Fuzzy Sets in Strong Sheffer Stroke NMV-Algebra with Respect to a Triangular Norm

Abstract

In this paper, we explore the application of fuzzy set theory in the context of triangular norms, with a focus on strong Sheffer stroke NMV-algebras. We introduce the concepts of $\mathfrak{T}$-fuzzy subalgebras and $\mathfrak{T}$-fuzzy filters, analyze their properties, and provide several illustrative examples. Our study demonstrates that $\mathfrak{T}$-fuzzy subalgebras and filters generalize classical subalgebras and filters, with level subsets preserving algebraic structures under t-norms. Notably, $\mathfrak{T}$-fuzzy sets exhibit strong closure properties, and homomorphisms between SSNMV-algebras extend naturally to fuzzy settings. Furthermore, we examine the relationships between $\mathfrak{T}$-fuzzy subalgebras (or filters) and their classical counterparts, as well as their corresponding level subsets and homomorphisms. These results contribute to refined uncertainty modeling in logical systems, with potential applications in fuzzy control and AI.

1. Introduction

Fuzzy set theory provides a robust framework for addressing uncertainty and imprecision across various domains. Fuzzy sets are employed to represent concepts that are inherently difficult to define precisely, capturing the degree to which elements belong to a set. Over time, this approach has been extended and refined through the integration of various logical systems and algebraic structures. One such system, the strong Sheffer stroke NMV-algebra, represents a generalization of classical Boolean algebra, offering a rich set of operations that are well-suited for modeling fuzzy relations and interactions.

The Sheffer operation, sometimes referred to as the Sheffer stroke or the NAND operator, was first introduced by H. M. Sheffer [1]. Its significance lies in its ability to operate independently, without reliance on any other logical operations, thereby forming a complete logical system. In other words, all the axioms of a logical framework can be expressed using solely the Sheffer operation, simplifying the analysis and manipulation of these systems. Furthermore, the axioms of Boolean algebra—the algebraic counterpart to classical propositional logic—can be fully articulated using only the Sheffer operation, thus emphasizing its flexibility and utility within logical systems.

The Sheffer stroke operation, due to its unique property of expressing all other logical connectives in Boolean logic, has numerous applications across diverse fields, including computer science, technology, and algebra. In the domain of mathematical logic and algebra, several significant applications have been explored. These include the use of Sheffer stroke operation reducts in basic algebras [2], an examination of state operators in Sheffer stroke basic algebras [3], and its role within orthoimplication algebras [4]. Moreover, the Sheffer stroke operation has been studied within the context of strong non-associative MV-algebras and their filters [5]. Recent research by Molkhasi et al. has explored representations of strongly algebraically closed algebras, Sheffer stroke algebras, and Visser algebras [6,7].

Triangular norms ($\mathfrak{t}$-norms) are fundamental operators in fuzzy logic, utilized to model the intersection or conjunction of fuzzy sets. These norms play a critical role in combining degrees of membership within fuzzy set theory, providing a formal mechanism to represent the “and” operation in a fuzzy context. $\mathfrak{t}$-norms are defined as binary operations that are associative, commutative, and monotonic, with 1 as the identity element. These properties render $\mathfrak{t}$-norms essential in various applications, including decision making, probability theory, and fuzzy logic, where they serve as key tools for managing uncertainty and vagueness in a structured manner. In recent years, numerous studies have investigated the applications of triangular norms [8,9,10,11,12].

In this paper, we examine the behavior of fuzzy sets within the context of a strong Sheffer stroke NMV-algebra, with a particular focus on their interaction with $\mathfrak{t}$-norms. $\mathfrak{t}$-norms, which generalize the classical concept of intersection in set theory, are critical for understanding the interactions between fuzzy sets. Their application within the strong Sheffer stroke NMV-algebra provides new insights into fuzzy reasoning and logical systems. We investigate the algebraic properties of fuzzy sets under the influence of $\mathfrak{t}$-norms and explore how these interactions enhance our understanding of uncertainty management within non-classical logical frameworks.

This section reviews the foundational concepts of Sheffer stroke operations [1], SSNMV-algebras [13], and t-norms [14], which are pivotal to our study. The Sheffer stroke, a universal logical operator, facilitates algebraic axiomatization [2], while SSNMV-algebras generalize non-associative MV-algebras by incorporating strong Sheffer properties [5]. Triangular norms (t-norms), which model fuzzy intersections, are central to our framework [8,14]. Key properties, such as associativity and boundary conditions, are discussed in Lemma 1 [15]. These tools provide the foundation for our definitions of $\mathfrak{T}$-fuzzy subalgebras and filters, effectively bridging algebraic and fuzzy set theories.

The objective of this study is to contribute to the theoretical advancement of fuzzy set theory by examining it through the lens of non-classical algebraic structures. Specifically, we aim to demonstrate how fuzzy sets behave within the strong Sheffer stroke NMV-algebra and how $\mathfrak{t}$-norms can refine logic and reasoning processes in fuzzy systems. This research opens new avenues for enhanced applications in fields where uncertainty and imprecision are prevalent, such as decision making, artificial intelligence, and fuzzy control systems.

2. Preliminaries

Definition 1

([1]). Let $(H,\mid )$ represent a groupoid. The operation ∣ is classified as a Sheffer stroke operation if it satisfies the following conditions:

• (S1) $\zeta \mid \eta =\eta \mid \zeta ,$
• (S2) $(\zeta \mid \zeta )\mid (\zeta \mid \eta )=\zeta ,$
• (S3) $\zeta \mid \left(\right(\eta \mid \varrho )\mid (\eta \mid \varrho \left)\right)=\left(\right(\zeta \mid \eta )\mid (\zeta \mid \eta \left)\right)\mid \varrho ,$
• (S4) $(\zeta \mid ((\zeta \mid \zeta )\mid (\eta \mid \eta )\left)\right)\mid (\zeta \mid ((\zeta \mid \zeta )\mid (\eta \mid \eta )\left)\right)=\zeta ,$

for all $\zeta ,\eta ,\varrho \in H.$

Definition 2

([13]). A strong Sheffer stroke NMV-algebra (abbreviated SSNMV-algebra) refers to an algebraic structure $(L,\mid ,1)$ of type $(2,0)$, which satisfies the following identities for all elements $\zeta ,\eta ,\varrho \in L$:

• (n1) $\zeta \mid \eta =\eta \mid \zeta ,$
• (n2) $\zeta \mid 0=1,$
• (n3) $(\zeta \mid 1)\mid 1=\zeta ,$
• (n4) $\left(\right(\zeta \mid 1)\mid \eta )\mid \eta =\left(\right(\eta \mid 1)\mid \zeta )\mid \zeta ,$
• (n5) $(\zeta \mid 1)\mid \left(\right(\zeta \mid \eta )\mid 1)=1,$
• (n6) $\zeta \mid \left(\right(\left(\right((\zeta \mid \eta )\mid \eta )\mid \varrho )\mid \varrho )\mid 1)=1,$

where 0 denotes the algebraic constant $1\mid 1$.

Proposition 1

([13]). Let $(L,\mid )$ be an SSNMV-algebra. Then, the binary relation $\zeta \le \eta$ is defined if and only if $\zeta \mid (\eta \mid 1)=1$, and this relation constitutes a partial order on L.

Definition 3

([13]). A nonempty subset G of an SSNMV-algebra L is said to be an SSNMV-subalgebra of L if $\zeta \mid (\eta \mid 1)\in G$ for all $\zeta ,\eta \in G$.

Definition 4

([13]). A nonempty subset G of an SSNMV-algebra L is termed an SSNMV-filter of L if, for all $\zeta ,\eta \in G$, the following conditions hold:

1. 

$1\in G$,

2. 

$\zeta \mid (\eta \mid 1)\in G$ and $\zeta \in G$ ⇒ $\eta \in G$.

Definition 5

([13]). Let $(\mathfrak{P},{|}_{\mathfrak{P}},1_{\mathfrak{P}})$ and $(\mathfrak{S},{|}_{\mathfrak{S}},1_{\mathfrak{S}})$ be SSNMV-algebras. A mapping $\mathfrak{f}:\mathfrak{P}\to \mathfrak{S}$ is called a homomorphism if, for all $\zeta ,\eta \in \mathfrak{P}$, it satisfies the condition $\mathfrak{f}\left(\zeta {\mid }_{\mathfrak{P}}\eta \right)=\mathfrak{f}\left(\zeta \right){\mid }_{\mathfrak{S}}\mathfrak{f}\left(\eta \right)$ and $\mathfrak{f}\left(1_{\mathfrak{P}}\right)=1_{\mathfrak{S}}$.

Theorem 1

([13]). Let $(\mathfrak{P},{|}_{\mathfrak{P}},1_{\mathfrak{P}})$ and $(\mathfrak{S},{|}_{\mathfrak{S}},1_{\mathfrak{S}})$ be SSNMV-algebras. The product $(\mathfrak{P}\times \mathfrak{S},{|}_{\mathfrak{P}\times \mathfrak{S}})$ forms an SSNMV-algebra, where $\mathfrak{P}\times \mathfrak{S}$ denotes the Cartesian product of $\mathfrak{P}$ and $\mathfrak{S}$, and the operation ${|}_{\mathfrak{P}\times \mathfrak{S}}$ is defined by

$$({\mathfrak{p}}_1,{\mathfrak{s}}_1){|}_{\mathfrak{P}\times \mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{s}}_2)=({\mathfrak{p}}_1{|}_{\mathfrak{P}}{\mathfrak{p}}_2,{\mathfrak{s}}_1{|}_{\mathfrak{S}}{\mathfrak{s}}_2)$$

for all $({\mathfrak{p}}_1,{\mathfrak{s}}_1),({\mathfrak{p}}_2,{\mathfrak{s}}_2)\in \mathfrak{P}\times \mathfrak{S}$. The identity element in this algebra is $(1_{\mathfrak{P}},1_{\mathfrak{S}})$.

Definition 6

([14]). A triangular norm (abbreviated as $\mathfrak{t}$-norm) is a binary operation $\mathfrak{T}$ on the unit interval $[0,1]$, meaning that $\mathfrak{T}:[0,1]\times [0,1]\to [0,1]$ is a function that satisfies the following properties:

1. 

Boundary condition: $(\forall \zeta \in [0,1\left]\right)\left(\mathfrak{T}\right(\zeta ,1)=\zeta )$,

2. 

Commutativity: $(\forall \zeta ,\eta \in [0,1\left]\right)\left(\mathfrak{T}\right(\zeta ,\eta )=\mathfrak{T}(\eta ,\zeta \left)\right)$,

3. 

Associativity: $(\forall \zeta ,\eta ,\varrho \in [0,1\left]\right)\left(\mathfrak{T}\right(\zeta ,\mathfrak{T}(\eta ,\varrho ))=\mathfrak{T}(\mathfrak{T}(\zeta ,\eta ),\varrho \left)\right)$,

4. 

Monotonicity: $(\forall \zeta ,\eta ,\varrho \in [0,1\left]\right)(\eta \le \varrho \Rightarrow \mathfrak{T}(\zeta ,\eta )\le \mathfrak{T}(\zeta ,\varrho \left)\right)$.

Definition 7

([16]). Let $\mathfrak{T}$ be a $\mathfrak{t}$-norm. The subset ${\Delta }_{\mathfrak{T}}$ of $[0,1]$ is defined as

$${\Delta }_{\mathfrak{T}}=\{\zeta \in [0,1]:\mathfrak{T}(\zeta ,\zeta )=\zeta \}.$$

A fuzzy set A on a nonempty set X is said to satisfy the imaginable property with respect to $\mathfrak{T}$ if $Im\left({\xi }_A\right)\subseteq {\Delta }_{\mathfrak{T}}$, which means that for all $\zeta \in X$, we have

$$(\forall \zeta \in X)(\mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\zeta \right))={\xi }_A\left(\zeta \right)),$$

where $Im\left({\xi }_A\right)=\{\rho \in [0,1]:\exists \zeta \in X,{\xi }_A\left(\zeta \right)=\rho \}$.

Lemma 1

([15]). Let $\mathfrak{T}$ be a $\mathfrak{t}$-norm. Then, the following properties are satisfied:

1. 

$(\forall \zeta ,\eta \in [0,1\left]\right)\left(\mathfrak{T}\right(\zeta ,\eta )\le \zeta and\mathfrak{T}(\zeta ,\eta )\le \eta )$),

2. 

$(\forall \zeta \in [0,1\left]\right)\left(\mathfrak{T}\right(\zeta ,0)=0)$,

3. 

$(\forall \hslash ,\ell ,\zeta ,\eta \in [0,1\left]\right)(\zeta \le \hslash ,\eta \le \ell \Rightarrow \mathfrak{T}(\zeta ,\eta )\le \mathfrak{T}(\hslash ,\ell \left)\right)$,

4. 

$(\forall \hslash ,\zeta ,\eta \in [0,1\left]\right)(\zeta \le \hslash ,\eta \le \hslash \Rightarrow \mathfrak{T}(\zeta ,\eta )\le \hslash )$.

3. Fuzzy Sets with Respect to a t-Norm in Strong Sheffer Stroke NMV-Algebras

In the following sections, let X denote an SSNMV-algebra and $\mathfrak{T}$ represent a $\mathfrak{t}$-norm, unless otherwise stated. We will introduce the concepts of $\mathfrak{T}$-fuzzy subalgebras and $\mathfrak{T}$-fuzzy filters, and explore their properties.

Definition 8.

A fuzzy set A in X is called a

• $\mathfrak{T}$-fuzzy subalgebra (briefly, ${\mathfrak{T}}_{FS}$) of X if

  $$\begin{array}{cc}\hfill & (\forall \zeta ,\eta \in X)\left({\xi }_A(\zeta \mid (\eta \mid 1))\ge \mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))\right).\hfill \end{array}$$

  (1)
• $\mathfrak{T}$-fuzzy filter (briefly, ${\mathfrak{T}}_{FF}$) of X if

  $$\begin{array}{cc}\hfill & (\forall \zeta \in X)\left({\xi }_A\left(1\right)\ge {\xi }_A\left(\zeta \right)\right),\hfill \end{array}$$

  (2)

  $$\begin{array}{cc}\hfill & (\forall \zeta ,\eta \in X)\left({\xi }_A\left(\eta \right)\ge \mathfrak{T}\left({\xi }_A(\zeta \mid (\eta \mid 1)),{\xi }_A\left(\zeta \right)\right)\right).\hfill \end{array}$$

  (3)

Example 1.

Consider the following SSNMV-algebra in

Table 1. Sheffer stroke NMV-algebra $(X,\mid )$.

∣	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
0	1	1	1	1	1	1	1	1
1	1	0	${\hslash }_{\mathfrak{b}}$	${\hslash }_{\mathfrak{a}}$	${\hslash }_{\mathfrak{d}}$	${\hslash }_{\mathfrak{c}}$	${\hslash }_{\mathfrak{f}}$	${\hslash }_{\mathfrak{e}}$
${\hslash }_{\mathfrak{a}}$	1	${\hslash }_{\mathfrak{b}}$	1	1	1	${\hslash }_{\mathfrak{e}}$	${\hslash }_{\mathfrak{d}}$	1
${\hslash }_{\mathfrak{b}}$	1	${\hslash }_{\mathfrak{a}}$	1	${\hslash }_{\mathfrak{b}}$	1	${\hslash }_{\mathfrak{d}}$	${\hslash }_{\mathfrak{e}}$	1
${\hslash }_{\mathfrak{c}}$	1	${\hslash }_{\mathfrak{d}}$	1	1	1	1	${\hslash }_{\mathfrak{d}}$	1
${\hslash }_{\mathfrak{d}}$	1	${\hslash }_{\mathfrak{c}}$	${\hslash }_{\mathfrak{e}}$	${\hslash }_{\mathfrak{d}}$	1	${\hslash }_{\mathfrak{b}}$	${\hslash }_{\mathfrak{a}}$	${\hslash }_{\mathfrak{e}}$
${\hslash }_{\mathfrak{e}}$	1	${\hslash }_{\mathfrak{f}}$	${\hslash }_{\mathfrak{d}}$	${\hslash }_{\mathfrak{e}}$	${\hslash }_{\mathfrak{d}}$	${\hslash }_{\mathfrak{a}}$	${\hslash }_{\mathfrak{b}}$	1
${\hslash }_{\mathfrak{f}}$	1	${\hslash }_{\mathfrak{e}}$	1	1	1	${\hslash }_{\mathfrak{e}}$	1	1

.

(i) Let ${\mathfrak{T}}_{\mathit{prod}}$ be the product $\mathfrak{t}$-norm defined by

$$(\forall \zeta ,\eta \in [0,1])\left({\mathfrak{T}}_{\mathit{prod}}(\zeta ,\eta )=\zeta \eta \right).$$

Let A be a fuzzy set in X given by

Table 2. Fuzzy set A of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\xi }_A\left(\zeta \right)$	$0.8$	$0.7$	$0.7$	$0.8$	$0.7$	$0.7$	$0.7$	$0.7$

:

Then, it can be easily verified that the fuzzy set A is a ${\mathfrak{T}}_{\mathit{prod}}$-fuzzy subalgebra of X.

(ii) Let ${\mathfrak{T}}_{\mathit{uk}}$ denote the Łukasiewicz $\mathfrak{t}$-norm, which is defined by

$$(\forall \zeta ,\eta \in [0,1])({\mathfrak{T}}_{\mathit{uk}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}).$$

Let A be a fuzzy set in X given by

Table 3. Fuzzyset A of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\xi }_A\left(\zeta \right)$	$0.7$	$0.6$	$0.5$	$0.5$	$0.5$	$0.7$	$0.2$	$0.5$

:

Then, it can be easily verified that the fuzzy set A is ${\mathfrak{T}}_{\mathit{uk}}$-fuzzy subalgebra of X.

(iii) Let ${\mathfrak{T}}_{\mathit{prod}}$ represent the product $\mathfrak{t}$-norm defined by

$$(\forall \zeta ,\eta \in [0,1])({\mathfrak{T}}_{\mathit{prod}}(\zeta ,\eta )=\zeta \eta ).$$

Let A be a fuzzy set in X given by

Table 4. Fuzzy set A of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\xi }_A\left(\zeta \right)$	$0.7$	$0.8$	$0.7$	$0.8$	$0.7$	$0.8$	$0.7$	$0.8$

:

Then, it can be easily verified that the fuzzy set A is ${\mathfrak{T}}_{\mathit{prod}}$-fuzzy filter of X.

(iv) Let ${\mathfrak{T}}_{\mathit{uk}}$ be the Łukasiewicz $\mathfrak{t}$-norm defined by

$$(\forall \zeta ,\eta \in [0,1])({\mathfrak{T}}_{\mathit{uk}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}).$$

Let A be a fuzzy set in X given by

Table 5. Fuzzy set A of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\xi }_A\left(\zeta \right)$	$0.6$	$0.7$	$0.6$	$0.6$	$0.7$	$0.7$	$0.5$	$0.7$

:

Then, it can be easily verified that the fuzzy set A is ${\mathfrak{T}}_{\mathit{uk}}$-fuzzy filter of X.

Definition 9.

Let A be a fuzzy set in X. Define the subset $I_{{\xi }_A}$ of X by

$$I_{{\xi }_A}=\{\zeta \in X:{\xi }_A\left(\zeta \right)={\xi }_A\left(1\right)\}.$$

Since ${\xi }_A\left(1\right)={\xi }_A\left(1\right)$, we have $1\in I_{{\xi }_A}$ ($\ne \varnothing$).

Definition 10.

A $\mathfrak{T}$-fuzzy subalgebra (resp. $\mathfrak{T}$-fuzzy filter) A of X is referred to as an imaginable $\mathfrak{T}$-fuzzy subalgebra (resp. imaginable $\mathfrak{T}$-fuzzy filter) of X if A satisfies the imaginable property with respect to $\mathfrak{T}$.

Example 2.

Consider the SSNMV-algebra given in Example 1 of Table 1.

(i) Let ${\mathfrak{T}}_{\mathit{uk}}$ be the Łukasiewicz $\mathfrak{t}$-norm defined by

$$(\forall \zeta ,\eta \in [0,1])({\mathfrak{T}}_{\mathit{uk}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}).$$

Let A be a fuzzy set in X given by

Table 6. Fuzzy set A of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\xi }_A\left(\zeta \right)$	0	1	0	0	0	0	0	0

:

Then, it can be easily verified that the fuzzy set A is imaginable ${\mathfrak{T}}_{\mathit{uk}}$-fuzzy subalgebra of X.

(ii) Let ${\mathfrak{T}}_{\mathit{uk}}$ be the Łukasiewicz $\mathfrak{t}$-norm defined by

$$(\forall \zeta ,\eta \in [0,1])({\mathfrak{T}}_{\mathit{uk}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}).$$

Let A be a fuzzy set in X given by

Table 7. Fuzzy set A of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\xi }_A\left(\zeta \right)$	0	1	0	1	0	0	0	0

:

Then, it can be easily verified that the fuzzy set A is imaginable ${\mathfrak{T}}_{\mathit{uk}}$-fuzzy filter of X.

Proposition 2.

If A is an imaginable ${\mathfrak{T}}_{FS}$ of X, then $(\forall \zeta \in X)({\xi }_A\left(1\right)\ge {\xi }_A\left(\zeta \right))$.

Proof. 

Assume that A is an imaginable ${\mathfrak{T}}_{FS}$ of X. Let $x\in X$. Then, we have ${\xi }_A\left(1\right)={\xi }_A(\zeta \mid (1\mid 1))\ge \mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\zeta \right))={\xi }_A\left(\zeta \right)$. □

Theorem 2.

Let μ be a ${\mathfrak{T}}_{FS}$ of X. If there is a sequence $\left\{{\zeta }_{\mathfrak{n}}\right\}$ in X such that $\underset{\mathfrak{n}\to \infty }{lim}\mathfrak{T}(\mu \left({\zeta }_{\mathfrak{n}}\right),\mu \left({\zeta }_{\mathfrak{n}}\right))=1$, then $\mu \left(1\right)=1$.

Proof. 

Let $\zeta \in X$. Then, we have

$$\mu \left(1\right)=\mu (\zeta \mid (1\mid 1\left)\right)\ge \mathfrak{T}\left(\mu \right(\zeta ),\mu (\zeta \left)\right).$$

Hence, for any $\mathfrak{n}\in \mathbb{N}$, we have

$$\mu \left(1\right)\ge \mathfrak{T}(\mu \left({\zeta }_{\mathfrak{n}}\right),\mu \left({\zeta }_{\mathfrak{n}}\right)).$$

Since $1\ge \mu \left(1\right)\ge {lim}_{\mathfrak{n}\to \infty }\mathfrak{T}(\mu \left({\zeta }_{\mathfrak{n}}\right),\mu \left({\zeta }_{\mathfrak{n}}\right))=1$, we conclude that

$$\mu \left(1\right)=1.$$

□

Theorem 3.

If A is an imaginable ${\mathfrak{T}}_{FS}$ of X, then $I_{{\xi }_A}$ is an SSNMV-subalgebra of X.

Proof. 

Assume that A is an imaginable ${\mathfrak{T}}_{FS}$ of X. Let $\zeta ,\eta \in I_{{\xi }_A}$. Then, we have ${\xi }_A\left(\zeta \right)={\xi }_A\left(1\right)$ and ${\xi }_A\left(\eta \right)={\xi }_A\left(1\right)$. Thus, we obtain the following inequality:

$$\begin{array}{ccc}\hfill {\xi }_A(\zeta \mid (\eta \mid 1))& \ge & \mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))\hfill \\ \hfill & =& \mathfrak{T}({\xi }_A\left(1\right),{\xi }_A\left(1\right))\hfill \\ \hfill & =& {\xi }_A\left(1\right)\hfill \\ \hfill & \ge & {\xi }_A(\zeta \mid (\eta \mid 1)).\hfill \end{array}$$

Thus, we conclude that

$${\xi }_A(\zeta \mid (\eta \mid 1))={\xi }_A\left(1\right),$$

which implies that $\zeta \mid (\eta \mid 1)\in I_{{\xi }_A}$. Therefore, $I_{{\xi }_A}$ is an SSNMV-subalgebra of X. □

Theorem 4.

If A is a ${\mathfrak{T}}_{FF}$ of X with $\mathfrak{T}({\xi }_A\left(1\right),{\xi }_A\left(1\right))={\xi }_A\left(1\right)$, then $I_{{\xi }_A}$ is an SSNMV-filter of X.

Proof. 

Assume that A is a ${\mathfrak{T}}_{FF}$ of X with $\mathfrak{T}({\xi }_A\left(1\right),{\xi }_A\left(1\right))={\xi }_A\left(1\right)$. By Definition 9, we know that $1\in I_{{\xi }_A}$. Let $\zeta ,\eta \in X$ be such that $\zeta \mid (\eta \mid 1)\in I_{{\xi }_A}$ and $\zeta \in I_{{\xi }_A}$. Then, by the definition of $I_{{\xi }_A}$, we have ${\xi }_A(\zeta \mid (\eta \mid 1))={\xi }_A\left(1\right)$ and ${\xi }_A\left(\zeta \right)={\xi }_A\left(1\right)$. Thus, we can write:

$$\begin{array}{ccc}\hfill {\xi }_A\left(\eta \right)& \ge & \mathfrak{T}({\xi }_A(\zeta \mid (\eta \mid 1)),{\xi }_A(\zeta \mid (\eta \mid 1)))\hfill \\ \hfill & =& \mathfrak{T}({\xi }_A\left(1\right),{\xi }_A\left(1\right))\hfill \\ \hfill & =& {\xi }_A\left(1\right)\hfill \\ \hfill & \ge & {\xi }_A\left(\eta \right).\hfill \end{array}$$

Thus, we conclude that ${\xi }_A\left(\eta \right)={\xi }_A\left(1\right)$, which implies that $\eta \in I_{{\xi }_A}$. Therefore, $I_{{\xi }_A}$ is an SSNMV-filter of X. □

Let S be a nonempty subset of X, and let $\nu ,\vartheta \in [0,1]$ with $\nu >\vartheta$. The $(\nu ,\vartheta )$-characteristic function ${\chi }_S^{(\nu ,\vartheta )}$ of X is a function from X into $\{\nu ,\vartheta \}$, defined as follows:

$$(\forall \zeta \in X){\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)=\left\{\begin{array}{cc}\nu \hfill & \mathrm{if}\zeta \in S,\hfill \\ \vartheta \hfill & \mathrm{if}\zeta \notin S.\hfill \end{array}\right.$$

By this definition, the $(\nu ,\vartheta )$-characteristic function ${\chi }_S^{(\nu ,\vartheta )}$ maps elements of X into $\{\nu ,\vartheta \}\subset [0,1]$. The fuzzy set $A_S^{(\nu ,\vartheta )}$ in X, associated with ${\chi }_S^{(\nu ,\vartheta )}$, is called the $(\nu ,\vartheta )$-characteristic fuzzy set of S in X.

Lemma 2.

Let S be a nonempty subset of X. Then, the following statements hold:

1. 

If the constant 1 of X belongs to S, then for all $\zeta \in X$, we have ${\chi }_S^{(\nu ,\vartheta )}\left(1\right)\ge {\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)$.

2. 

If there exists an element $\zeta \in S$ such that ${\chi }_S^{(\nu ,\vartheta )}\left(1\right)\ge {\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)$, then the constant 1 of X must be an element of S.

Proof. 

(1) If $1\in S$, then ${\chi }_S^{(\nu ,\vartheta )}\left(1\right)=\nu \ge {\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)$ for all $\zeta \in X$.

(2) Assume that there exists an element $\zeta \in S$ such that ${\chi }_S^{(\nu ,\vartheta )}\left(1\right)\ge {\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)$. Then, ${\chi }_S^{(\nu ,\vartheta )}\left(1\right)\ge \nu$, so it follows that ${\chi }_S^{(\nu ,\vartheta )}\left(1\right)=\nu$. Hence, the constant element $1\in S$. □

Theorem 5.

If S is an SSNMV-subalgebra of X, then ${\chi }_S^{(\nu ,\vartheta )}$ is a ${\mathfrak{T}}_{FS}$ of X.

Proof. 

Assume that S is an SSNMV-subalgebra of X.

Case 1: Let $\zeta ,\eta \in S$. Then, ${\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)=\nu ={\chi }_S^{(\nu ,\vartheta )}\left(\eta \right)$. Since S is a subalgebra of X, we have $\zeta \left|\right(\eta \left|1\right)\in S$, and consequently, ${\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1))=\nu$. By Lemma 1 (1), we obtain

$$\mathfrak{T}({\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right),{\chi }_S^{(\nu ,\vartheta )}\left(\eta \right))=\mathfrak{T}(\nu ,\nu )\le \nu ={\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1)).$$

Case 2: Let $\zeta \notin S$ or $\eta \notin S$. In this case, either ${\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)=\vartheta$ or ${\chi }_S^{(\nu ,\vartheta )}\left(\eta \right)=\vartheta$. By Lemma 1 (1), we have

$$\mathfrak{T}({\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right),{\chi }_S^{(\nu ,\vartheta )}\left(\eta \right))\le \vartheta \le {\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1)).$$

Therefore, ${\chi }_S^{(\nu ,\vartheta )}$ is a ${\mathfrak{T}}_{FS}$ of X. □

Theorem 6.

Let S be a nonempty subset of X such that ${\chi }_S^{(\nu ,\vartheta )}$ is a ${\mathfrak{T}}_{FS}$ of X and satisfies $\mathfrak{T}(\nu ,\nu )=\nu$. Then, S is an SSNMV-subalgebra of X.

Proof. 

Assume that S is a nonempty subset of X such that ${\chi }_S^{(\nu ,\vartheta )}$ is a ${\mathfrak{T}}_{FS}$ of X with $\mathfrak{T}(\nu ,\nu )=\nu$. Let $\zeta ,\eta \in S$. Then, by the definition of the characteristic function, we have

$${\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)=\nu ={\chi }_S^{(\nu ,\vartheta )}\left(\eta \right).$$

By the properties of the ${\mathfrak{T}}_{FS}$, specifically using (1), we obtain the following:

$$\begin{array}{ccc}{\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1))& \ge & \mathfrak{T}({\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right),{\chi }_S^{(\nu ,\vartheta )}\left(\eta \right))\hfill \\ & =& \mathfrak{T}(\nu ,\nu )\hfill \\ & =& \nu \hfill \\ & \ge & {\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1)).\hfill \end{array}$$

Therefore, we conclude that

$${\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1))=\nu ,$$

which implies that $\zeta \mid (\eta \mid 1)\in S$. Hence, S is an SSNMV-subalgebra of X. □

If $\mathfrak{T}(\nu ,\nu )\ne \nu$, then Theorem 6 is not applicable, as demonstrated by the following example.

Example 3.

Consider the SSNMV-algebra defined in Example 1 of Table 1. Let $S=\{0,{\hslash }_{\mathfrak{d}}\}$, and define a fuzzy set ${\chi }_S^{(0.7,0.5)}$ in X as given by the table in

Table 8. Fuzzy set ${\chi }_S^{(0.7,0.5)}$ of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\chi }_S^{(0.7,0.5)}\left(\zeta \right)$	$0.7$	$0.5$	$0.5$	$0.5$	$0.5$	$0.7$	$0.5$	$0.5$

:

Let ${\mathfrak{T}}_{\mathrm{Luk}}$ be the Łukasiewicz $\mathfrak{t}$-norm, defined as

$$(\forall \zeta ,\eta \in [0,1])({\mathfrak{T}}_{\mathrm{Luk}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}).$$

Then, ${\chi }_S^{(0.7,0.5)}$ is a ${\mathfrak{T}}_{\mathrm{Luk}}$-fuzzy subalgebra of X. However, we observe that

$${\mathfrak{T}}_{\mathrm{Luk}}(0.7,0.7)=0.4\ne 0.7.$$

Therefore, the condition ${\mathfrak{T}}_{\mathrm{Luk}}(0.7,0.7)=0.7$ does not hold, and Theorem 6 cannot be applied. Additionally, S is not an SSNMV-subalgebra of X because $0,{\hslash }_{\mathfrak{d}}\in S$, but we have

$$0\mid ({\hslash }_{\mathfrak{d}}\mid 1)=0\mid {\hslash }_{\mathfrak{c}}=1\notin S.$$

Corollary 1.

A nonempty subset S of X is an SSNMV-subalgebra of X if and only if ${\chi }_S^{(1,\vartheta )}$ is a ${\mathfrak{T}}_{FS}$ of X.

Proof. 

The result follows directly from Theorems 5 and 6. □

Theorem 7.

Let P be an SSNMV-subalgebra of X and A be a fuzzy set in X defined by

$${\xi }_A\left(\zeta \right)=\left\{\begin{array}{ccc}\nu & if\zeta \in P& \\ \theta & otherwise& \end{array}\right.$$

for all $\nu ,\theta \in [0,1]$ with $\nu \ge \theta$. Then, A is a ${\mathfrak{T}}_{\mathfrak{m}}$-FS of X, where ${\mathfrak{T}}_m$ is the $\mathfrak{t}$-norm defined by

$$(\forall \zeta ,\eta \in [0,1])({\mathfrak{T}}_{\mathfrak{m}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}).$$

In particular, if $\nu =1$ and $\theta =0$, then A is an imaginable ${\mathfrak{T}}_{\mathfrak{m}}$-fuzzy subalgebra of X. Moreover, $I_{{\xi }_A}=P$.

Proof. 

Let $\zeta ,\eta \in X$. We consider three cases:

• Case i: If $\zeta ,\eta \in P$, then

  $$\begin{array}{ccc}{\mathfrak{T}}_{\mathfrak{m}}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))& =& {\mathfrak{T}}_{\mathfrak{m}}(\nu ,\nu )\hfill \\ & =& max(2\nu -1,0)\hfill \\ & =& \left\{\begin{array}{ccc}2\nu -1& \mathrm{if}\nu \ge {\displaystyle \frac{1}{2}}& \\ 0& otherwise& \end{array}\right.\hfill \\ & \le & \nu \hfill \\ & =& {\xi }_A(\zeta \mid (\eta \mid 1)).\hfill \end{array}$$
• Case ii: If $\zeta \in P$ and $\eta \notin P$ (or, $\zeta \notin P$ and $\eta \in P$), then

  $$\begin{array}{ccc}{\mathfrak{T}}_{\mathfrak{m}}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))& =& {\mathfrak{T}}_{\mathfrak{m}}(\nu ,\theta )\hfill \\ & =& max(\nu +\theta -1,0)\hfill \\ & =& \left\{\begin{array}{ccc}\nu +\theta -1& \mathrm{if}\nu +\theta \ge 1& \\ 0& otherwise& \end{array}\right.\hfill \\ & \le & \theta \hfill \\ & =& {\xi }_A(\zeta \mid (\eta \mid 1)).\hfill \end{array}$$
• Case iii: If $\zeta ,\eta \notin P$, then

  $$\begin{array}{ccc}{\mathfrak{T}}_{\mathfrak{m}}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))& =& {\mathfrak{T}}_{\mathfrak{m}}(\theta ,\theta )\hfill \\ & =& max(2\theta -1,0)\hfill \\ & =& \left\{\begin{array}{ccc}2\theta -1& \mathrm{if}\theta \ge {\displaystyle \frac{1}{2}}& \\ 0& otherwise& \end{array}\right.\hfill \\ & \le & \theta \hfill \\ & =& {\xi }_A(\zeta \mid (\eta \mid 1)).\hfill \end{array}$$

Hence, A is a ${\mathfrak{T}}_{\mathfrak{m}}$-fuzzy subalgebra of X.

Assume that $\nu =1$ and $\theta =0$. Then,

$${\mathfrak{T}}_{\mathfrak{m}}(\nu ,\nu )=max(\nu +\nu -1,0)=1=\nu ,$$

and

$${\mathfrak{T}}_{\mathfrak{m}}(\theta ,\theta )=max(\theta +\theta -1,0)=0=\theta .$$

Thus, $\nu ,\theta \in {\Delta }_{{\mathfrak{T}}_{\mathfrak{m}}}$, that is, $I_m\left({\xi }_A\right)\subseteq {\Delta }_{{\mathfrak{T}}_{\mathfrak{m}}}$. Therefore, A is an imaginable ${\mathfrak{T}}_{\mathfrak{m}}$-fuzzy subalgebra of X. Also,

$$I_{{\xi }_A}=\{\zeta \in X\mid {\xi }_A\left(\zeta \right)={\xi }_A\left(1\right)\}=\{\zeta \in X\mid {\xi }_A\left(\zeta \right)=1\}=P.$$

Hence, $I_{{\xi }_A}=P$. □

Theorem 8.

If S is an SSNMV-filter of X, then ${\chi }_S^{(\nu ,\vartheta )}$ is a ${\mathfrak{T}}_{FF}$ of X.

Proof. 

Assume that S is an SSNMV-filter of X. Since $1\in S$, it follows from Lemma 2 (1) that ${\chi }_S^{(\nu ,\vartheta )}\left(1\right)\ge {\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)$ for all $\zeta \in X$. Next, let $\zeta ,\eta \in X$.

Case 1: If $\zeta \mid (\eta \mid 1)\in S$ and $\zeta \in S$, then

$${\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1))=\nu \mathrm{and}{\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)=\nu .$$

Since S is an SSNMV-filter of X, we have $\eta \in S$ and thus ${\chi }_S^{(\nu ,\vartheta )}\left(\eta \right)=\nu$. By Lemma 1 (1), we have

$$\mathfrak{T}({\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1)),{\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right))=\mathfrak{T}(\nu ,\nu )\le \nu ={\chi }_S^{(\nu ,\vartheta )}\left(\eta \right).$$

Case 2: If $\zeta \mid (\eta \mid 1)\notin S$ or $\zeta \notin S$, then

$${\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1))=\vartheta \mathrm{or}{\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)=\vartheta .$$

By Lemma 1 (1), we have

$$\mathfrak{T}({\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1)),{\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right))\le \vartheta \le {\chi }_S^{(\nu ,\vartheta )}\left(\eta \right).$$

Therefore, ${\chi }_S^{(\nu ,\vartheta )}$ is a ${\mathfrak{T}}_{FF}$ of X. □

Theorem 9.

If S is a nonempty subset of X such that ${\chi }_S^{(\nu ,\vartheta )}$ is a ${\mathfrak{T}}_{FF}$ of X with $\mathfrak{T}(\nu ,\nu )=\nu$, then S is an SSNMV-filter of X.

Proof. 

Assume that S is a nonempty subset of X such that ${\chi }_S^{(\nu ,\vartheta )}$ is a ${\mathfrak{T}}_{FF}$ of X with $T(\nu ,\nu )=\nu$. Then, by Lemma 2 (2), we have ${\chi }_S^{(\nu ,\vartheta )}\left(1\right)\ge {\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right)$ for all $\zeta \in X$, which implies that $1\in S$.

Let $\zeta ,\eta \in X$ be such that $\zeta \mid (\eta \mid 1)\in S$ and $\zeta \in S$. Then, we have

$${\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1))=\nu ={\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right).$$

By Equation (3), we obtain

$${\chi }_S^{(\nu ,\vartheta )}\left(\eta \right)\ge \mathfrak{T}({\chi }_S^{(\nu ,\vartheta )}(\zeta \mid (\eta \mid 1)),{\chi }_S^{(\nu ,\vartheta )}\left(\zeta \right))=\mathfrak{T}(\nu ,\nu )=\nu \ge {\chi }_S^{(\nu ,\vartheta )}\left(\eta \right).$$

Therefore, ${\chi }_S^{(\nu ,\vartheta )}\left(\eta \right)=\nu$, which implies that $\eta \in S$. Hence, S is an SSNMV-filter of X. □

If $\mathfrak{T}(\nu ,\nu )\ne \nu$, then Theorem 9 is not applicable, as demonstrated in the following example:

Example 4.

Consider the SSNMV-algebra given in Example 1 of Table 1. Let $S=\{1,{\hslash }_{\mathfrak{c}},{\hslash }_{\mathfrak{d}},{\hslash }_{\mathfrak{f}}\}$. Define a fuzzy set ${\chi }_S^{(0.7,0.6)}$ in X by the table in

Table 9. Fuzzy set ${\chi }_S^{(0.7,0.6)}$ of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\chi }_S^{(0.7,0.6)}\left(\zeta \right)$	$0.6$	$0.7$	$0.6$	$0.6$	$0.7$	$0.7$	$0.6$	$0.7$

:

Let ${\mathfrak{T}}_{\mathit{uk}}$ be the Lukasiewicz $\mathfrak{t}$-norm defined by

$$(\forall \zeta ,\eta \in [0,1]){\mathfrak{T}}_{\mathit{uk}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}.$$

Then, ${\chi }_S^{(0.7,0.6)}$ is a ${\mathfrak{T}}_{\mathit{uk}}$-fuzzy filter of X. However, ${\mathfrak{T}}_{\mathit{uk}}(0.7,0.7)=0.4\ne 0.7$. Furthermore, S is not an SSNMV-filter of X, since ${\hslash }_{\mathfrak{c}}\mid ({\hslash }_{\mathfrak{b}}\mid 1)={\hslash }_{\mathfrak{c}}\mid {\hslash }_{\mathfrak{a}}=1\in S$, but ${\hslash }_{\mathfrak{b}}\notin S$ while ${\hslash }_{\mathfrak{c}}\in S$.

Corollary 2.

A nonempty subset S of X is an SSNMV-filter of X if and only if ${\chi }_S^{(1,\vartheta )}$ is a ${\mathfrak{T}}_{FF}$ of X.

Proof. 

The result follows immediately from Theorems 8 and 9. □

4. Level Subsets of a Fuzzy Set with Respect to a $\mathfrak{t}$-Norm

Fuzzy sets have become a fundamental concept in various fields, and the analysis of their level subsets plays a crucial role in understanding their properties and applications. This section specifically focuses on the level subsets of a fuzzy set with respect to a $\mathfrak{t}$-norm, a mathematical framework that enables a more nuanced analysis of fuzzy relations and operations.

Definition 11.

Let A be a fuzzy set in a nonempty set X. For any $\mathfrak{s}\in [0,1]$, we define the following sets:

$$\begin{array}{c}\mathfrak{U}(A,\mathfrak{s})=\{\zeta \in X:{\xi }_A\left(\zeta \right)\ge \mathfrak{s}\},\\ \mathfrak{L}(A,\mathfrak{s})=\{\zeta \in X:{\xi }_A\left(\zeta \right)\le \mathfrak{s}\},\\ \mathfrak{E}(A,\mathfrak{s})=\{\zeta \in X:{\xi }_A\left(\zeta \right)=\mathfrak{s}\}.\end{array}$$

These sets are referred to as the upper $\mathfrak{s}$-level subset, lower $\mathfrak{s}$-level subset, and equal $\mathfrak{s}$-level subset of A, respectively. For a fuzzy set A in X, it follows that $\mathfrak{U}(A,1)=\mathfrak{E}(A,1)$, $\mathfrak{L}(A,1)=\mathfrak{E}(A,1)$, and $\mathfrak{U}(A,1)=\mathfrak{L}(A,1)=X$.

Lemma 3.

Let A be a fuzzy set in X. The following statements hold:

1. 

If the constant element 1 of X belongs to $\mathfrak{E}(A,1)$, then ${\xi }_A\left(1\right)\ge {\xi }_A\left(\zeta \right)$ for all $\zeta \in X$.

2. 

For any $\mathfrak{s}\in [0,1]$, if there exists an element $\zeta \in \mathfrak{U}(A,\mathfrak{s})$ such that ${\xi }_A\left(1\right)\ge {\xi }_A\left(\zeta \right)$, then the constant element 1 of X is in $\mathfrak{U}(A,\mathfrak{s})$.

Proof. 

(1) If $1\in \mathfrak{E}(A,1)$, then ${\xi }_A\left(1\right)=1\ge {\xi }_A\left(\zeta \right)$ for all $\zeta \in X$.

(2) Suppose that ${\xi }_A\left(1\right)\ge {\xi }_A\left(\zeta \right)$ for some $\zeta \in \mathfrak{U}(A,\mathfrak{s})$. Since $\zeta \in \mathfrak{U}(A,\mathfrak{s})$, it follows that ${\xi }_A\left(\zeta \right)\ge \mathfrak{s}$. Therefore, we have ${\xi }_A\left(1\right)\ge {\xi }_A\left(\zeta \right)\ge \mathfrak{s}$, which implies that $1\in \mathfrak{U}(A,\mathfrak{s})$. □

Theorem 10.

If A is a ${\mathfrak{T}}_{FS}$ of X, then for every $\mathfrak{s}\in [0,1]$ such that $\mathfrak{T}(\mathfrak{s},\mathfrak{s})=\mathfrak{s}$, the set $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-subalgebra of X whenever it is nonempty. In particular, $\mathfrak{E}(A,1)$ is an SSNMV-subalgebra of X if $\mathfrak{E}(A,1)$ is nonempty.

Proof. 

Let A be a ${\mathfrak{T}}_{FS}$ of X. Consider any $\mathfrak{s}\in [0,1]$ such that $\mathfrak{T}(\mathfrak{s},\mathfrak{s})=\mathfrak{s}$, and assume that $\mathfrak{U}(A,\mathfrak{s})\ne \varnothing$. Let $\zeta ,\eta \in \mathfrak{U}(A,\mathfrak{s})$. Then, by definition, we have ${\xi }_A\left(\zeta \right)\ge \mathfrak{s}$ and ${\xi }_A\left(\eta \right)\ge \mathfrak{s}$. Therefore,

$${\xi }_A(\zeta \mid (\eta \mid 1))\ge \mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))\ge \mathfrak{T}(\mathfrak{s},\mathfrak{s})=\mathfrak{s}.$$

Hence, $\zeta \mid (\eta \mid 1)\in \mathfrak{U}(A,\mathfrak{s})$, implying that $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-subalgebra of X. □

Theorem 11.

If a fuzzy set A in X is such that for every $\mathfrak{s}\in [0,1]$, the set $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-subalgebra of X whenever it is nonempty, then A is a ${\mathfrak{T}}_{FS}$ of X.

Proof. 

Suppose that for each $\mathfrak{s}\in [0,1]$, the set $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-subalgebra of X whenever it is nonempty. We aim to show that A is a ${\mathfrak{T}}_{FS}$ of X.

Assume, for the sake of contradiction, that there exist $\zeta ,\eta \in X$ such that

$${\xi }_A(\zeta \mid (\eta \mid 1))<\mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right)).$$

Define

$${\mathfrak{s}}_0={\displaystyle \frac{1}{2}}\left[{\xi }_A(\zeta \mid (\eta \mid 1))+\mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))\right].$$

Then, ${\mathfrak{s}}_0\in [0,1]$, and we have the inequalities

$${\xi }_A(\zeta \mid (\eta \mid 1))<{\mathfrak{s}}_0<\mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right)).$$

This implies that $\zeta \mid (\eta \mid 1)\notin \mathfrak{U}(A,{\mathfrak{s}}_0)$.

By Lemma 3 (1), since ${\xi }_A(\zeta \mid (\eta \mid 1))<{\mathfrak{s}}_0$, we know that if $\zeta ,\eta \in \mathfrak{U}(A,{\mathfrak{s}}_0)$, then $\mathfrak{U}(A,{\mathfrak{s}}_0)$ is nonempty.

Hence, the assumption that $\zeta ,\eta \in \mathfrak{U}(A,{\mathfrak{s}}_0)$ leads to a contradiction, as it implies that $\mathfrak{U}(A,{\mathfrak{s}}_0)$ cannot be an SSNMV-subalgebra of X, contradicting our assumption that all nonempty $\mathfrak{U}(A,\mathfrak{s})$ are SSNMV-subalgebras.

Therefore, we conclude that ${\xi }_A(\zeta \mid (\eta \mid 1))\ge \mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))$ for all $\zeta ,\eta \in X$, which implies that A is a ${\mathfrak{T}}_{FS}$ of X. □

The converse of Theorem 11 need not be true, as demonstrated in the following example.

Example 5.

Consider the SSNMV-algebra given in Example 1 of Table 1. Let ${\mathfrak{T}}_{\mathit{uk}}$ be the Łukasiewicz $\mathfrak{t}$-norm, defined by

$$\forall \zeta ,\eta \in [0,1],{\mathfrak{T}}_{\mathit{uk}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}.$$

Let A be a fuzzy set in X, as specified in

Table 10. Fuzzy set of A of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\xi }_A\left(\zeta \right)$	$0.7$	$0.6$	$0.6$	$0.5$	$0.5$	$0.7$	$0.4$	$0.7$

:

The fuzzy set A is a ${\mathfrak{T}}_{\mathit{uk}}$-fuzzy subalgebra of X. Let $\mathfrak{s}=0.7$. Then, the upper level subset is

$$\mathfrak{U}(A,0.7)=\{0,{\hslash }_{\mathfrak{d}},{\hslash }_{\mathfrak{f}}\}\ne \varnothing .$$

However, $\mathfrak{U}(A,0.7)$ is not an SSNMV-subalgebra of X. This is because, while $0,{\hslash }_{\mathfrak{d}}\in \mathfrak{U}(A,0.7)$, we have the following:

$$0\mid ({\hslash }_{\mathfrak{d}}\mid 1)=0\mid {\hslash }_{\mathfrak{c}}=1\notin \mathfrak{U}(A,0.7).$$

Therefore, the set $\mathfrak{U}(A,0.7)$ is not closed under the operations required for it to be an SSNMV-subalgebra of X, thereby illustrating that the converse of Theorem 11 does not hold in this case.

Theorem 12.

Let A be a ${\mathfrak{T}}_{FF}$ of X. For every $\mathfrak{s}\in [0,1]$ such that $\mathfrak{T}(\mathfrak{s},\mathfrak{s})=\mathfrak{s}$, the set $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-filter of X whenever it is nonempty. In particular, if $\mathfrak{U}(A,1)$ is nonempty, then $\mathfrak{E}(A,1)$ is an SSNMV-filter of X.

Proof. 

Assume that A is a ${\mathfrak{T}}_{FF}$ of X. Let $\mathfrak{s}\in [0,1]$ such that $\mathfrak{T}(\mathfrak{s},\mathfrak{s})=\mathfrak{s}$, and suppose that $\mathfrak{U}(A,\mathfrak{s})\ne \varnothing$. Since A is a ${\mathfrak{T}}_{FF}$ of X, we know that ${\xi }_A\left(1\right)\ge {\xi }_A\left(\zeta \right)$ for all $\zeta \in X$. By Lemma 3 (2), we conclude that $1\in \mathfrak{U}(A,\mathfrak{s})$.

Next, let $\zeta ,\eta \in X$ be such that $\zeta \mid (\eta \mid 1)\in \mathfrak{U}(A,\mathfrak{s})$ and $\zeta \in \mathfrak{U}(A,\mathfrak{s})$. From the assumption, we know that

$${\xi }_A(\zeta \mid (\eta \mid 1))\ge \mathfrak{s}\mathrm{and}{\xi }_A\left(\zeta \right)\ge \mathfrak{s}.$$

By the property of the $\mathfrak{t}$-norm, we have

$${\xi }_A\left(\eta \right)\ge \mathfrak{T}({\xi }_A(\zeta \mid (\eta \mid 1)),{\xi }_A\left(\zeta \right))\ge \mathfrak{T}(\mathfrak{s},\mathfrak{s})=\mathfrak{s}.$$

Therefore, $\eta \in \mathfrak{U}(A,\mathfrak{s})$, which shows that $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-filter of X. □

Theorem 13.

Let A be a fuzzy set in X. If for all $\mathfrak{s}\in [0,1]$, the set $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-filter of X whenever it is nonempty, then A is a ${\mathfrak{T}}_{FF}$ of X.

Proof. 

Assume that for every $\mathfrak{s}\in [0,1]$, the set $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-filter of X whenever it is nonempty. For any $\zeta \in X$, let ${\xi }_A\left(\zeta \right)=\mathfrak{s}$. Then, $\zeta \in \mathfrak{U}(A,\mathfrak{s})\ne \varnothing$. By assumption, $\mathfrak{U}(A,\mathfrak{s})$ is an SSNMV-filter of X, so $1\in \mathfrak{U}(A,\mathfrak{s})$. This implies that ${\xi }_A\left(1\right)\ge \mathfrak{s}={\xi }_A\left(\zeta \right)$.

Now, assume that there exist $\mathfrak{a},\mathfrak{b}\in X$ such that

$${\xi }_A\left(\mathfrak{b}\right)<\mathfrak{T}({\xi }_A(\mathfrak{a}\mid (\mathfrak{b}\mid 1)),{\xi }_A\left(\mathfrak{a}\right)).$$

Define

$${\mathfrak{s}}_0={\displaystyle \frac{1}{2}}\left[{\xi }_A\left(\mathfrak{b}\right)+\mathfrak{T}({\xi }_A(\mathfrak{a}\mid (\mathfrak{b}\mid 1)),{\xi }_A\left(\mathfrak{a}\right))\right].$$

Thus, ${\mathfrak{s}}_0\in [0,1]$ and we have

$${\xi }_A\left(\mathfrak{b}\right)<{\mathfrak{s}}_0<\mathfrak{T}({\xi }_A(\mathfrak{a}\mid (\mathfrak{b}\mid 1)),{\xi }_A\left(\mathfrak{a}\right)).$$

This implies that $\mathfrak{b}\notin \mathfrak{U}(A,{\mathfrak{s}}_0)$. By Lemma 3 (1), we know that $\mathfrak{a}\mid (\mathfrak{b}\mid 1),\mathfrak{a}\in \mathfrak{U}(A,{\mathfrak{s}}_0)\ne \varnothing$. Therefore, $\mathfrak{U}(A,{\mathfrak{s}}_0)$ cannot be an SSNMV-filter of X, which leads to a contradiction, since $\mathfrak{U}(A,{\mathfrak{s}}_0)$ was assumed to be an SSNMV-filter of X. Hence, we conclude that

$${\xi }_A\left(\eta \right)\ge \mathfrak{T}({\xi }_A(\zeta \mid (\eta \mid 1)),{\xi }_A\left(\zeta \right))$$

for all $\zeta ,\eta \in X$. Therefore, A is a ${\mathfrak{T}}_{FF}$ of X. □

The converse of Theorem 13 need not be true, as shown in the following example.

Example 6.

Consider the SSNMV-algebra given in Example 1 of Table 1. Let ${\mathfrak{T}}_{\mathit{uk}}$ be the Łukasiewicz $\mathfrak{t}$-norm defined by

$$(\forall \zeta ,\eta \in [0,1]){\mathfrak{T}}_{\mathit{uk}}(\zeta ,\eta )=max\{\zeta +\eta -1,0\}.$$

Let A be a fuzzy set in X given by the

Table 11. Fuzzy set A of X.

X	0	1	${\mathit{\hslash }}_{\mathfrak{a}}$	${\mathit{\hslash }}_{\mathfrak{b}}$	${\mathit{\hslash }}_{\mathfrak{c}}$	${\mathit{\hslash }}_{\mathfrak{d}}$	${\mathit{\hslash }}_{\mathfrak{e}}$	${\mathit{\hslash }}_{\mathfrak{f}}$
${\xi }_A\left(\zeta \right)$	$0.6$	$0.7$	$0.6$	$0.6$	$0.7$	$0.6$	$0.6$	$0.7$

:

Then, the fuzzy set A is a ${\mathfrak{T}}_{\mathit{uk}}$-fuzzy filter of X. If $\mathfrak{s}=0.7$, then $\mathfrak{U}(A,0.7)=\{1,{\hslash }_{\mathfrak{c}},{\hslash }_{\mathfrak{f}}\}\ne \varnothing$. However, $\mathfrak{U}(A,0.7)$ is not an SSNMV-filter of X since

$${\hslash }_{\mathfrak{c}}\mid ({\hslash }_{\mathfrak{b}}\mid 1)={\hslash }_{\mathfrak{c}}\mid {\hslash }_{\mathfrak{a}}=1\in \mathfrak{U}(A,0.7),$$

and

$${\hslash }_{\mathfrak{c}}\in \mathfrak{U}(A,0.7)but{\hslash }_{\mathfrak{b}}\notin \mathfrak{U}(A,0.7).$$

Definition 12.

Let $\mathfrak{f}:X\to Y$ be a function, where X and Y are nonempty sets. If B is a fuzzy set in Y, the inverse image of B under $\mathfrak{f}$ is a fuzzy set in X, denoted by ${\mathfrak{f}}^{-1}\left(B\right)$. This fuzzy set is defined as follows:

$$\forall \zeta \in X,{\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(\zeta \right)={\xi }_B\left(\mathfrak{f}\left(\zeta \right)\right),$$

where ${\xi }_B$ denotes the membership function of the fuzzy set B in Y.

Definition 13.

Let $(X,{\mid }_X,1_X)$ and $(Y,{\mid }_Y,1_Y)$ be SSNMV-algebras. A mapping $\mathfrak{f}:X\to Y$ is called a homomorphism if

$$(\forall \zeta ,\eta \in X)(\mathfrak{f}\left(\zeta {\mid }_X\eta \right)=\mathfrak{f}\left(\zeta \right){\mid }_Y\mathfrak{f}\left(\eta \right)).$$

A homomorphism $\mathfrak{f}:X\to Y$ is called an epimorphism if $\mathfrak{f}$ is surjective.

In the following, let $\mathfrak{f}$ represent a homomorphism from an SSNMV-algebra $(X,{|}_X,1_X)$ to an SSNMV-algebra $(Y,{|}_Y,1_Y)$, unless stated otherwise. It is important to note that for every $\zeta \in X$, the condition $\mathfrak{f}\left(1_X\right)\ge \mathfrak{f}\left(\zeta \right)$ holds.

Theorem 14.

Let $\mathfrak{f}:(X,{\mid }_X,1_X)\to (Y,{\mid }_Y,1_Y)$ be a surjective homomorphism between SSNMV-algebras. If B is a $\mathfrak{T}$-$FS$ in Y, then the inverse image of B under $\mathfrak{f}$, denoted by ${\mathfrak{f}}^{-1}\left(B\right)$, is a ${\mathfrak{T}}_{FS}$ in X.

Proof. 

Let $\mathfrak{f}:(X,{\mid }_X,1_X)\to (Y,{\mid }_Y,1_Y)$ be a surjective homomorphism, and assume that B is a ${\mathfrak{T}}_{FS}$ in Y. For any $\zeta ,\eta \in X$, we observe the following:

$$\begin{array}{ccc}{\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(\zeta {\mid }_X\left(\eta {\mid }_{X}1\right)\right)& =& {\xi }_B\left(\mathfrak{f}\left(\zeta {\mid }_X\left(\eta {\mid }_{X}1\right)\right)\right)\hfill \\ & =& {\xi }_B\left(\mathfrak{f}\left(\zeta \right){\mid }_Y\left(\mathfrak{f}\left(\eta \right){\mid }_Y\mathfrak{f}\left(1\right)\right)\right)\hfill \\ & \ge & \mathfrak{T}({\xi }_B\left(\mathfrak{f}\left(x\right)\right),{\xi }_B\left(\mathfrak{f}\left(\eta \right)\right))\hfill \\ & =& \mathfrak{T}({\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(\zeta \right),{\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(\eta \right)).\hfill \end{array}$$

Thus, ${\mathfrak{f}}^{-1}\left(B\right)$ is a ${\mathfrak{T}}_{FS}$ in X. □

Theorem 15.

Let $\mathfrak{f}:(X,{\mid }_X,1_X)\to (Y,{\mid }_Y,1_Y)$ be a surjective homomorphism between SSNMV-algebras. If B is a ${\mathfrak{T}}_{FF}$ in Y such that the mapping ${\xi }_B$ is order-preserving, then ${\mathfrak{f}}^{-1}\left(B\right)$ is a ${\mathfrak{T}}_{FF}$ in X.

Proof. 

Let $\mathfrak{f}:(X,{\mid }_X,1_X)\to (Y,{\mid }_Y,1_Y)$ be a surjective homomorphism, and suppose that B is a ${\mathfrak{T}}_{FF}$ in Y with ${\xi }_B$ being an order-preserving mapping. For every $\zeta \in X$, we have

$${\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(1_X\right)={\xi }_B\left(\mathfrak{f}\left(1_X\right)\right)\ge {\xi }_B\left(\mathfrak{f}\left(\zeta \right)\right)={\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(\zeta \right).$$

Now, for any $\zeta ,\eta \in X$, we observe

$$\begin{array}{ccc}\hfill {\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(\eta \right)& =& {\xi }_B\left(\mathfrak{f}\left(\eta \right)\right)\hfill \\ & \ge & \mathfrak{T}\left({\xi }_B\left(\mathfrak{f}\left(\zeta {\mid }_X\left(\eta {\mid }_{X}1\right)\right)\right),{\xi }_B\left(\mathfrak{f}\left(\zeta \right)\right)\right)\hfill \\ & =& \mathfrak{T}\left({\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(\zeta {\mid }_X\left(\eta {\mid }_{X}1\right)\right),{\xi }_{{\mathfrak{f}}^{-1}\left(B\right)}\left(\zeta \right)\right).\hfill \end{array}$$

Thus, ${\mathfrak{f}}^{-1}\left(B\right)$ is a ${\mathfrak{T}}_{FF}$ in X. □

Definition 14

([17]). Let X and Y be nonempty sets, and let $\mathfrak{f}:X\to Y$ be a function. A fuzzy set A in X is said to be invariant under $\mathfrak{f}$ if the following condition holds for all $\zeta ,\eta \in X$:

$$\forall \zeta ,\eta \in X,\left(\mathfrak{f}\left(\zeta \right)=\mathfrak{f}\left(\eta \right)\Rightarrow {\xi }_A\left(\zeta \right)={\xi }_A\left(\eta \right)\right).$$

Definition 15.

Let $\mathfrak{f}$ be a function from a nonempty set X to a nonempty set Y. For a fuzzy set A in X, the image of A under $\mathfrak{f}$, denoted $\mathfrak{f}\left(A\right)$, is a fuzzy set in Y defined as follows:

$$\forall \eta \in Y,{\xi }_{\mathfrak{f}\left(A\right)}\left(\eta \right)=\left\{\begin{array}{cc}\underset{\zeta \in {\mathfrak{f}}^{-1}\left(\eta \right)}{sup}\left\{{\xi }_A\left(\zeta \right)\right\}\hfill & if{\mathfrak{f}}^{-1}\left(\eta \right)\ne \varnothing ,\hfill \\ 0\hfill & if{\mathfrak{f}}^{-1}\left(\eta \right)=\varnothing .\hfill \end{array}\right.$$

Definition 16

([18]). A fuzzy set A in X has the sup property if for any nonempty subset S of X, there exists ${\mathfrak{s}}_0\in S$ such that

$${\xi }_A\left({\mathfrak{s}}_0\right)=\underset{\mathfrak{s}\in S}{sup}\left\{{\xi }_A\left(\mathfrak{s}\right)\right\}.$$

Lemma 4

([19]). Assume that $\mathfrak{f}$ is surjective. Let A be a $\mathfrak{f}$-invariant fuzzy set in X with the sup property. For any $\zeta ,\eta \in Y$, there exist ${\zeta }_0\in {\mathfrak{f}}^{-1}\left(\zeta \right)$ and ${\eta }_0\in {\mathfrak{f}}^{-1}\left(\eta \right)$ such that ${\xi }_{\mathfrak{f}\left(A\right)}\left(\zeta \right)={\xi }_A\left({\zeta }_0\right)$ and ${\xi }_{\mathfrak{f}\left(A\right)}\left(\eta \right)={\xi }_A\left({\eta }_0\right)$.

Theorem 16.

Let $\mathfrak{f}$ be a surjective function. If A is a ${\mathfrak{T}}_{FS}$ in X that is invariant under $\mathfrak{f}$ and satisfies the sup property, then the image of A under $\mathfrak{f}$, denoted by $\mathfrak{f}\left(A\right)$, in Y is also a ${\mathfrak{T}}_{FS}$.

Proof. 

Let A be a $\mathfrak{f}$-invariant ${\mathfrak{T}}_{FS}$ in X that satisfies the sup property. Consider $\zeta ,\eta \in Y$. Since $\mathfrak{f}$ is surjective, the inverse images ${\mathfrak{f}}^{-1}\left(\zeta \right)$ and ${\mathfrak{f}}^{-1}\left(\eta \right)$ are nonempty subsets of X. By Lemma 4, there exist elements ${\zeta }_0\in {\mathfrak{f}}^{-1}\left(\zeta \right)$ and ${\eta }_0\in {\mathfrak{f}}^{-1}\left(\eta \right)$ such that ${\xi }_{\mathfrak{f}\left(A\right)}\left(\zeta \right)={\xi }_A\left({\zeta }_0\right)$ and ${\xi }_{\mathfrak{f}\left(A\right)}\left(\eta \right)={\xi }_A\left({\eta }_0\right)$. Thus,

$$\begin{array}{ccc}\hfill {\xi }_{\mathfrak{f}\left(A\right)}\left(\zeta {\mid }_Y\left(\eta {\mid }_Y{1}_Y\right)\right)& =& {\xi }_A\left({\zeta }_0{\mid }_X\left({\eta }_0{\mid }_X{1}_X\right)\right)\hfill \\ & \ge & \mathfrak{T}({\xi }_A\left({\zeta }_0\right),{\xi }_A\left({\eta }_0\right))\hfill \\ & =& \mathfrak{T}({\xi }_{\mathfrak{f}\left(A\right)}\left(\zeta \right),{\xi }_{\mathfrak{f}\left(A\right)}\left(\eta \right)).\hfill \end{array}$$

Hence, $\mathfrak{f}\left(A\right)$ is a ${\mathfrak{T}}_{FS}$ in Y. □

Theorem 17.

Let $\mathfrak{f}$ be a surjective function. If A is a ${\mathfrak{T}}_{FF}$ in X that is $\mathfrak{f}$-invariant and possesses the sup property, then the image of A under $\mathfrak{f}$, denoted $\mathfrak{f}\left(A\right)$, is a ${\mathfrak{T}}_{FF}$ in Y.

Proof. 

Let A be a $\mathfrak{f}$-invariant ${\mathfrak{T}}_{FF}$ in X that satisfies the sup property. Since A is a ${\mathfrak{T}}_{FF}$, it follows that ${\xi }_A\left(1_X\right)\ge {\xi }_A\left(\zeta \right)$ for every $\zeta \in X$. Given that $\mathfrak{f}\left(1_X\right)=1_Y$, we know that $1_X\in {\mathfrak{f}}^{-1}\left(1_Y\right)$, implying that ${\mathfrak{f}}^{-1}\left(1_Y\right)$ is nonempty. Therefore, we have

$${\xi }_A\left(1_X\right)\ge \underset{\mathfrak{t}\in {\mathfrak{f}}^{-1}\left(1_Y\right)}{sup}\left\{{\xi }_A\left(\mathfrak{t}\right)\right\}={\xi }_{\mathfrak{f}\left(A\right)}\left(1_Y\right).$$

Thus, we obtain

$${\xi }_{\mathfrak{f}\left(A\right)}\left(1_Y\right)\ge {\xi }_{\mathfrak{f}\left(A\right)}\left(\eta \right).$$

Next, for $\zeta ,\eta \in Y$, by Lemma 4, there exist ${\zeta }^{\prime }\in {\mathfrak{f}}^{-1}\left(\zeta \right)$ and ${\eta }^{\prime }\in {\mathfrak{f}}^{-1}\left(\eta \right)$ such that ${\xi }_{\mathfrak{f}\left(A\right)}\left(\zeta \right)={\xi }_A\left({\zeta }^{\prime }\right)$ and ${\xi }_{\mathfrak{f}\left(A\right)}\left(\eta \right)={\xi }_A\left({\eta }^{\prime }\right)$. Thus, we have

$$\begin{array}{ccc}\hfill {\xi }_{\mathfrak{f}\left(A\right)}\left(\eta \right)& =& {\xi }_A\left({\eta }^{\prime }\right)\hfill \\ & \ge & \mathfrak{T}\left({\xi }_A\left({\zeta }^{\prime }{\mid }_X\left({\eta }^{\prime }{\mid }_X{1}_X\right)\right),{\xi }_A\left({\zeta }^{\prime }\right)\right)\hfill \\ & =& \mathfrak{T}\left({\xi }_{\mathfrak{f}\left(A\right)}\left(\zeta {\mid }_Y\left(\eta {\mid }_Y{1}_Y\right)\right),{\xi }_{\mathfrak{f}\left(A\right)}\left(\zeta \right)\right).\hfill \end{array}$$

Hence, $\mathfrak{f}\left(A\right)$ is a ${\mathfrak{T}}_{FF}$ in Y. □

Definition 17

([20]). Let $\mathcal{A}$ be a nonempty collection of fuzzy sets in a non-empty set X. The intersection of the family $\mathcal{A}$ in X is defined by the membership function ${\xi }_{\cap \mathcal{A}}$ as follows:

$$\forall \zeta \in X,{\xi }_{\cap \mathcal{A}}\left(\zeta \right)=inf\{{\xi }_A\left(\zeta \right)\mid A\in \mathcal{A}\}.$$

Similarly, the union of the family $\mathcal{A}$ in X is defined by its membership function ${\xi }_{\cup \mathcal{A}}$ as follows:

$$\forall \zeta \in X,{\xi }_{\cup \mathcal{A}}\left(\zeta \right)=sup\{{\xi }_A\left(\zeta \right)\mid A\in \mathcal{A}\}.$$

Theorem 18.

Let $\mathcal{A}$ be a nonempty collection of $\mathfrak{T}$-fuzzy subalgebras of X. Then, the intersection $\cap \mathcal{A}$ is also a ${\mathfrak{T}}_{FS}$ of X.

Proof. 

Let $\zeta ,\eta \in X$. For each $A\in \mathcal{A}$, we know that

$${\xi }_A(\zeta \mid (\eta \mid 1))\ge \mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right)).$$

Since for each $A\in \mathcal{A}$, the infimum of the membership functions satisfies

$$inf\left\{{\xi }_A\left(\zeta \right)\right\}\le {\xi }_A\left(\zeta \right)\mathrm{and}inf\left\{{\xi }_A\left(\eta \right)\right\}\le {\xi }_A\left(\eta \right),$$

applying Lemma 1 (3) yields

$$\mathfrak{T}\left(inf{\left\{{\xi }_A\left(\zeta \right)\right\}}_{A\in \mathcal{A}},inf{\left\{{\xi }_A\left(\eta \right)\right\}}_{A\in \mathcal{A}}\right)\le \mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_A\left(\eta \right))\le {\xi }_A(\zeta \mid (\eta \mid 1)),$$

for all $A\in \mathcal{A}$. Therefore, we obtain the following inequality:

$$\begin{array}{ccc}\hfill \mathfrak{T}({\xi }_{\cap \mathcal{A}}\left(\zeta \right),{\xi }_{\cap \mathcal{A}}\left(\eta \right))& =& \mathfrak{T}\left(inf{\left\{{\xi }_A\left(\zeta \right)\right\}}_{A\in \mathcal{A}},inf{\left\{{\xi }_A\left(\eta \right)\right\}}_{A\in \mathcal{A}}\right)\hfill \\ & \le & inf{\left\{{\xi }_A(\zeta \mid (\eta \mid 1))\right\}}_{A\in \mathcal{A}}\hfill \\ & =& {\xi }_{\cap \mathcal{A}}(\zeta \mid (\eta \mid 1)).\hfill \end{array}$$

Thus, we conclude that $\cap \mathcal{A}$ is a ${\mathfrak{T}}_{FS}$ of X. □

Theorem 19.

Let $A=\{(\zeta ,{\xi }_A\left(\zeta \right)):\zeta \in X\}$ and $B=\{(\eta ,{\xi }_B\left(\eta \right)):\eta \in Y\}$ be $\mathfrak{T}$-fuzzy subalgebras of the sets X and Y, respectively. Then, the Cartesian product

$$A\times B=\{((\zeta ,\eta ),{\xi }_{A\times B}(\zeta ,\eta )):(\zeta ,\eta )\in X\times Y\},$$

where the membership function ${\xi }_{A\times B}$ is defined as

$${\xi }_{A\times B}(\zeta ,\eta )=\mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_B\left(\eta \right)),$$

for all $(\zeta ,\eta )\in X\times Y$ is a ${\mathfrak{T}}_{FS}$ of $X\times Y$.

Proof. 

Let $({\zeta }_1,{\eta }_1),({\zeta }_2,{\eta }_2)\in X\times Y$. Then,

$$\begin{array}{cc}\hfill {\xi }_{A\times B}(({\zeta }_1,{\eta }_1){\mid }_{A\times B}& \left(({\zeta }_2,{\eta }_2){\mid }_{A\times B}(1,1)\right))\hfill \\ & ={\xi }_{A\times B}({\zeta }_1{\mid }_A\left({\zeta }_2{\mid }_{A}1\right),{\eta }_1{\mid }_B\left({\eta }_2{\mid }_{B}1\right)))\hfill \\ & =\mathfrak{T}\left({\xi }_A({\zeta }_1{\mid }_A\left({\zeta }_2{\mid }_{A}1\right),{\xi }_B\left({\eta }_1{\mid }_B\left({\eta }_2{\mid }_{B}1\right)\right))\right)\hfill \\ & \ge \mathfrak{T}(\mathfrak{T}({\xi }_A\left({\zeta }_1\right),{\xi }_A\left({\zeta }_2\right)),\mathfrak{T}({\xi }_B\left({\eta }_1\right),{\xi }_B\left({\eta }_2\right)))\hfill \\ & =\mathfrak{T}k(\mathfrak{T}({\xi }_A\left({\zeta }_1\right),{\xi }_B\left({\eta }_1\right)),\mathfrak{T}({\xi }_A\left({\zeta }_2\right),{\xi }_B\left({\eta }_2\right)))\hfill \\ & =\mathfrak{T}({\xi }_{A\times B}({\zeta }_1,{\eta }_1),{\xi }_{A\times B}({\zeta }_2,{\eta }_2)).\hfill \end{array}$$

Hence, $A\times B$ is a ${\mathfrak{T}}_{FS}$ of $X\times Y$. □

Theorem 20.

Let $A=\{(\zeta ,{\xi }_A\left(\zeta \right)):\zeta \in X\}$ and $B=\{(\eta ,{\xi }_B\left(\eta \right)):\eta \in Y\}$ be $\mathfrak{T}$-fuzzy filters of the sets X and Y, respectively. Then, the product

$$A\times B=\{((\zeta ,\eta ),{\xi }_{A\times B}(\zeta ,\eta )):(\zeta ,\eta )\in X\times Y\}$$

is a ${\mathfrak{T}}_{FF}$ of $X\times Y$, where the membership function ${\xi }_{A\times B}(\zeta ,\eta )$ is given by

$${\xi }_{A\times B}(\zeta ,\eta )=\mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_B\left(\eta \right)),$$

for every pair $(\zeta ,\eta )\in X\times Y$.

Proof. 

Let $(\zeta ,\eta )\in X\times Y$. Then,

$$\begin{array}{ccc}{\xi }_{A\times B}(1,1)& =& \mathfrak{T}({\xi }_A\left(1\right),{\xi }_B\left(1\right))\hfill \\ & \ge & \mathfrak{T}({\xi }_A\left(\zeta \right),{\xi }_B\left(\eta \right))\hfill \\ & =& {\xi }_{A\times B}(\zeta ,\eta ).\hfill \end{array}$$

Let $({\zeta }_1,{\eta }_1),({\zeta }_2,{\eta }_2)\in X\times Y$. Then,

${\xi }_{A\times B}({\zeta }_2,{\eta }_2)$

$$\begin{array}{ccc}& =& \mathfrak{T}({\xi }_A\left({\zeta }_2\right),{\xi }_B\left({\eta }_2\right))\hfill \\ & \ge & \mathfrak{T}(\mathfrak{T}({\xi }_A\left({\zeta }_1\right),{\xi }_A({\zeta }_1\mid ({\zeta }_2\mid 1))),\mathfrak{T}({\xi }_B\left({\eta }_1\right),{\xi }_B({\eta }_1\mid ({\eta }_2\mid 1))))\hfill \\ & =& \mathfrak{T}(\mathfrak{T}({\xi }_A\left({\zeta }_1\right),{\xi }_B\left({\eta }_1\right)),\mathfrak{T}({\xi }_{A_1}({\zeta }_1\mid ({\zeta }_2\mid 1)),{\xi }_{A_2}({\eta }_1\mid ({\eta }_2\mid 1))))\hfill \\ & =& \mathfrak{T}({\xi }_{A\times B}({\zeta }_1,{\eta }_1),{\xi }_{A\times B}\left(({\zeta }_1,{\eta }_1){\mid }_{A\times B}\left(({\zeta }_2,{\eta }_2){\mid }_{A\times B}(1,1)\right)\right)).\hfill \end{array}$$

Hence, $A\times B$ is a ${\mathfrak{T}}_{FF}$ of $X\times Y$. □

Theorem 21.

Consider a descending chain of SSNMV-filters $S_1\supseteq S_2\supseteq S_3\supseteq \cdots$ on X that eventually terminates after a finite number of steps. Let A be a ${\mathfrak{T}}_{FF}$ on X. If the sequence of values in the image of ${\xi }_A$ is strictly increasing, then A must be finite-valued.

Proof. 

Assume, for the sake of contradiction, that A is infinite-valued. Let $\left\{{\vartheta }_n\right\}$ denote a strictly increasing sequence of values from the image of ${\xi }_A$, such that $0\le {\vartheta }_1<{\vartheta }_2<\cdots <1$. Consider the SSNMV-filter $\mathfrak{U}({\xi }_A:{\psi }_{\mathfrak{T}})$ on X for each $\mathfrak{T}=1,2,3,\cdots$.

If $\zeta \in \mathfrak{U}({\xi }_A:{\vartheta }_{\mathfrak{T}})$ for $\mathfrak{T}=2,3,\cdots$, it follows that ${\xi }_A\left(\zeta \right)\ge {\vartheta }_{\mathfrak{T}}>{\vartheta }_{\mathfrak{T}-1}$, which implies that $\zeta \in \mathfrak{U}({\xi }_A:{\vartheta }_{\mathfrak{T}-1})$. Thus, we obtain the inclusion

$$\mathfrak{U}({\xi }_A:{\vartheta }_{\mathfrak{T}})\subseteq \mathfrak{U}({\xi }_A:{\vartheta }_{\mathfrak{T}-1})\mathrm{for}\mathfrak{T}=2,3,\cdots .$$

Since ${\vartheta }_{\mathfrak{T}-1}$ is in the image of ${\xi }_A$, there exists some ${\zeta }_{\mathfrak{T}-1}$ such that ${\xi }_A\left({\zeta }_{\mathfrak{T}-1}\right)={\vartheta }_{\mathfrak{T}-1}$. Therefore, ${\zeta }_{\mathfrak{T}-1}\in \mathfrak{U}({\xi }_A:{\vartheta }_{\mathfrak{T}-1})$, but ${\zeta }_{\mathfrak{T}-1}\notin \mathfrak{U}({\xi }_A:{\vartheta }_{\mathfrak{T}})$. This implies the strict inclusion:

$$\mathfrak{U}({\xi }_A:{\vartheta }_{\mathfrak{T}})⊊\mathfrak{U}({\xi }_A:{\vartheta }_{\mathfrak{T}-1})\mathrm{for}\mathfrak{T}=2,3,\cdots .$$

Consequently, we obtain a strictly descending chain of SSNMV-filters on X:

$$\mathfrak{U}({\xi }_A:{\vartheta }_1)⊋\mathfrak{U}({\xi }_A:{\vartheta }_2)⊋\cdots ,$$

which does not terminate. This is a contradiction to the assumption that the chain eventually terminates. Therefore, A must be finite-valued. □

Theorem 22.

If every ${\mathfrak{T}}_{FF}$ on X has a finite image, then any descending sequence of SSNMV-filters of X must eventually terminate after a finite number of steps.

Proof. 

Assume that there exists an infinite strictly descending chain of SSNMV-filters on X:

$$S_0⊋S_1⊋S_2⊋\cdots ,$$

which does not terminate after a finite number of steps. Now, define a fuzzy set ${\xi }_A$ on X as follows:

$${\xi }_A\left(\zeta \right)=\left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{n}}{\mathfrak{n}+1}}& \mathrm{if}\zeta \in S_{\mathfrak{n}}\setminus S_{\mathfrak{n}+1},\\ 1& \mathrm{if}\zeta \in \bigcap _{\mathfrak{n}=0}^{\infty }{S}_{\mathfrak{n}},\end{array}\right.$$

where $\mathfrak{n}=0,1,2,\cdots$, and $S_0=X$. This construction defines ${\xi }_A$ as a fuzzy set on X where the membership values decrease as the SSNMV-filters decrease in the sequence.

Now, assume that $\zeta ,\eta \in X$. Suppose that $\zeta \mid (\eta \mid 1)\in S_{\mathfrak{n}}\setminus S_{\mathfrak{n}+1}$ for some $\mathfrak{n}\in \{0,1,2,\cdots \}$ and $\zeta \in S_{\mathfrak{k}}\setminus S_{\mathfrak{k}+1}$ for some $\mathfrak{k}\in \{0,1,2,\cdots \}$, with $\mathfrak{n}\le \mathfrak{k}$ without loss of generality.

Clearly, since $\zeta \mid (\eta \mid 1)\in S_{\mathfrak{n}}$, we have $\eta \in S_{\mathfrak{n}}$ because $S_{\mathfrak{n}}$ is an SSNMV-filter. Hence, we have:

$${\xi }_A\left(\eta \right)\ge {\displaystyle \frac{\mathfrak{n}}{\mathfrak{n}+1}}=\mathfrak{T}({\xi }_A(\zeta \mid (\eta \mid 1)),{\xi }_A\left(\zeta \right)).$$

If $\zeta \mid (\eta \mid 1)\in {\bigcap }_{\mathfrak{n}=0}^{\infty }{S}_{\mathfrak{n}}$, then $\eta \in {\bigcap }_{\mathfrak{n}=0}^{\infty }{S}_{\mathfrak{n}}$, and thus,

$${\xi }_A\left(\eta \right)=1=\mathfrak{T}({\xi }_A(\zeta \mid (\eta \mid 1)),{\xi }_A\left(\zeta \right)).$$

If $\zeta \mid (\eta \mid 1)\notin {\bigcap }_{\mathfrak{n}=0}^{\infty }{S}_{\mathfrak{n}}$ but $\zeta \in {\bigcap }_{\mathfrak{n}=0}^{\infty }{S}_{\mathfrak{n}}$, there exists an integer $\mathfrak{r}$ such that $\zeta \mid (\eta \mid 1)\in S_{\mathfrak{r}}\setminus S_{\mathfrak{r}+1}$. This implies that $\eta \in S_{\mathfrak{r}}$, and therefore,

$${\xi }_A\left(\eta \right)\ge {\displaystyle \frac{\mathfrak{r}}{\mathfrak{r}+1}}=\mathfrak{T}({\xi }_A(\zeta \mid (\eta \mid 1)),{\xi }_A\left(\zeta \right)).$$

Lastly, if $\zeta \mid (\eta \mid 1)\in {\bigcap }_{\mathfrak{n}=0}^{\infty }{S}_{\mathfrak{n}}$ but $\zeta \notin {\bigcap }_{\mathfrak{n}=0}^{\infty }{S}_{\mathfrak{n}}$, there exists an integer $\mathfrak{s}$ such that $\eta \in S_{\mathfrak{s}}\setminus S_{\mathfrak{s}+1}$, leading to:

$${\xi }_A\left(\eta \right)\ge {\displaystyle \frac{\mathfrak{s}}{\mathfrak{s}+1}}=\mathfrak{T}({\xi }_A(\zeta \mid (\eta \mid 1)),{\xi }_A\left(\zeta \right)).$$

Therefore, we conclude that A is a ${\mathfrak{T}}_{FF}$ with an infinite number of distinct values, which contradicts the assumption that every ${\mathfrak{T}}_{FF}$ on X has a finite image. Hence, the assumption of an infinite strictly descending chain of SSNMV-filters must be incorrect, and the chain must eventually terminate after a finite number of steps. □

Theorem 23.

Let $\mathfrak{T}$ be a $\mathfrak{t}$-norm defined on I, and let μ be a fuzzy set in an SSNMV-algebra X with the image of μ given by $Im\left(\mu \right)=\{{\xi }_1,{\xi }_2,\cdots ,{\xi }_{\mathfrak{n}}\}$, where ${\xi }_{\mathfrak{i}}<{\xi }_{\mathfrak{j}}$ for $\mathfrak{i}>\mathfrak{j}$. Assume that there exists a chain of SSNMV-subalgebras of X:

$$G_0\subset G_1\subset \cdots \subset G_{\mathfrak{n}}=X,$$

such that for each $\mathfrak{k}$, $\mu \left(\overline{G_{\mathfrak{k}}}\right)={\xi }_{\mathfrak{k}}$, where $\overline{G_{\mathfrak{k}}}=G_{\mathfrak{k}}\setminus G_{\mathfrak{k}-1}$ and $G_{-1}=\varnothing$. Then, μ is a ${\mathfrak{T}}_{FS}$ on X.

Proof. 

Let $\zeta ,\eta \in X$. If both $\zeta$ and $\eta$ are elements of the same $\overline{G_{\mathfrak{k}}}$, then we have $\mu \left(\zeta \right)=\mu \left(\eta \right)={\xi }_{\mathfrak{k}}$ and $\zeta \mid (\eta \mid 1)\in G_{\mathfrak{k}}$. This leads to

$$\mu (\zeta \mid (\eta \mid 1))\ge {\xi }_{\mathfrak{k}}=min\{\mu \left(\zeta \right),\mu \left(\eta \right)\}\ge \mathfrak{T}(\mu \left(\zeta \right),\mu \left(\eta \right)).$$

Now, suppose that $\zeta \in \overline{G_{\mathfrak{i}}}$ and $\eta \in \overline{G_{\mathfrak{j}}}$ for some $\mathfrak{i}\ne \mathfrak{j}$. Without loss of generality, assume that $\mathfrak{i}>\mathfrak{j}$. In this case, $\mu \left(\zeta \right)={\xi }_{\mathfrak{i}}$ and $\mu \left(\eta \right)={\xi }_{\mathfrak{j}}$, where ${\xi }_{\mathfrak{i}}<{\xi }_{\mathfrak{j}}$. Since $\zeta \mid (\eta \mid 1)\in G_{\mathfrak{i}}$, we have

$$\mu (\zeta \mid (\eta \mid 1))\ge {\xi }_{\mathfrak{i}}=min\{\mu \left(\zeta \right),\mu \left(\eta \right)\}\ge \mathfrak{T}(\mu \left(\zeta \right),\mu \left(\eta \right)).$$

Thus, we conclude that $\mu$ is a ${\mathfrak{T}}_{FS}$ on X. □

5. Conclusions

In this paper, we explore the concept of fuzzy sets in the context of triangular norms, specifically applied to strong Sheffer stroke NMV-algebras. We introduce the notions of $\mathfrak{T}$-fuzzy subalgebras and $\mathfrak{T}$-fuzzy filters, and investigate their properties, providing several illustrative examples. Additionally, we examine the relationships between $\mathfrak{T}$-fuzzy subalgebras (filters) and their classical counterparts, as well as their level subsets and homomorphisms.

The results presented contribute to a deeper understanding of fuzzy structures within NMV-algebras and their potential applications. The interplay between fuzzy subalgebras, filters, and classical counterparts opens new avenues for further research.

Theoretical Implications: This paper generalizes fuzzy set theory to non-classical algebraic structures, namely SSNMV-algebras, thereby enriching the mathematical foundation of fuzzy logic. The introduction of $\mathfrak{T}$-fuzzy subalgebras and $\mathfrak{T}$-fuzzy filters provides a unified framework for studying algebraic and logical properties under uncertainty.

Practical Implications: This work enhances tools for artificial intelligence (AI) and decision making by modeling imprecise logical operations (e.g., via Sheffer stroke and $\mathfrak{t}$-norms). It offers potential applications in areas such as control systems, database systems, and multi-valued logic design, where uncertainty management is crucial.

Future work could focus on extending the concepts introduced in this paper to more general classes of algebras, particularly those arising in other areas of mathematics and logic. Specifically, investigating the implications of $\mathfrak{T}$-fuzzy subalgebras and $\mathfrak{T}$-fuzzy filters in the context of multi-valued logic or lattice theory represents a promising direction. Furthermore, exploring the computational aspects of these structures—such as algorithms for constructing or analyzing $\mathfrak{T}$-fuzzy subalgebras and filters—could provide valuable practical insights. Additionally, the study of homomorphisms between fuzzy algebras and their applications in optimization or decision-making processes presents an exciting area for future research.