the investigation of BCI/BCK-algebras in 1966 as a consensus of set-hypothetical contrast and propositional mathematics. Iséki [3,4] presented two major classes of sensible algebras: BCK/BCI-algebras. Zadeh [5] was quick to present the hypothesis of fuzzy sets and different operations on them. The membership degrees of items in typical fuzzy sets range between 0 and 1. The level of the belongings of factors in a fuzzy set is expressed by the associate grade. An associate grade of 1 shows that a component has a spot totally through its related fuzzy set, while an associate grade of 0 demonstrates that a component does not have a factor with the fuzzy set. The partial belongings in the fuzzy set are indicated by the associate grades on the interval (0, 1). The associate grade by a factor of the property or imperative related to a fuzzy set is sometimes referred to as the associate grade (see [6]). In the fuzzy set demonstration, components with participation degree 0 are ordinarily seen as having similar highlights. By the way, it has been pointed out that a portion of these elements are antithetical to the things related to a fuzzy set, while others have characteristics that are contradictory to the things. The standard fuzzy set portrayal is incapable of distinguishing between contradictory and unimportant elements.

Fuzzy sets identify the grade to which a factor is the belonging of a precise set, whereas bipolar fuzzy sets provide together optimistic and destructive membership grades. The range [0, 1] encompasses the positive associate grade, whereas [−1, 0] encompasses the negative associate grade. For bipolar fuzzy sets, the range of an associate grade is extended from [0, 1] to [−1, 1]. Numerous processes and associations over bipolar uncertain sets [7] were familiarized. The principle of bipolar uncertain sets has recently gained popularity in several disciplines, including group (such as semi-group, ring, semi-ring, graph) theory, manufacturing, astronomy, statics, medicinal and social science, artificial intelligence, computer nets, expert classifications, and decision making.

Meng et al. [8] constructed BCK-algebras with uncertain implicative ideals. In BCI-algebras, Jun [9] developed closed ambiguous ideals. Recently, an accumulative quantity of academics have dedicated their time to researching certain bipolar fuzzy arithmetical constructions based on the output of bipolar uncertain sets. Lee [10] offered the idea of bipolar ambiguous ideals/subalgebras of a BCI/BCK-algebra and looked into a few properties. He established that engineering, economics, sociology, environmental science, and information science methodologies have failed to model uncertain data adequately due to inherent constraints, generating issues with traditional theoretical approaches.

Lee [11,12] provided a pragmatic analysis of a bipolar-valued ambiguous set and a comparison of interval-valued fuzzy sets. Molodtsov [13] studied soft set ideology as a brand-new mathematical tool for handling uncertainty that can resolve these problems. Soft set theory research is now moving at a breakneck pace. Maji et al. [14,15] looked at a variety of efforts on the ideology of soft sets and explained how the soft set principle may be used in a decision-making situation. As a result of this firm foundation, numerous academics have discovered wonderful works employing soft set hypotheses.

Jana et al. [16] described how bipolar intuitionistic ambiguous soft sets can be utilized by means of decision-making problems. Mursaleen et al. [17] familiarized $(\in ,\in \vee \widehat{q})$-bipolar ambiguous ideals of BCK/BCI-algebras. Muhiuddin et al. [18,19] acquaint with tripolar picture fuzzy ideals and bipolar fuzzy implicative ideals of BCK-algebras. Abbott [20] introduced the concept of sets, Lattices, and Boolean algebras. Zaronzy et al. [21] introduced the concept of a bipolar query, meant as a database query that involves both negative and positive conditions. Hamidi et al. [22,23] extended BCK-ideal based on single-valued neutrosophic hyper BCK-ideals. The motivation and contributions of this study are as follows (

Table 1. Motivations and Contributions of Bipolar Intuitionistic Fuzzy Soft Ideals.

Authors	Motivations	Contributions
Lee [12]	Bipolar-valued fuzzy sets	Bipolar intuitionistic fuzzy soft sets
Lee [10]	Bipolar fuzzy subalgebras and bipolar fuzzy ideals	Bipolar intuitionistic fuzzy soft ideals
Jana et al. [16,24]	Bipolar intuitionistic fuzzy soft sets and subalgebras	Operations of $\mathcal{BIFSI}s$

):

This manuscript presents a merge of the thoughts of soft sets and bipolar intuitionistic fuzzy sets. Combining the above concepts, a bipolar intuitionistic fuzzy soft ideal in a BCK-algebra is defined as a soft set over the set of components in the BCK-algebra, where each element is associated with an intuitionistic fuzzy set. This association allows for capturing uncertainty, hesitancy, and non-membership degrees within the context of BCK-algebras. The essential operations on the bipolar intuitionistic fuzzy soft ideal with intersection, union, AND, and OR are investigated. Furthermore, we demonstrate the use of bipolar intuitionistic fuzzy soft sets in a decision-making problem and provide a common algorithm for solving such problems.

Throughout this paper, we frequently utilize various symbols, along with their corresponding meanings. These symbols and their explanations are summarized in

Table 2. List of symbols used in this article.

Symbols	Representations
X	BCK/BCI-algebra
$\mathcal{P}\left(\mathcal{U}\right)$	Power set
$\mathcal{BFS}$	Bipolar fuzzy set
$\mathcal{BF}\left(\mathcal{U}\right)$	Bipolar fuzzy subset
$\mathcal{BFSS}$	Bipolar fuzzy soft set
$\mathcal{BIFS}$	Bipolar intuitionistic fuzzy set
$\mathcal{BIF}\left(\mathcal{U}\right)$	Bipolar intuitionistic fuzzy subset
$\mathcal{BIFSS}$	Bipolar intuitionistic fuzzy soft set
$\mathcal{BFSI}$	Bipolar fuzzy soft ideal
$\mathcal{BIFSI}$	Bipolar intuitionistic fuzzy soft ideal

below:

2. Preliminaries

In this section, we recall some concepts that are relevant to our work.

Definition 1

([1]). An algebra $(X,\ast ,0)$ of kind $(2,0)$ could be a BCK-algebra if

$\left(K_1\right)$$((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\ast (\dot{\varsigma }\ast \dot{\omega })=0$,

$\left(K_2\right)$$(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\ast \dot{\omega }=0$,

$\left(K_3\right)$$\dot{\phi }\ast \dot{\phi }=0$,

$\left(K_4\right)$$0\ast \dot{\phi }=0$,

$\left(K_5\right)$$\dot{\phi }\ast \dot{\omega }=0$ and $\dot{\omega }\ast \dot{\phi }=0\Rightarrow \dot{\phi }=\dot{\omega }$, for all $\dot{\phi },\dot{\omega },\dot{\varsigma }\in X$.

Within a BCK-algebra X, the ensuing conditions hold: $\left(P_1\right)\dot{\phi }\ast 0=\dot{\phi },$ $\left(P_2\right)(\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }=(\dot{\phi }\ast \dot{\varsigma })\ast \dot{\omega }$.

Definition 2

([4]). If a nonempty subset $\underset{¸}{A}$ of $\underset{¸}{X}$ satisfies $\left(I_1\right)0\in \underset{¸}{A}$, $\left(I_2\right)\forall \dot{\phi },\dot{\omega }\in X,\dot{\phi }\ast \dot{\omega }\in \underset{¸}{A},\dot{\omega }\in \underset{¸}{A}\Rightarrow \dot{\phi }\in \underset{¸}{A}$, then $\underset{¸}{A}$ is an ideal of X:

Definition 3

([13]). Let X be an ubiquitous set and $\underset{¸}{E}$ be a list of specifications. Let $\mathcal{P}\left(\mathcal{U}\right)$ express the power set of X, and $\underset{¸}{A}\subset \underset{¸}{E}$. A couple $(\underset{¸}{F},\underset{¸}{A})$ is a soft set over X, where $\underset{¸}{F}:\underset{¸}{A}\to \mathcal{P}\left(\mathcal{U}\right)$.

Definition 4

([12]). An item of the system $\underset{¸}{A}=(X;{\tilde{\vartheta }}^{-},{\tilde{\vartheta }}^{+})$ is a bipolar fuzzy set ($\mathcal{BFS}$) over X, where ${\tilde{\vartheta }}^{-}:X\to [-1,0]$ and ${\tilde{\vartheta }}^{+}:X\to [0,1]$.

Definition 5

([25]). Let X be an ubiquitous set, $\underset{¸}{E}$ be a list of specifications, and $\underset{¸}{A}\subseteq \underset{¸}{E}$. Define $\underset{¸}{F}:\underset{¸}{A}\to \mathcal{BF}\left(\mathcal{U}\right)$. Then, the pair $(\underset{¸}{F},\underset{¸}{A})$ is referred to as bipolar fuzzy soft set ($\mathcal{BFSS}$) over X, and it stands for $(\underset{¸}{F},\underset{¸}{A})=\underset{¸}{F}\left(\dot{\dot{\eta }}\right)$ and detailed by $\underset{¸}{F}\left(\dot{\eta }\right)=\left\{(\dot{\phi },{\tilde{\vartheta }}^{-}\left(\dot{\phi }\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right))\right\}$, where $\dot{\eta }\in X$ and $\dot{\eta }\in \underset{¸}{A}$.

Definition 6

([16]). A bipolar intuitionistic fuzzy set ($\mathcal{BIFS}$) $\underset{¸}{A}$ over X is an element of the system ${\underset{¸}{A}}^{*}=(X;{\tilde{\vartheta }}^{-},{\tilde{\vartheta }}^{+},{\tilde{\varpi }}^{-},{\tilde{\varpi }}^{+})$, where ${\tilde{\vartheta }}^{-}:X\to [-1,0],{\tilde{\vartheta }}^{+}:X\to [0,1],{\tilde{\varpi }}^{-}:X\to [-1,0]$ and ${\tilde{\varpi }}^{+}:X\to [0,1]$ are such that $-1\le {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)+{\tilde{\varpi }}^{-}\left(\dot{\phi }\right)\le 0$ and $0\le {\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)+{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\le 1$.

Definition 7

([16]). Let X be an ubiquitous set, $\underset{¸}{E}$ be a list of specifications, and ${\underset{¸}{A}}^{*}\subseteq \underset{¸}{E}$. Define $\underset{¸}{F}:{\underset{¸}{A}}^{*}\to \mathcal{BIF}\left(\mathcal{U}\right)$. Then, the pair $(\underset{¸}{F},{\underset{¸}{A}}^{*})$ is referred to as bipolar intuitionistic fuzzy soft set ($\mathcal{BIFSS}$) over X, and it stands for $(\underset{¸}{F},{\underset{¸}{A}}^{*})=\underset{¸}{F}\left(\dot{\eta }\right)$ and detailed by $\underset{¸}{F}\left(\dot{\eta }\right)=\left\{(\dot{\phi },{\tilde{\vartheta }}^{-}\left(\dot{\phi }\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right),{\tilde{\varpi }}^{-}\left(\dot{\phi }\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right))\right\}$.

3. Bipolar Intuitionistic Fuzzy Soft Ideal of BCK/BCI-Algebras

In this section, we present the concept of the bipolar intuitionistic fuzzy soft ideal of BCK/BCI-algebras as an extension of bipolar fuzzy soft ideals of BCK/BCI-algebras and seek some of their properties.

Definition 8.

A bipolar fuzzy soft set ($\mathcal{BFSS}$) $\underset{¸}{F}\left(\dot{\eta }\right)$ in X is called a bipolar fuzzy soft ideal ($\mathcal{BFSI}$) of X if

$\left({\mathcal{BFSI}}_1\right)(\forall \dot{\phi }\in X)\left({\tilde{\vartheta }}^{-}\left(0\right)\le {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right),{\tilde{\vartheta }}^{+}\left(0\right)\ge {\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\right)$,

$\left({\mathcal{BFSI}}_2\right)(\forall \dot{\phi },\dot{\omega }\in X)\left({\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)\le {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })\vee {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })\wedge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)\right)$.

Definition 9.

A bipolar intuitionistic fuzzy soft set ($\mathcal{BIFSS}$) $\underset{¸}{F}\left(\dot{\eta }\right)$ in X is called a bipolar intuitionistic fuzzy soft ideal ($\mathcal{BIFSI})$ of X if

$\left({\mathcal{BIFSI}}_1\right)(\forall \dot{\phi }\in X)\left({\tilde{\vartheta }}^{-}\left(0\right)\le {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right),{\tilde{\vartheta }}^{+}\left(0\right)\ge {\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\right)$,

$\left({\mathcal{BIFSI}}_2\right)(\forall \dot{\phi },\dot{\omega }\in X)\left({\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)\le {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })\vee {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })\wedge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)\right)$,

$\left({\mathcal{BIFSI}}_3\right)(\forall \dot{\phi }\in X)\left({\tilde{\varpi }}^{-}\left(0\right)\ge {\tilde{\varpi }}^{-}\left(\dot{\phi }\right),{\tilde{\varpi }}^{+}\left(0\right)\le {\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\right)$,

$\left({\mathcal{BIFSI}}_4\right)(\forall \dot{\phi },\dot{\omega }\in X)\left({\tilde{\varpi }}^{-}\left(\dot{\phi }\right)\ge {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })\wedge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })\vee {\tilde{\varpi }}^{+}\left(\dot{\omega }\right)\right)$.

Example 1.

Take a BCK-algebra $(X;\ast ,0)$, as provided in

Table 3. Cayley table for BCK-algebra ($\mathcal{BIFSI}$).

∗	0	$\dot{\mathit{\phi }}$	$\dot{\mathit{\omega }}$	$\dot{\mathit{\varsigma }}$
0	0	0	0	0
$\dot{\phi }$	$\dot{\phi }$	0	0	$\dot{\phi }$
$\dot{\omega }$	$\dot{\omega }$	$\dot{\phi }$	0	$\dot{\omega }$
$\dot{\varsigma }$	$\dot{\varsigma }$	$\dot{\varsigma }$	$\dot{\varsigma }$	0

.

Define

$\underset{¸}{F}\left(\dot{\eta }\right)=\{(0,-0.3,0.2,-0.2,0.5),(\dot{\phi },-0.5,0.4,-0.6,0.2),(\dot{\omega },-0.2,0.5,-0.3,0.3),(\dot{\varsigma },-0.1,0.3,-0.4,0.4)\}$.

Thus, $\underset{¸}{F}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$ of X.

Theorem 1.

If $\underset{¸}{F}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$ in X and $\underset{¸}{I}\left(0\right)=\{\dot{\phi }\in X:{\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)={\tilde{\vartheta }}^{-}\left(0\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)={\tilde{\vartheta }}^{+}\left(0\right),and{\tilde{\varpi }}^{-}\left(\dot{\phi }\right)={\tilde{\varpi }}^{-}\left(0\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)={\tilde{\varpi }}^{+}\left(0\right)\},$ then $\underset{¸}{I}\left(0\right)$ is an ideal of X.

Proof. 

Let $\dot{\phi },\dot{\omega }\in X$ be chosen that $\dot{\phi }\ast \dot{\omega }\in \underset{¸}{I}\left(0\right)$ and $\dot{\omega }\in \underset{¸}{I}\left(0\right)$. Then, we have

$${\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })={\tilde{\vartheta }}^{-}\left(0\right),{\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })={\tilde{\vartheta }}^{+}\left(0\right),{\tilde{\vartheta }}^{-}\left(\dot{\omega }\right)={\tilde{\vartheta }}^{-}\left(0\right),{\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)={\tilde{\vartheta }}^{+}\left(0\right)$$

and

$${\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })={\tilde{\varpi }}^{-}\left(0\right),{\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })={\tilde{\varpi }}^{+}\left(0\right),{\tilde{\varpi }}^{-}\left(\dot{\omega }\right)={\tilde{\varpi }}^{-}\left(0\right),{\tilde{\varpi }}^{+}\left(\dot{\omega }\right)={\tilde{\varpi }}^{+}\left(0\right).$$

Thus, using Definition 9, $\left({\mathcal{BIFSI}}_2\right)$ and $\left({\mathcal{BIFSI}}_4\right)$, we obtain

$${\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)\le {\tilde{\vartheta }}^{-}\left(0\right)\vee {\tilde{\vartheta }}^{-}\left(0\right)={\tilde{\vartheta }}^{-}\left(0\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{-}\left(0\right)\wedge {\tilde{\vartheta }}^{-}\left(0\right)={\tilde{\vartheta }}^{-}\left(0\right),$$

and

$${\tilde{\varpi }}^{-}\left(\dot{\phi }\right)\ge {\tilde{\varpi }}^{-}\left(0\right)\wedge {\tilde{\varpi }}^{-}\left(0\right)={\tilde{\varpi }}^{-}\left(0\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{-}\left(0\right)\vee {\tilde{\varpi }}^{-}\left(0\right)={\tilde{\varpi }}^{-}\left(0\right).$$

On the other hand, we know from Definition 9, $\left({\mathcal{BIFSI}}_1\right)$ and $\left({\mathcal{BIFSI}}_3\right)$ that

$${\tilde{\vartheta }}^{-}\left(0\right)\le {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right),{\tilde{\vartheta }}^{+}\left(0\right)\ge {\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\Rightarrow {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)={\tilde{\vartheta }}^{-}\left(0\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)={\tilde{\vartheta }}^{+}\left(0\right),$$

and

$${\tilde{\varpi }}^{-}\left(0\right)\ge {\tilde{\varpi }}^{-}\left(\dot{\phi }\right),{\tilde{\varpi }}^{+}\left(0\right)\le {\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\Rightarrow {\tilde{\varpi }}^{-}\left(\dot{\phi }\right)={\tilde{\varpi }}^{-}\left(0\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)={\tilde{\varpi }}^{+}\left(0\right).$$

Henceforth, $\dot{\phi }\in \underset{¸}{I}\left(0\right)$. It is evident that $0\in \underset{¸}{I}\left(0\right)$. Thence, $\underset{¸}{I}\left(0\right)$ is an ideal of X.    □

If $\underset{¸}{F}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSS}$ in X that satisfies Definition 9, $\left({\mathcal{BIFSI}}_1\right)$ and $\left({\mathcal{BIFSI}}_3\right)$, then the ensuing example shows that in Definition 9, $\left({\mathcal{BIFSI}}_2\right)$ and $\left({\mathcal{BIFSI}}_4\right)$ are plentiful for $\underset{¸}{I}\left(0\right)$ not to be an ideal of X.

Example 2

(see [19]). Take a BCK-algebra $(X;\ast ,0)$, provided in

Table 4. Cayley table for BCK-algebra ($\underset{¸}{I}(0)$ is not an ideal).

∗	0	$\dot{\mathit{\phi }}$	$\dot{\mathit{\omega }}$	$\dot{\mathit{\varsigma }}$	$\dot{\mathit{\eta }}$
0	0	0	0	0	0
$\dot{\phi }$	$\dot{\phi }$	0	$\dot{\phi }$	0	0
$\dot{\omega }$	$\dot{\omega }$	$\dot{\omega }$	0	$\dot{\omega }$	0
$\dot{\varsigma }$	$\dot{\varsigma }$	$\dot{\varsigma }$	$\dot{\varsigma }$	0	$\dot{\varsigma }$
$\dot{\eta }$	$\dot{\eta }$	$\dot{\eta }$	$\dot{\eta }$	$\dot{\eta }$	0

.

Define

$\underset{¸}{F}\left(\dot{\eta }\right)=\{(0,-0.6,0.7,-0.5,0.6),(\dot{\phi },-0.6,0.7,-0.5,0.6),(\dot{\omega },-0.2,0.1,-0.3,0),(\dot{\varsigma },-0.6,0.7,-0.5,0.6),(\dot{\eta },-0.6,0.7,-0.5,0.6)\}$.

Note that $\underset{¸}{F}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$ in X that satisfies Definition 9, $\left({\mathcal{BIFSI}}_1\right)$, $\left({\mathcal{BIFSI}}_3\right)$, $\left({\mathcal{BIFSI}}_4\right)$ and not $\left({\mathcal{BIFSI}}_2\right)$ as there exists $\dot{\omega },\dot{\eta }\in X$, where

$${\tilde{\vartheta }}^{-}\left(\dot{\omega }\right)=-0.2\nleqq -0.6={\tilde{\vartheta }}^{-}(\dot{\omega }\ast \dot{\eta })\vee {\tilde{\vartheta }}^{-}\left(\dot{\eta }\right)$$

and

$${\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)=0.1\ngeqq 0.7={\tilde{\vartheta }}^{+}(\dot{\omega }\ast \dot{\eta })\wedge {\tilde{\vartheta }}^{+}\left(\dot{\eta }\right).$$

Then, $\underset{¸}{I}\left(0\right)=\{0,\dot{\phi },\dot{\eta }\}$ is not an ideal of X as $\dot{\omega }\ast \dot{\eta }=0\in \underset{¸}{I}\left(0\right)$ and $\dot{\eta }\in \underset{¸}{I}\left(0\right)$, but $\dot{\omega }\notin \underset{¸}{I}\left(0\right)$.

Theorem 2.

Let $\theta \in X$. If $\underset{¸}{F}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$ of X, then $\underset{¸}{I}\left(\theta \right)$ is an ideal of X.

Proof. 

Remember that 0 belongs to the ideal $\underset{¸}{I}\left(\theta \right)$. Now, let $\dot{\phi }$ and $\dot{\omega }$ be components of X such that $\dot{\phi }\ast \dot{\omega }$ is a component of $\underset{¸}{I}\left(\theta \right)$ and $\dot{\omega }$ is also an element of $\underset{¸}{I}\left(\theta \right)$. In this instance, the following holds:

$${\tilde{\vartheta }}^{-}\left(\theta \right)\ge {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega }),{\tilde{\vartheta }}^{+}\left(\theta \right)\le {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega }),{\tilde{\vartheta }}^{-}\left(\theta \right)\ge {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right),{\tilde{\vartheta }}^{+}\left(\theta \right)\le {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right),$$

and

$${\tilde{\varpi }}^{-}\left(\theta \right)\le {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega }),{\tilde{\varpi }}^{+}\left(\theta \right)\ge {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega }),{\tilde{\varpi }}^{-}\left(\theta \right)\le {\tilde{\varpi }}^{-}\left(\dot{\omega }\right),{\tilde{\varpi }}^{+}\left(\theta \right)\ge {\tilde{\varpi }}^{+}\left(\dot{\omega }\right).$$

As $\underset{¸}{F}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$ of X, it derives from Definition 9, $\left({\mathcal{BIFSI}}_2\right)$ and $\left({\mathcal{BIFSI}}_4\right)$ that

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)& \le {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })\vee {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right)\hfill \\ \hfill & \le {\tilde{\vartheta }}^{-}\left(\theta \right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)& \ge {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })\wedge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)\hfill \\ \hfill & \ge {\tilde{\vartheta }}^{+}\left(\theta \right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{-}\left(\dot{\phi }\right)& \ge {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })\wedge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right)\hfill \\ \hfill & \ge {\tilde{\varpi }}^{-}\left(\theta \right),\hfill \\ \hfill {\tilde{\varpi }}^{+}\left(\dot{\phi }\right)& \le {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })\vee {\tilde{\varpi }}^{+}\left(\dot{\omega }\right)\hfill \\ \hfill & \le {\tilde{\varpi }}^{+}\left(\theta \right),\hfill \end{array}$$

so $\dot{\phi }\in \underset{¸}{I}\left(\theta \right)$. Thence, $\underset{¸}{I}\left(\theta \right)$ is an ideal of X.    □

Example 3

(see [19]). Take a BCK-algebra $(X;\ast ,0)$ provided in

Table 5. Cayley table for BCK-algebra ($\underset{¸}{I}\left(\theta \right)$ is an ideal).

∗	0	$\dot{\mathit{\phi }}$	$\dot{\mathit{\omega }}$	$\dot{\mathit{\varsigma }}$	$\dot{\mathit{\eta }}$
0	0	0	0	0	0
$\dot{\phi }$	$\dot{\phi }$	0	$\dot{\phi }$	0	0
$\dot{\omega }$	$\dot{\omega }$	$\dot{\omega }$	0	$\dot{\omega }$	0
$\dot{\varsigma }$	$\dot{\varsigma }$	$\dot{\varsigma }$	$\dot{\varsigma }$	0	$\dot{\varsigma }$
$\dot{\eta }$	$\dot{\eta }$	$\dot{\eta }$	$\dot{\eta }$	$\dot{\eta }$	0

.

Define

$\underset{¸}{F}\left(\dot{\eta }\right)=\{(0,-0.6,0.7,-0.2,0.1),(\dot{\phi },-0.4,0.6,-0.4,0.2),(\dot{\omega },-0.2,0.3,-0.6,0.5),(\dot{\varsigma },0,0.1,-0.8,0.7),(\dot{\eta },-0.3,0.5,-0.5,0.3)\}$.

Note that $\underset{¸}{F}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$ in X that satisfies Definition 9. Then, $\underset{¸}{I}\left(\theta \right)=\{0,\dot{\phi },\dot{\omega },\dot{\eta }\}$ is an ideal of X.

Theorem 3.

Consider $\underset{¸}{F}\left(\dot{\eta }\right)$ as a $\mathcal{BIFSS}$ in X, and let θ be an element of X.

$\left(1\right)$ If $\underset{¸}{I}\left(\theta \right)$ is an ideal of X, then the following condition holds: $\underset{¸}{F}\left(\dot{\eta }\right)$ delight the ensuing assumptions for all $\dot{\phi },\dot{\omega },\dot{\varsigma }\in X$,

$$\begin{array}{c}\hfill {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{-}(\dot{\omega }\ast \dot{\varsigma })\vee {\tilde{\vartheta }}^{-}\left(\dot{\varsigma }\right)\Rightarrow {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{-}\left(\theta \right),\end{array}$$

(1)

$$\begin{array}{c}\hfill {\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{+}(\dot{\omega }\ast \dot{\varsigma })\wedge {\tilde{\vartheta }}^{+}\left(\dot{\varsigma }\right)\Rightarrow {\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\le {\tilde{\vartheta }}^{+}\left(\theta \right)\end{array}$$

(2)

and

$$\begin{array}{c}\hfill {\tilde{\varpi }}^{-}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{-}(\dot{\omega }\ast \dot{\varsigma })\wedge {\tilde{\varpi }}^{-}\left(\dot{\varsigma }\right)\Rightarrow {\tilde{\varpi }}^{-}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{-}\left(\theta \right),\end{array}$$

(3)

$$\begin{array}{c}\hfill {\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{+}(\dot{\omega }\ast \dot{\varsigma })\vee {\tilde{\varpi }}^{+}\left(\dot{\varsigma }\right)\Rightarrow {\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\varpi }}^{+}\left(\theta \right).\end{array}$$

(4)

$\left(2\right)$ If $\underset{¸}{F}\left(\dot{\eta }\right)$ satisfies Definition 9, $\left({\mathcal{BIFSI}}_1\right)$, $\left({\mathcal{BIFSI}}_2\right)$, $\left({\mathcal{BIFSI}}_3\right)$, and $\left({\mathcal{BIFSI}}_4\right)$, then $\underset{¸}{I}\left(\theta \right)$ is an ideal of X.

Proof. 

(1) Assume that $\underset{¸}{I}\left(\theta \right)$ is an ideal of X. Suppose that

$${\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{-}(\dot{\omega }\ast \dot{\varsigma })\vee {\tilde{\vartheta }}^{-}\left(\dot{\varsigma }\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{+}(\dot{\omega }\ast \dot{\varsigma })\wedge {\tilde{\vartheta }}^{+}\left(\dot{\varsigma }\right)$$

and

$${\tilde{\varpi }}^{-}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{-}(\dot{\omega }\ast \dot{\varsigma })\wedge {\tilde{\varpi }}^{-}\left(\dot{\varsigma }\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{+}(\dot{\omega }\ast \dot{\varsigma })\vee {\tilde{\varpi }}^{+}\left(\dot{\varsigma }\right)$$

for all $\dot{\phi },\dot{\omega },\dot{\varsigma }\in X$. Then, $\dot{\omega }\ast \dot{\varsigma }\in \underset{¸}{I}\left(\theta \right)$ and $\dot{\varsigma }\in \underset{¸}{I}\left(\theta \right)$. Since $\underset{¸}{I}\left(\theta \right)$ is an ideal of X, it ensue that $\dot{\omega }\in \underset{¸}{I}\left(\theta \right)$, that is,

$${\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\le {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)$$

and

$${\tilde{\varpi }}^{-}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{-}\left(\dot{\omega }\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\varpi }}^{+}\left(\dot{\omega }\right).$$

(2) Suppose that $\underset{¸}{F}\left(\dot{\eta }\right)$ satisfies Definition 9, $\left({\mathcal{BIFSI}}_1\right)$, $\left({\mathcal{BIFSI}}_2\right)$, $\left({\mathcal{BIFSI}}_3\right)$, and $\left({\mathcal{BIFSI}}_4\right)$. For each $\theta \in X$, let $\dot{\phi },\dot{\omega }\in X$ be such a thing that $\dot{\phi }\ast \dot{\omega }\in \underset{¸}{I}\left(\theta \right)$ and $\dot{\omega }\in \underset{¸}{I}\left(\theta \right)$. Then,

$${\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })\le {\tilde{\vartheta }}^{-}\left(\theta \right),{\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })\ge {\tilde{\vartheta }}^{+}\left(\theta \right),{\tilde{\vartheta }}^{-}\left(\dot{\omega }\right)\le {\tilde{\vartheta }}^{-}\left(\theta \right),{\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)\ge {\tilde{\vartheta }}^{+}\left(\theta \right)$$

and

$${\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })\ge {\tilde{\varpi }}^{-}\left(\theta \right),{\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })\le {\tilde{\varpi }}^{+}\left(\theta \right),{\tilde{\varpi }}^{-}\left(\dot{\omega }\right)\ge {\tilde{\varpi }}^{-}\left(\theta \right),{\tilde{\varpi }}^{+}\left(\dot{\omega }\right)\le {\tilde{\varpi }}^{+}\left(\theta \right),$$

which implies that

$${\tilde{\vartheta }}^{-}\left(\theta \right)\ge {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })\vee {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\le {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })\wedge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)$$

and

$${\tilde{\varpi }}^{-}\left(\theta \right)\le {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })\wedge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })\vee {\tilde{\varpi }}^{+}\left(\dot{\omega }\right).$$

Using $\left({\mathcal{BIFSI}}_2\right)$ and $\left({\mathcal{BIFSI}}_4\right)$,

$${\tilde{\vartheta }}^{-}\left(\theta \right)\ge {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right),{\tilde{\vartheta }}^{+}\left(\theta \right)\le {\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)$$

and

$${\tilde{\varpi }}^{-}\left(\theta \right)\le {\tilde{\varpi }}^{-}\left(\dot{\phi }\right),{\tilde{\varpi }}^{+}\left(\theta \right)\ge {\tilde{\varpi }}^{+}\left(\dot{\phi }\right),$$

and so $\dot{\phi }\in \underset{¸}{I}\left(\theta \right)$. Since $\underset{¸}{F}\left(\dot{\eta }\right)$ satisfies Definition 9, $\left({\mathcal{BIFSI}}_1\right)$ and $\left({\mathcal{BIFSI}}_3\right)$, it ensues that $0\in \underset{¸}{I}\left(\theta \right)$. Thence $\underset{¸}{I}\left(\theta \right)$ is an ideal of X.    □

Lemma 1.

Every $\mathcal{BIFSI}$$\underset{¸}{F}\left(\dot{\eta }\right)$ of X is fulfilled the subsequent implication: $(\forall \dot{\phi },\dot{\omega }\in X)(\dot{\phi }\le \dot{\omega }\Rightarrow {\tilde{\vartheta }}^{-}\left(\dot{\phi }\right)\le {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right),{\tilde{\vartheta }}^{+}\left(\dot{\phi }\right)\ge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)and{\tilde{\varpi }}^{-}\left(\dot{\phi }\right)\ge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right),{\tilde{\varpi }}^{+}\left(\dot{\phi }\right)\le {\tilde{\varpi }}^{+}\left(\dot{\omega }\right))$.

Theorem 4.

Let $\underset{¸}{F}\left(\dot{\eta }\right)$ be a $\mathcal{BIFSI}$ of X. Then,

$${\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })\le {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\varsigma })\vee {\tilde{\vartheta }}^{-}(\dot{\varsigma }\ast \dot{\omega }),{\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })\ge {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\varsigma })\wedge {\tilde{\vartheta }}^{+}(\dot{\varsigma }\ast \dot{\omega }),$$

and

$${\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })\ge {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\varsigma })\wedge {\tilde{\varpi }}^{-}(\dot{\varsigma }\ast \dot{\omega }),{\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })\le {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\varsigma })\vee {\tilde{\varpi }}^{+}(\dot{\varsigma }\ast \dot{\omega })$$

for all $\dot{\phi },\dot{\omega },\dot{\varsigma }\in X$.

Proof. 

Note that $((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\le (\dot{\varsigma }\ast \dot{\omega })$ for all $\dot{\phi },\dot{\omega },\dot{\varsigma }\in X$. From Lemma 1,

$${\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\le {\tilde{\vartheta }}^{-}(\dot{\varsigma }\ast \dot{\omega }),{\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\ge {\tilde{\vartheta }}^{+}(\dot{\varsigma }\ast \dot{\omega })$$

and

$${\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\ge {\tilde{\varpi }}^{-}(\dot{\varsigma }\ast \dot{\omega }),{\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\le {\tilde{\varpi }}^{+}(\dot{\varsigma }\ast \dot{\omega }).$$

Now, by Definition 9, we have

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })& \le {\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\vee {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\varsigma })\hfill \\ \hfill & \le {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\varsigma })\vee {\tilde{\vartheta }}^{-}(\dot{\varsigma }\ast \dot{\omega }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })& \ge {\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\wedge {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\varsigma })\hfill \\ \hfill & \ge {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\varsigma })\wedge {\tilde{\vartheta }}^{+}(\dot{\varsigma }\ast \dot{\omega }),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })& \ge {\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\wedge {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\varsigma })\hfill \\ \hfill & \ge {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\varsigma })\wedge {\tilde{\varpi }}^{-}(\dot{\varsigma }\ast \dot{\omega }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })& \le {\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\varsigma }))\vee {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\varsigma })\hfill \\ \hfill & \le {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\varsigma })\vee {\tilde{\varpi }}^{+}(\dot{\varsigma }\ast \dot{\omega }),\hfill \end{array}$$

for all $\dot{\phi },\dot{\omega },\dot{\varsigma }\in X$.    □

Theorem 5.

Let $\underset{¸}{F}\left(\dot{\eta }\right)\in X$ be a $\mathcal{BIFSI}$ of X. Then, for all $\dot{\phi },\dot{\omega }\in X$,

$\left(1\right){\tilde{\vartheta }}^{-}(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\le {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right).$

$\left(2\right){\tilde{\vartheta }}^{+}(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\ge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)$.

$\left(3\right){\tilde{\varpi }}^{-}(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\ge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right).$

$\left(4\right){\tilde{\varpi }}^{+}(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\le {\tilde{\varpi }}^{+}\left(\dot{\omega }\right).$

Proof. 

Let $\underset{¸}{F}\left(\dot{\eta }\right)\in X$ be a $\mathcal{BIFSI}$ of X. Then, for all $\dot{\phi },\dot{\omega }\in X$, we have

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{-}(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))& \le {\tilde{\vartheta }}^{-}((\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\ast \dot{\omega })\vee {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right)\hfill \\ \hfill & ={\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\omega }))\vee {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right)\hfill \\ \hfill & ={\tilde{\vartheta }}^{-}\left(0\right)\vee {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right)\hfill \\ \hfill & \le {\tilde{\vartheta }}^{-}\left(\dot{\omega }\right),\hfill \end{array}$$

which proves Condition (1).

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{+}(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))& \ge {\tilde{\vartheta }}^{+}((\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\ast \dot{\omega })\wedge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)\hfill \\ \hfill & ={\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\omega }))\wedge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)\hfill \\ \hfill & ={\tilde{\vartheta }}^{+}\left(0\right)\wedge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right)\hfill \\ \hfill & \ge {\tilde{\vartheta }}^{+}\left(\dot{\omega }\right),\hfill \end{array}$$

which proves Condition (2).

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{-}(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))& \ge {\tilde{\varpi }}^{-}((\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\ast \dot{\omega })\wedge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right)\hfill \\ \hfill & ={\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\omega }))\wedge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right)\hfill \\ \hfill & ={\tilde{\varpi }}^{-}\left(0\right)\wedge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right)\hfill \\ \hfill & \ge {\tilde{\varpi }}^{-}\left(\dot{\omega }\right),\hfill \end{array}$$

which proves Condition (3).

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{+}(\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))& \le {\tilde{\varpi }}^{+}((\dot{\phi }\ast (\dot{\phi }\ast \dot{\omega }))\ast \dot{\omega })\vee {\tilde{\varpi }}^{+}\left(\dot{\omega }\right)\hfill \\ \hfill & ={\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\phi }\ast \dot{\omega }))\vee {\tilde{\varpi }}^{+}\left(\dot{\omega }\right)\hfill \\ \hfill & ={\tilde{\varpi }}^{+}\left(0\right)\vee {\tilde{\varpi }}^{+}\left(\dot{\omega }\right)\hfill \\ \hfill & \le {\tilde{\varpi }}^{+}\left(\dot{\omega }\right),\hfill \end{array}$$

which proves Condition (4).    □

Proposition 1.

Let $\underset{¸}{F}\left(\dot{\eta }\right)\in X$ be a $\mathcal{BIFSI}$ of X. Consequently, the assertions that follow are equal.

(1)

$(\forall \dot{\phi },\dot{\omega }\in X)({\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })\le {\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\omega }),{\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })\ge {\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\omega }))$.

(2)

$(\forall \dot{\phi },\dot{\omega },\dot{\varsigma }\in X)({\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))\le {\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }),{\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))\ge$ ${\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma })\left)\right)$.

(3)

$(\forall \dot{\phi },\dot{\omega }\in X)({\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })\ge {\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\omega }),{\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })\le {\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\omega }))$.

(4)

$(\forall \dot{\phi },\dot{\omega },\dot{\varsigma }\in X)({\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))\ge {\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }),{\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))\le$ ${\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma })\left)\right)$.

Proof. 

Assume that Condition (1) is valid. Note that

$$((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\ast \dot{\varsigma }=((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma }\le (\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }$$

for all $\dot{\phi },\dot{\omega },\dot{\varsigma }\in X$ by using $\left({\mathcal{BIFSI}}_1\right),\left(K_3\right)$, and $\left({\mathcal{BIFSI}}_3\right)$. From Lemma 1, we have that

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma })& \ge {\tilde{\vartheta }}^{-}(((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma })\ast \dot{\varsigma })\ast \dot{\varsigma }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }))& \le {\tilde{\vartheta }}^{+}(((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma })\ast \dot{\varsigma })\ast \dot{\varsigma })\hfill \end{array}$$

so from $\left(P_2\right)$ and (1) that

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))& ={\tilde{\vartheta }}^{-}((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\hfill \\ \hfill & \le {\tilde{\vartheta }}^{-}(((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\ast \dot{\varsigma })\hfill \\ \hfill & \le {\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))& ={\tilde{\vartheta }}^{+}((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\hfill \\ \hfill & \ge {\tilde{\vartheta }}^{+}(((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\ast \dot{\varsigma })\hfill \\ \hfill & \ge {\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }).\hfill \end{array}$$

Thus, if (2) holds, then the following statement can be made:

If we replace $\dot{\varsigma }$ with $\dot{\omega }$ in (2), then

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{-}(\dot{\phi }\ast \dot{\omega })& ={\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast 0)\hfill \\ \hfill & ={\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\omega }\ast \dot{\omega }))\hfill \\ \hfill & \le {\tilde{\vartheta }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\omega }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\vartheta }}^{+}(\dot{\phi }\ast \dot{\omega })& ={\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast 0)\hfill \\ \hfill & ={\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\omega }\ast \dot{\omega }))\hfill \\ \hfill & \ge {\tilde{\vartheta }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\omega })\hfill \end{array}$$

which proves (1).

Assume that Condition (3) is valid. Note that

$$((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\ast \dot{\varsigma }=((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma }\le (\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }$$

for all $\dot{\phi },\dot{\omega },\dot{\varsigma }\in X$ by using $\left({\mathcal{BIFSI}}_2\right),\left(K_3\right)$, and $\left({\mathcal{BIFSI}}_4\right)$. From Lemma 1, we obtain that

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma })& \le {\tilde{\varpi }}^{-}(((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma })\ast \dot{\varsigma })\ast \dot{\varsigma }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }))& \ge {\tilde{\varpi }}^{+}(((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma })\ast \dot{\varsigma })\ast \dot{\varsigma })\hfill \end{array}$$

so from $\left(P_2\right)$ and (3) that

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))& ={\tilde{\varpi }}^{-}((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\hfill \\ \hfill & \ge {\tilde{\varpi }}^{-}(((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\ast \dot{\varsigma })\hfill \\ \hfill & \ge {\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\varsigma })\ast (\dot{\omega }\ast \dot{\varsigma }))& ={\tilde{\varpi }}^{+}((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\hfill \\ \hfill & \le {\tilde{\varpi }}^{+}(((\dot{\phi }\ast (\dot{\omega }\ast \dot{\varsigma }))\ast \dot{\varsigma })\ast \dot{\varsigma })\hfill \\ \hfill & \le {\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\varsigma }).\hfill \end{array}$$

Thus, if (4) holds, then the following statement can be made:

If we replace $\dot{\varsigma }$ with $\dot{\omega }$ in (4), then

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{-}(\dot{\phi }\ast \dot{\omega })& ={\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast 0)\hfill \\ \hfill & ={\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\omega }\ast \dot{\omega }))\hfill \\ \hfill & \ge {\tilde{\varpi }}^{-}((\dot{\phi }\ast \dot{\omega })\ast \dot{\omega }),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{\varpi }}^{+}(\dot{\phi }\ast \dot{\omega })& ={\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast 0)\hfill \\ \hfill & ={\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast (\dot{\omega }\ast \dot{\omega }))\hfill \\ \hfill & \le {\tilde{\varpi }}^{+}((\dot{\phi }\ast \dot{\omega })\ast \dot{\omega })\hfill \end{array}$$

which proves (3).    □

4. Operations on Bipolar Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras

In this section, we present the operations of bipolar intuitionistic fuzzy soft ideal of BCK/BCI and seek some of their examples.

Definition 10.

Let $\underset{¸}{F}\left(\dot{\eta }\right),\underset{¸}{G}\left(\dot{\eta }\right)\in \mathcal{BIF}\left(\mathcal{U}\right)$. Then, the “$intersection$" of $\underset{¸}{F}\left(\dot{\eta }\right)$ and $\underset{¸}{G}\left(\dot{\eta }\right)$ is expressed by $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\cap }\underset{¸}{G}\left(\dot{\eta }\right)$ and is demarcated as

$$\begin{array}{c}\hfill \underset{¸}{F}\left(\dot{\eta }\right)\tilde{\cap }\underset{¸}{G}\left(\dot{\eta }\right)=\underset{¸}{H}\left(\dot{\eta }\right),\end{array}$$

(5)

where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\cap {\underset{¸}{B}}^{*}$, and for all $\dot{\eta }\in {\underset{¸}{D}}^{*}$.

Corollary 1.

The intersection of $\underset{¸}{F}\left(\dot{\eta }\right)$ and $\underset{¸}{G}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$.

Proof. 

Straightforward.    □

Example 4.

Take a BCI-algebra $(X;\ast ,0)$ as provided in

Table 6. Cayley table for BCI-algebra ($\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\cap }\underset{¸}{G}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$).

∗	0	$\dot{\mathit{\phi }}$	$\dot{\mathit{\omega }}$	$\dot{\mathit{\varsigma }}$
0	0	$\dot{\phi }$	$\dot{\omega }$	$\dot{\varsigma }$
$\dot{\phi }$	$\dot{\phi }$	0	$\dot{\varsigma }$	$\dot{\omega }$
$\dot{\omega }$	$\dot{\omega }$	$\dot{\varsigma }$	0	$\dot{\phi }$
$\dot{\varsigma }$	$\dot{\varsigma }$	$\dot{\omega }$	$\dot{\phi }$	0

.

Let $\underset{¸}{E}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3,{\dot{\eta }}_4\},{\underset{¸}{A}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2\},$ and ${\underset{¸}{B}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3\}$. Then

$\underset{¸}{F}\left({\dot{\eta }}_1\right)=\{(0,-0.8,0.7,-0.1,0.2),(\dot{\phi },-0.3,0.6,-0.6,0.3),(\dot{\omega },-0.7,0.4,-0.2,0.5),(\dot{\varsigma },-0.3,0.4,-0.6,0.5)\}$

$\underset{¸}{F}\left({\dot{\eta }}_2\right)=\{(0,-0.7,0.6,-0.2,0.3),(\dot{\phi },-0.2,0.5,-0.7,0.4),(\dot{\omega },-0.6,0.3,-0.3,0.6),(\dot{\varsigma },-0.2,0.3,-0.7,0.6)\}$

and

$\underset{¸}{G}\left({\dot{\eta }}_1\right)=\{(0,-0.6,0.5,-0.3,0.4),(\dot{\phi },-0.1,0.4,-0.8,0.5),(\dot{\omega },-0.5,0.2,-0.4,0.7),(\dot{\varsigma },-0.1,0.2,-0.8,0.7)\}$

$\underset{¸}{G}\left({\dot{\eta }}_2\right)=\{(0,-0.5,0.4,-0.4,0.5),(\dot{\phi },0,0.3,-0.9,0.6),(\dot{\omega },-0.4,0.1,-0.5,0.8),(\dot{\varsigma },0,0.1,-0.9,0.8)\}$

$\underset{¸}{G}\left({\dot{\eta }}_3\right)=\{(0,-0.9,0.8,0,0.1),(\dot{\phi },-0.4,0.6,-0.5,0.2),(\dot{\omega },-0.8,0.5,-0.1,0.4),(\dot{\varsigma },-0.4,0.5,-0.5,0.4)\}$.

Therefore, $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\cap }\underset{¸}{G}\left(\dot{\eta }\right)=\underset{¸}{H}\left(\dot{\eta }\right)$, where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\cap {\underset{¸}{B}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2\}$ is given by

$\underset{¸}{H}\left({\dot{\eta }}_1\right)=\{(0,-0.6,0.5,-0.3,0.4),(\dot{\phi },-0.1,0.4,-0.8,0.5),(\dot{\omega },-0.5,0.2,-0.4,0.7),(\dot{\varsigma },-0.1,0.2,-0.8,0.7)\}$

$\underset{¸}{H}\left({\dot{\eta }}_2\right)=\{(0,-0.5,0.4,-0.4,0.5),(\dot{\phi },0,0.3,-0.9,0.6),(\dot{\omega },-0.4,0.1,-0.5,0.8),(\dot{\varsigma },0,0.1,-0.9,0.8)\}$.

Hence, $\underset{¸}{H}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$.

Definition 11.

The union $\underset{¸}{F}\left(\dot{\eta }\right),\underset{¸}{G}\left(\dot{\eta }\right)\in \mathcal{BIF}\left(\mathcal{U}\right)$ is a $\mathcal{BIFSS}$ $\underset{¸}{H}\left(\dot{\eta }\right)$, where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\cup {\underset{¸}{B}}^{*}$ and for all $\dot{\eta }\in {\underset{¸}{D}}^{*}$ is defined by

$$\underset{¸}{H}\left(\dot{\eta }\right)=\left\{\begin{array}{c}\underset{¸}{F}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{A}}^{*}-{\underset{¸}{B}}^{*},\hfill \\ \underset{¸}{G}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{B}}^{*}-{\underset{¸}{A}}^{*},\hfill \\ \underset{¸}{F}\left(\dot{\eta }\right)\cup \underset{¸}{G}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{A}}^{*}\cap {\underset{¸}{B}}^{*}\hfill \end{array}\right.$$

Example 5.

Let $X=\{\dot{{\phi }_1},\dot{{\phi }_2},\dot{{\phi }_3},\dot{{\phi }_4}\}$ be a set, and let $\underset{¸}{E}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3,{\dot{\eta }}_4\}$ be a set of factors and ${\underset{¸}{A}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3\}\subseteq \underset{¸}{E},{\underset{¸}{B}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3,{\dot{\eta }}_4\}\subseteq \underset{¸}{E}$. Then, $\underset{¸}{F}\left(\dot{\eta }\right)$ is defined by

$\underset{¸}{F}\left({\dot{\eta }}_1\right)=\{(\dot{{\phi }_1},-0.4,0.0,-0.3,0.2),(\dot{{\phi }_2},-0.3,0.2,-0.2,0.1),(\dot{{\phi }_3},-0.4,0.3,-0.2,0.2),(\dot{{\phi }_4},-0.2,0.2,-0.3,0.1)\}$

$\underset{¸}{F}\left({\dot{\eta }}_2\right)=\{(\dot{{\phi }_1},-0.3,0.1,-0.6,0.2),(\dot{{\phi }_2},-0.3,0.2,-0.5,0.1),(\dot{{\phi }_3},-0.3,0.1,-0.4,0.3),(\dot{{\phi }_4},-0.3,0.2,-0.5,0.2)\}$

$\underset{¸}{F}\left({\dot{\eta }}_3\right)=\{(\dot{{\phi }_1},-0.2,0.1,-0.1,0.4),(\dot{{\phi }_2},-0.3,0.2,-0.2,0.3),(\dot{{\phi }_3},-0.1,0.3,-0.3,0.2),(\dot{{\phi }_4},-0.3,0.4,-0.1,0.1)\}$

and $\underset{¸}{G}\left(\dot{\eta }\right)$ defined as

$\underset{¸}{G}\left({\dot{\eta }}_1\right)=\{(\dot{{\phi }_1},-0.3,0.1,-0.1,0.4),(\dot{{\phi }_2},-0.3,0.2,-0.1,0.5),(\dot{{\phi }_3},0.0,0.3,-0.1,0.3),(\dot{{\phi }_4},-0.2,0.5,0.0,0.1)\}$

$\underset{¸}{G}\left({\dot{\eta }}_2\right)=\{(\dot{{\phi }_1},-0.3,0.2,-0.3,0.5),(\dot{{\phi }_2},-0.1,0.3,-0.2,0.3),(\dot{{\phi }_3},-0.2,0.4,-0.1,0.3),(\dot{{\phi }_4},-0.2,0.3,0.0,0.4)\}$

$\underset{¸}{G}\left({\dot{\eta }}_3\right)=\{(\dot{{\phi }_1},-0.3,0.5,-0.2,0.2),(\dot{{\phi }_2},-0.2,0.3,-0.2,0.1),(\dot{{\phi }_3},-0.2,0.4,-0.1,0.3),(\dot{{\phi }_4},-0.1,0.2,-0.3,0.4)\}$

$\underset{¸}{G}\left({\dot{\eta }}_4\right)=\{(\dot{{\phi }_1},-0.1,0.3,-0.2,0.1),(\dot{{\phi }_2},-0.2,0.1,-0.1,0.2),(\dot{{\phi }_3},-0.3,0.1,-0.1,0.3),(\dot{{\phi }_4},-0.1,0.4,-0.3,0.2)\}$.

Therefore, $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\cup }\underset{¸}{G}\left(\dot{\eta }\right)=\underset{¸}{H}\left(\dot{\eta }\right)$, where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\cup ={\underset{¸}{B}}^{*}\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3,{\dot{\eta }}_4\}$ is given by

$\underset{¸}{H}\left({\dot{\eta }}_1\right)=\{(\dot{{\phi }_1},-0.4,0.1,-0.1,0.2),(\dot{{\phi }_2},-0.3,0.2,-0.1,0.1),(\dot{{\phi }_3},-0.4,0.3,-0.1,0.2),(\dot{{\phi }_4},-0.2,0.5,0.1,0.0)\}$

$\underset{¸}{H}\left({\dot{\eta }}_2\right)=\{(\dot{{\phi }_1},-0.3,0.2,-0.3,0.2),(\dot{{\phi }_2},-0.3,0.3,-0.1,0.1),(\dot{{\phi }_3},-0.3,0.4,-0.1,0.3),(\dot{{\phi }_4},-0.3,0.3,0.0,0.1)\}$

$\underset{¸}{H}\left({\dot{\eta }}_3\right)=\{(\dot{{\phi }_1},-0.3,0.5,-0.1,0.2),(\dot{{\phi }_2},-0.3,0.3,-0.2,0.1),(\dot{{\phi }_3},-0.2,0.4,-0.1,0.2),(\dot{{\phi }_4},-0.3,0.4,-0.1,0.1)\}$

$\underset{¸}{H}\left({\dot{\eta }}_4\right)=\{(\dot{{\phi }_1},-0.1,0.3,-0.2,0.1),(\dot{{\phi }_2},-0.2,0.1,-0.1,0.2),(\dot{{\phi }_3},-0.3,0.1,-0.1,0.3),(\dot{{\phi }_4},-0.1,0.4,-0.3,0.2)\}$.

Hence, $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\cup }\underset{¸}{G}\left(\dot{\eta }\right)=\underset{¸}{H}\left(\dot{\eta }\right)$, where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\cup {\underset{¸}{B}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3,{\dot{\eta }}_4\}$.

Theorem 6.

The union of two $\mathcal{BIFSI}$ is a $\mathcal{BIFSI}$.

Proof. 

By Definition 11, we know that $\underset{¸}{H}\left(\dot{\eta }\right)$, where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\cup {\underset{¸}{B}}^{*}$ and for all $\dot{\eta }\in {\underset{¸}{D}}^{*}$.

$$\begin{array}{c}\hfill \underset{¸}{H}\left(\dot{\eta }\right)=\underset{¸}{F}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{A}}^{*}-{\underset{¸}{B}}^{*}\end{array}$$

(6)

$$\begin{array}{c}\hfill \underset{¸}{H}\left(\dot{\eta }\right)=\underset{¸}{G}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{B}}^{*}-{\underset{¸}{A}}^{*}\end{array}$$

(7)

$$\begin{array}{c}\hfill \underset{¸}{H}\left(\dot{\eta }\right)=\underset{¸}{F}\left(\dot{\eta }\right)\cup \underset{¸}{G}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{A}}^{*}\cap {\underset{¸}{B}}^{*}.\end{array}$$

(8)

Now, there are three cases:

Case 1: If $\dot{\eta }\in {\underset{¸}{A}}^{*}-{\underset{¸}{B}}^{*}$, then

$$\begin{array}{cc}\hfill \underset{¸}{H}\left(\dot{\eta }\right)& =\underset{¸}{F}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{A}}^{*}-{\underset{¸}{B}}^{*}\hfill \\ \hfill & =\underset{¸}{F}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{A}}^{*}\hfill \\ \hfill & =\underset{¸}{F}\left(\dot{\eta }\right)\left[from\left(6\right)\right].\hfill \end{array}$$

Case 2: If $\dot{\eta }\in {\underset{¸}{A}}^{*}-{\underset{¸}{B}}^{*}$, then

$$\begin{array}{cc}\hfill \underset{¸}{H}\left(\dot{\eta }\right)& =\underset{¸}{G}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{B}}^{*}-{\underset{¸}{A}}^{*}\hfill \\ \hfill & =\underset{¸}{G}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{B}}^{*}\hfill \\ \hfill & =\underset{¸}{G}\left(\dot{\eta }\right)\left[from\left(7\right)\right].\hfill \end{array}$$

Case 3: If $\dot{\eta }\in {\underset{¸}{A}}^{*}\cap {\underset{¸}{B}}^{*}$, then

$$\begin{array}{cc}\hfill \underset{¸}{H}\left(\dot{\eta }\right)& =\underset{¸}{F}\left(\dot{\eta }\right)\cup \underset{¸}{G}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{A}}^{*}\cap {\underset{¸}{B}}^{*}\hfill \\ \hfill & =\underset{¸}{F}\left(\dot{\eta }\right)\cup \underset{¸}{G}\left(\dot{\eta }\right)if\dot{\eta }\in {\underset{¸}{B}}^{*}\cap {\underset{¸}{A}}^{*}\left[from\left(8\right)\right].\hfill \end{array}$$

Combining Case 1, 2, and 3, we obtain, $\underset{¸}{H}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$.    □

Example 6.

Let $(X;\ast ,0)$ be a BCI-algebra, as in Example 4.

Let $\underset{¸}{E}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3,{\dot{\eta }}_4\},{\underset{¸}{A}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2\},$ and ${\underset{¸}{B}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3\}$. Then

$\underset{¸}{F}\left({\dot{\eta }}_1\right)=\{(0,-0.8,0.7,-0.1,0.2),(\dot{\phi },-0.3,0.6,-0.6,0.3),(\dot{\omega },-0.7,0.4,-0.2,0.5),(\dot{\varsigma },-0.3,0.4,-0.6,0.5)\}$

$\underset{¸}{F}\left({\dot{\eta }}_2\right)=\{(0,-0.7,0.6,-0.2,0.3),(\dot{\phi },-0.2,0.5,-0.7,0.4),(\dot{\omega },-0.6,0.3,-0.3,0.6),(\dot{\varsigma },-0.2,0.3,-0.7,0.6)\}$

and

$\underset{¸}{G}\left({\dot{\eta }}_1\right)=\{(0,-0.6,0.5,-0.3,0.4),(\dot{\phi },-0.1,0.4,-0.8,0.5),(\dot{\omega },-0.5,0.2,-0.4,0.7),(\dot{\varsigma },-0.1,0.2,-0.8,0.7)\}$

$\underset{¸}{G}\left({\dot{\eta }}_2\right)=\{(0,-0.5,0.4,-0.4,0.5),(\dot{\phi },0,0.3,-0.9,0.6),(\dot{\omega },-0.4,0.1,-0.5,0.8),(\dot{\varsigma },0,0.1,-0.9,0.8)\}$

$\underset{¸}{G}\left({\dot{\eta }}_3\right)=\{(0,-0.9,0.8,0,0.1),(\dot{\phi },-0.4,0.6,-0.5,0.2),(\dot{\omega },-0.8,0.5,-0.1,0.4),(\dot{\varsigma },-0.4,0.5,-0.5,0.4)\}$

Therefore, $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\cup }\underset{¸}{G}\left(\dot{\eta }\right)=\underset{¸}{H}\left(\dot{\eta }\right)$, where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\cup ={\underset{¸}{B}}^{*}\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3\}$ is given by

$\underset{¸}{H}\left({\dot{\eta }}_1\right)=\{(0,-0.8,0.7,-0.1,0.2),(\dot{\phi },-0.3,0.6,-0.6,0.3),(\dot{\omega },-0.7,0.4,-0.2,0.5),(\dot{\varsigma },-0.3,0.4,-0.6,0.5)\}$

$\underset{¸}{H}\left({\dot{\eta }}_2\right)=\{(0,-0.7,0.6,-0.2,0.3),(\dot{\phi },-0.2,0.5,-0.7,0.4),(\dot{\omega },-0.6,0.3,-0.3,0.6),(\dot{\varsigma },-0.2,0.3,-0.7,0.6)\}$

$\underset{¸}{H}\left({\dot{\eta }}_3\right)=\{(0,-0.9,0.8,0,0.1),(\dot{\phi },-0.4,0.6,-0.5,0.2),(\dot{\omega },-0.8,0.5,-0.1,0.4),(\dot{\varsigma },-0.4,0.5,-0.5,0.4)\}$

Hence, $\underset{¸}{H}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$.

Definition 12.

The AND of $\underset{¸}{F}\left(\dot{\eta }\right),\underset{¸}{G}\left(\dot{\eta }\right)\in \mathcal{BIF}\left(\mathcal{U}\right)$ is a $\mathcal{BIFSS}$ defined by $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\wedge }\underset{¸}{G}\left(\dot{\delta }\right)=\underset{¸}{H}(\dot{\eta },\dot{\delta })$, where $\dot{\eta }\in {\underset{¸}{A}}^{*},\dot{\delta }\in {\underset{¸}{B}}^{*}$, and $(\dot{\eta },\dot{\delta })\in {\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\times {\underset{¸}{B}}^{*}$.

Example 7.

Let $(X;\ast ,0)$ be a BCI-algebra, as in Example 4.

Let $\underset{¸}{E}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3,{\dot{\eta }}_4\},{\underset{¸}{A}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2\},$ and ${\underset{¸}{B}}^{*}=\{{\dot{\eta }}_3,{\dot{\eta }}_4\}$. Then

$\underset{¸}{F}\left({\dot{\eta }}_1\right)=\{(0,-0.8,0.7,-0.1,0.2),(\dot{\phi },-0.3,0.6,-0.6,0.3),(\dot{\omega },-0.7,0.4,-0.2,0.5),(\dot{\varsigma },-0.3,0.4,-0.6,0.5)\}$

$\underset{¸}{F}\left({\dot{\eta }}_2\right)=\{(0,-0.7,0.6,-0.2,0.3),(\dot{\phi },-0.2,0.5,-0.7,0.4),(\dot{\omega },-0.6,0.3,-0.3,0.6),(\dot{\varsigma },-0.2,0.3,-0.7,0.6)\}$,

and

$\underset{¸}{G}\left({\dot{\eta }}_3\right)=\{(0,-0.6,0.5,-0.3,0.4),(\dot{\phi },-0.1,0.4,-0.8,0.5),(\dot{\omega },-0.5,0.2,-0.4,0.7),(\dot{\varsigma },-0.1,0.2,-0.8,0.7)\}$

$\underset{¸}{G}\left({\dot{\eta }}_4\right)=\{(0,-0.5,0.4,-0.4,0.5),(\dot{\phi },0,0.3,-0.9,0.6),(\dot{\omega },-0.4,0.1,-0.5,0.8),(\dot{\varsigma },0,0.1,-0.9,0.8)\}$.

Therefore, $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\wedge }\underset{¸}{G}\left(\dot{\eta }\right)=\underset{¸}{H}\left(\dot{\eta }\right)$,

where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\times {\underset{¸}{B}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2\}\times \{{\dot{\eta }}_3,{\dot{\eta }}_4\}=\{({\dot{\eta }}_1,{\dot{\eta }}_3),({\dot{\eta }}_1,{\dot{\eta }}_4),({\dot{\eta }}_2,{\dot{\eta }}_3),({\dot{\eta }}_2,{\dot{\eta }}_4)\}$. Then

$\underset{¸}{H}({\dot{\eta }}_1,{\dot{\eta }}_3)=\{(0,-0.6,0.5,-0.3,0.4),(\dot{\phi },-0.1,0.4,-0.8,0.5),(\dot{\omega },-0.5,0.2,-0.4,0.7),(\dot{\varsigma },-0.1,0.2,-0.8,0.7)\}$

$\underset{¸}{H}({\dot{\eta }}_1,{\dot{\eta }}_4)=\{(0,-0.5,0.4,-0.4,0.5),(\dot{\phi },0,0.3,-0.9,0.6),(\dot{\omega },-0.4,0.1,-0.5,0.8),(\dot{\varsigma },0,0.1,-0.9,0.8)\}$

$\underset{¸}{H}({\dot{\eta }}_2,{\dot{\eta }}_3)=\{(0,-0.6,0.5,-0.3,0.4),(\dot{\phi },-0.1,0.4,-0.8,0.5),(\dot{\omega },-0.5,0.2,-0.4,0.7),(\dot{\varsigma },-0.1,0.2,-0.8,0.7)\}$

$\underset{¸}{H}({\dot{\eta }}_2,{\dot{\eta }}_4)=\{(0,-0.5,0.4,-0.4,0.5),(\dot{\phi },0,0.3,-0.9,0.6),(\dot{\omega },-0.4,0.1,-0.5,0.8),(\dot{\varsigma },0,0.1,-0.9,0.8)\}$

Hence, $\underset{¸}{H}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$.

Definition 13.

The OR of $\underset{¸}{F}\left(\dot{\eta }\right),\underset{¸}{G}\left(\dot{\eta }\right)\in \mathcal{BIF}\left(\mathcal{U}\right)$ is a $\mathcal{BIFSS}$ defined by $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\vee }\underset{¸}{G}\left(\dot{\delta }\right)=\underset{¸}{H}(\dot{\eta },\dot{\delta })$, where $\dot{\eta }\in {\underset{¸}{A}}^{*},\dot{\delta }\in {\underset{¸}{B}}^{*}$, and $(\dot{\eta },\dot{\delta })\in {\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\times {\underset{¸}{B}}^{*}$.

Example 8.

Let $(X;\ast ,0)$ be a BCI-algebra, as in Example 4.

Let $\underset{¸}{E}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_3,{\dot{\eta }}_4\},{\underset{¸}{A}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2\},$ and ${\underset{¸}{B}}^{*}=\{{\dot{\eta }}_3,{\dot{\eta }}_4\}$. Then

$\underset{¸}{F}\left({\dot{\eta }}_1\right)=\{(0,-0.8,0.7,-0.1,0.2),(\dot{\phi },-0.3,0.6,-0.6,0.3),(\dot{\omega },-0.7,0.4,-0.2,0.5),(\dot{\varsigma },-0.3,0.4,-0.6,0.5)\}$

$\underset{¸}{F}\left({\dot{\eta }}_2\right)=\{(0,-0.7,0.6,-0.2,0.3),(\dot{\phi },-0.2,0.5,-0.7,0.4),(\dot{\omega },-0.6,0.3,-0.3,0.6),(\dot{\varsigma },-0.2,0.3,-0.7,0.6)\}$,

and

$\underset{¸}{G}\left({\dot{\eta }}_3\right)=\{(0,-0.6,0.5,-0.3,0.4),(\dot{\phi },-0.1,0.4,-0.8,0.5),(\dot{\omega },-0.5,0.2,-0.4,0.7),(\dot{\varsigma },-0.1,0.2,-0.8,0.7)\}$

$\underset{¸}{G}\left({\dot{\eta }}_4\right)=\{(0,-0.5,0.4,-0.4,0.5),(\dot{\phi },0,0.3,-0.9,0.6),(\dot{\omega },-0.4,0.1,-0.5,0.8),(\dot{\varsigma },0,0.1,-0.9,0.8)\}$.

Therefore, $\underset{¸}{F}\left(\dot{\eta }\right)\tilde{\wedge }\underset{¸}{G}\left(\dot{\eta }\right)=\underset{¸}{H}\left(\dot{\eta }\right)$,

where ${\underset{¸}{D}}^{*}={\underset{¸}{A}}^{*}\times {\underset{¸}{B}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2\}\times \{{\dot{\eta }}_3,{\dot{\eta }}_4\}=\{({\dot{\eta }}_1,{\dot{\eta }}_3),({\dot{\eta }}_1,{\dot{\eta }}_4),({\dot{\eta }}_2,{\dot{\eta }}_3),({\dot{\eta }}_2,{\dot{\eta }}_4)\}$. Then

$\underset{¸}{H}({\dot{\eta }}_1,{\dot{\eta }}_3)=\{(0,-0.8,0.7,-0.1,0.2),(\dot{\phi },-0.3,0.6,-0.6,0.3),(\dot{\omega },-0.7,0.4,-0.2,0.5),(\dot{\varsigma },-0.3,0.4,-0.6,0.5)\}$

$\underset{¸}{H}({\dot{\eta }}_1,{\dot{\eta }}_4)=\{(0,-0.8,0.7,-0.1,0.2),(\dot{\phi },-0.3,0.6,-0.6,0.3),(\dot{\omega },-0.7,0.4,-0.2,0.5),(\dot{\varsigma },-0.3,0.4,-0.6,0.5)\}$

$\underset{¸}{H}({\dot{\eta }}_2,{\dot{\eta }}_3)=\{(0,-0.7,0.6,-0.2,0.3),(\dot{\phi },-0.2,0.5,-0.7,0.4),(\dot{\omega },-0.6,0.3,-0.3,0.6),(\dot{\varsigma },-0.2,0.3,-0.7,0.6)\}$

$\underset{¸}{H}({\dot{\eta }}_2,{\dot{\eta }}_3)=\{(0,-0.7,0.6,-0.2,0.3),(\dot{\phi },-0.2,0.5,-0.7,0.4),(\dot{\omega },-0.6,0.3,-0.3,0.6),(\dot{\varsigma },-0.2,0.3,-0.7,0.6)\}$.

Hence, $\underset{¸}{H}\left(\dot{\eta }\right)$ is a $\mathcal{BIFSI}$.

5. An Application of Bipolar Intuitionistic Fuzzy Soft Sets in Decision Making

The bipolar intuitionistic fuzzy soft set provides a variety of applications for dealing with uncertainties in our daily lives. In this paper, we look at how such an application can be used to work out a prevailing decision-making problem. We use the theory of a bipolar intuitionistic fuzzy soft set to describe a prevailing decision-making problem, and subsequently, we provide a set of rules for selecting the best item based on the given data sets (see Algorithm 1).

Assume $\mathcal{U}=\{{\dot{\varkappa }}_1,{\dot{\varkappa }}_2,{\dot{\varkappa }}_3,{\dot{\varkappa }}_4\}$ is the set of four trucks under investigation, $\underset{¸}{E}$ = {${\dot{\eta }}_1$ = gross vehicle weight rating, ${\dot{\eta }}_2$ = pay load capacity, ${\dot{\eta }}_3$ = fuel efficient, ${\dot{\eta }}_4$ = engine Specifications, ${\dot{\eta }}_5$ = cargo bed} is a set of factors.

Assume that Mr. Joseph wishes to purchase a truck based on the criteria given above. Our goal is to locate the most appealing vehicle for Mr. Joseph. Assume Mr. Joseph’s desiring parameters are ${\underset{¸}{A}}^{*}\subset \underset{¸}{E}$, where ${\underset{¸}{A}}^{*}=\{{\dot{\eta }}_1,{\dot{\eta }}_2,{\dot{\eta }}_5\}$.

Let us consider the following bipolar intuitionistic fuzzy soft set as below:

$\underset{¸}{F}\left({\dot{\eta }}_1\right)=\{({\dot{\varkappa }}_1,-0.4,0.0,-0.3,0.2),({\dot{\varkappa }}_2,-0.3,0.2,-0.2,0.1),({\dot{\varkappa }}_3,-0.4,0.3,-0.2,0.2),({\dot{\varkappa }}_4,-0.2,0.2,-0.3,0.1)\}$

$\underset{¸}{F}\left({\dot{\eta }}_2\right)=\{({\dot{\varkappa }}_1,-0.3,0.1,-0.6,0.2),({\dot{\varkappa }}_2,-0.3,0.2,-0.5,0.1),({\dot{\varkappa }}_3,-0.3,0.1,-0.4,0.3),({\dot{\varkappa }}_4,-0.3,0.2,-0.5,0.1)\}$

$\underset{¸}{F}\left({\dot{\eta }}_5\right)=\{({\dot{\varkappa }}_1,-0.3,0.1,-0.1,0.4),({\dot{\varkappa }}_2,-0.3,0.2,-0.2,0.3),({\dot{\varkappa }}_3,-0.1,0.3,-0.3,0.2),({\dot{\varkappa }}_4,-0.3,0.4,-0.1,0.1)\}$.

Definition 14.

(Comparison table). It is a square table by the same number of rows, and columns as the ubiquitous object names, such as ${\dot{\varkappa }}_1,{\dot{\varkappa }}_2,{\dot{\varkappa }}_3,{\dot{\varkappa }}_4,...,{\dot{\varkappa }}_n$ and the possibilities $c_{ij}$, where $c_{ij}$ is the number of factors for which specific the value of $c_i$ be greater than or equals the value of $c_j$.

Algorithm 1: To find a best choice of truck according to interest made by Mr. John

The set ${\underset{¸}{A}}^{*}\subset \underset{¸}{E}$ should be used to input the parameters of Mr. John. Take a look at the tabulated bipolar intuitionistic fuzzy soft set. Create a comparison Table 7. Tabular representation of ${\tilde{\vartheta }}^{-}$. ${\dot{\mathit{\eta }}}_1$ ${\dot{\mathit{\eta }}}_2$ ${\dot{\mathit{\eta }}}_5$ ${\dot{\varkappa }}_1$ −0.4 −0.3 −0.2 ${\dot{\varkappa }}_2$ −0.3 −0.3 −0.3 ${\dot{\varkappa }}_3$ −0.4 −0.3 −0.1 ${\dot{\varkappa }}_4$ −0.2 −0.3 −0.3 , Table 8. Comparison table of the above table. ${\dot{\mathit{\varkappa }}}_1$ ${\dot{\mathit{\varkappa }}}_2$ ${\dot{\mathit{\varkappa }}}_3$ ${\dot{\mathit{\varkappa }}}_4$ ${\dot{\varkappa }}_1$ 5 3 4 3 ${\dot{\varkappa }}_2$ 4 5 3 4 ${\dot{\varkappa }}_3$ 4 4 5 4 ${\dot{\varkappa }}_4$ 4 4 3 5 , Table 9. Table for ${\tilde{\vartheta }}^{-}$ score. Row Sum $\left(\mathit{a}\right)$ Column Sum $\left(\mathit{b}\right)$ ${\tilde{\mathit{\vartheta }}}^{-}$ Score $\left(\mathit{a}\right)-\left(\mathit{b}\right)$ ${\dot{\varkappa }}_1$ 15 17 −2 ${\dot{\varkappa }}_2$ 16 16 0 ${\dot{\varkappa }}_3$ 17 15 2 ${\dot{\varkappa }}_4$ 16 16 0 , Table 10. Tabular representation of ${\tilde{\vartheta }}^{+}$. ${\dot{\mathit{\eta }}}_1$ ${\dot{\mathit{\eta }}}_2$ ${\dot{\mathit{\eta }}}_5$ ${\dot{\varkappa }}_1$ 0.0 0.1 0.1 ${\dot{\varkappa }}_2$ 0.2 0.2 0.2 ${\dot{\varkappa }}_3$ 0.3 0.1 0.3 ${\dot{\varkappa }}_4$ 0.2 0.2 0.4 , Table 11. Comparison table of the above table. ${\dot{\mathit{\varkappa }}}_1$ ${\dot{\mathit{\varkappa }}}_2$ ${\dot{\mathit{\varkappa }}}_3$ ${\dot{\mathit{\varkappa }}}_4$ ${\dot{\varkappa }}_1$ 5 3 4 5 ${\dot{\varkappa }}_2$ 4 5 4 5 ${\dot{\varkappa }}_3$ 3 3 5 5 ${\dot{\varkappa }}_4$ 2 2 4 5 , Table 12. Table for ${\tilde{\vartheta }}^{+}$ score. Row Sum $\left(\dot{\mathit{\phi }}\right)$ Column Sum $\left(\dot{\mathit{\varsigma }}\right)$ ${\tilde{\mathit{\vartheta }}}^{+}$ Score $(\dot{\mathit{\phi }}-\dot{\mathit{\varsigma }})$ ${\dot{\varkappa }}_1$ 17 14 3 ${\dot{\varkappa }}_2$ 18 13 5 ${\dot{\varkappa }}_3$ 16 17 −1 ${\dot{\varkappa }}_4$ 13 20 −7 , Table 13. Tabular representation of ${\tilde{\varpi }}^{-}$. ${\dot{\mathit{\eta }}}_1$ ${\dot{\mathit{\eta }}}_2$ ${\dot{\mathit{\eta }}}_5$ ${\dot{\varkappa }}_1$ −0.3 −0.6 −0.1 ${\dot{\varkappa }}_2$ −0.2 −0.5 −0.2 ${\dot{\varkappa }}_3$ −0.2 −0.4 −0.3 ${\dot{\varkappa }}_4$ −0.3 −0.5 −0.1 , Table 14. Comparison table of the above table. ${\dot{\mathit{\varkappa }}}_1$ ${\dot{\mathit{\varkappa }}}_2$ ${\dot{\mathit{\varkappa }}}_3$ ${\dot{\mathit{\varkappa }}}_4$ ${\dot{\varkappa }}_1$ 5 4 4 4 ${\dot{\varkappa }}_2$ 3 5 2 3 ${\dot{\varkappa }}_3$ 4 5 5 4 ${\dot{\varkappa }}_4$ 3 4 3 5 , Table 15. Table for ${\tilde{\varpi }}^{-}$ score. Row Sum $\left(\dot{\mathit{\phi }}\right)$ Column Sum $\left(\dot{\mathit{\varsigma }}\right)$ Negative Non-Membership Score $(\dot{\mathit{\phi }}-\dot{\mathit{\varsigma }})$ ${\dot{\varkappa }}_1$ 17 15 2 ${\dot{\varkappa }}_2$ 13 18 −5 ${\dot{\varkappa }}_3$ 18 14 4 ${\dot{\varkappa }}_4$ 15 16 −1 , Table 16. Tabular representation of ${\tilde{\varpi }}^{+}$. ${\dot{\mathit{\eta }}}_1$ ${\dot{\mathit{\eta }}}_2$ ${\dot{\mathit{\eta }}}_5$ ${\dot{\varkappa }}_1$ 0.2 0.2 0.4 ${\dot{\varkappa }}_2$ 0.1 0.1 0.3 ${\dot{\varkappa }}_3$ 0.2 0.3 0.2 ${\dot{\varkappa }}_4$ 0.1 0.1 0.1 , Table 17. Comparison table of the above table. ${\dot{\mathit{\varkappa }}}_1$ ${\dot{\mathit{\varkappa }}}_2$ ${\dot{\mathit{\varkappa }}}_3$ ${\dot{\mathit{\varkappa }}}_4$ ${\dot{\varkappa }}_1$ 5 3 3 3 ${\dot{\varkappa }}_2$ 5 5 4 5 ${\dot{\varkappa }}_3$ 5 3 5 3 ${\dot{\varkappa }}_4$ 4 4 4 5 , Table 18. Table for ${\tilde{\varpi }}^{+}$ score. Row Sum $\left(\dot{\mathit{\phi }}\right)$ Column Sum $\left(\dot{\mathit{\varsigma }}\right)$ ${\tilde{\mathit{\varpi }}}^{+}$ Score $(\dot{\mathit{\phi }}-\dot{\mathit{\varsigma }})$ ${\dot{\varkappa }}_1$ 14 19 −5 ${\dot{\varkappa }}_2$ 19 15 4 ${\dot{\varkappa }}_3$ 16 16 0 ${\dot{\varkappa }}_4$ 17 16 1 and Table 19. Table for final Score. Positive Score $\left(\mathfrak{p}\right)$ Negative Score $\left(\mathfrak{n}\right)$ Final Score $(\mathfrak{p}-\mathfrak{n})$ ${\dot{\varkappa }}_1$ 8 −4 12 ${\dot{\varkappa }}_2$ 1 5 −4 ${\dot{\varkappa }}_3$ 1 −2 3 ${\dot{\varkappa }}_4$ 8 1 7 for the negative associates function ${\tilde{\vartheta }}^{-}$, the positive associates function ${\tilde{\vartheta }}^{+}$, the negative non-associates function ${\tilde{\varpi }}^{-}$, and the positive non-associates function ${\tilde{\varpi }}^{+}$. Calculate the score for both positive and negative associates. Subtract the positive associate score from the negative associate score to determine the final score. Find the maximums score, if it appears in position ith, then Mr. John will purchase to ${\dot{\varkappa }}_i,1\le i\le 4$. The truck ${\dot{\varkappa }}_1$ scored 12, which is obviously the maximum score.

Decision: Mr. John will purchase ${\dot{\varkappa }}_1$ due to the fact that ${\dot{\varkappa }}_4$ will be his second choice if he decides against purchasing ${\dot{\varkappa }}_1$ for some reason, and his third choice is ${\dot{\varkappa }}_3$.

6. Advantages of Bipolar Intuitionistic Fuzzy Soft Ideals

Bipolar intuitionistic fuzzy soft ideals in the context of BCK/BCI-algebras are a specialized concept in the field of mathematical algebra and decision-making. These structures combine several mathematical and fuzzy set-based approaches to enhance decision-making processes. Here are some advantages of using bipolar intuitionistic fuzzy soft sets in BCK/BCI-algebras and their applications in decision making:

• Enhanced Modeling of Uncertainty: Bipolar intuitionistic fuzzy soft sets capture this uncertainty more effectively by allowing for both membership and non-membership values, as well as a degree of hesitation.
• Flexibility in Decision-Making: Bipolar intuitionistic fuzzy soft sets provide a range of possibilities for making decisions. This flexibility allows decision-makers to express their preferences and uncertainties more accurately.
• Applicability in BCK/BCI-Algebras: Bipolar intuitionistic fuzzy soft sets can be applied to BCK/BCI-algebraic structures to solve problems related to algebraic reasoning, propositional logic, and more.
• Robust Decision-Making: Bipolar intuitionistic fuzzy soft sets allow decision-makers to consider the worst-case and best-case scenarios by incorporating both membership and non-membership values. This leads to more robust and reliable decisions that can withstand variations in input data.
• Handling Incomplete Information: Bipolar intuitionistic fuzzy soft sets can handle incomplete information by considering both known and unknown aspects of a problem, making them valuable in decision making under uncertainty.
• Multi-Criteria Decision Analysis: bipolar intuitionistic fuzzy soft sets are particularly useful in multi-criteria decision analysis (MCDA) where multiple conflicting criteria need to be considered.
• Application in Real-World Decision Problems: Bipolar intuitionistic fuzzy soft sets have been used in a wide range of practical applications, including medical diagnosis, environmental assessment, financial decision making, and more. They help decision-makers make informed choices when dealing with complex and uncertain situations.

7. Limitations of Bipolar Intuitionistic Fuzzy Soft Ideals

Bipolar intuitionistic fuzzy soft ideals of BCK/BCI-algebras and their applications in decision making can be a complex and specialized area of research. These structures are extensions of classical algebraic systems with the addition of fuzzy and bipolar elements. While they have their merits, they also come with several limitations and challenges, both in their theoretical development and practical applications. Here are some limitations and considerations:

• Lack of standardization: Different researchers may define these concepts in slightly different ways, making it difficult to compare and generalize results.
• Computational Complexity: Solving problems involving these structures may require significant computational resources and time.
• Limited Practical Data: Decision making typically relies on real-world data.
• Interpretability: The interpretation of results derived from bipolar intuitionistic fuzzy soft sets may be challenging for decision-makers who are not familiar with the intricacies of these mathematical structures.
• Data Collection and Aggregation: In practice, it may be challenging to collect sufficient and accurate data to make informed decisions using these models.
• Limited Real-World Applications: The applications of bipolar intuitionistic fuzzy soft sets in BCK/BCI-algebras in real-world decision-making contexts are limited.
• Lack of Case Studies: There may be a shortage of case studies and practical examples demonstrating the effectiveness of these models in real decision-making scenarios.

8. Conclusions

It is confirmed. Our method combines a bipolar intuitionistic fuzzy soft set with a soft set to produce a theoretical study of bipolar intuitionistic fuzzy soft ideals. A bipolar intuitionistic fuzzy soft set is applied to a problem of decision making. Additionally, we provided a general algorithm for solving bipolar intuitionistic fuzzy soft set decision-making problems. In the future, we will concentrate on applications in artificial intelligence, general systems, and bipolar intuitionistic fuzzy soft (p, q, positive implicative, and implicative) ideals in BCK-algebras. This approach is highly effective for analyzing data and can be applied to medical diagnosis with great success.