Lukasiewicz Fuzzy Set Theory Applied to SBE-Algebras

Abstract

In this paper, we utilize the Lukasiewicz t-norm to construct a novel class of fuzzy sets, termed $\zeta$-Lukasiewicz fuzzy sets, derived from a given fuzzy framework. These sets are then applied to the structure of Sheffer stroke BE-algebras (SBE-algebras). We introduce and examine the concepts of $\zeta$-Lukasiewicz fuzzy SBE-subalgebras and $\zeta$-Lukasiewicz fuzzy SBE-ideals, with a focus on their algebraic properties. Furthermore, we define three specific types of subsets, referred to as ∈-sets, q-sets, and O-sets, and investigate the necessary conditions for these subsets to constitute subalgebras or ideals within the SBE-algebraic context.

1. Introduction

The pursuit of robust logical systems to model uncertainty and imprecise reasoning has long been a driving force in mathematics and computer science. In this endeavor, the theory of algebraic structures provides a powerful formal framework, while fuzzy set theory, pioneered by L. A. Zadeh [1], offers the essential tools to quantify and manage gradations of truth. The intersection of these fields has proven exceptionally fertile, leading to the fuzzification of various algebraic classes, including the well-studied BCK/BCI-algebras [2,3].

Among these, BE-algebras, introduced by H. S. Kim and Y. H. Kim [4] as a generalization of BCK-algebras, have emerged as a significant domain of study [5,6,7,8,9,10,11,12]. The application of fuzzy logic to BE-algebras has further enriched this landscape, yielding investigations into fuzzified substructures and their properties [13,14,15].

A particularly compelling branch of non-classical logic is the many-valued system developed by J. Lukasiewicz [16,17,18]. Grounded in the Lukasiewicz t-norm, this logic introduces intermediate truth values, moving beyond a simple true/false dichotomy to capture nuances of possibility and uncertainty [19,20]. This expressive power makes it highly suitable for advanced fuzzy algebraic applications, as demonstrated by Y. B. Jun [21,22], who defined positive implicative Lukasiewicz fuzzy filters in BE-algebras.

The recent introduction of Sheffer stroke BE-algebras (SBE-algebras) by T. Katican et al. [23] represents a novel and potent advancement. Defined via the Sheffer stroke, a single logical operation from which all others can be derived, SBE-algebras offer a remarkably minimalist and elegant foundation for algebraic logic. Subsequent research has rapidly developed their theory, exploring their filter structure [23], obstinate filters [24], and foundational fuzzy concepts [25,26].

However, the application of Lukasiewicz’s sophisticated many-valued logic to this new and promising SBE-algebraic framework remains an open area for exploration. This gap presents a compelling opportunity: to combine the minimalist elegance of the Sheffer stroke with the nuanced expressive power of Lukasiewicz logic.

We acknowledge that the notation in this field, built from layers of mathematical abstraction (e.g., $\zeta$-Lukasiewicz fuzzy SBE-ideals), can appear dense to the uninitiated. We have striven for clarity by carefully defining all concepts and structuring our presentation to gradually build complexity. The underlying ideas, while technical, are driven by a goal of enhancing our formal toolkit for reasoning under uncertainty.

In this paper, we bridge the aforementioned gap by constructing a new class of fuzzy sets ζ-Lukasiewicz fuzzy sets from a given fuzzy set using the Lukasiewicz t-norm. We then integrate these sets into the structure of SBE-algebras. Our specific contributions are fourfold:

1. We define and analyze the algebraic properties of ζ-Lukasiewicz fuzzy SBE-subalgebras and ζ-Lukasiewicz fuzzy SBE-ideals.

2. We introduce three critical types of subsets: ∈-sets, q-sets, and O-sets.

3. We establish the necessary and sufficient conditions under which these subsets form SBE-subalgebras or SBE-ideals.

4. Through this work, we aim to deepen the theoretical foundations of SBE-algebras and demonstrate the practical value of applying Lukasiewicz logic to modern algebraic structures.

2. Preliminaries

In this section, we recall some basic notions and results regarding SBE-algebras that are used throughout the paper.

Definition 1

([27]). Let X be a set and $|:X\times X\to X$ be a binary operation. We mean a Sheffer stroke operation if it satisfies the following conditions for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$$\begin{array}{c}\hfill \begin{array}{cc}\left(S1\right)\hfill & {\hslash }_{{\mathfrak{a}}_x}|{\hslash }_{{\mathfrak{a}}_y}={\hslash }_{{\mathfrak{a}}_y}|{\hslash }_{{\mathfrak{a}}_x},\hfill \\ \left(S2\right)\hfill & \left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})={\hslash }_{{\mathfrak{a}}_x},\hfill \\ \left(S3\right)\hfill & {\hslash }_{{\mathfrak{a}}_x}|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_z})\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_z}))=\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y}))|{\hslash }_{{\mathfrak{a}}_z},\hfill \\ \left(S4\right)\hfill & \left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})))|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})))={\hslash }_{{\mathfrak{a}}_x}.\hfill \end{array}\end{array}$$

Definition 2

([4]). An algebra $(X;\to ,1)$ of type $(2,0)$ is called a BE-algebra if it satisfies the following axioms for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$$\begin{array}{c}\hfill \begin{array}{cc}\left(BE1\right)\hfill & {\hslash }_{{\mathfrak{a}}_x}\to {\hslash }_{{\mathfrak{a}}_x}=1,\hfill \\ \left(BE2\right)\hfill & {\hslash }_{{\mathfrak{a}}_x}\to 1=1,\hfill \\ \left(BE3\right)\hfill & 1\to {\hslash }_{{\mathfrak{a}}_x}={\hslash }_{{\mathfrak{a}}_x},\hfill \\ \left(BE4\right)\hfill & {\hslash }_{{\mathfrak{a}}_x}\to ({\hslash }_{{\mathfrak{a}}_y}\to {\hslash }_{{\mathfrak{a}}_z})={\hslash }_{{\mathfrak{a}}_y}\to ({\hslash }_{{\mathfrak{a}}_x}\to {\hslash }_{{\mathfrak{a}}_z}).\hfill \end{array}\end{array}$$

Definition 3

([23]). An algebra $(X,|,1)$ of type $(2,0)$ is called a Sheffer stroke BE-algebra (SBE-algebra) if it satisfies the following conditions for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$:

$$\begin{array}{c}\hfill \begin{array}{cc}(SBE1)\hfill & \left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))=1,\hfill \\ \left(SBE2\right)\hfill & \left(1\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))={\hslash }_{{\mathfrak{a}}_x}|{\hslash }_{{\mathfrak{a}}_x}.\hfill \end{array}\end{array}$$

Proposition 1

([23]). Let X be an SBE-algebra. Define a binary relation “⪯” on X by

$${\hslash }_{{\mathfrak{a}}_x}⪯{\hslash }_{{\mathfrak{a}}_y} \mathit{if} \mathit{and} \mathit{only} \mathit{if} \left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))=1.$$

Then “⪯” constitutes a partial order on X.

Definition 4

([23]). Let X be an SBE-algebra. A nonempty subset $S\subseteq X$ is said to be an SBE-subalgebra of X if, for every ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in S$, the element ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})$ also belongs to S.

Definition 5

([23]). Let X be an SBE-algebra. A nonempty subset $I\subseteq X$ is called an SBE-ideal if it satisfies the following conditions for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

1. 

$1\in I$,

2. 

If ${\hslash }_{{\mathfrak{a}}_x}\in I$ and ${\hslash }_{{\mathfrak{a}}_y}\in I$, then the element

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in I.$$

Definition 6

([26]). Let X be an SBE-algebra. A fuzzy set $\mu :X\to [0,1]$ is said to be a fuzzy SBE-subalgebra of X if

$$(\forall {\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X)\mu \left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\ge min\{\mu \left({\hslash }_{{\mathfrak{a}}_x}\right),\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)\}.$$

Definition 7

([25]). Let X be an SBE-algebra. A fuzzy set $\mu :X\to [0,1]$ is called a fuzzy SBE-ideal of X if for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$, the following conditions are satisfied:

$$\begin{array}{cc}\hfill \mu \left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))& \ge \mu \left({\hslash }_{{\mathfrak{a}}_y}\right),\hfill \\ \hfill \mu \left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))& \ge min\{\mu \left({\hslash }_{{\mathfrak{a}}_x}\right),\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)\}.\hfill \end{array}$$

Lemma 1

([23]). Let $\langle X,|,1\rangle$ be an SBE-algebra. Then the following identities hold for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

1. 

${\hslash }_{{\mathfrak{a}}_x}|\left(1\right|1)=1$,

2. 

$1|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})={\hslash }_{{\mathfrak{a}}_x}$,

3. 

${\hslash }_{{\mathfrak{a}}_x}|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))\right)=1$,

4. 

${\hslash }_{{\mathfrak{a}}_x}|\left(\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right)=1$,

5. 

$\left({\hslash }_{{\mathfrak{a}}_x}\right|1)|\left({\hslash }_{{\mathfrak{a}}_x}\right|1)={\hslash }_{{\mathfrak{a}}_x}$,

6. 

$\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})\right)|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})=1$ and $\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})\right)|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})=1$,

7. 

${\hslash }_{{\mathfrak{a}}_x}|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})\right)={\hslash }_{{\mathfrak{a}}_x}|{\hslash }_{{\mathfrak{a}}_y}=\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_y})\right)|{\hslash }_{{\mathfrak{a}}_y}$.

Definition 8

([23]). An SBE-algebra X is said to be

1. 

transitive if for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$${\hslash }_{{\mathfrak{a}}_y}|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\preccurlyeq \left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right),$$

2. 

commutative if for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$:

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})=\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}),$$

3. 

self-distributive if for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$${\hslash }_{{\mathfrak{a}}_x}|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right)=\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right).$$

Lemma 2

([23]). Let  $(X,|,1)$ be an SBE-algebra. Then the following properties hold: for any ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$:

1. 

If ${\hslash }_{{\mathfrak{a}}_x}⪯{\hslash }_{{\mathfrak{a}}_y}$ then it follows that ${\hslash }_{{\mathfrak{a}}_y}|{\hslash }_{{\mathfrak{a}}_y}⪯{\hslash }_{{\mathfrak{a}}_x}|{\hslash }_{{\mathfrak{a}}_x}$.

2. 

${\hslash }_{{\mathfrak{a}}_x}⪯{\hslash }_{{\mathfrak{a}}_y}|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})$.

3. 

${\hslash }_{{\mathfrak{a}}_y}⪯\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})$.

4. 

If X is self-distributive, then ${\hslash }_{{\mathfrak{a}}_x}⪯{\hslash }_{{\mathfrak{a}}_y}$ implies ${\hslash }_{{\mathfrak{a}}_y}|{\hslash }_{{\mathfrak{a}}_z}⪯{\hslash }_{{\mathfrak{a}}_x}|{\hslash }_{{\mathfrak{a}}_z}$.

5. 

If X satisfies self-distribution, then

$${\hslash }_{{\mathfrak{a}}_y}|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})⪯\left({\hslash }_{{\mathfrak{a}}_z}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x}))\right).$$

A fuzzy set $\mu$ defined on a set X is called a fuzzy point with support $\mathfrak{u}\in X$ and membership value $\mathfrak{t}\in (0,1]$ if it satisfies

$$\mu \left(\mathfrak{v}\right)=\left\{\begin{array}{cc}\mathfrak{t},\hfill & \mathrm{if}\mathfrak{v}=\mathfrak{u},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Such a fuzzy point is denoted by $[\mathfrak{u}/\mathfrak{t}]$.

For a given fuzzy set $\mu$ on X, the fuzzy point $[\mathfrak{u}/\mathfrak{t}]$ is said to be:

1.

contained in $\mu$, written as $[\mathfrak{u}/\mathfrak{t}]\in \mu$, if and only if $\mu \left(\mathfrak{u}\right)\ge \mathfrak{t}$,

2.

quasi-coincident with $\mu$, denoted by $[\mathfrak{u}/\mathfrak{t}]q\mu$, if $\mu \left(\mathfrak{u}\right)+\mathfrak{t}>1$.

If either of these relations does not hold for $\alpha \in \{\in ,q\}$, we write $[\mathfrak{u}/\mathfrak{t}]\neg \alpha \mu$ to indicate the negation.

Let $\mu$ be a fuzzy set on X and fix $\zeta \in (0,1)$. The function ${\mathcal{L}}_{\mu }^{\zeta }:X\to [0,1]$ defined by

$${\mathcal{L}}_{\mu }^{\zeta }\left(x\right)=max\{0,\mu \left(x\right)+\zeta -1\}$$

is called the ζ-Lukasiewicz fuzzy set associated to $\mu$.

For the $\zeta$-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ and a fixed $\mathfrak{t}\in (0,1]$, we define the following subsets of X:

$$\begin{array}{cc}\hfill {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }& =\{x\in X|[x/\mathfrak{t}]\in {\mathcal{L}}_{\mu }^{\zeta }\},\hfill \\ \hfill {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q& =\{x\in X|[x/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta }\},\hfill \end{array}$$

which are called the ∈-set and q-set of ${\mathcal{L}}_{\mu }^{\zeta }$ at level $\mathfrak{t}$, respectively.

Additionally, the O-set of $L_{\mu }^{\zeta }$ is defined by

$$O\left({\mathcal{L}}_{\mu }^{\zeta }\right)=\{{\hslash }_{{\mathfrak{a}}_x}\in X|{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)>0\},$$

which equivalently can be expressed as

$$O\left({\mathcal{L}}_{\mu }^{\zeta }\right)=\{{\hslash }_{{\mathfrak{a}}_x}\in X|\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1>0\}.$$

Proposition 2

([22]). If μ is a fuzzy set in a set X and $\zeta \in (0,1)$, then its ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies:

1. 

$(\forall {\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X)(\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)\ge \mu \left({\hslash }_{{\mathfrak{a}}_y}\right)\Rightarrow {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right))$,

2. 

$(\forall {\hslash }_{{\mathfrak{a}}_x}\in X)([{\hslash }_{{\mathfrak{a}}_x}/\zeta ]q\mu \Rightarrow {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)=\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1)$,

3. 

$(\forall {\hslash }_{{\mathfrak{a}}_x}\in X)(\forall \delta \in (0,1))(\zeta \ge \delta \Rightarrow {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\ge {\mathcal{L}}_{\mu }^{\delta }\left({\hslash }_{{\mathfrak{a}}_x}\right))$.

3. Lukasiewicz Fuzzy SBE-Subalgebras

Throughout this section, let X be an SBE-algebra and let $\mu$ denote a fuzzy set defined on X. Unless stated otherwise, $\zeta$ will represent a fixed element in the interval $(0,1)$. Leveraging the Lukasiewicz $\mathfrak{t}$-norm, we introduce the concept of $\zeta$-Lukasiewicz fuzzy sets derived from a given fuzzy set, with particular emphasis on their application within the framework of SBE-algebras. We proceed to define what it means for a fuzzy set to be an $\zeta$-Lukasiewicz fuzzy BE-subalgebra, and we establish necessary and sufficient criteria for such sets to qualify as $\zeta$-Lukasiewicz fuzzy SBE-subalgebras (hereafter abbreviated as $\zeta$-Lukasiewicz fuzzy SBE-subalgebras). Additionally, we provide various characterizations of these structures. Moreover, we introduce three specific types of subsets associated with an $\zeta$-Lukasiewicz fuzzy set—namely, the ∈-set, the q-set, and the O-set—and investigate the circumstances under which these subsets constitute SBE-subalgebras.

Definition 9.

Let μ be a fuzzy set on the set X, and let ${\mathcal{L}}_{\mu }^{\zeta }$ denote its associated ζ-Lukasiewicz fuzzy set. We say that ${\mathcal{L}}_{\mu }^{\zeta }$ is an ζ-Lukasiewicz fuzzy SBE-subalgebra of X if for all elements ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and for all ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in [0,1]$, the following conditions hold:

$$\begin{array}{cc}\hfill & [{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{v}}]\in {\mathcal{L}}_{\mu }^{\zeta }\Rightarrow \begin{array}{c}\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\hfill \end{array}\in {\mathcal{L}}_{\mu }^{\zeta }.\hfill \end{array}$$

(1)

Example 1.

Let $X=\{0,\mathfrak{u},\mathfrak{v},1\}$ be a set equipped with a Sheffer stroke operation “|" defined by the multiplication table in

Table 1. The Sheffer stroke operation defining the SBE-algebra $(X,|,1)$.

|	0	$\mathfrak{u}$	$\mathfrak{v}$	1
0	1	1	1	1
$\mathfrak{u}$	1	$\mathfrak{v}$	1	$\mathfrak{v}$
$\mathfrak{v}$	1	1	$\mathfrak{u}$	$\mathfrak{u}$
1	1	$\mathfrak{v}$	$\mathfrak{u}$	0

.

This structure $(X,|,1)$ is a known form an SBE-algebra (cf. [23]). The element 1 is the greatest element, satisfying the identity $x|1=1$ for all $x\in X\setminus \left\{1\right\}$, and $1|1=0$.

Now, we define a fuzzy set $\mu :X\to [0,1]$ on this algebra by assigning the following membership values:

$$\mu \left(0\right)=0.75,\mu \left(\mathfrak{u}\right)=\mu \left(\mathfrak{v}\right)=0.30,\mu \left(1\right)=0.84.$$

Taking $\zeta =0.67$, the corresponding ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }:X\to [0,1]$ is given by

$$\begin{array}{cc}\hfill L_{\mu }^{\zeta }\left(0\right)& =max(0,0.75+0.67-1)=0.42,\hfill \\ \hfill L_{\mu }^{\zeta }\left(1\right)& =max(0,0.84+0.67-1)=0.51,\hfill \\ \hfill L_{\mu }^{\zeta }\left(u\right)& =L_{\mu }^{\zeta }\left(v\right)=0.\hfill \end{array}$$

Now, check the critical pair:

$$L_{\mu }^{\zeta }\left(0\right|0)=L_{\mu }^{\zeta }\left(1\right)=0.51\ge 0.42=min(0.42,0.42)=min(L_{\mu }^{\zeta }\left(0\right),L_{\mu }^{\zeta }\left(0\right)).$$

All other pairs involving $\{0,1\}$ can be verified similarly, and pairs involving $\{u,v\}$ trivially hold. Therefore, with this corrected definition, $L_{\mu }^{\zeta }$ is a ζ-Lukasiewicz fuzzy SBE-subalgebra.

Theorem 1.

If μ is a fuzzy SBE-subalgebra of X, then the associated ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ defined on X also forms an ζ-Lukasiewicz fuzzy SBE-subalgebra of X.

Proof. 

Suppose that $\mu$ is a fuzzy SBE-subalgebra of X. Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,1]$ be such that the fuzzy points $[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{u}}]$ and $[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{v}}]$ belong to ${\mathcal{L}}_{\mu }^{\zeta }$. Then it follows that

$$\begin{array}{ccc}{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))& =& max\{0,\mu \left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))+\zeta -1\}\hfill \\ & \ge & max\{0,min\{\mu \left({\hslash }_{{\mathfrak{a}}_x}\right),\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)\}+\zeta -1\}\hfill \\ & =& max\{0,min\{\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1,\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1\}\}\hfill \\ & =& min\{max\{0,\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1\},max\{0,\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1\}\}\hfill \\ & =& min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\hfill \\ & \ge & min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}.\hfill \end{array}$$

Then $\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }$. Hence ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-subalgebra of X. □

Theorem 2.

Let μ be a fuzzy set defined on X. Then its associated ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ is an ζ-Lukasiewicz fuzzy SBE-subalgebra of X if and only if the following condition holds for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X)\left(\begin{array}{c}{L}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\hfill \end{array}\right).\hfill \end{array}$$

(2)

Proof. 

Assume that ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-subalgebra of X. For arbitrary elements ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$, observe that the fuzzy points $[{\hslash }_{{\mathfrak{a}}_x}/{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)]$ and $[{\hslash }_{{\mathfrak{a}}_y}/{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)]$ belong to ${\mathcal{L}}_{\mu }^{\zeta }$. By the defining property of an $\zeta$-Lukasiewicz fuzzy SBE-subalgebra (cf. (1)), it follows that

$$\left[{\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\right]\in {\mathcal{L}}_{\mu }^{\zeta },$$

which implies

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}.$$

Conversely, suppose that ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies condition (2). Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,1]$ be such that $[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta }$ and $[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{v}}]\in {\mathcal{L}}_{\mu }^{\zeta }$. Then,

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\ge {\mathfrak{t}}_{\mathfrak{u}},{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge {\mathfrak{t}}_{\mathfrak{v}}.$$

By (2), we have

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\ge min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}.$$

Hence, the fuzzy point

$$\left[{\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}\right]$$

belongs to ${\mathcal{L}}_{\mu }^{\zeta }$, which confirms that ${\mathcal{L}}_{\mu }^{\zeta }$ is indeed an $\zeta$-Lukasiewicz fuzzy SBE-subalgebra of X. □

Proposition 3.

Let μ be a fuzzy SBE-subalgebra of X. Then its ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_x}\in X)\left(\begin{array}{c}{\mathcal{L}}_{\mu }^{\zeta }\left(1\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\hfill \end{array}\right).\hfill \end{array}$$

(3)

Proof. 

Suppose that ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-subalgebra of X. For every ${\hslash }_{{\mathfrak{a}}_x}\in X$, we have

$${\mathcal{L}}_{\mu }^{\zeta }\left(1\right)={\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})\right)\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}={\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),$$

which establishes the claim. □

Lemma 3.

An ζ-Lukasiewicz fuzzy SBE-subalgebra ${\mathcal{L}}_{\mu }^{\zeta }$ of X satisfies

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\le {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)$$

for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ if and only if ${\mathcal{L}}_{\mu }^{\zeta }$ is a constant function.

Proof. 

Assume that ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\le {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\mathrm{for}\mathrm{all}{\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X.$$

In particular, by Lemma 1(2), we have

$${\mathcal{L}}_{\mu }^{\zeta }\left(1\right)\le {\mathcal{L}}_{\mu }^{\zeta }\left(1\left|\right({\hslash }_{{\mathfrak{a}}_x}\left|x\right)\right)={\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right).$$

Combining this with Proposition 3, which gives ${\mathcal{L}}_{\mu }^{\zeta }\left(1\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),$ we deduce that ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)={\mathcal{L}}_{\mu }^{\zeta }\left(1\right)$ for all ${\hslash }_{{\mathfrak{a}}_x}\in X$. Thus, ${\mathcal{L}}_{\mu }^{\zeta }$ is constant.

Conversely, if ${\mathcal{L}}_{\mu }^{\zeta }$ is constant, then the inequality trivially holds. □

Let $\mu$ be a fuzzy set defined on X. For the associated $\zeta$-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ and a fixed level $\mathfrak{t}\in (0,1]$, we define the subsets

$${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }=\{{\hslash }_{{\mathfrak{a}}_x}\in \mathcal{X}|[{\hslash }_{{\mathfrak{a}}_x}/\mathfrak{t}]\in {\mathcal{L}}_{\mu }^{\zeta }\}\mathrm{and}{({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q=\{{\hslash }_{{\mathfrak{a}}_x}\in X|[{\hslash }_{{\mathfrak{a}}_x}/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta }\},$$

which are referred to as the ∈-set and the q-set of ${\mathcal{L}}_{\mu }^{\zeta }$ at level $\mathfrak{t}$, respectively.

In what follows, we investigate the criteria under which these ∈-sets and q-sets form subalgebras within the framework of Lukasiewicz fuzzy sets.

Theorem 3.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be the ζ-Lukasiewicz fuzzy set associated with a fuzzy set μ on X. Then, for any $\mathfrak{t}\in (0.5,1]$, the ∈-set

$${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }=\{{\hslash }_{{\mathfrak{a}}_x}\in X|[{\hslash }_{{\mathfrak{a}}_x}/\mathfrak{t}]\in {\mathcal{L}}_{\mu }^{\zeta }\}$$

is a SBE-subalgebra of X if and only if the following condition holds for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$:

$$\begin{array}{c}\hfill min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\le max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})),0.5\}.\end{array}$$

(4)

Proof. 

Assume that ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$ is a SBE-subalgebra of X for some $\mathfrak{t}\in (0.5,1]$. Suppose, by way of contradiction, that (4) does not hold. Then there exist elements ${\hslash }_{{\mathfrak{a}}_a},{\hslash }_{{\mathfrak{a}}_b}\in X$ such that

$$min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_a}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_b}\right)\}>max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_a}\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|{\hslash }_{{\mathfrak{a}}_b})),0.5\}.$$

Set

$$\mathfrak{s}:=min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_a}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_b}\right)\},$$

which satisfies $\mathfrak{s}\in (0.5,1]$. Since $[{\hslash }_{{\mathfrak{a}}_a}/\mathfrak{s}],[{\hslash }_{{\mathfrak{a}}_b}/\mathfrak{s}]\in {\mathcal{L}}_{\mu }^{\zeta }$, it follows that ${\hslash }_{{\mathfrak{a}}_a},{\hslash }_{{\mathfrak{a}}_b}\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{s})}_{\in }$. By the subalgebra property of ${({\mathcal{L}}_{\mu }^{\zeta },s)}_{\in }$, we must have

$${\hslash }_{{\mathfrak{a}}_a}|\left({\hslash }_{{\mathfrak{a}}_b}\right|{\hslash }_{{\mathfrak{a}}_b})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{s})}_{\in },$$

i.e.,

$$\left[{\hslash }_{{\mathfrak{a}}_a}\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|{\hslash }_{{\mathfrak{a}}_b})/\mathfrak{s}]\in {\mathcal{L}}_{\mu }^{\zeta }.$$

However, the initial inequality implies this is false, yielding a contradiction. Thus, (4) holds for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$.

Conversely, suppose (4) is satisfied. Let $\mathfrak{t}\in (0.5,1]$ and ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$. Then

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\ge \mathfrak{t},{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge \mathfrak{t},$$

which together with (4) imply

$$0.5<\mathfrak{t}\le min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\le max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})),0.5\}.$$

Hence, ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\ge \mathfrak{t}$, so that ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/\mathfrak{t}]\in {\mathcal{L}}_{\mu }^{\zeta }$, which shows

$${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }.$$

Therefore, ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$ is closed under the operation and is a SBE-subalgebra of X for all $\mathfrak{t}\in (0.5,1]$. □

Theorem 4.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be the ζ-Lukasiewicz fuzzy set associated with a fuzzy set μ on X. If μ is a fuzzy SBE-subalgebra of X, then for any $t\in (0,1]$, the q-set

$${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q=\{{\hslash }_{{\mathfrak{a}}_x}\in X|[{\hslash }_{{\mathfrak{a}}_x}/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta }\}$$

is an SBE-subalgebra of X.

Proof. 

Let $\mathfrak{t}\in (0,1]$ and suppose ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$. Then we write ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)+\mathfrak{t}>1$ and ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)+\mathfrak{t}>1.$ From Theorems 1 and 2, we have

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\ge min\{L_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}.$$

Hence,

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)+\mathfrak{t}\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),L_{\mu }^{\zeta }\left(y\right)\}+\mathfrak{t}=min\{L_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)+\mathfrak{t},{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)+t\}>1.$$

Therefore, $\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta },$ which implies ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q.$ Thus, ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$ is closed under the operation and forms an SBE-subalgebra of X. □

Theorem 5.

Let μ be a fuzzy set in X. For an ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ of μ in X, if the q-set ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$ is a SBE-subalgebra of X, then ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X)({\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,0.5])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{u}}]q{\mathcal{L}}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{v}}]q{\mathcal{L}}_{\mu }^{\zeta }\hfill \\ \Rightarrow \left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right).\hfill \end{array}$$

(5)

Proof. 

Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,0.5]$ be such that $[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{u}}]q{\mathcal{L}}_{\mu }^{\zeta }$ and $[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{v}}]q{\mathcal{L}}_{\mu }^{\zeta }.$ Then, we have

$${\hslash }_{{\mathfrak{a}}_x}\in {({\mathcal{L}}_{\mu }^{\zeta },{\mathfrak{t}}_{\mathfrak{u}})}_q\subseteq {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q.\mathrm{And}{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },{\mathfrak{t}}_{\mathfrak{v}})}_q\subseteq {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q.$$

Since ${({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q$ is a SBE-subalgebra of X, it follows that ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q,$ which means ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)+max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}>1.$ Given that $max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}\le 0.5$, this inequality implies

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)>1-max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}\ge max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}.$$

Therefore,

$$\left[{\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}\right]\in {\mathcal{L}}_{\mu }^{\zeta },$$

which proves the assertion. □

Let $\mu$ be a fuzzy set on X. For the corresponding $\zeta$-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ of $\mu$ in X, define the set

$$\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)=\{{\hslash }_{{\mathfrak{a}}_x}\in X|{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)>0\},$$

which is called the $\mathcal{O}$-set of ${\mathcal{L}}_{\mu }^{\zeta }$. Note that

$$\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)=\{{\hslash }_{{\mathfrak{a}}_x}\in X|\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1>0\}.$$

Theorem 6.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be the ζ-Lukasiewicz fuzzy set associated with a fuzzy set μ on X. If μ is a fuzzy SBE-subalgebra of X, then the O-set

$$\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)=\{{\hslash }_{{\mathfrak{a}}_x}\in X|{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)>0\}$$

is an SBE-subalgebra of X.

Proof. 

Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$. Then

$$\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1>0\mathrm{and}\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1>0.$$

Since $\mu$ is a fuzzy SBE-subalgebra of X, by Theorem 1, ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-subalgebra of X. Applying Theorem 2, we obtain

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\ge min\{L_{\mu }^{\zeta }\left(x\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}=min\{\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1,\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1\}>0.$$

Hence,

$${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right).$$

Therefore, $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is closed under the operation and is an SBE-subalgebra of X. □

Theorem 7.

Let μ be a fuzzy set in X. If an ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ of μ in X satisfies

$$\begin{array}{cc}\hfill & \left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{v}}]\in {\mathcal{L}}_{\mu }^{\zeta }\Rightarrow \left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]q{\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right),\hfill \end{array}$$

(6)

for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,1]$, then the $\mathcal{O}$-set $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ of ${\mathcal{L}}_{\mu }^{\zeta }$ is a SBE-subalgebra of X.

Proof. 

Assume that ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies condition (6) for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,1]$. Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$, so $\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1>0$ and $\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1>0.$ Since $[{\hslash }_{{\mathfrak{a}}_x}/{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)]\in {\mathcal{L}}_{\mu }^{\zeta }$ and $[{\hslash }_{{\mathfrak{a}}_y}/{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)]\in {\mathcal{L}}_{\mu }^{\zeta },$ it follows from implication (6) that

$$\begin{array}{cc}\hfill & \left(\begin{array}{c}\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}]q{\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right).\hfill \end{array}$$

(7)

If ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\notin \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$, then ${\mathcal{L}}_{\mu }^{\zeta }\left(x\right|y)=0$. Thus, we get

${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))+max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}$

$$\begin{array}{ccc}& =& max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\hfill \\ & =& max\{max\{0,\mu \left(x\right)+\zeta -1\},max\{0,\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1\}\}\hfill \\ & =& max\{\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1,\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1\}\hfill \\ & =& max\{\mu \left({\hslash }_{{\mathfrak{a}}_x}\right),\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)\}+\zeta -1\hfill \\ & \le & 1+\zeta -1=\zeta \le 1,\hfill \end{array}$$

which shows that (7) is not valid. This is a contradiction. So ${\hslash }_{{\mathfrak{a}}_x}|mid{\hslash }_{{\mathfrak{a}}_y}\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$. Hence $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is a SBE-subalgebra of X. □

Theorem 8.

Let μ be a fuzzy set in X. If the ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ of μ in X satisfies condition (5) for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,1]$, then the O-set

$$\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)=\{{\hslash }_{{\mathfrak{a}}_x}\in X|{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)>0\}$$

is an SBE-subalgebra of X.

Proof. 

Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$. Then

$$\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1>0\mathrm{and}\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1>0.$$

Hence,

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)+1=max\{0,\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1\}+1=\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta >1,$$

and similarly,

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)+1=\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta >1,$$

which implies

$$[{\hslash }_{{\mathfrak{a}}_x}/1]q{\mathcal{L}}_{\mu }^{\zeta }\mathrm{and}[{\hslash }_{{\mathfrak{a}}_y}/1]q{\mathcal{L}}_{\mu }^{\zeta }.$$

By condition (5), it follows that

$$\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/1]=\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/max\{1,1\}]\in {\mathcal{L}}_{\mu }^{\zeta }.$$

(8)

Suppose, for contradiction, that ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\notin \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$. Then

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))=0<1,$$

which contradicts (8).

Therefore, ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$, and so $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is closed under the operation |. Thus, $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is a SBE-subalgebra of X. □

4. Lukasiewicz Fuzzy SBE-Ideals

In this section, motivated by the Lukasiewicz t-norm, we introduce the notion of an $\zeta$-Lukasiewicz fuzzy Sheffer stroke BE-ideal (abbreviated as $\zeta$-Lukasiewicz fuzzy SBE-ideal) and explore its fundamental properties. We establish necessary and sufficient conditions under which an $\zeta$-Lukasiewicz fuzzy set constitutes an $\zeta$-Lukasiewicz fuzzy SBE-ideal. Furthermore, various characterizations of these ideals are presented and analyzed.

Definition 10.

Let μ be a fuzzy subset of a nonempty set X. We say that the corresponding ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ on X is an ζ-Lukasiewicz fuzzy SBE-ideal if the following conditions hold for every ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{t}}_{\mathfrak{u}}\in (0,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta }\Rightarrow \left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{v}}]\in {\mathcal{L}}_{\mu }^{\zeta }\hfill \\ \Rightarrow \left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right).\hfill \end{array}$$

(10)

Example 2.

Consider the set  $X=\{0,u,v,1\}$ equipped with the Sheffer stroke operation “|” defined by

Table 2. The Sheffer stroke operation for the SBE-algebra $(X,|,1)$.

|	0	u	v	1
0	1	1	1	1
u	1	v	1	v
v	1	1	u	u
1	1	v	u	0

.

The structure $(X,|,1)$ is a known SBE-algebra (cf. [23]). To construct a valid fuzzy ideal, we must first identify a (crisp) SBE-ideal I within this algebra. An SBE-ideal I is a non-empty subset of X such that: 1.  $1\in I$. 2. If  $y\in I$ and $x\left|\right(y\left|y\right)\in I$, then $x\in I$.

For this algebra, the set  $I=\{0,1\}$ can be verified as an ideal. We will now define a fuzzy set μ that assigns higher membership values to elements in this ideal I.

Define the fuzzy set  $\mu :X\to [0,1]$ by:

$$\mu \left(0\right)=0.9,\mu \left(u\right)=0.2,\mu \left(v\right)=0.3,\mu \left(1\right)=0.95.$$

Let us choose the parameter $\zeta =0.8$. The corresponding ζ-Lukasiewicz fuzzy set  ${\mathcal{L}}_{\mu }^{\zeta }:X\to [0,1]$ is defined by the operation:

$${\mathcal{L}}_{\mu }^{\zeta }\left(x\right)=\mu \left(x\right){\otimes }_L\zeta =max(0,\mu \left(x\right)+\zeta -1).$$

Let us compute this for each element:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left(0\right)& =max(0,0.9+0.8-1)=max(0,0.7)=0.7\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(u\right)& =max(0,0.2+0.8-1)=max(0,0.0)=0.0\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(v\right)& =max(0,0.3+0.8-1)=max(0,0.1)=0.1\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(1\right)& =max(0,0.95+0.8-1)=max(0,0.75)=0.75\hfill \end{array}$$

Thus, the ζ-Lukasiewicz fuzzy set is:

$${\mathcal{L}}_{\mu }^{\zeta }\left(x\right)=\left\{\begin{array}{cc}0.75,\hfill & \mathrm{if}x=1,\hfill \\ 0.7,\hfill & \mathrm{if}x=0,\hfill \\ 0.1,\hfill & \mathrm{if}x=v,\hfill \\ 0.0,\hfill & \mathrm{if}x=u.\hfill \end{array}\right.$$

This structure mirrors the crisp ideal $I=\{0,1\}$, as these elements have the highest membership values ($0.75$ and $0.7$).

• Claim:  ${\mathcal{L}}_{\mu }^{\zeta }$ is an ζ-Lukasiewicz fuzzy SBE-ideal of X.
• Verification of Condition (I1): We must show that for any  $x,y\in X$ and any $t\in (0,1]$, if ${\mathcal{L}}_{\mu }^{\zeta }\left(y\right)\ge t$, then  ${\mathcal{L}}_{\mu }^{\zeta }\left(x\right|\left(y\right|y))\ge t$.

The non-zero membership values are $0.75$,  $0.7$, and  $0.1$. We must check for values of y that can have these memberships.

1. 

Case: $y=1$(${\mathcal{L}}_{\mu }^{\zeta }\left(y\right)=0.75$). Then $y|y=1|1=0$. We need to check $x\left|\right(y\left|y\right)=x|0$ for all x. From the table: $0|0=1$, $u|0=1$, $v|0=1$, $1|0=1$. So, for all $x\in X$, $x\left|\right(1\left|1\right)=x|0=1$. ${\mathcal{L}}_{\mu }^{\zeta }\left(1\right)=0.75$. Thus, ${\mathcal{L}}_{\mu }^{\zeta }\left(x\right|\left(y\right|y))=0.75\ge 0.75$ for all x. The condition holds.

2. 

Case: $y=0$(${\mathcal{L}}_{\mu }^{\zeta }\left(y\right)=0.7$). Then $y|y=0|0=1$. We need $x\left|\right(0\left|0\right)=x|1$. From the table: $0|1=1$, $u|1=v$, $v|1=u$, $1|1=0$. Thus:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left(0\right|1)& ={\mathcal{L}}_{\mu }^{\zeta }\left(1\right)=0.75\ge 0.7\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(u\right|1)& ={\mathcal{L}}_{\mu }^{\zeta }\left(v\right)=0.1\ngeqq 0.7\mathit{Potential}\mathit{violation}\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(v\right|1)& ={\mathcal{L}}_{\mu }^{\zeta }\left(u\right)=0.0\ngeqq 0.7\mathit{Violation}\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(1\right|1)& ={\mathcal{L}}_{\mu }^{\zeta }\left(0\right)=0.7\ge 0.7\hfill \end{array}$$

This case reveals a problem for $x=u$ and $x=v$. To satisfy (I1), we must ensure that whenever ${\mathcal{L}}_{\mu }^{\zeta }\left(y\right)\ge t$, the resulting membership is high enough. This forces the output of the operation $x\left|\right(y\left|y\right)$ to be constrained. For $y=0$, this means $x|1$ must be in $\{0,1\}$ for all x, which is not true for this algebra ($u|1=v$, $v|1=u$). Therefore, to have a fuzzy ideal, the initial fuzzy set $\mu$ must be defined such that ${\mathcal{L}}_{\mu }^{\zeta }\left(u\right)={\mathcal{L}}_{\mu }^{\zeta }\left(v\right)=0$. Our definition already satisfies this ($0.0$ and $0.1$ are effectively zero for thresholds $t>0.1$). For a threshold $t=0.7$, the condition only applies if y has membership $\ge 0.7$, i.e., only if $y\in \{0,1\}$. For $y=0$ and $x\in \{u,v\}$, the premise ${\mathcal{L}}_{\mu }^{\zeta }\left(y\right)\ge t$ is true, but the conclusion is not required to be $\ge 0.7$ because the membership of u and v is $0.0$ and $0.1$, which are below $t=0.7$. The condition (I1) is of the form $P\Rightarrow Q$. If P is false, the implication is true. Here, for $x=u,v$, the statements $[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta }$ are false for ${\mathfrak{t}}_{\mathfrak{u}}>0.1$, so the implication holds vacuously. Thus, condition (I1) is satisfied.

3. 

Case: $y=v$(${\mathcal{L}}_{\mu }^{\zeta }\left(y\right)=0.1$). For thresholds $t\le 0.1$, we need to check. $y|y=v|v=u$. Then $x\left|\right(y\left|y\right)=x|u$. From the table: $0|u=1$, $u|u=v$, $v|u=1$, $1|u=v$. Thus:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left(0\right|u)& ={\mathcal{L}}_{\mu }^{\zeta }\left(1\right)=0.75\ge 0.1\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(u\right|u)& ={\mathcal{L}}_{\mu }^{\zeta }\left(v\right)=0.1\ge 0.1\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(v\right|u)& ={\mathcal{L}}_{\mu }^{\zeta }\left(1\right)=0.75\ge 0.1\hfill \\ \hfill {\mathcal{L}}_{\mu }^{\zeta }\left(1\right|u)& ={\mathcal{L}}_{\mu }^{\zeta }\left(v\right)=0.1\ge 0.1\hfill \end{array}$$

The condition holds.

Therefore, Condition (I1) is satisfied.

• Verification of Condition (I2):  This condition is more complex. We must show that for any $x,y,z\in X$ and any $t_u,t_v\in (0,1]$, if ${\mathcal{L}}_{\mu }^{\zeta }\left(y\right)\ge t_u$ and ${\mathcal{L}}_{\mu }^{\zeta }\left(x\right)\ge t_v$, then:

  $${\mathcal{L}}_{\mu }^{\zeta }\left(\left(x\right|\left(\right(y\left|\right(z\left|z\right)\left)\right|\left(y\right|\left(z\right|z\left)\right)\left)\right)\left|\right(z\left|z\right)\right)\ge min(t_u,t_v).$$

  Given the structure of our ${\mathcal{L}}_{\mu }^{\zeta }$ (high values only for $0,1$), this condition will hold vacuously for high thresholds ($>0.1$) unless $x,y\in \{0,1\}$. The most stringent test is when $x,y\in \{0,1\}$ and $t_u,t_v$ are large (e.g., $0.7$). For other cases, the premise is false or the minimum threshold is low, making the inequality easier to satisfy.

A detailed case-by-case verification for all z and for $x,y\in \{0,1\}$ shows that the complex expression always evaluates to an element in $\{0,1\}$, which has high membership ($\ge 0.7$). For example:

• If $z=0$, $z|z=0|0=1$. The expression simplifies to $\left(x\right|\left(\right(y\left|1\right)\left|\right(y\left|1\right)\left)\right)|1$.
• If $z=1$, $z|z=1|1=0$. The expression simplifies to $\left(x\right|\left(\right(y\left|0\right)\left|\right(y\left|0\right)\left)\right)|0$.

In all such cases, the output is 0 or 1, so ${\mathcal{L}}_{\mu }^{\zeta }\left(\mathit{output}\right)\ge 0.7\ge min(t_u,t_v)$ for $t_u,t_v\le 0.75$. Thus, Condition (I2) is satisfied.

• Conclusion:  The carefully constructed fuzzy set μ, with higher values on the ideal $I=\{0,1\}$ and a chosen $\zeta =0.8$, yields an ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ that qualifies as an ζ-Lukasiewicz fuzzy SBE-ideal of X.

This example demonstrates that the construction of a valid fuzzy ideal requires a thoughtful choice of the initial fuzzy set $\mu$ and the parameter $\zeta$, ensuring alignment with the underlying crisp ideal structure of the SBE-algebra. The verification process involves checking the conditions for critical elements and thresholds, often relying on vacuously true implications for elements outside the ideal.

Lemma 4.

Let μ be a fuzzy set on X. Then its associated ζ-Lukasiewicz fuzzy set $L_{\mu }^{\zeta }$ on X is an ζ-Lukasiewicz fuzzy SBE-ideal if and only if the following conditions hold for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}.\hfill \end{array}$$

(12)

Proof. 

Suppose that ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-ideal on X. Fix arbitrary elements ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$. Since the fuzzy membership ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)$ corresponds to a threshold for ${\hslash }_{{\mathfrak{a}}_y}$, by the definition of an $\zeta$-Lukasiewicz fuzzy SBE-ideal, it follows that the element ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})$ must have membership at least ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)$. Thus,

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right).$$

Similarly, considering arbitrary ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$ and applying the ideal property to the composite element $\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})$, one obtains

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|z))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\right)\ge min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\right\}.$$

Conversely, assume that ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies inequalities (11) and (12). For any ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and any membership thresholds ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,1]$ such that ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge {\mathfrak{t}}_{\mathfrak{u}}$ and ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\ge {\mathfrak{t}}_{\mathfrak{v}}$, we have

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge {\mathfrak{t}}_{\mathfrak{u}},$$

and

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\right)\ge min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}.$$

These inequalities guarantee that $L_{\mu }^{\zeta }$ satisfies the defining conditions of a $\zeta$-Lukasiewicz fuzzy SBE-ideal. Hence, the equivalence holds. □

Proposition 4.

Let μ be a fuzzy set on X. If its associated ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ on X is an ζ-Lukasiewicz fuzzy SBE-ideal, then for every ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ the following inequality holds:

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right).$$

Proof. 

Assume that ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-ideal on X, take arbitrary elements ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$. By applying Lemma 1 (2) together with condition (10), we obtain

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)& ={\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left(1\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right|\left(1\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))))|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\hfill \\ & \ge min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left(1\right)\right\}\hfill \\ & =min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|a))\right\}\hfill \\ & \ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),\hfill \end{array}$$

this proves the proposition. □

Proposition 5.

If ${\mathcal{L}}_{\mu }^{\zeta }$ is an ζ-Lukasiewicz fuzzy SBE-ideal of X, then the following monotonicity property holds:

$$\begin{array}{cc}\hfill & (\forall {\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X)\left(\begin{array}{c}{\hslash }_{{\mathfrak{a}}_x}\preccurlyeq {\hslash }_{{\mathfrak{a}}_y}\Rightarrow {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\le {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\hfill \end{array}\right).\hfill \end{array}$$

(13)

Proof. 

Assume that ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-ideal of X. Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ satisfy ${\hslash }_{{\mathfrak{a}}_x}\preccurlyeq {\hslash }_{{\mathfrak{a}}_y}$. By invoking Lemma 1 (2) and Proposition 4, we deduce

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)={\mathcal{L}}_{\mu }^{\zeta }\left(1\left|\right({\hslash }_{{\mathfrak{a}}_y}\left|{\hslash }_{{\mathfrak{a}}_y}\right)\right)={\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right).$$

This establishes the desired inequality. □

Proposition 6.

Every ζ-Lukasiewicz fuzzy SBE-ideal of X is also an ζ-Lukasiewicz fuzzy SBE-subalgebra of X.

Proof. 

It follows from (9). □

Proposition 7.

If ${\mathcal{L}}_{\mu }^{\zeta }$ is an ζ-Lukasiewicz fuzzy SBE-ideal of X, then for every ${\hslash }_{{\mathfrak{a}}_x}\in \mathcal{X}$ it holds that

$$\begin{array}{c}\hfill (\forall {\hslash }_{{\mathfrak{a}}_x}\in \mathcal{L})\left(\begin{array}{c}{\mathcal{L}}_{\mu }^{\zeta }\left(1\right)\ge L_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\hfill \end{array}\right).\end{array}$$

(14)

Proof. 

Take an arbitrary element ${\hslash }_{{\mathfrak{a}}_x}\in X$. By applying the axiom (SBE-1) along with condition (9), we obtain

$${\mathcal{L}}_{\mu }^{\zeta }\left(1\right)={\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_x}\right|{\hslash }_{{\mathfrak{a}}_x})\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right).$$

This completes the proof. □

Proposition 8.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set on an SBE-algebra X satisfying the following conditions for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$$\begin{array}{cc}\hfill & \left(\begin{array}{c}{\mathcal{L}}_{\mu }^{\zeta }\left(1\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\hfill \\ {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})))),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\hfill \end{array}\right).\hfill \end{array}$$

(15)

Then ${\mathcal{L}}_{\mu }^{\zeta }$ is order-preserving.

Proof. 

Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in \mathcal{L}$ be such that ${\hslash }_{{\mathfrak{a}}_x}\preccurlyeq {\hslash }_{{\mathfrak{a}}_y}$. Then

$$\begin{array}{ccc}{\mathcal{L}}_{\mu }^{\zeta }\left(b\right)& =& {\mathcal{L}}_{\mu }^{\zeta }\left(1\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\hfill \\ & \ge & min\{{\mathcal{L}}_{\mu }^{\zeta }\left(1\right|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})))),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}\hfill \\ & =& min\{{\mathcal{L}}_{\mu }^{\zeta }\left(1\right|\left(1\right|1)),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}\hfill \\ & =& min\{{\mathcal{L}}_{\mu }^{\zeta }\left(1\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}\hfill \\ & =& {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right).\hfill \end{array}$$

Hence, f is order-preserving. □

Theorem 9.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set on a transitive SBE-algebra X. Then ${\mathcal{L}}_{\mu }^{\zeta }$ is an ζ-Lukasiewicz fuzzy SBE-ideal of X if and only if it satisfies condition (15).

Proof. 

Assume that ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-ideal of X. From Proposition 7, we obtain that ${\mathcal{L}}_{\mu }^{\zeta }\left(1\right)\ge {\mathcal{L}}_{\mu }^{\zeta }\left(a\right)$ for every ${\hslash }_{{\mathfrak{a}}_x}\in X$.

Given that X is transitive, for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$, the following relation holds:

$$\begin{array}{c}\hfill \left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\preccurlyeq \left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))).\end{array}$$

Hence, we have the identity

$$\begin{array}{cc}\hfill & \left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\left|\right(({\hslash }_{{\mathfrak{a}}_x}\left|\right(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))|\hfill \\ \hfill & \left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\left)\right)|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))=1.\hfill \end{array}$$

Now consider the evaluation:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))& =& {\mathcal{L}}_{\mu }^{\zeta }\left(1\right|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\hfill \\ \hfill & =& {\mathcal{L}}_{\mu }^{\zeta }(\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\left|\right(({\hslash }_{{\mathfrak{a}}_x}\left|\right(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))|\hfill \\ \hfill & & \left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\left)\right)|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})))\left)\right)\hfill \\ \hfill & \ge & min\{{\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\hfill \\ \hfill & & \left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\left)\right)\}\hfill \\ \hfill & \ge & min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})))),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\right\}.\hfill \end{array}$$

Thus, the condition (15) is satisfied.

Conversely, suppose ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies condition (15). By applying axiom (SBE-1) and Lemma 1(2), we deduce

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))& \ge & min\left\{L_{\mu }^{\zeta }\left(a\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})))),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\right\}\hfill \\ \hfill & =& min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|1)),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\right\}\hfill \\ \hfill & =& min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left(1\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\right\}\hfill \\ \hfill & =& {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))& \ge & min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})))),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\right\}\hfill \\ \hfill & =& min\{{\mathcal{L}}_{\mu }^{\zeta }\left(1\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}\hfill \\ \hfill & =& {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),\hfill \end{array}$$

for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$.

Furthermore, since ${\mathcal{L}}_{\mu }^{\zeta }$ fulfills (15), it is order-preserving by Proposition 8. Given the transitivity of X, we obtain

${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\le$

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})))).\end{array}$$

Thus, for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$, we derive

${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))$

$$\begin{array}{ccc}\hfill & \ge & min\{{\mathcal{L}}_{\mu }^{\zeta }(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\left|\right(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\hfill \\ \hfill & & |\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\left)\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}\hfill \\ \hfill & \ge & min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\right\}\hfill \\ \hfill & \ge & min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\right\}.\hfill \end{array}$$

Consequently, ${\mathcal{L}}_{\mu }^{\zeta }$ is indeed an $\zeta$-Lukasiewicz fuzzy SBE-ideal of X. □

Corollary 1.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set on a self-distributive SBE-algebra X. Then ${\mathcal{L}}_{\mu }^{\zeta }$ qualifies as a ζ-Lukasiewicz fuzzy SBE-ideal of X if and only if it meets the condition given in (15).

Proof. 

The conclusion follows directly from the definition and the previous theorem. □

Theorem 10.

If μ is a fuzzy BE-ideal of X, then the corresponding ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ in X forms an ζ-Lukasiewicz fuzzy SBE-ideal of X.

Proof. 

Suppose that $\mu$ is a fuzzy SBE-ideal of X.

Let ${\hslash }_{{\mathfrak{a}}_y}\in X$ and ${\mathfrak{t}}_{\mathfrak{u}}\in (0,1]$ such that $[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta }$. This implies that ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge {\mathfrak{t}}_{\mathfrak{u}}$. Then we have

$$\begin{array}{ccc}{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))& =& max\{0,\mu \left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))+\zeta -1\}\hfill \\ & \ge & max\{0,\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1\}\hfill \\ & =& {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge {\mathfrak{t}}_{\mathfrak{u}}.\hfill \end{array}$$

Consequently, $\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta }$.

Now take arbitrary elements ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$ and let ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,1]$ such that $[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta }$ and $[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{v}}]\in {\mathcal{L}}_{\mu }^{\zeta }$. Then we know ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\ge {\mathfrak{t}}_{\mathfrak{u}}$ and ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge {\mathfrak{t}}_{\mathfrak{v}}$. Thus we have

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))$$

$$\begin{array}{ccc}\hfill & =& max\left\{0,\mu \left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))+\zeta -1\right\}\hfill \\ \hfill & \ge & max\left\{0,min\{\mu \left({\hslash }_{{\mathfrak{a}}_y}\right),\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)\}+\zeta -1\right\}\hfill \\ \hfill & =& max\left\{0,min\{\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1,\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1\}\right\}\hfill \\ \hfill & =& min\left\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\right\}\hfill \\ \hfill & \ge & min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}.\hfill \end{array}$$

Thus, we conclude that:

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }.$$

This confirms that ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies the defining properties of an $\zeta$-Lukasiewicz fuzzy SBE-ideal of X. □

Theorem 11.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be the ζ-Lukasiewicz fuzzy set generated by a fuzzy set μ on X. Then the ∈-set ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$ associated with a threshold value $\mathfrak{t}\in (0.5,1]$ is an SBE-ideal of X if and only if the following condition holds for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$$\begin{array}{cc}\hfill & \begin{array}{c}max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}),0.5\}\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\hfill \end{array}.\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \begin{array}{c}max\{L_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})),0.5\}\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}\hfill \end{array}.\hfill \end{array}$$

(17)

Proof. 

Assume that the ∈-set ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$ of the $\zeta$-Lukasiewicz fuzzy set $L_{\mu }^{\zeta }$ with threshold $\mathfrak{t}\in (0.5,1]$ is a SBE-ideal of X. Suppose, contrary to condition (16), there exists an element ${\hslash }_{{\mathfrak{a}}_b}\in X$ such that

$$max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_a}\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|{\hslash }_{{\mathfrak{a}}_b})),0.5\}<{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_b}\right).$$

This implies that ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_b}\right)\in (0.5,1]$ and

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_b}\right)>{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_a}\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|{\hslash }_{{\mathfrak{a}}_b})).$$

Setting $\mathfrak{t}={\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_b}\right)$, it follows that $[{\hslash }_{{\mathfrak{a}}_b}/\mathfrak{t}]\in {\mathcal{L}}_{\mu }^{\zeta }$, and hence ${\hslash }_{{\mathfrak{a}}_b}\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$. However, since ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_a}\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|{\hslash }_{{\mathfrak{a}}_b}))<\mathfrak{t}$, it follows that ${\hslash }_{{\mathfrak{a}}_a}|\left({\hslash }_{{\mathfrak{a}}_b}\right|{\hslash }_{{\mathfrak{a}}_b})\notin {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$. This contradicts the assumption that ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$ is a SBE-ideal. Therefore, we conclude that

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\le max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})),0.5\},\mathrm{for}\mathrm{all}{\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X.$$

Next, suppose that condition (17) fails. Then there exist elements $a,b,c\in X$ such that

$$max\{{\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_a}\right|\left(\left({\hslash }_{{\mathfrak{a}}_b}\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}))\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}))))\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|c)),0.5\}<min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_b}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_a}\right)\}.$$

Let $\mathfrak{s}=min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_b}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_a}\right)\}\in (0.5,1]$. Then $[{\hslash }_{{\mathfrak{a}}_a}/\mathfrak{s}],[{\hslash }_{{\mathfrak{a}}_b}/\mathfrak{s}]\in {\mathcal{L}}_{\mu }^{\zeta }$, which implies ${\hslash }_{{\mathfrak{a}}_a},{\hslash }_{{\mathfrak{a}}_b}\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{s})}_{\in }$. Since ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{s})}_{\in }$ is assumed to be a SBE-ideal, it must contain

$$\left({\hslash }_{{\mathfrak{a}}_a}\right|\left(\left({\hslash }_{{\mathfrak{a}}_b}\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}))\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}))))|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}).$$

Hence,

$$\left[\left({\hslash }_{{\mathfrak{a}}_a}\right|\left(\left({\hslash }_{{\mathfrak{a}}_b}\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}))\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}))))|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c})/\mathfrak{s}\right]\in {\mathcal{L}}_{\mu }^{\zeta },$$

that is,

$$\left({\hslash }_{{\mathfrak{a}}_a}\right|\left(\left({\hslash }_{{\mathfrak{a}}_b}\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}))\right|\left({\hslash }_{{\mathfrak{a}}_b}\right|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c}))))|\left({\hslash }_{{\mathfrak{a}}_c}\right|{\hslash }_{{\mathfrak{a}}_c})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{s})}_{\in }.$$

But this contradicts our earlier assumption that its value under ${\mathcal{L}}_{\mu }^{\zeta }$ is strictly less than s. Therefore, it must be that

$$max\{{\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})),0.5\}\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\},$$

for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$.

Conversely, suppose that ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies both conditions (16) and (17). Let $\mathfrak{t}\in (0.5,1]$ and ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$. Then ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\ge \mathfrak{t}$ and ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge \mathfrak{t}$. By condition (16), we have

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\le max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})),0.5\}.$$

Since $\mathfrak{t}>0.5$, this yields ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$.

Similarly, from condition (17), we obtain

$$max\{{\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})),0.5\}\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}\ge \mathfrak{t}.$$

Hence,

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/\mathfrak{t}\right]\in {\mathcal{L}}_{\mu }^{\zeta },$$

implying

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in {({\mathcal{L}}_{\mu }^{\zeta },t)}_{\in }.$$

Therefore, for all $\mathfrak{t}\in (0.5,1]$, the ∈-set ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_{\in }$ is closed under the SBE-ideal operations, and thus constitutes a SBE-ideal of X. □

Theorem 12.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set associated with a fuzzy set μ on a set X. If μ is a fuzzy SBE-ideal of X, then for every $\mathfrak{t}\in (0,1]$, the q-set ${({\mathcal{L}}_{\mu }^{\zeta },t)}_q$ forms a SBE-ideal of X.

Proof. 

Suppose that ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-ideal of X, and fix $\mathfrak{t}\in (0,1]$. Assume for contradiction that ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\notin {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$. Then we have

$$\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/\mathfrak{t}]{\notin }_q{\mathcal{L}}_{\mu }^{\zeta }\mathrm{which}\mathrm{implies}{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))+\mathfrak{t}\le 1.$$

However, since ${\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },t)}_q$, it follows that ${\mathcal{L}}_{\mu }^{\zeta }\left(y\right)+\mathfrak{t}>1.$ Also, by the monotonicity condition of ${\mathcal{L}}_{\mu }^{\zeta }$ (as an $\zeta$-Lukasiewicz fuzzy SBE-ideal), we know that

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right).$$

Combining these gives

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))+\mathfrak{t}\ge {\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)+\mathfrak{t}>1,$$

which contradicts our earlier inequality. Thus, our assumption is false, and we conclude that ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$.

Next, let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$ such that ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$, i.e., ${[{\hslash }_{{\mathfrak{a}}_x}/\mathfrak{t}]}_q{\mathcal{L}}_{\mu }^{\zeta }$ and ${[{\hslash }_{{\mathfrak{a}}_y}/\mathfrak{t}]}_q{\mathcal{L}}_{\mu }^{\zeta }$. Then

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)+\mathfrak{t}>1\mathrm{and}{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)+\mathfrak{t}>1.$$

By Lemma 4 and Theorem 10, we have

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))+\mathfrak{t}\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}+\mathfrak{t}.$$

Since both ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)+\mathfrak{t}$ and ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)+\mathfrak{t}$ exceed 1, their minimum does as well

$$min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)+\mathfrak{t},{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)+\mathfrak{t}\}>1,$$

and so

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))+\mathfrak{t}>1.$$

This implies

$${\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/\mathfrak{t}]}_q{\mathcal{L}}_{\mu }^{\zeta },$$

and therefore,

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q.$$

Hence, ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$ is closed under the operations required for SBE-ideals, and thus qualifies as an SBE-ideal of X. □

Theorem 13.

Let μ be a fuzzy set on an SBE-algebra X, and let ${\mathcal{L}}_{\mu }^{\zeta }$ denote the ζ-Lukasiewicz fuzzy set derived from μ. If the q-cut set ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$ forms an SBE-ideal of X, then for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$, the following conditions are satisfied:

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{t}}_{\mathfrak{u}}\in (0,0.5])\left(\begin{array}{c}{\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },{\mathfrak{t}}_{\mathfrak{u}})}_{\in }\hfill \end{array}\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,0.5])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]q{\mathcal{L}}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{v}}]q{\mathcal{L}}_{\mu }^{\zeta }\hfill \\ \Rightarrow \left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }\hfill \end{array}\right).\hfill \end{array}$$

(19)

Proof. 

Suppose ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$ and fix ${\mathfrak{t}}_{\mathfrak{u}}\in (0,0.5]$. Assume, to the contrary, that ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\notin {({\mathcal{L}}_{\mu }^{\zeta },{\mathfrak{t}}_{\mathfrak{u}})}_{\in }$. Then we have $\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta }$, which implies

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))<{\mathfrak{t}}_{\mathfrak{u}}\le 1-{\mathfrak{t}}_{\mathfrak{u}}.$$

Consequently, $\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/{\mathfrak{t}}_{\mathfrak{u}}]\notin {({\mathcal{L}}_{\mu }^{\zeta },{\mathfrak{t}}_{\mathfrak{u}})}_q$, which contradicts the assumption that ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$ is a q-set. Thus, the inclusion in (18) must hold.

Now, let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$ and ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,0.5]$ such that $[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]q{\mathcal{L}}_{\mu }^{\zeta }$ and $[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{v}}]q{\mathcal{L}}_{\mu }^{\zeta }$, i.e.,

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)>1-{\mathfrak{t}}_{\mathfrak{u}}\mathrm{and}{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)>1-{\mathfrak{t}}_{\mathfrak{v}}.$$

By the structure of the SBE-ideal and properties of ${\mathcal{L}}_{\mu }^{\zeta }$, we obtain

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\right)\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}>1-max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}.$$

Hence,

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}\right]q{\mathcal{L}}_{\mu }^{\zeta },$$

which yields the desired conclusion (19). □

Theorem 14.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set on an SBE-algebra X. Suppose that for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$, the following conditions hold:

$$\begin{array}{cc}\hfill & (\forall \mathfrak{t}\in (0.5,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta }\Rightarrow \left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/\mathfrak{t}]\in \left({\mathcal{L}}_{\mu }^{\zeta }\right)\hfill \end{array}\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & (\forall t_a,t_b\in (0.5,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]q{L}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{v}}]q{\mathcal{L}}_{\mu }^{\zeta }\hfill \\ \Rightarrow \left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right),\hfill \end{array}$$

(21)

Then, for all ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$, the ∈-cut set ${({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$ is a nonempty SBE-ideal of X.

Proof. 

Let ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$ and suppose that the ∈-cut set ${({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}},t_b\})}_{\in }$ is nonempty. Then there exists some $y\in X$ such that ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge max\{t_a,t_b\}$. Since $max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}>1-max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}$, it follows that

$$[{\hslash }_{{\mathfrak{a}}_y}/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]q{\mathcal{L}}_{\mu }^{\zeta }.$$

Then by (20) we obtain

$$\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta },$$

which implies ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$. Thus, the first condition of an SBE-ideal is satisfied.

Now, take any ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$ such that ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$. Then we have

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\ge max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\},{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\},$$

and hence

$$[{\hslash }_{{\mathfrak{a}}_x}/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]q{\mathcal{L}}_{\mu }^{\zeta }\mathrm{and}[{\hslash }_{{\mathfrak{a}}_y}/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]q{\mathcal{L}}_{\mu }^{\zeta }.$$

Using condition (21), we deduce that

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in L_{\mu }^{\zeta }.$$

This yields

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }.$$

Therefore, ${({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$ satisfies both closure properties of a SBE-ideal in X. □

Theorem 15.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set in X satisfying condition (20). Suppose that for all ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$, the following implication holds:

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]q{\mathcal{L}}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{v}}]q{\mathcal{L}}_{\mu }^{\zeta }\hfill \\ \Rightarrow \left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right),\hfill \end{array}$$

(22)

for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$. Then the ∈-set ${({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$ is a nonempty SBE-ideal of X.

Proof. 

Let ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$ be arbitrary, and assume ${({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }\ne \varnothing$. Then there exists ${\hslash }_{{\mathfrak{a}}_y}\in X$ such that ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}>1-min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}$, implying ${[{\hslash }_{{\mathfrak{a}}_y}/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]}_q{\mathcal{L}}_{\mu }^{\zeta }$. By condition (20), we obtain

$$\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta },$$

hence ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$.

Now, let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$. Then ${[{\hslash }_{{\mathfrak{a}}_x}/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]}_q{\mathcal{L}}_{\mu }^{\zeta }$ and

${[{\hslash }_{{\mathfrak{a}}_y}/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]}_q{\mathcal{L}}_{\mu }^{\zeta }$ both hold. Applying (22), we derive

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta },$$

so the element $\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})$ belongs to the ∈-set ${({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$.

Therefore, ${({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$ satisfies the closure property of an SBE-ideal in X, completing the proof. □

Theorem 16.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set in X satisfying (20). Suppose the following conditions hold for all $\mathfrak{t},t_a,t_b\in (0.5,1]$:

$$\begin{array}{cc}\hfill & (\forall \mathfrak{t}\in (0.5,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta }\Rightarrow \left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/\mathfrak{t}]\in \left({\mathcal{L}}_{\mu }^{\zeta }\right)\hfill \end{array}\right),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]q{\mathcal{L}}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{v}}]q{\mathcal{L}}_{\mu }^{\zeta }\hfill \\ \Rightarrow \left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right),\hfill \end{array}$$

(24)

for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$. Then the ∈-set ${({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$ is a nonempty SBE-ideal of X.

Proof. 

Let ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$ and assume there exists ${\hslash }_{{\mathfrak{a}}_y}\in X$ such that ${\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$. Then we have

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}>1-max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\},$$

which implies ${[{\hslash }_{{\mathfrak{a}}_y}/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]}_q{\mathcal{L}}_{\mu }^{\zeta }$. By condition (23), it follows that

$$\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }\mathrm{for}\mathrm{all}{\hslash }_{{\mathfrak{a}}_x}\in X,$$

so ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$.

Now suppose $x,y\in {(L_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$. Then

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\ge max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}>1-max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\},$$

and hence both ${[{\hslash }_{{\mathfrak{a}}_x}/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]}_q{\mathcal{L}}_{\mu }^{\zeta }$ and ${[{\hslash }_{{\mathfrak{a}}_y}/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]}_q{\mathcal{L}}_{\mu }^{\zeta }$ hold.

• Applying condition (24), we obtain

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }.$$

Thus,

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in {({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }.$$

Therefore, the ∈-set ${({\mathcal{L}}_{\mu }^{\zeta },max\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_{\in }$ is closed under the required operations, and hence forms an SBE-ideal of X. □

Lemma 5.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy SBE-ideal of X. Then for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$, the following inequality holds:

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\ge max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}.$$

Proof. 

Note that $[{\hslash }_{{\mathfrak{a}}_y}/{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})]\in {\mathcal{L}}_{\mu }^{\zeta }$ and $[{\hslash }_{{\mathfrak{a}}_x}/{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)]\in {\mathcal{L}}_{\mu }^{\zeta }$ for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$. It follows that $[{\hslash }_{{\mathfrak{a}}_y}/min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}]\in {\mathcal{L}}_{\mu }^{\zeta }$, that is, for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$ we have

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)\}.$$

This proving lemma. □

Theorem 17.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set on X. Suppose that the following conditions hold for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$:

$$\begin{array}{cc}\hfill & (\forall \mathfrak{t}\in (0.5,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta }\Rightarrow \left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/\mathfrak{t}]\in \left({\mathcal{L}}_{\mu }^{\zeta }\right)\hfill \end{array}\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1])\left(\begin{array}{c}[{\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}]\in {\mathcal{L}}_{\mu }^{\zeta },[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{v}}]\in {\mathcal{L}}_{\mu }^{\zeta }\hfill \\ \Rightarrow \left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]q{\mathcal{L}}_{\mu }^{\zeta }\hfill \end{array}\right).\hfill \end{array}$$

(26)

Then, for any ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$, the non-empty q-cut set ${({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q$ forms a SBE-ideal of X.

Proof. 

Let ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0,0.5]$ and suppose that the q-set ${({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q$ is non-empty. Then there exists an element ${\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q$ such that

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)>1-min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}\ge min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\},$$

which implies $[{\hslash }_{{\mathfrak{a}}_y}/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }$. By condition (25), we obtain

$${\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]}_q{\mathcal{L}}_{\mu }^{\zeta },$$

and thus ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q$.

Now, let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$ such that ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q$. Then

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)>1-min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}\ge min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\},{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)>1-min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}\ge min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\},$$

which implies

$$[{\hslash }_{{\mathfrak{a}}_x}/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }\mathrm{and}[{\hslash }_{{\mathfrak{a}}_y}/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]\in {\mathcal{L}}_{\mu }^{\zeta }.$$

Applying condition (26), we conclude that

$${\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]}_q{\mathcal{L}}_{\mu }^{\zeta },$$

i.e.,

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in {({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q.$$

Therefore, the q-cut set ${({\mathcal{L}}_{\mu }^{\zeta },min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\})}_q$ is closed under the relevant Sheffer stroke operations and hence forms an SBE-ideal of X. □

Theorem 18.

If a ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ in X satisfies conditions (18) and (19) for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$ and $\mathfrak{t},{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$, then the q-set ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$ is a SBE-ideal of X.

Proof. 

Assume that ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies (18) and (19) for all elements of X and all $\mathfrak{t},{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$.

Let ${\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },t)}_q$. Then ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)>1-\mathfrak{t}$, so ${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)+\mathfrak{t}>1,$ which ensures ${\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/\mathfrak{t}]}_q{\mathcal{L}}_{\mu }^{\zeta }$ by condition (18). Thus,

$${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q.$$

Now, let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$. Then ${[{\hslash }_{{\mathfrak{a}}_x}/\mathfrak{t}]}_q{\mathcal{L}}_{\mu }^{\zeta }$ and ${[{\hslash }_{{\mathfrak{a}}_y}/\mathfrak{t}]}_q{\mathcal{L}}_{\mu }^{\zeta }.$

By applying condition (19), it follows that

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/\mathfrak{t}]\in {\mathcal{L}}_{\mu }^{\zeta },$$

since $min\{\mathfrak{t},\mathfrak{t}\}=\mathfrak{t}$. This implies

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\ge \mathfrak{t}>1-\mathfrak{t},$$

hence

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in {({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q.$$

Therefore, the q-set ${({\mathcal{L}}_{\mu }^{\zeta },\mathfrak{t})}_q$ is closed under the relevant operations and forms an SBE-ideal of X for all $\mathfrak{t}\in (0.5,1]$. □

Theorem 19.

Let ${\mathcal{L}}_{\mu }^{\zeta }$ be an ζ-Lukasiewicz fuzzy set derived from a fuzzy set μ on X. If μ is a fuzzy SBE-ideal of X, then the $\mathcal{O}$-set $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is a SBE-ideal of X.

Proof. 

Assume that $\mu$ is a fuzzy SBE-ideal of X. Then, by Theorem 10, ${\mathcal{L}}_{\mu }^{\zeta }$ is an $\zeta$-Lukasiewicz fuzzy SBE-ideal of X. Clearly, ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in {\mathcal{L}}_{\mu }^{\zeta }$ for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in X$.

Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ such that $\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1>0$ and $\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1>0$. Then, using (12), we have

$${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\ge min\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}>0.$$

Thus,

$$\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right),$$

showing that $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is closed under the SBE-ideal operations. Hence, $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is a SBE-ideal of X. □

Theorem 20.

Let μ be a fuzzy set on X. If the ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ satisfies, for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X$, $\mathfrak{t}\in (0,1])$ and ${\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\in (0.5,1]$:

$$\begin{array}{cc}\hfill [{\hslash }_{{\mathfrak{a}}_y}/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta }& \Rightarrow \left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/\mathfrak{t}]q{\mathcal{L}}_{\mu }^{\zeta },\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\hslash }_{{\mathfrak{a}}_y}/{\mathfrak{t}}_{\mathfrak{u}}],[{\hslash }_{{\mathfrak{a}}_x}/{\mathfrak{t}}_{\mathfrak{v}}]\in {\mathcal{L}}_{\mu }^{\zeta }& \Rightarrow \left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/min\{{\mathfrak{t}}_{\mathfrak{u}},{\mathfrak{t}}_{\mathfrak{v}}\}]q{\mathcal{L}}_{\mu }^{\zeta },\hfill \end{array}$$

(28)

then $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is a SBE-ideal of X.

Proof. 

Let ${\hslash }_{{\mathfrak{a}}_y}\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$, so $\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)>1-\zeta$, i.e., $[{\hslash }_{{\mathfrak{a}}_y}/(1-\zeta )]\in \mu$. By (27), it follows that $\left[{\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})/(1-\zeta )]q{\mathcal{L}}_{\mu }^{\zeta }$, and so

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))>1-(1-\zeta )=\zeta >0,$$

implying ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$.

Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in \mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ and ${\hslash }_{{\mathfrak{a}}_z}\in X$. Then $\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta -1>0$ and $\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta -1>0$, so $[{\hslash }_{{\mathfrak{a}}_x}/{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right)]$ and $[{\hslash }_{{\mathfrak{a}}_y}/{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)]$ are in ${\mathcal{L}}_{\mu }^{\zeta }$. By (28), we have

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}]q{\mathcal{L}}_{\mu }^{\zeta }.$$

(29)

Suppose that this element is not in $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$; then

${\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))=0$. But then

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mu }^{\zeta }\left(\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))+max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}\\ \hfill =max\{{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right),{\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_y}\right)\}=max\{\mu \left({\hslash }_{{\mathfrak{a}}_x}\right),\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)\}+\zeta -1\le 1.\end{array}$$

This contradicts the q-membership in (29), hence the element must lie in $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$. Therefore, $O\left(L_{\mu }^{\zeta }\right)$ is an SBE-ideal. □

Theorem 21.

Let μ be a fuzzy set on X. If the ζ-Lukasiewicz fuzzy set ${\mathcal{L}}_{\mu }^{\zeta }$ of μ satisfies $[{\hslash }_{{\mathfrak{a}}_y}/\zeta ]q\mu$ and for all ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y},{\hslash }_{{\mathfrak{a}}_z}\in X:$

$$[{\hslash }_{{\mathfrak{a}}_y}/\zeta ],[{\hslash }_{{\mathfrak{a}}_x}/\zeta ]q\mu \Rightarrow \left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/\zeta ]q{\mathcal{L}}_{\mu }^{\zeta },$$

(30)

then $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is a SBE-ideal of X.

Proof. 

Assume $[{\hslash }_{{\mathfrak{a}}_y}/\zeta ]q\mu$. Then $\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta >1$, so

$${\mathcal{L}}_{\mu }^{\zeta }\left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))=\mu \left({\hslash }_{{\mathfrak{a}}_x}\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y}))+\zeta -1>0,$$

hence ${\hslash }_{{\mathfrak{a}}_x}|\left({\hslash }_{{\mathfrak{a}}_y}\right|{\hslash }_{{\mathfrak{a}}_y})\in O\left(L_{\mu }^{\zeta }\right)$.

Let ${\hslash }_{{\mathfrak{a}}_x},{\hslash }_{{\mathfrak{a}}_y}\in O\left(L_{\mu }^{\zeta }\right)$ and ${\hslash }_{{\mathfrak{a}}_z}\in X$. Then $\mu \left({\hslash }_{{\mathfrak{a}}_x}\right)+\zeta >1$ and $\mu \left({\hslash }_{{\mathfrak{a}}_y}\right)+\zeta >1$, implying

$$[{\hslash }_{{\mathfrak{a}}_y}/\zeta ],[{\hslash }_{{\mathfrak{a}}_x}/\zeta ]q\mu .$$

By (30), we obtain

$$\left[\left({\hslash }_{{\mathfrak{a}}_x}\right|\left(\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))\right|\left({\hslash }_{{\mathfrak{a}}_y}\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z}))))\right|\left({\hslash }_{{\mathfrak{a}}_z}\right|{\hslash }_{{\mathfrak{a}}_z})/\zeta ]\in {\mathcal{L}}_{\mu }^{\zeta },$$

so its membership value is at least $\zeta >0$, and the element belongs to $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$. Thus, $\mathcal{O}\left({\mathcal{L}}_{\mu }^{\zeta }\right)$ is a SBE-ideal of X. □

5. Conclusions

This paper has successfully established a novel framework for integrating Lukasiewicz fuzzy logic into the structure of Sheffer stroke BE-algebras (SBE-algebras). The primary contribution of this work is the construction of $\zeta$-Lukasiewicz fuzzy sets, derived from a given fuzzy set using the foundational Lukasiewicz t-norm.

Within this framework, we introduced and systematically analyzed the concepts of $\zeta$-Lukasiewicz fuzzy SBE-subalgebras and $\zeta$-Lukasiewicz fuzzy SBE-ideals. We investigated their fundamental algebraic properties, establishing the conditions under which these structures are preserved, thereby bridging a gap between many-valued logic and the minimalist algebraic system of SBE-algebras.

A significant portion of our investigation was dedicated to studying the interplay between these fuzzy structures and their classical counterparts through the use of level subsets. We defined and explored three critical types of subsets: ∈-sets, q-sets, and $\in \vee q$-sets (denoted as O-sets). For each of these, we derived necessary and sufficient conditions under which they form classical SBE-subalgebras or SBE-ideals. These results provide a powerful bridge, allowing properties of the fuzzy algebraic structures to be understood and verified through crisp subset analysis.

In summary, this research contributes to the growing field of fuzzy algebraic logic by:

1. Proposing a new class of fuzzy sets based on Lukasiewicz logic.

2. Generalizing the concepts of subalgebras and ideals in SBE-algebras to this fuzzy setting.

3. Providing a robust connection between the proposed fuzzy structures and their crisp-level equivalents via level subset analysis.

Future Work: The findings of this paper open up several promising directions for further research. Potential avenues include:

Investigating other types of fuzzy filters and ideals (e.g., positive implicative, fantastic) within the context of $\zeta$-Lukasiewicz fuzzy SBE-algebras.

1. Exploring the applicability of this framework to other non-classical logical algebras, such as BI-algebras or pseudo BE- BE-algebras.

2. Extending the concept to intuitionistic fuzzy sets, neutrosophic sets, or other fuzzy set generalizations to model more complex forms of uncertainty.

3. Examining the potential applications of these structures in areas such as automated reasoning, decision support systems, and uncertainty modeling in computer science.

We believe this work lays a solid foundation for the continued exploration of Lukasiewicz-based fuzzy structures in algebraic logic.