A New Perspective on Intuitionistic Fuzzy Structures in Sheffer Stroke BCK-Algebras

Abstract

This study introduces the concept of an intuitionistic fuzzy SBCK-subalgebra (SBCK-ideal) and explores the level set of an intuitionistic fuzzy set within the context of Sheffer stroke BCK-algebras. These newly defined concepts are crucial for understanding the interaction between intuitionistic logic and Sheffer stroke BCK-algebras. The paper establishes a connection between subalgebras and level sets in the framework of Sheffer stroke BCK-algebras, demonstrating that the level set of intuitionistic fuzzy SBCK-subalgebras corresponds precisely to their subalgebras, and conversely. Additionally, the study provides novel results regarding the structural properties of Sheffer stroke BCK-algebras under intuitionistic fuzzy logic, specifically focusing on the conditions under which fuzzy sets become SBCK-subalgebras or SBCK-ideals. This work contributes to the theoretical foundations of fuzzy logic in algebraic structures, offering a deeper understanding of the interplay between intuitionistic fuzzy sets and the algebraic operations within Sheffer stroke BCK-algebras.

1. Introduction

The concept of a fuzzy set was first introduced by Zadeh [1] in 1965, marking the beginning of a new era in set theory and logic. Following this, various extensions of the fuzzy set were proposed to address different complexities. The first extension was the L-fuzzy set, introduced by Goguen [2] in 1967, which expanded the scope of fuzzy sets. The second extension, introduced by Zadeh [3], was the interval-valued fuzzy set, which allowed for more nuanced representations of uncertainty. The third extension, known as the rough set, was defined by Pawlak in 1981 [4,5], providing a framework for handling approximation and uncertainty in data. The fourth extension, and perhaps one of the most influential, was the intuitionistic fuzzy set, introduced by Atanassov in 1983 [6,7,8], which added an additional layer of membership and non-membership, providing a more flexible approach to handling vagueness and uncertainty in various applications.

The Sheffer operation, also called the Sheffer stroke or NAND operator, was introduced by H. M. Sheffer [9]. It is highly significant because it can define an entire logical system by itself, enabling any axiom within that system to be expressed solely in terms of the Sheffer operation. This ability provides a distinct advantage in controlling and simplifying the logical structure. Interestingly, even the axioms of Boolean algebra, the algebraic counterpart of classical propositional calculus, can be fully expressed using only the Sheffer operation, highlighting its essential role in logical and algebraic frameworks.

The Sheffer stroke is critical because it allows all Boolean functions to be represented with a single operation, making it a powerful and efficient tool in Boolean algebra. This simplification reduces the need for multiple operations, streamlining logical systems. Beyond Boolean algebra, the Sheffer stroke has been applied in numerous other algebraic structures, such as MV-algebras, BL-algebras, BCK-algebras, BE-algebras, ortholattices, and Hilbert algebras, demonstrating its broad applicability across different areas of mathematics.

The motivation for this study arises from the intersection of two powerful concepts in mathematical logic: the Sheffer stroke operation and intuitionistic fuzzy set theory. While Sheffer stroke BCK-algebras have been studied for their foundational role in algebraic logic, the incorporation of intuitionistic fuzzy structures into these algebras remains relatively unexplored. Intuitionistic fuzzy sets provide a nuanced way to represent uncertainty and partial truth, which is crucial for modern applications in decision making, control theory, and artificial intelligence. By defining and analyzing intuitionistic fuzzy SBCK-subalgebras and ideals, this paper aims to establish a novel theoretical framework that enhances the expressive power of BCK-algebras under the Sheffer operation. This contributes to both the enrichment of algebraic theory and the potential for practical applications in areas requiring fuzzy logic with structured algebraic underpinnings.

Initially presented by Sheffer, the Sheffer stroke has since become a cornerstone in Boolean algebra due to its ability to replace all standard Boolean operations. Its usefulness extends beyond Boolean algebras, with applications in basic algebras and ortholattices, among others (see [10,11]). One of the relevant studies on this topic is Intuitionistic Fuzzy Structures on Sheffer Stroke UP-algebras [12].

Recent developments in mathematical logic and dynamical systems, such as studies on bi-center problems, isochronous centers in switching systems, and nilpotent center conditions [13,14,15], offer valuable insights into complex structures. Although focused on dynamical systems, these works share a common goal with the present study: advancing the understanding of structured mathematical models through detailed analysis.

In this paper, we introduce the concept of an intuitionistic fuzzy SBCK-subalgebra and examine the level set of an intuitionistic fuzzy set within the context of Sheffer stroke BCK-algebras. These newly defined concepts play a crucial role in understanding the behavior of intuitionistic logic within the framework of Sheffer stroke BCK-algebras. The study establishes a connection between subalgebras and level sets, demonstrating that the level set of an intuitionistic fuzzy SBCK-subalgebra in this algebra is precisely its subalgebra, and vice versa. This relationship underscores the close connection between these two concepts within the given algebraic structure. Furthermore, the paper defines the notion of an intuitionistic fuzzy SBCK-ideal in a Sheffer stroke BCK-algebra and explores its properties. It is also shown that every intuitionistic fuzzy SBCK-ideal of a Sheffer stroke BCK-algebra is, in fact, an intuitionistic fuzzy SBCK-subalgebra; however, the reverse is generally not true. This distinction highlights the unique characteristics and behavior of intuitionistic fuzzy SBCK-ideals within the context of Sheffer stroke BCK-algebras.

The rest of this paper is organized as follows: Section 2 introduces the basic definitions and preliminaries related to Sheffer stroke BCK-algebras and intuitionistic fuzzy sets. In Section 3, we define the concept of intuitionistic fuzzy SBCK-subalgebras and explore their structural properties, followed by the examination of their corresponding level sets. Section 4 focuses on the notion of intuitionistic fuzzy SBCK-ideals, providing a comprehensive analysis of their relationship with SBCK-subalgebras. Finally, Section 5 concludes the paper by summarizing the findings and suggesting potential areas for future research.

The list of acronyms is given in

Table 1. List of acronyms.

Acronyms	Representation
$SBCK$-algebra	Sheffer stroke BCK-algebra
$I{F}_{SBCK}$-subalgebra	intuitionistic fuzzy SBCK-subalgebra
$I{F}_{SBCK}$-ideal	intuitionistic fuzzy SBCK-ideal
IFS	intuitionistic fuzzy set
IFI	intuitionistic fuzzy ideal

.

2. Preliminaries

Definition 1 

([9]). Let $(G_S,\mid )$ be a groupoid. The operation | is said to be a Sheffer stroke operation if it satisfies the following conditions:

(S1) 

${\mathfrak{g}}_{s_1}\mid {\mathfrak{g}}_{s_2}={\mathfrak{g}}_{s_2}\mid {\mathfrak{g}}_{s_1},$

(S2) 

$({\mathfrak{g}}_{s_1}\mid {\mathfrak{g}}_{s_1})|({\mathfrak{g}}_{s_1}\mid {\mathfrak{g}}_{s_2})={\mathfrak{g}}_{s_1},$

(S3) 

${\mathfrak{g}}_{s_1}\mid (({\mathfrak{g}}_{s_2}\mid {\mathfrak{g}}_{s_3})\mid ({\mathfrak{g}}_{s_2}\mid {\mathfrak{g}}_{s_3}))=(({\mathfrak{g}}_{s_1}\mid {\mathfrak{g}}_{s_2})\mid ({\mathfrak{g}}_{s_1}\mid {\mathfrak{g}}_{s_2}))\mid {\mathfrak{g}}_{s_3},$

(S4) 

$({\mathfrak{g}}_{s_1}\mid (({\mathfrak{g}}_{s_1}\mid {\mathfrak{g}}_{s_1})\mid ({\mathfrak{g}}_{s_2}\mid {\mathfrak{g}}_{s_2})))|({\mathfrak{g}}_{s_1}\mid (({\mathfrak{g}}_{s_1}\mid {\mathfrak{g}}_{s_1})\mid ({\mathfrak{g}}_{s_2}\mid {\mathfrak{g}}_{s_2})))={\mathfrak{g}}_{s_1},$

for all ${\mathfrak{g}}_{s_1},{\mathfrak{g}}_{s_2},{\mathfrak{g}}_{s_3}\in G_S$.

As anotational convenience, we denote

$$\begin{array}{c}\hfill {\mathfrak{S}}_{\mathfrak{g}}\left({\mathfrak{g}}_s\right)=\mathfrak{g}\mid ({\mathfrak{g}}_s\mid {\mathfrak{g}}_s).\end{array}$$

Definition 2 

([16]). A Sheffer stroke BCK-algebra (briefly, SBCK-algebra) is a structure $(X,\mid )$ of type (2) such that 0 is the fixed element in X and the following conditions are satisfied for all ${\mathfrak{b}}_{s_1},{\mathfrak{b}}_{s_2},{\mathfrak{b}}_{s_3}\in X$:

$$\begin{array}{c}\hfill \begin{array}{c}\left(SBC{K}_1\right)(({\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right))\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_3}\right))\mid (({\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right))\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_3}\right))=0\mid 0,\hfill \\ \left(SBC{K}_2\right){\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)=0and{\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)=0\Rightarrow {\mathfrak{b}}_{s_1}={\mathfrak{b}}_{s_2}.\hfill \end{array}\end{array}$$

Proposition 1 

([16]). Let $(X,\mid )$ be an SBCK-algebra. Then, the binary relation ${\mathfrak{b}}_{s_1}\preccurlyeq {\mathfrak{b}}_{s_2}$ if and only if ${\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)=0$ is a partial order on X.

Definition 3 

([16]). Let $(X,\mid )$ be an SBCK-algebra. A nonempty subset S of X is called an SBCK-subalgebra of X if $({\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right))\in S$ for all ${\mathfrak{b}}_{s_1},{\mathfrak{b}}_{s_2}\in S$.

Proposition 2 

([16]). et $(X,\mid )$ be an SBCK-algebra. A nonempty subset I of X is called an SBCK-ideal of X if for all ${\mathfrak{b}}_{s_1},{\mathfrak{b}}_{s_2}\in I$.

1. 

$0\in I$,

2. 

$({\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right))\in I$ and ${\mathfrak{b}}_{s_2}\in I$⇒${\mathfrak{b}}_{s_1}\in I$.

Lemma 1 

([17]). Let μ be a fuzzy set in a nonempty set H and ${\mathfrak{x}}_1,{\mathfrak{x}}_2\in H$. Then,

1. 

$1-max\{\mu \left({\mathfrak{x}}_1\right),\mu \left({\mathfrak{x}}_2\right)\}=min\{1-\mu \left({\mathfrak{x}}_1\right),1-\mu \left({\mathfrak{x}}_2\right)\}$,

2. 

$1-min\{\mu \left({\mathfrak{x}}_1\right),\mu \left({\mathfrak{x}}_2\right)\}=max\{1-\mu \left({\mathfrak{x}}_1\right),1-\mu \left({\mathfrak{x}}_2\right)\}$.

Intuitionistic fuzzy sets (IFSs), introduced by Atanassov, extend classical fuzzy sets by incorporating both a membership function $\varrho \left(x\right)$ and a non-membership function $\zeta \left(x\right)$, where $\varrho \left(x\right),\zeta \left(x\right)\in [0,1]$ for all $x\in X$. The degree of indeterminacy, $\gamma \left(x\right)=1-\varrho \left(x\right)-\zeta \left(x\right)$, quantifies uncertainty or indecision. The primary condition is that $\varrho \left(x\right)+\zeta \left(x\right)\le 1$, ensuring a degree of indeterminacy.

Definition 4 

([6]). Let H be a nonempty set. The intutionstic fuzzy set A on H is defined to be a structure

$$\begin{array}{c}\hfill A:=\{\langle \mathfrak{a},\varrho (\mathfrak{a}),\zeta (\mathfrak{a})\mid \mathfrak{a}\in H\},\end{array}$$

(1)

where $\varrho :H\to [0,1]$ is the degree of membership of $\mathfrak{a}$ to A and $\zeta :H\to [0,1]$ is the degree of non-membership of $\mathfrak{a}$ to A such that $0\le \varrho \left(\mathfrak{a}\right)+\zeta \left(\mathfrak{a}\right)\le 1$.

IFSs have been widely applied in areas such as decision making, pattern recognition, and control systems, where uncertainty plays a significant role. Recently, the integration of IFSs into algebraic structures like BCK-algebras has led to the development of new concepts such as intuitionistic fuzzy SBCK-subalgebras and SBCK-ideals, which are explored in the following sections.

Lemma 2 

([17]). Let $\mathfrak{p},\mathfrak{q},\mathfrak{r}\in \mathbb{R}$. Then the following statements hold:

1. 

$\mathfrak{p}-min\{\mathfrak{q},\mathfrak{r}\}=max\{\mathfrak{p}-\mathfrak{q},\mathfrak{p}-\mathfrak{r}\}$,

2. 

$\mathfrak{p}-max\{\mathfrak{q},\mathfrak{r}\}=min\{\mathfrak{p}-\mathfrak{q},\mathfrak{p}-\mathfrak{r}\}$.

3. Intuitionistic Fuzzy SBCK-Subalgebras

In this section, the study introduces the concept of intuitionistic fuzzy SBCK-subalgebras within the context of $SBCK$-algebras. It is important to note that, unless explicitly stated otherwise, X refers to a Sheffer stroke BCK-algebra.

Definition 5. 

An $IFS$ $(X,\varrho ,\zeta )$ in X is called an intuitionistic fuzzy SBCK-subalgebra (briefly, $I{F}_{SBCK}$-subalgebra) of X if

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{b}}_{s_1},{\mathfrak{b}}_{s_2}\in X)\left(\begin{array}{c}\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right))\ge min\{\varrho \left({\mathfrak{b}}_{s_1}\right),\varrho \left({\mathfrak{b}}_{s_2}\right)\}\hfill \\ \zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{s_1}}\left({\mathfrak{b}}_{s_2}\right))\le max\{\zeta \left({\mathfrak{b}}_{s_1}\right),\zeta \left({\mathfrak{b}}_{s_2}\right)\}\hfill \end{array}\right).\hfill \end{array}$$

(2)

Example 1. 

Let $X=\{0,1,2,3,4,5,6,7\}$ be a set with the binary operation “$\mid$” given in the following table:

|	0	1	2	3	4	5	6	7
0	1	1	1	1	1	1	1	1
1	1	0	3	2	5	4	7	6
2	1	3	3	1	1	3	3	1
3	1	2	1	2	5	6	5	6
4	1	5	1	5	5	1	5	1
5	1	4	3	6	1	4	3	6
6	1	7	3	5	5	3	7	1
7	1	6	1	6	1	6	1	6

Then, $(X,\mid )$ is a $SBCK$-algebra. Define $IFS$ $(X,\varrho ,\zeta )$ in X by the table below:

X	0	1	2	3	4	5	6	7
$\varrho$	$0.2$	$0.1$	$0.2$	$0.1$	$0.2$	$0.1$	$0.1$	$0.2$
$\zeta$	$0.7$	$0.9$	$0.7$	$0.7$	$0.7$	$0.7$	$0.8$	$0.7$

It is routine to verify that the $IFS$ $(X,\varrho ,\zeta )$ in X is a $I{F}_{SBCK}$-subalgebra of X.

Definition 6. 

Let μ be a fuzzy set on an $SBCK$-algebra X and let $\mathfrak{t}\in [0,1]$. We define the following two subsets of X:

• The upper level set of μ at level $\mathfrak{t}$ is defined as

  $$\mathfrak{U}(\mu ,\mathfrak{t})=\{\mathfrak{x}\in X:\mu (\mathfrak{x})\ge \mathfrak{t}\}.$$
• The lower level set of μ at level $\mathfrak{t}$ is defined as

  $$\mathfrak{L}(\mu ,\mathfrak{t})=\{\mathfrak{x}\in X:\mu (\mathfrak{x})\le \mathfrak{t}\}.$$

These sets are referred to as the level sets of the fuzzy set μ and play an important role in the study of fuzzy subalgebras in $SBCK$-algebras.

Example 2. 

Let $(X,\varrho ,\zeta )$ be the $IFS$ given in Example 1. For $\mathfrak{t}=0.8$, we have

$$\mathfrak{L}(\zeta ,0.8)=\{\mathfrak{x}\in X\mid \zeta (x)\le 0.8\}=\{0,2,3,4,5,6,7\}.$$

For $\mathfrak{t}=0.2$, we have

$$\mathfrak{U}(\varrho ,0.2)=\{\mathfrak{x}\in X\mid \varrho (\mathfrak{x})\ge 0.2\}=\{0,2,4,7\}.$$

Theorem 1. 

An $IFS$ $(X,\varrho ,\zeta )$ in X is an $I{F}_{SBCK}$-subalgebra of X if and only if the sets $\mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ and $\mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ are SBCK-subalgebras of X whenever they are nonempty for all ${\mathfrak{l}}_s,{\mathfrak{u}}_{t}in[0,1]$.

Proof. 

Assume that $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-subalgebra of X and $\mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\ne \varnothing \ne \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ for all ${\mathfrak{l}}_s,{\mathfrak{u}}_t\in [0,1]$. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2},{\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2}\in X$ be such that $({\mathfrak{b}}_{x_1},{\mathfrak{b}}_{y_1})\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\times \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ and $({\mathfrak{b}}_{x_2},{\mathfrak{b}}_{y_2})\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\times \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$. Then, $\zeta \left({\mathfrak{b}}_{x_1}\right)\le {\mathfrak{l}}_s$, $\zeta \left({\mathfrak{b}}_{x_2}\right)\le {\mathfrak{l}}_s$, $\varrho \left({\mathfrak{b}}_{y_1}\right)\ge {\mathfrak{u}}_t$ and $\varrho \left({\mathfrak{b}}_{y_2}\right)\ge {\mathfrak{u}}_t$. Then, $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\le max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}\le {\mathfrak{l}}_s$ and $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\ge min\{\varrho \left({\mathfrak{b}}_{y_1}\right),\varrho \left({\mathfrak{b}}_{y_2}\right)\}\ge {\mathfrak{u}}_t$ and so $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\times \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$. Therefore, $\mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ and $\mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ are SBCK-subalgebras of X.

Conversely, let $(X,\varrho ,\zeta )$ be an $IFS$ in X for which $\mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ and $\mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ are SBCK-subalgebras of X whenever they are nonempty for all ${\mathfrak{l}}_s,{\mathfrak{u}}_t\in [0,1]$. Suppose that $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))>max\{\zeta \left({\mathfrak{b}}_{y_1}\right),\zeta \left({\mathfrak{b}}_{y_2}\right)\}$ or $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))<min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}$ for some ${\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2},{\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then, we have ${\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2}\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ or ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in \mathfrak{U}(\varrho ,{\mathfrak{u}}_t),$ where ${\mathfrak{l}}_s=max\{\zeta \left({\mathfrak{b}}_{y_1}\right),\zeta \left({\mathfrak{b}}_{y_2}\right)\}$ and ${\mathfrak{u}}_t=min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}$. However, $({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\notin \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ or $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\notin \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$, a contradiction. Therefore, we obtain $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\le max\{\zeta \left({\mathfrak{b}}_{y_1}\right),\zeta \left({\mathfrak{b}}_{y_2}\right)\}$ and $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\ge min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}$ for all ${\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2},{\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Consequently, $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-subalgebra of X. □

Theorem 2. 

An $IFS$ $(X,\varrho ,\zeta )$ in X is an $I{F}_{SBCK}$-subalgebra of X if and only if the fuzzy sets ${\zeta }_{\mathfrak{c}}$ and ϱ are fuzzy SBCK-subalgebras of X, where $\zeta :X\to [0,1]$ is defined by ${\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_x\right)=1-\zeta \left({\mathfrak{b}}_x\right)$ for all ${\mathfrak{b}}_x\in X$.

Proof. 

Assume that $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-subalgebra of X. It is clear that $\varrho$ is a fuzzy SBCK-subalgebra of X. For every ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$,

$$\begin{array}{ccc}{\zeta }_{\mathfrak{c}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& =& 1-\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\hfill \\ & \ge & 1-max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& min\{1-\zeta \left({\mathfrak{b}}_{x_1}\right),1-\zeta \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& min\{{\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_1}\right),{\zeta }_c\left({\mathfrak{b}}_{x_2}\right)\}.\hfill \end{array}$$

Hence, ${\zeta }_c$ is a fuzzy SBCK-subalgebra of X.

Conversely, let $(X,\varrho ,\zeta )$ be an $IFS$ of X for which ${\zeta }_c$ and $\varrho$ are fuzzy SBCK-subalgebras of X. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then

$$\begin{array}{ccc}1-\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& =& {\zeta }_{\mathfrak{c}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\hfill \\ & \ge & min\{{\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_1}\right),{\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& min\{1-\zeta \left({\mathfrak{b}}_{x_1}\right),1-\zeta \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& 1-max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ \zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)))& \le & max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}.\hfill \end{array}$$

Hence, $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-subalgebra of X. □

Theorem 3. 

Given a nonempty subset Y of X, let $(X,{\varrho }_Y,{\zeta }_Y)$ be an IFS in X defined as follows:

$$\varrho \left({\mathfrak{b}}_x\right)=\left\{\begin{array}{cc}{\varrho }_0& if{\mathfrak{b}}_x\in Y\\ {\varrho }_1& otherwise,& \end{array}\right.$$

and

$$\zeta \left({\mathfrak{b}}_x\right)=\left\{\begin{array}{cc}{\zeta }_0& if{\mathfrak{b}}_x\in Y\\ {\zeta }_1& otherwise& \end{array}\right.$$

for all ${\mathfrak{b}}_x\in X$ and ${\varrho }_{\mathfrak{k}},{\zeta }_{\mathfrak{k}}\in [0,1]$ such that ${\varrho }_0>{\varrho }_1$, ${\zeta }_0<{\zeta }_1$, and ${\varrho }_{\mathfrak{k}}+{\zeta }_{\mathfrak{k}}\le 1$ for $\mathfrak{k}=0,1$. Then, $(X,{\varrho }_Y,{\zeta }_Y)$ be an $I{F}_{SBCK}$-subalgebra of X if and only if Y is a SBCK-subalgebra of X.

Proof. 

Assume that $(X,{\varrho }_Y,{\zeta }_Y)$ is an $I{F}_{SBCK}$-subalgebra of X. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$ be such that ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in Y$. Then,

$$\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\ge min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}={\varrho }_0,$$

and

$$\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\le max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}={\zeta }_0,$$

and so $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\varrho }_0$ and $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\zeta }_0$. This shows that ${\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\in Y$. Therefore, Y is a SBCK-subalgebra of X.

Conversely, let Y be a SBCK-subalgebra of X. For every ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$, if ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in Y$, then, ${\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\in Y$, which implies that $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\varrho }_0=min\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}$ and $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\zeta }_0=max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}$. If ${\mathfrak{b}}_{x_1}\notin Y$ of ${\mathfrak{b}}_{x_2}\notin Y$, then we have $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\ge {\varrho }_1=min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}$ and $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\le {\zeta }_1=max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}$. Therefore, $(X,{\varrho }_Y,{\zeta }_Y)$ is an $I{F}_{SBCK}$-subalgebra of X. □

Theorem 4. 

An IFS $(X,\varrho ,\zeta )$ in X is an $I{F}_{SBCK}$-subalgebra of X, then, $\varrho \left(0\right)\ge \varrho \left({\mathfrak{b}}_x\right)$ and $\zeta \left(0\right)\le \zeta \left({\mathfrak{b}}_x\right)$ for all ${\mathfrak{b}}_x\in X$.

Proof. 

For any ${\mathfrak{b}}_x\in X$,

$$\varrho \left(0\right)=\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_x}\left({\mathfrak{b}}_x\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_x}\left({\mathfrak{b}}_x\right))\ge min\{\varrho \left({\mathfrak{b}}_x\right),\varrho \left({\mathfrak{b}}_x\right))\}=\varrho \left({\mathfrak{b}}_x\right),$$

and

$$\zeta \left(0\right)=\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_x}\left({\mathfrak{b}}_x\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_x}\left({\mathfrak{b}}_x\right))\le max\{\zeta \left({\mathfrak{b}}_x\right),\zeta \left({\mathfrak{b}}_x\right)\}=\zeta \left({\mathfrak{b}}_x\right).$$

□

Proposition 3. 

For an $I{F}_{SBCK}$-subalgebra of X satisfies

$$\begin{array}{c}\hfill (\forall {\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X)\left(\begin{array}{c}\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\ge \varrho \left({\mathfrak{b}}_{x_2}\right)\hfill \\ \zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\le \zeta \left({\mathfrak{b}}_{x_2}\right)\hfill \end{array}\right)\end{array}$$

(3)

if and only if $\varrho \left(0\right)=\varrho \left({\mathfrak{b}}_{x_1}\right)$ and $\zeta \left(0\right)=\zeta \left({\mathfrak{b}}_{x_1}\right)$.

Proof. 

We have

$$\begin{array}{ccc}\varrho \left({\mathfrak{b}}_x\right)& =& \varrho (({\mathfrak{b}}_x\mid {\mathfrak{b}}_x)\mid ({\mathfrak{b}}_x\mid {\mathfrak{b}}_x))\hfill \\ & =& \varrho ({\mathfrak{S}}_1({\mathfrak{b}}_x\mid {\mathfrak{b}}_x)\mid {\mathfrak{S}}_1({\mathfrak{b}}_x\mid {\mathfrak{b}}_x))\hfill \\ & =& \varrho (({\mathfrak{b}}_x\mid 1)\mid ({\mathfrak{b}}_x\mid 1))\hfill \\ & =& \varrho ({\mathfrak{S}}_{{\mathfrak{b}}_x}\left(0\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_x}\left(0\right))\hfill \\ & \ge & \varrho \left(0\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\zeta \left({\mathfrak{b}}_x\right)& =& \zeta (({\mathfrak{b}}_x\mid {\mathfrak{b}}_x)\mid ({\mathfrak{b}}_x\mid {\mathfrak{b}}_x))\hfill \\ & =& \zeta ({\mathfrak{S}}_1({\mathfrak{b}}_x\mid {\mathfrak{b}}_x)\mid {\mathfrak{S}}_1({\mathfrak{b}}_x\mid {\mathfrak{b}}_x))\hfill \\ & =& \zeta (({\mathfrak{b}}_x\mid 1)\mid ({\mathfrak{b}}_x\mid 1))\hfill \\ & =& \zeta ({\mathfrak{S}}_{{\mathfrak{b}}_x}\left(0\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_x}\left(0\right))\hfill \\ & \le & \zeta \left(0\right)\hfill \end{array}$$

Then, by Theorem 4, we have $\varrho \left(0\right)=\varrho \left({\mathfrak{b}}_x\right)$ and $\zeta \left(0\right)=\zeta \left({\mathfrak{b}}_x\right)$. The converse is clear. □

4. Intuitionistic Fuzzy Ideals in SBCK-Algebras

In this section, the study presents the introduction of intuitionistic fuzzy SBCK-ideals within the framework of Sheffer stroke BCK-algebras. By defining these new concepts, we aim to expand the understanding of the interplay between intuitionistic fuzzy logic and $SBCK$-algebraic structures. The exploration of intuitionistic fuzzy SBCK-ideals provides a foundational perspective for further analysis of their properties and their role in the broader context of algebraic logic.

Definition 7. 

An $IFS$ $(X,\varrho ,\zeta )$ on X is called an intuitionistic fuzzy SBCK-ideal (briefly, $I{F}_{SBCK}$-ideal) of X if

$$\begin{array}{c}\hfill (\forall {\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X)\left(\begin{array}{c}\varrho \left(0\right)\ge \varrho \left({\mathfrak{b}}_{x_1}\right)\ge min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\varrho \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ \zeta \left(0\right)\le \zeta \left({\mathfrak{b}}_{x_1}\right)\le max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\zeta \left({\mathfrak{b}}_{x_2}\right)\}\hfill \end{array}\right).\end{array}$$

(4)

Example 3. 

Let $X=\{0,1,2,3,4,5,6,7\}$ be a set with the binary operation “∣” given in the following table:

|	0	1	2	3	4	5	6	7
0	1	1	1	1	1	1	1	1
1	1	0	3	2	5	4	7	6
2	1	3	3	1	6	4	4	6
3	1	2	1	2	2	1	2	1
4	1	5	6	2	5	1	2	6
5	1	4	4	1	1	4	4	1
6	1	7	4	2	2	4	7	1
7	1	6	6	1	6	1	1	6

Then, $(X,\mid )$ is a $SBCK$-algebra. Define the $IFS$ $(X,\varrho ,\zeta )$ in X by the table below:

X	0	1	2	3	4	5	6	7
$\varrho$	$0.7$	$0.5$	$0.5$	$0.5$	$0.5$	$0.5$	$0.5$	$0.5$
$\zeta$	$0.2$	$0.4$	$0.4$	$0.4$	$0.4$	$0.4$	$0.4$	$0.4$

It is routine to verify that the $IFS$ $\mathcal{X}=(X,\varrho ,\zeta )$ in X is an $I{F}_{SBCK}$-ideal of X.

Lemma 3. 

If $(X,\varrho ,\zeta )$ is an $IFS$ of X, then

$$\begin{array}{c}\hfill (\forall {\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X)\left(\begin{array}{c}x\preccurlyeq y\Rightarrow \left\{\begin{array}{c}\varrho \left({\mathfrak{b}}_{x_1}\right)\ge \varrho \left({\mathfrak{b}}_{x_2}\right)\hfill \\ \zeta \left({\mathfrak{b}}_{x_1}\right)\le \zeta \left({\mathfrak{b}}_{x_2}\right)\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$

(5)

Proof. 

Let $(X,\varrho ,\zeta )$ be an $IFI$ of X and ${\mathfrak{b}}_{x_1}\preccurlyeq {\mathfrak{b}}_{x_2}$. Then, by Theorem 4, we have

$$\begin{array}{ccc}\varrho \left({\mathfrak{b}}_{x_1}\right)& \ge & min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\varrho \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& min\{\varrho \left(0\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& \varrho \left({\mathfrak{b}}_{x_2}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\zeta \left({\mathfrak{b}}_{x_1}\right)& \le & max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\zeta \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& max\{\zeta \left(0\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& \zeta \left({\mathfrak{b}}_{x_2}\right),\hfill \end{array}$$

for all ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. □

Theorem 5. 

Let $(X,\varrho ,\zeta )$ be an $IFS$ of X. Then, $(X,\varrho ,\zeta )$ is an $IFI$ of X if and only if

$$\begin{array}{c}\left(\begin{array}{c}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\preccurlyeq {\mathfrak{b}}_{x_3}\Rightarrow \left\{\begin{array}{c}\varrho \left({\mathfrak{b}}_{x_1}\right)\ge min\{\varrho \left({\mathfrak{b}}_{x_2}\right),\varrho \left({\mathfrak{b}}_{x_3}\right)\}\hfill \\ \zeta \left({\mathfrak{b}}_{x_1}\right)\le max\{\zeta \left({\mathfrak{b}}_{x_2}\right),\zeta \left({\mathfrak{b}}_{x_3}\right)\}\hfill \end{array}\right.\hfill \end{array}\right).\hfill \end{array}$$

(6)

for all ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2},{\mathfrak{b}}_{x_3}\in X.$

Proof. 

Let $(X,\varrho ,\zeta )$ be an $IFI$ of X and $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\preccurlyeq {\mathfrak{b}}_{x_3}$. Then,

$${\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_3}\right)\mid {\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_3}\right)=1\mid 1=0.$$

$$\begin{array}{ccc}\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& \ge & min\{\varrho ({\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_3}\right)\mid {\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_3}\right)),\varrho \left({\mathfrak{b}}_{x_3}\right)\}\hfill \\ & =& min\{\varrho \left(0\right),\varrho \left({\mathfrak{b}}_{x_3}\right)\}\hfill \\ & =& \varrho \left({\mathfrak{b}}_{x_3}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& \le & max\{\zeta ({\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_3}\right)\mid {\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_3}\right)),\zeta \left(z\right)\}\hfill \\ & =& max\{\zeta \left(0\right),\zeta \left({\mathfrak{b}}_{x_3}\right)\}\hfill \\ & =& \zeta \left({\mathfrak{b}}_{x_3}\right).\hfill \end{array}$$

We have

$$\varrho \left({\mathfrak{b}}_{x_1}\right)\ge min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\varrho \left({\mathfrak{b}}_{x_2}\right)\}\ge min\{\varrho \left({\mathfrak{b}}_{x_2}\right),\varrho \left({\mathfrak{b}}_{x_3}\right)\}$$

and

$$\zeta \left({\mathfrak{b}}_{x_1}\right)\le \zeta \left({\mathfrak{b}}_{x_1}\right)\le max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\zeta \left({\mathfrak{b}}_{x_2}\right)\}\le max\{\zeta \left({\mathfrak{b}}_{x_2}\right),\zeta \left({\mathfrak{b}}_{x_3}\right)\}.$$

Conversely, let $\mathcal{X}=(X,\varrho ,\zeta )$ be an $IFS$ of X satisfies the condition (6). Since

$${\mathfrak{S}}_0\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_0\left({\mathfrak{b}}_{x_1}\right)={\mathfrak{S}}_{{\mathfrak{b}}_{x_1}\mid {\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}\mid {\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_1}\right)=1\mid 1=0\preccurlyeq {\mathfrak{b}}_{x_3},$$

we have $\varrho \left(0\right)\ge \varrho \left({\mathfrak{b}}_{x_1}\right),$ and $\zeta \left(0\right)\le \zeta \left({\mathfrak{b}}_{x_1}\right)$ for all ${\mathfrak{b}}_{x_1}\in X$. Since $(({\mathfrak{b}}_{x_1}\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\mid ({\mathfrak{b}}_{x_1}\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))))\mid ({\mathfrak{b}}_{x_2}\mid {\mathfrak{b}}_{x_2})={\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))=1$, we obtain $({\mathfrak{b}}_{x_1}\mid ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\preccurlyeq {\mathfrak{b}}_{x_2}$. Since $\left({\mathfrak{b}}_{x_1}\right|((({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\mid ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)))\mid (({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\mid ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)))))$ $\mid \left({\mathfrak{b}}_{x_1}\right|((({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\mid ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)))\mid (({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\mid ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)))))=\left({\mathfrak{b}}_{x_1}\right|({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)))$ $\mid \left({\mathfrak{b}}_{x_1}\right|({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)))\preccurlyeq {\mathfrak{b}}_{x_2}$, we obtain $\varrho \left({\mathfrak{b}}_{x_1}\right)\ge min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\varrho \left({\mathfrak{b}}_{x_2}\right)\}$ and $\zeta \left({\mathfrak{b}}_{x_1}\right)\ge min\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\zeta \left({\mathfrak{b}}_{x_2}\right)\}$ for all ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Thus, $(X,\varrho ,\zeta )$ is an $IFI$ of X. □

Theorem 6. 

Every $I{F}_{SBCK}$-ideal of X is an $I{F}_{SBCK}$-subalgebra of X.

Proof. 

Let $(X,\varrho ,\zeta )$ be an $I{F}_{SBCK}$-ideal of X. Then,

$$\begin{array}{ccc}\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& \ge & min\{\varrho ({\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_1}\right)),\varrho \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& min\{\varrho \left({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}\mid {\mathfrak{b}}_{x_2}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_1}\right))\left({\mathfrak{b}}_{x_1}\right)\right),\varrho \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}\mid {\mathfrak{b}}_{x_2}}\left(1\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_2}\mid {\mathfrak{b}}_{x_2}}\left(1\right)),\varrho \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & \ge & min\{\varrho (1\mid 1),\varrho \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& min\{\varrho \left(0\right),\varrho \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& \varrho \left({\mathfrak{b}}_{x_1}\right)\hfill \\ & \ge & min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& \le & max\{\zeta ({\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)}\left({\mathfrak{b}}_{x_1}\right)),\zeta \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& max\{\zeta \left({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}\mid {\mathfrak{b}}_{x_2}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_1}\right))\left({\mathfrak{b}}_{x_1}\right)\right),\zeta \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}\mid {\mathfrak{b}}_{x_2}}\left(1\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_2}\mid {\mathfrak{b}}_{x_2}}\left(1\right)),\zeta \left(kx\right)\}\hfill \\ & =& max\{\zeta (1\mid 1),\zeta \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& max\{\zeta \left(0\right),\zeta \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& \zeta \left({\mathfrak{b}}_{x_1}\right)\hfill \\ & \le & max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}.\hfill \end{array}$$

Hence, $(X,\varrho ,\zeta )$ is an $IFI$ of X. □

Theorem 7. 

An $IFS$ $(X,\varrho ,\zeta )$ in X is an $I{F}_{SBCK}$-ideal of X if and only if the sets $\mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ and $\mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ are SBCK-ideals of X whenever they are nonempty for all ${\mathfrak{l}}_s,{\mathfrak{u}}_t\in [0,1]$.

Proof. 

Assume that $(X,\varrho ,\zeta )$ is an $IFS$ of X and $\mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\ne \varnothing \ne \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ for all ${\mathfrak{l}}_s,{\mathfrak{u}}_t\in [0,1]$. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2},{\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2}\in X$ be such that $({\mathfrak{b}}_{x_2},{\mathfrak{b}}_{y_2})\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\times \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$. Then $\zeta \left({\mathfrak{b}}_{x_2}\right)\le {\mathfrak{l}}_s$ and $\varrho \left({\mathfrak{b}}_{y_2}\right)\ge {\mathfrak{u}}_t$. Then, $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right))\le \zeta \left({\mathfrak{b}}_{x_2}\right)\le {\mathfrak{l}}_s$ and $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\ge \varrho \left({\mathfrak{b}}_{y_2}\right)\ge {\mathfrak{u}}_t$ and so $({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)),({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\times \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2},{\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2}\in X$ be such that $({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)),({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\times \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ and $({\mathfrak{b}}_{x_1},{\mathfrak{b}}_{y_1})\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\times \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$. Then, $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right))\le {\mathfrak{l}}_s$, $\zeta \left({\mathfrak{b}}_{x_1}\right)\le {\mathfrak{l}}_s$, $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\ge {\mathfrak{u}}_t$ and $\varrho \left({\mathfrak{b}}_{y_1}\right)\ge {\mathfrak{u}}_t$. Then, $\zeta \left({\mathfrak{b}}_{x_2}\right)\le max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)),\zeta \left({\mathfrak{b}}_{x_1}\right)\}\le {\mathfrak{l}}_s$ and $\varrho \left({\mathfrak{b}}_{y_2}\right)\ge min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)),\varrho \left({\mathfrak{b}}_{y_1}\right)\}\ge {\mathfrak{u}}_t$ and so $({\mathfrak{b}}_{x_2},{\mathfrak{b}}_{y_2})\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)\times \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$. Therefore, $\mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ and $\mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ are SBCK-ideals of X.

Conversely, let $(X,\varrho ,\zeta )$ be an $IFS$ in X for which its negative ${\mathfrak{l}}_s$-cut and positive ${\mathfrak{u}}_t$-cut are SBCK-ideals of X whenever they are nonempty for all ${\mathfrak{l}}_s,{\mathfrak{u}}_t\in [0,1]$. Suppose that $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))>\zeta \left({\mathfrak{b}}_{y_2}\right)$ for some ${\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2}\in X$. Then, ${\mathfrak{b}}_{y_2}\in \mathfrak{L}(\zeta ,\zeta \left({\mathfrak{b}}_{y_2}\right))$ but $({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\notin \mathfrak{L}(\zeta ,\zeta \left({\mathfrak{b}}_{y_2}\right))$, a contradiction. Hence, $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}\left({\mathfrak{b}}_{x_1}\right))\le \zeta \left({\mathfrak{b}}_{x_2}\right)$ for all ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Suppose that $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))<\varrho \left({\mathfrak{b}}_{x_2}\right)$ for some ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then, ${\mathfrak{b}}_{x_2}\in \mathfrak{U}(\varrho ,\varrho \left({\mathfrak{b}}_{x_2}\right))$ but $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\notin \mathfrak{U}(\varrho ,\varrho \left({\mathfrak{b}}_{x_2}\right))$, a contradiction. Hence, $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right))\ge \zeta \left({\mathfrak{b}}_{y_2}\right)$ for all ${\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2}\in X$. Assume that $\zeta \left(b\right)>max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)),\zeta \left(a\right)\}$ or $\varrho \left(y\right)<min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\varrho \left({\mathfrak{b}}_{x_1}\right)\}$ for some ${\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2},{\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then, $({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)),{\mathfrak{b}}_{y_1}\in \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ or $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),{\mathfrak{b}}_{x_1}\in \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$ where $s=max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{y_1}}\left({\mathfrak{b}}_{y_2}\right)),\zeta \left({\mathfrak{b}}_{y_1}\right)\}$ and $t=min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\varrho \left({\mathfrak{b}}_{x_1}\right)\}$. However, ${\mathfrak{b}}_{y_2}\notin \mathfrak{L}(\zeta ,{\mathfrak{l}}_s)$ or ${\mathfrak{b}}_{x_2}\notin \mathfrak{U}(\varrho ,{\mathfrak{u}}_t)$, a contradiction. Therefore, $\zeta \left({\mathfrak{b}}_{x_2}\right)\le max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\zeta \left({\mathfrak{b}}_{x_1}\right)\}$ and $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\ge min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}$ for all ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Consequently, $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-ideal of X. □

Theorem 8. 

An $IFS$ $(X,\varrho ,\zeta )$ in X is an $I{F}_{SBCK}$-ideal of X if and only if the fuzzy sets ${\zeta }_{\mathfrak{c}}$ and ϱ are fuzzy SBCK-ideals of X, where $\zeta :X\to [0,1]$ defined by ${\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_x\right)=1-\zeta \left({\mathfrak{b}}_x\right)$.

Proof. 

Assume that $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-ideal of X. It is clear that $\varrho$ is a fuzzy SBCK-ideal of X. For every ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$,

$$\begin{array}{ccc}{\zeta }_{\mathfrak{c}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& =& 1-\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\hfill \\ & \ge & 1-\zeta \left({\mathfrak{b}}_{x_2}\right)\hfill \\ & =& 1-\zeta \left({\mathfrak{b}}_{x_2}\right)\hfill \\ & =& {\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_2}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}{\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_2}\right)& =& 1-\zeta \left({\mathfrak{b}}_{x_2}\right)\hfill \\ & \ge & 1-max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& min\{1-\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),1-\zeta \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& min\{{\zeta }_{\mathfrak{c}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),{\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_1}\right)\}.\hfill \end{array}$$

Hence, ${\zeta }_c$ is a fuzzy SBCK-ideal of X.

Conversely, let $(X,\varrho ,\zeta )$ be an $IFS$ of X for which ${\zeta }_c$ and $\varrho$ are fuzzy SBCK-ideals of X. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then,

$$\begin{array}{ccc}1-\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& =& {\zeta }_{\mathfrak{c}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\hfill \\ & \ge & {\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_2}\right)\hfill \\ & =& 1-\zeta \left({\mathfrak{b}}_{x_2}\right),\hfill \end{array}$$

so

$$\begin{array}{ccc}\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))& \le & \zeta \left({\mathfrak{b}}_{x_2}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}1-\zeta \left(y\right)& =& {\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_2}\right)\hfill \\ & \ge & min\{{\zeta }_{\mathfrak{c}}\left(x\right),{\zeta }_{\mathfrak{c}}\left({\mathfrak{b}}_{x_2}\right)\}\hfill \\ & =& min\{1-\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),1-\zeta \left({\mathfrak{b}}_{x_1}\right)\}\hfill \\ & =& 1-max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\zeta \left({\mathfrak{b}}_{x_1}\right)\},\hfill \end{array}$$

so

$$\begin{array}{ccc}\zeta \left(y\right)& \le & max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\zeta \left({\mathfrak{b}}_{x_1}\right)\}.\hfill \end{array}$$

Hence, $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-ideal of X. □

Theorem 9. 

Given a nonempty subset Y of X, let $(X,{\varrho }_Y,{\zeta }_Y)$ be an $IFS$ in X defined as follows:

$${\varrho }_Y:X\to [0,1],{\mathfrak{b}}_x\mapsto \left\{\begin{array}{cc}{\varrho }_0\hfill & if{\mathfrak{b}}_x\in Y,\hfill \\ {\varrho }_1\hfill & otherwise\hfill \end{array}\right.$$

and

$${\zeta }_Y:X\to [0,1],{\mathfrak{b}}_y\mapsto \left\{\begin{array}{cc}{\zeta }_0\hfill & if{\mathfrak{b}}_y\in Y,\hfill \\ {\zeta }_1\hfill & otherwise,\hfill \end{array}\right.$$

where ${\zeta }_0<{\zeta }_1$ in $[0,1]$ and ${\varrho }_0>{\varrho }_1$ in $[0,1]$. Then $(X,{\varrho }_Y,{\zeta }_Y)$ be an $I{F}_{SBCK}$-ideal of X if and only if Y is a SBCK-ideal of X.

Proof. 

Assume that $(X,{\varrho }_Y,{\zeta }_Y)$ is an $I{F}_{SBCK}$-ideal of X. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$ be such that ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in Y$. Then,

$$\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\ge \varrho \left({\mathfrak{b}}_{x_2}\right)={\varrho }_0,$$

$$\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\le \zeta \left({\mathfrak{b}}_{x_2}\right)={\zeta }_0,$$

and so $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\varrho }_0$ and $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\zeta }_0$. This shows that $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\in Y$. Then,

$$\varrho \left({\mathfrak{b}}_{x_2}\right)\ge min\{\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\varrho \left({\mathfrak{b}}_{x_1}\right)\}={\varrho }_0,$$

$$\zeta \left({\mathfrak{b}}_{x_2}\right)\le max\{\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)),\zeta \left({\mathfrak{b}}_{x_1}\right)\}={\zeta }_0,$$

and so $\varrho \left({\mathfrak{b}}_{x_2}\right)={\varrho }_0$ and $\zeta \left({\mathfrak{b}}_{x_2}\right)={\zeta }_0$. This shows that $y\in F$. Therefore, F is a SBCK-ideal of X.

Conversely, let Y be a SBCK-ideal of X. For every ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$, if $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\in Y$, then ${\mathfrak{b}}_{x_2}\in Y$ which implies that $\varrho \left({\mathfrak{b}}_{x_2}\right)={\varrho }_0=\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))$ and $\zeta \left({\mathfrak{b}}_{x_2}\right)={\zeta }_0=\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))$. If $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\notin Y$, then $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\varrho }_1<\zeta \left({\mathfrak{b}}_{x_2}\right)$ and $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\zeta }_1>\zeta \left({\mathfrak{b}}_{x_2}\right)$. For every ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$, if ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in Y$, then $({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\in Y$, which implies that

$$\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\varrho }_0=min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}$$

and

$$\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))={\zeta }_0=max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}.$$

If ${\mathfrak{b}}_{x_1}\notin Y$ of ${\mathfrak{b}}_{x_2}\notin Y$, then $\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\ge {\varrho }_1=min\{\varrho \left({\mathfrak{b}}_{x_1}\right),\varrho \left({\mathfrak{b}}_{x_2}\right)\}$ and $\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right)\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}\left({\mathfrak{b}}_{x_2}\right))\le {\zeta }_1=max\{\zeta \left({\mathfrak{b}}_{x_1}\right),\zeta \left({\mathfrak{b}}_{x_2}\right)\}$. Therefore, $(X,{\varrho }_Y,{\zeta }_Y)$ is an $I{F}_{SBCK}$-ideal of X. □

Proposition 4. 

If ${\mathcal{X}}_{\mathfrak{k}}=\{(X,{\varrho }_{\mathfrak{k}},{\zeta }_{\mathfrak{k}}):\mathfrak{k}\in \Delta \}$, where Δ is an arbitrary index set, is a family of $I{F}_{SBCK}$-ideals of X, then ${\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\mathcal{X}}_{\mathfrak{k}}$ is an $I{F}_{SBCK}$-ideal of X.

Proof. 

Let ${\mathcal{X}}_{\mathfrak{k}}=\{(X,{\varrho }_{\mathfrak{k}},{\zeta }_{\mathfrak{k}}):\mathfrak{k}\in \Delta \}$ be a family of $I{F}_{SBCK}$-ideals of a SBCK-algebra X.

Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then,

$$\begin{array}{ccc}({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\varrho }_{\mathfrak{k}})({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2}))& =& \underset{\mathfrak{k}\in \Delta }{inf}\{{\varrho }_{\mathfrak{k}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2}))\}\hfill \\ & \ge & \underset{\mathfrak{k}\in \Delta }{inf}\{{\varrho }_{\mathfrak{k}}({\mathfrak{b}}_{x_2})\}\hfill \\ & =& ({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\varrho }_{\mathfrak{k}})({\mathfrak{b}}_{x_2}),\hfill \end{array}$$

and

$$\begin{array}{ccc}({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\zeta }_{\mathfrak{k}})({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2}))& =& \underset{\mathfrak{k}\in \Delta }{sup}\{{\zeta }_{\mathfrak{k}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2}))\}\hfill \\ & \le & ({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\zeta }_{\mathfrak{k}})(({\mathfrak{b}}_{x_2})\hfill \\ & =& ({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\varrho }_{\mathfrak{k}})({\mathfrak{b}}_{x_2}).\hfill \end{array}$$

Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then

$$\begin{array}{ccc}({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\varrho }_{\mathfrak{k}})({\mathfrak{b}}_{x_2})& =& \underset{\mathfrak{k}\in \Delta }{inf}\{{\varrho }_{\mathfrak{k}}({\mathfrak{b}}_{x_2})\}\hfill \\ & \ge & \underset{\mathfrak{k}\in \Delta }{inf}\{min\{{\varrho }_{\mathfrak{k}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2}))),{\varrho }_{\mathfrak{k}}({\mathfrak{b}}_{x_1})\}\}\hfill \\ & =& min\{\underset{\mathfrak{k}\in \Delta }{inf}{\varrho }_{\mathfrak{k}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})),\underset{\mathfrak{k}\in \Delta }{inf}{\varrho }_{\mathfrak{k}}({\mathfrak{b}}_{x_1})\}\hfill \\ & =& min\{({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\varrho }_{\mathfrak{k}})({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})),({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\varrho }_{\mathfrak{k}})({\mathfrak{b}}_{x_1})\}\hfill \end{array}$$

and

$$\begin{array}{ccc}({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\zeta }_{\mathfrak{k}})({\mathfrak{b}}_{x_2})& =& \underset{\mathfrak{k}\in \Delta }{sup}\{{\zeta }_{\mathfrak{k}}({\mathfrak{b}}_{x_2})\}\hfill \\ & \le & \underset{\mathfrak{k}\in \Delta }{sup}\{max\{{\zeta }_{\mathfrak{k}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})),{\zeta }_{\mathfrak{k}}({\mathfrak{b}}_{x_1})\}\}\hfill \\ & =& max\{\underset{\mathfrak{k}\in \Delta }{sup}{\zeta }_{\mathfrak{k}}({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})),\underset{\mathfrak{k}\in \Delta }{sup}{\zeta }_{\mathfrak{k}}({\mathfrak{b}}_{x_1})\}\hfill \\ & =& max\{({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\zeta }_{\mathfrak{k}})({\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})\mid {\mathfrak{S}}_{{\mathfrak{b}}_{x_1}}({\mathfrak{b}}_{x_2})),({\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\zeta }_{\mathfrak{k}})({\mathfrak{b}}_{x_1})\}.\hfill \end{array}$$

Hence, ${\displaystyle \underset{\mathfrak{k}\in \Delta }{\bigwedge }}{\mathcal{X}}_{\mathfrak{k}}$ is an $I{F}_{SBCK}$-ideal of a SBCK-algebra X. □

Definition 8 

([16]). Let $(X,{\mid }_X,0_X)$ and $(Y,{\mid }_Y,0_Y)$ be SBCK-algebras. Then, a mapping $\hslash :X\to Y$ is called a homomorphism if $\hslash ({\mathfrak{b}}_{x_1}{\mid }_X{\mathfrak{b}}_{x_2})=\hslash ({\mathfrak{b}}_{x_1}){\mid }_Y\hslash ({\mathfrak{b}}_{x_2})$ for all ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$ and $\hslash (0_X)=0_Y$.

Let $(G_S,{\mid }_A)$ be a groupoid. For every element $g\in G_S$, consider the following mapping:

$$\begin{array}{c}\hfill {\mathfrak{S}}_{\mathfrak{g}}^A:G_S\to G_S,{\mathfrak{g}}_s\mapsto \mathfrak{g}{\mid }_A({\mathfrak{g}}_s{\mid }_A{\mathfrak{g}}_s).\end{array}$$

Theorem 10. 

Let $(X,{\mid }_X,0_X)$ and $(Y,{\mid }_Y,0_Y)$ be SBCK-algebras, $\hslash :X\to Y$ be a surjective homomorphism and $(X,\varrho ,\zeta )$ be an $IFS$ on Y. Then, $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-ideal of Y if and only if $(X,{\varrho }^{\hslash },{\zeta }^{\hslash })$ is an $I{F}_{SBCK}$-ideal of X.

Proof. 

Let $(X,{\mid }_X,0_X)$ and $(Y,{\mid }_Y,0_Y)$ be SBCK-algebras, $\hslash :X\to Y$ be a surjective homomorphism and $(X,\varrho ,\zeta )$ be an $I{F}_{SBCK}$-ideal of Y. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then,

$$\begin{array}{ccc}{\varrho }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))& =& \varrho (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\hfill \\ & =& \varrho ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1}))\hfill \\ & \ge & \varrho (\hslash ({\mathfrak{b}}_{x_2}))\hfill \\ & =& {\varrho }^{\hslash }({\mathfrak{b}}_{x_2}),\hfill \end{array}$$

and

$$\begin{array}{ccc}{\varrho }^{\hslash }({\mathfrak{b}}_{x_2})& =& \varrho (\hslash ({\mathfrak{b}}_{x_2}))\hfill \\ & \ge & min\{\varrho (\hslash ({\mathfrak{b}}_{x_1})),\varrho ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1}))\}\hfill \\ & =& min\{\varrho (\hslash ({\mathfrak{b}}_{x_1})),\varrho (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\}\hfill \\ & =& min\{{\varrho }^{\hslash }({\mathfrak{b}}_{x_1}),{\varrho }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\},\hfill \end{array}$$

and

$$\begin{array}{ccc}{\zeta }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))& =& \zeta (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\hfill \\ & =& \zeta ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1}))\hfill \\ & \le & \zeta (\hslash ({\mathfrak{b}}_{x_2}))\hfill \\ & =& {\zeta }^{\hslash }({\mathfrak{b}}_{x_2}),\hfill \end{array}$$

and

$$\begin{array}{ccc}{\zeta }^{\hslash }({\mathfrak{b}}_{x_2})& =& \zeta (\hslash ({\mathfrak{b}}_{x_2}))\hfill \\ & \le & max\{\zeta (\hslash ({\mathfrak{b}}_{x_1})),\zeta ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})))\}\hfill \\ & =& max\{\zeta (\hslash ({\mathfrak{b}}_{x_1})),\zeta (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1})))\}\hfill \\ & =& max\{{\zeta }^{\hslash }({\mathfrak{b}}_{x_1}),{\zeta }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\}.\hfill \end{array}$$

Hence, $(X,{\varrho }^{\hslash },{\zeta }^{\hslash })$ is an $I{F}_{SBCK}$-ideal of X.

Conversely, let $(X,{\varrho }^{\hslash },{\zeta }^{\hslash })$ be an $I{F}_{SBCK}$-ideall of X. Let ${\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2}\in Y$ such that $\hslash ({\mathfrak{b}}_{x_1})={\mathfrak{b}}_{y_1}$ and $\hslash ({\mathfrak{b}}_{x_2})={\mathfrak{b}}_{y_2}$ for ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then,

$$\begin{array}{ccc}\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}){\mid }_Y{\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}))& =& \varrho ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})))\hfill \\ & =& {\varrho }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\hfill \\ & \ge & {\varrho }^{\hslash }({\mathfrak{b}}_{x_2})\hfill \\ & =& \varrho (\hslash ({\mathfrak{b}}_{x_2}))\hfill \\ & =& \varrho ({\mathfrak{b}}_{y_2}),\hfill \end{array}$$

and

$$\begin{array}{ccc}\varrho ({\mathfrak{b}}_{y_2})& =& \varrho (\hslash ({\mathfrak{b}}_{x_2}))\hfill \\ & =& {\varrho }^{\hslash }({\mathfrak{b}}_{x_2})\hfill \\ & \ge & min\{{\varrho }^{\hslash }({\mathfrak{b}}_{x_1}),{\varrho }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\}\hfill \\ & =& min\{\varrho (\hslash ({\mathfrak{b}}_{x_1})),\varrho (f({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1})))\}\hfill \\ & =& min\{\varrho (\hslash ({\mathfrak{b}}_{x_1})),\varrho ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})))\}\hfill \\ & =& min\{\varrho ({\mathfrak{b}}_{y_1}),\varrho (({\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}){\mid }_Y{\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}))\},\hfill \end{array}$$

and

$$\begin{array}{ccc}\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}){\mid }_Y{\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}))& =& \zeta ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})))\hfill \\ & =& {\zeta }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\hfill \\ & \le & {\zeta }^{\hslash }({\mathfrak{b}}_{x_2})\hfill \\ & =& \zeta (\hslash ({\mathfrak{b}}_{x_2}))\hfill \\ & =& \zeta ({\mathfrak{b}}_{y_2}),\hfill \end{array}$$

and

$$\begin{array}{ccc}\zeta ({\mathfrak{b}}_{y_2})& =& \zeta (\hslash ({\mathfrak{b}}_{x_2}))\hfill \\ & =& {\zeta }^{\hslash }({\mathfrak{b}}_{x_2})\hfill \\ & \le & max\{{\zeta }^{\hslash }({\mathfrak{b}}_{x_1}),{\zeta }^f({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\}\hfill \\ & =& max\{\zeta (\hslash ({\mathfrak{b}}_{x_1})),\zeta (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1})))\}\hfill \\ & =& max\{\zeta (\hslash ({\mathfrak{b}}_{x_1})),\zeta ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})))\}\hfill \\ & =& max\{\zeta ({\mathfrak{b}}_{y_1}),\zeta (({\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}){\mid }_Y{\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}))\}.\hfill \end{array}$$

Hence, $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-ideal of Y. □

Theorem 11. 

Let $(X,{\mid }_X,0_X)$ and $(Y,{\mid }_Y,0_Y)$ be SBCK-algebras, $\hslash :X\to Y$ be a surjective homomorphism and $(X,\varrho ,\zeta )$ be an $IFS$ on Y. Then, $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-subalgebra of Y if and only if $(X,{\varrho }^{\hslash },{\zeta }^{\hslash })$ is an $I{F}_{SBCK}$-subalgebra of X.

Proof. 

Let $(X,{\mid }_X,0_X)$ and $(Y,{\mid }_Y,0_Y)$ be SBCK-algebras, $\hslash :X\to Y$ be a surjective homomorphism and $(X,\varrho ,\zeta )$ be an $I{F}_{SBCK}$-subalgebra of Y. Let ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then,

${\varrho }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))$

$$\begin{array}{ccc}& =& \varrho (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1})))\hfill \\ & =& \varrho ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})))\hfill \\ & \ge & min\{\varrho (\hslash ({\mathfrak{b}}_{x_1})),\varrho (\hslash ({\mathfrak{b}}_{x_2}))\}\hfill \\ & =& min\{{\varrho }^{\hslash }({\mathfrak{b}}_{x_1}),{\varrho }^{\hslash }({\mathfrak{b}}_{x_2})\},\hfill \end{array}$$

and ${\zeta }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))$

$$\begin{array}{ccc}& =& \zeta (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1})))\hfill \\ & =& \zeta ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})))\hfill \\ & \le & max\{\zeta (\hslash ({\mathfrak{b}}_{x_1})),\zeta (\hslash ({\mathfrak{b}}_{x_2}))\}\hfill \\ & =& max\{{\zeta }^{\hslash }({\mathfrak{b}}_{x_1}),{\zeta }^{\hslash }({\mathfrak{b}}_{x_2})\}.\hfill \end{array}$$

Hence, $(X,{\varrho }^{\hslash },{\zeta }^{\hslash })$ is an $I{F}_{SBCK}$-subalgebra of X.

Conversely, let $(X,{\varrho }^{\hslash },{\zeta }^{\hslash })$ be an $I{F}_{SBCK}$-subalgebra of X. Let ${\mathfrak{b}}_{y_1},{\mathfrak{b}}_{y_2}\in Y$ such that $\varrho ({\mathfrak{b}}_{x_1})={\mathfrak{b}}_{y_1}$ and $\varrho ({\mathfrak{b}}_{x_2})={\mathfrak{b}}_{y_2}$ for ${\mathfrak{b}}_{x_1},{\mathfrak{b}}_{x_2}\in X$. Then,

$$\begin{array}{ccc}\varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}){\mid }_Y{\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}))& =& \varrho ({\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}){\mid }_Y{\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}))\hfill \\ & =& \varrho (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1})))\hfill \\ & =& {\varrho }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\hfill \\ & \ge & min\{{\varrho }^{\hslash }(x_1),{\varrho }^{\hslash }(x_2)\}\hfill \\ & =& min\{\varrho (x_1),\varrho (x_2)\}\hfill \\ & =& min\{\varrho (y_1),\varrho (y_2)\},\hfill \end{array}$$

$$\begin{array}{ccc}\zeta ({\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}){\mid }_Y{\mathfrak{S}}_{{\mathfrak{b}}_{y_2}}^Y({\mathfrak{b}}_{y_1}))& =& \zeta ({\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})){\mid }_Y{\mathfrak{S}}_{\hslash ({\mathfrak{b}}_{x_2})}^Y(\hslash ({\mathfrak{b}}_{x_1})))\hfill \\ & =& \zeta (\hslash ({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1})))\hfill \\ & =& {\zeta }^{\hslash }({\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}){\mid }_X{\mathfrak{S}}_{{\mathfrak{b}}_{x_2}}^X({\mathfrak{b}}_{x_1}))\hfill \\ & \le & max\{{\zeta }^{\hslash }({\mathfrak{b}}_{x_1}),{\zeta }^{\hslash }({\mathfrak{b}}_{x_2})\}\hfill \\ & =& max\{\zeta (\hslash ({\mathfrak{b}}_{y_1})),\zeta (\hslash ({\mathfrak{b}}_{x_2}))\}\hfill \\ & =& max\{\zeta ({\mathfrak{b}}_{y_1}),\zeta ({\mathfrak{b}}_{y_2})\}.\hfill \end{array}$$

Hence, $(X,\varrho ,\zeta )$ is an $I{F}_{SBCK}$-subalgebra of Y. □

5. Conclusions

In this study, we have introduced and systematically examined the structure of intuitionistic fuzzy SBCK-subalgebras and SBCK-ideals within the framework of SBCK-algebras. Our primary contributions center on characterizing these intuitionistic fuzzy structures through their corresponding level sets and associated fuzzy subsets.

We first established that an intuitionistic fuzzy set (IFS) in an SBCK-algebra is an $I{F}_{SBCK}$-subalgebra (or $I{F}_{SBCK}$-ideal) if and only if its upper and lower level sets form SBCK-subalgebras (or ideals), providing a fundamental link between intuitionistic fuzzy structures and classical algebraic substructures. Additionally, we demonstrated that such an IFS can be equivalently described in terms of its component fuzzy sets, particularly the transformed non-membership function ${\zeta }_{\mathfrak{c}}(x)=1-\zeta (x)$, which must itself satisfy the fuzzy subalgebra (or ideal) conditions.

A further key result showed that if an IFS is defined via characteristic functions on a subset $Y\subseteq X$, then it forms an $I{F}_{SBCK}$-subalgebra (or ideal) if and only if Y is itself an SBCK-subalgebra (or ideal). This provides a constructive method for generating intuitionistic fuzzy substructures. We also identified several necessary conditions that such fuzzy structures must satisfy, including bounds on the membership and non-membership functions at the zero element of the algebra and their behavior under the algebra’s partial order.

Moreover, we confirmed that every $I{F}_{SBCK}$-ideal is necessarily an $I{F}_{SBCK}$-subalgebra, reflecting a hierarchical structure within these intuitionistic fuzzy systems. Closure properties under intersection were also established, ensuring the internal consistency of the class of $I{F}_{SBCK}$-ideals.

Finally, we extended our findings to the homomorphic image of intuitionistic fuzzy SBCK-subalgebras and ideals, showing that such structures are preserved under surjective homomorphisms. This result broadens the applicability of our framework to categorical and structural studies in algebra.

Future research could also investigate the application of intuitionistic fuzzy SBCK-subalgebras to problems such as center and isochronous center conditions in switching systems [18], complex isochronous centers in ${\mathbb{Z}}_2$-equivariant planar systems [19], and the center-focus classification in generalized cubic Kukles systems with nilpotent singular points [20]. These connections may open new pathways for applying fuzzy algebraic methods to the qualitative analysis of dynamical systems.

Future research could focus on extending the current results to more general algebraic systems or different classes of non-classical logics. One potential avenue is to investigate the properties of intuitionistic fuzzy SBCK-subalgebras within other algebraic structures, such as lattice-based or non-commutative algebras. Moreover, computational approaches for handling these structures in practical applications, such as fuzzy decision making and reasoning, could be explored further. Additionally, a deeper investigation into the relationships between intuitionistic fuzzy sets and other logical operations within the broader context of BCK-algebras may yield new insights and open up new avenues for research in both algebraic theory and fuzzy logic applications.