Journal of Mathematics, Physics and Mechanics

INTRODUCTION

Background and Context

Pawlak tackled uncertain knowledge in the fields of AI information systems, and cognitive sciences in areas including ML, knowledge acquisition, and decision analysis at the start of the 1980s, among others, by utilizing the concept of RSs and its approximations (LAs and UAs) spaces in his papers [1-5]. The relationship between RSs and algebraic systems (structures) was formed via investigation and creativity. As a result, several concepts were introduced by various authors.

Numerous arbitrary binary relations, motivated by Pawlak's rough set and induced by an equivalence relation, have been used to study vast concepts. $𝑆𝑁𝜋\left(𝑥\right)=\{{𝑥}^{\prime }\in 𝑋:(𝑥,{𝑥}^{\prime })\in 𝜋\}$ indicates a successor neighborhood of u produced by a binary relation on a universal set U in which u is an element of U. Yao, suggested roughness models based on binary-induced successor neighborhoods [6]. A framework based on binary relations was given for analyzing neighborhood systems and RS approximations, and he explored the features of neighborhood and approximation operators and their relationships. If u is an element in X, then $𝑆𝐶𝑁𝜋\left(𝑥\right)=\{{𝑥}^{\prime }\in 𝑋:𝑆𝑁𝜋(𝑥)=𝑆𝑁𝜋(𝑥\left)\right\}$ denotes the core successor neighborhood generated by a binary relation on a universal set X. RSs were proposed as core successor neighborhoods resulting from binary relations by Mareay [7]. If the property of being an equivalence relation is present in a binary relation on a universal set, then Mareay's RSs and Yao's RSs are extensions of Pawlak's RSs.

In what may be regarded as the notion of roughness in classical algebras, Kuroki, presented roughness in semigroups and its ideals [8]. The concept of rough groups and rough subgroups was later established by Biswas and Nanda [9]. Li and Yin, proposed $\vartheta -LA$ and T-UA fuzzy rough approximation operators on a semigroup, whereas Qi and Liu, investigated the ideas of rough approximation in Boolean algebras [10-11]. It was Davvaz, who first proposed the idea of roughness in rings [12].

Rough set theory (RST) was studied by Ameri, in relation to hyper BCK-algebras and by Dudek, in relation to hyper BCI-algebras [13-14]. Roughness concepts in BCC algebras have also been studied by Jun [15]. The origin of FSs goes back to 1965, when they were first developed by Zadeh [16]. Since then, it has been the subject of numerous studies by many authors in several scientific, technical, and engineering fields. Fuzziness, and forms of logical algebras are recent developments and topics of interest for many writers and researchers [17-20].

A number of authors are among them. In recent years, Rough fuzzy ideals in BCK/BCI algebras were investigated by Ahn, and others. Understanding the many kinds and orientations of applied algebras has been largely based on logical and classical algebras [21]. This is particularly true for artificial intelligence related computer systems that simulate how a person would handle the certainty and uncertainty of information using logical approaches [22]. Using these notions to tackle challenges is made easy by these tactics. BCI/BCK algebras and BL-algebras are two important categories of fundamental logical algebras [23].

Perfect BL-algebras, local BL-algebras, SBL-algebras, and all of its variations have been the subject of several investigations. Whether or not they have hyperstructures, these ideas relate to roughness and softness in an intriguing way. Impan introduced the roughness notion lately, and it has been used to UA-algebras [24]. UA-algebras are subject to a number of traditional theorems from RST, and related findings have been analyzed using this idea. A UA-algebra's rough ideal characteristics have been demonstrated using lower approximations (LAs) and upper approximations (UAs). It is demonstrated that a strong UA-ideal is once again a strong ideal in terms of a UA- algebra's UAs and Las.

Motivation and Research Gap

The motivation for this study is to link the fuzzy relational uncertainty with algebraic roughness closer. The RSs approach is successfully applied in various algebraic systems, but the FR induced CSCs and JU-algebraic systems have not been systematically studied.

The primary research gaps, as identified in this study are:

Approximation to the JU-Algebra Using the CSC Approach

The present research deals with the problem of constructing rough JU-substructures and analyzing their algebraic attributes of CSCs based on FRs and its RS approaches, which were developed for a few logical algebras.

Preservation Of JU-Algebraic Structures

It is not sufficient to define lower and upper approximations on a JU-algebra. It also needs to be determined whether they are compatible with the algebraic operation. Thus, the aim of this study is to investigate the preservation of JU-subalgebras, JU-ideals, weak JU-ideals and strong JU-ideals under the proposed approximations.

Approximation by Fuzzy Relations and Dual-Universe

Fuzzy relationships allow to directly address the consideration of uncertain relationships between the two universes in the approximation process. CSCs give the corresponding information granules to build the lower and upper approximations.

Extension to Quantales

The framework is then extended to quantales, to see if the algebraic structure proposed above is compatible with a richer complete-lattice based algebraic structure. The quantale extension is concerned with rough JU-ideals, weak JU-ideals and strong JU-ideals.

For that reason, the main research question of this paper is:

Under what conditions do FR induced CSC-based lower and upper approximations preserve important JU-algebraic structures, and how can this framework be extended to quantales?

This question is different from transferring an already known rough-set definition to another algebraic structure. Rather, the mathematical relationships and interactions between fuzzy relational granulation, the CSC-based approximation operators and the algebraic preservation properties are emphasized.

Key Contributions

Our novel contributions are:

CSC-Based Fuzzy Rough Approximations: We use the CSCs induced by FRs to build the lower and upper approximations, instead of classical equivalence relations.

Interaction with JU-Structures: We investigate that these approximations preserve JU- subalgebras, JU-ideals, weak JU-ideals and strong JU-ideals.

Extension to Quantales: The framework is extended to quantales and results on rough JU- ideals and variants are given.

Dual-Universe Fuzzy Relational Model: The model includes fuzzy relations and CSCs in order to represent the uncertainty existing between two related universes.

New Structural Preservation Results: New results on interaction between CSC-based approximations and algebraic operations and ideal structures.

Summary of The Work

This is a summary of the paper organization. The preliminary result and some of the basic definitions necessary for the subsequent analysis are given in Materials and Methods. The principal theoretical results are extended to quantales, the development of the CSC-based rough JU-substructures under fuzzy relations is carried out in Results. 4. Comparative analysis and Discussion, discusses how the proposed framework compares with the current approaches. Finally, the paper ends with Conclusion that summarizes the findings and provides some directions for future research.

MATERIAL AND METHODS

JU-algebras, JU-subalgebras, and JU-ideals are some of the fundamental definitions that will be covered in this part, along with some associated fundamental findings and examples. These discoveries will support our efforts.

Definition 2.1

When any $𝔞,𝔟,𝔠\in 𝒳,$ fulfill the following identities, $𝕁𝕌-$algebra is defined as an algebra $(X,⊛,⊺)$ with a single binary operation [25] $⊛.$

$$\left({\mathrm{𝕁𝕌}}_1\right)(𝔟⊛𝔠)\left[\right(𝔠⊛𝔞)⊛(𝔟⊛𝔞\left)\right]=⊺,$$

$$\left({\mathrm{𝕁𝕌}}_2\right)⊺⊛𝔞=𝔞,$$

$$\left({\mathrm{𝕁𝕌}}_3\right)𝔞⊛𝔟=𝔟⊛𝔞=⊺\Rightarrow 𝔞=𝔟.$$

The fixed element of $𝒳$ is the constant $⊺.$ For convenience, we express a $𝕁𝕌-$algebra by writing $𝒳$ rather than $(𝒳,⊛,⊺).$

Example 2.1: .In the case of $𝒳=\{0,𝔞,𝔟,𝔠,𝔡\},$ where $⊛$ is established by the subsequent in Table 2 [25].

The fact that $𝒳$ is $𝕁𝕌-$algebra is obvious.

Definition 2.2 [26]

Fuzzy set (FS) of $𝒳$ is the mapping $𝜂:𝒳\to [0,1]$ when $𝒳$ is a nonempty universal set.

Definition 2.3 [16]

Fuzzy relation (FR) from $𝒳$ to $𝒴$ is defined as mapping $𝜂:𝒳\times 𝒴\to [0,1]$ given two nonempty universal sets $𝒳$ and $𝒴.$ A mapping on $𝒳$ is termed FR if it is $𝜂:𝒳\times 𝒳\to [0,1].$ An FR represented as a matrix is represented by

Definition 2.4 [26]

Assume that $𝜂$ represents a FR between $𝒳$ to $𝒴.$ If $𝔟$ is in $𝒳$ and for every $𝔞$ in $𝒴,𝜂(𝔟,𝔞)=1$ then $𝜂$ is serial FR (SFR).

Definition 2.5 [26]

Let $𝜂$ be a FR on $𝒳,$ then:

1.It is said to be a reflexive FR if $𝜂(𝔟,𝔟)=1,\forall q\in 𝒳;$

2.It is said to as a symmetric FR if $𝜂({𝔟}_1,{𝔟}_2)=𝜂({𝔟}_2,{𝔟}_1),\forall {𝔟}_1,{𝔟}_2\in 𝒳;$

3.Transitive FR if $𝜂({𝔟}_1,{𝔟}_2)\ge {\vee }_{𝔟3\in 𝒳}\left(𝜂\right({𝔟}_1,{𝔟}_3)\wedge 𝜂({𝔟}_3,{𝔟}_2\left)\right)\forall {𝔟}_1,{𝔟}_2\in 𝒳.$

Definition 2.6 [26]

$𝜂$ is referred to as similarity FR if it is reflexive, symmetric, and transitive FR.

Definition 2.7 [25]

If $𝔞⊛𝔟\in 𝒮,\forall 𝔞,𝔟\in 𝒮,$ then $𝒮\subseteq 𝒳$ is $𝕁𝕌-$subalgebra of $𝒳.$

Definition 2.8 [25]

If $\varphi \ne ℐ\subseteq 𝒳$ fulfills the following specifications, it is referred to as the $𝕁𝕌-$ideal of $𝒳:$

(1) 0 ∈ℐ,

$$\left(2\right)𝔞⊛(𝔟⊛𝔠)\in ℐ,𝔟\in ℐ\Rightarrow 𝔞⊛𝔠\in ℐ,\forall 𝔞,𝔟,𝔠\in 𝒳.$$

Example 2.2: [6] The 3 that follows defines $𝒳=\{0,𝔞,𝔟,𝔠,𝔡,𝔢\}$ and $⊛.$

It is obvious that $(𝒳,⊛,⊺)$ is a $𝕁𝕌-$algebra. It is easy to demonstrate that $𝔄=\{0,𝔞\}$ and $𝔙=$ $\{0,𝔞,𝔟,𝔠,𝔡\}$ are $𝕁𝕌-$ideals of $𝒳.$

Definition 2.9 [25]

Given a $𝕁𝕌-$algebra $𝒳$ and $0\in ℐ,$ let $\varphi \ne ℐ\subseteq 𝒳.$

Then

$ℐ$ is said to be a weak $𝕁𝕌-$ideal of $𝒳$ if $𝔟⊛𝔞\in ℐ$ and $𝔟\in ℐ\Rightarrow 𝔞\in ℐ,\forall 𝔞,𝔟\in 𝒳;$

If $𝔟⊛𝔞\cap ℐ\ne \varphi$ and $𝔟\in ℐ\Rightarrow 𝔞\in ℐ,\forall 𝔞,𝔟\in 𝒳,$ then $ℐ$ is said to be a strong $𝕁𝕌-$ideal of $𝒳.$

Definition 2.10 [27]

Let $\varsigma$ be in the interval [0,1] and $𝜂$ be a FR from $𝒴$ to $𝒳,$ then for each $𝔟$ in $𝒴:$

$$𝑆{𝐶}_{\eta }(𝔟;\varsigma ):=\{𝔞\in 𝒳:\eta (𝔟,𝔞)\ge \varsigma \}\text{(1)}$$

It is known that a successor class (SC) of $𝔟$ related to $\varsigma -$level under $\eta .$

Definition 2.11 [27]

Assume $\varsigma \in [0,1]$ and that $𝜂$ is a FR from $𝒴$ to $𝒳,$ then for ${𝔟}_1\in 𝒴:$ $𝐶𝑆{𝐶}_{\eta }({𝔟}_1;\varsigma ):=\{{𝔟}_2\in 𝒴:𝑆{𝐶}_{\eta }({𝔟}_1;\varsigma )=𝑆{𝐶}_{\eta }({𝔟}_2;\varsigma \left)\right\}\text{(2)}$

is referred to as a core successor class (CSC) of ${𝔟}_1$ associated with $\varsigma -$level under $𝜂.$ For any $𝔟$ in $𝒴,$

we represent the collection of $𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )$ by $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma ).$

Proposition 2.1

Let $\varsigma$ be in the interval [0,1] and $\eta$ be a FR from $𝒴$ to $𝒳.$

Then the properties listed below holds:

$𝔟\in 𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )$ for all $𝔟\in 𝒴$

For all ${𝔟}_1,{𝔟}_2\in 𝒴,{𝔟}_2\in 𝐶𝑆{𝐶}_{\eta }({𝔟}_1;\varsigma )\iff 𝐶𝑆{𝐶}_{\eta }({𝔟}_1;\varsigma )=𝐶𝑆{𝐶}_{\eta }({𝔟}_2;\varsigma ).$

Definition 2.12 [27]

Let $\varsigma$ be in the interval [0,1] and $\eta$ be a FR from $𝒴$ to $𝒳.$ A triple $(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ is referred to as an approximation space (AS) based on $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma ).$ If $𝒴=𝒳,$ then $(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ can be written as $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right).$

Definition 2.13 [28]

Let $(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS and $\varnothing \ne 𝒮\subseteq 𝒴,$ then we define UA of $𝒮$ in $(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ as:

$$\overline{\eta }(𝒮;\varsigma )≔\{𝔟\in 𝒴:𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\cap 𝒮\ne \varnothing \},\text{(3)}$$

$$\text{and}LAof𝒮in(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)as:$$

$$\underset{\_}{\eta }(𝒮;\varsigma )≔\{𝔟\in 𝒴:𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\subseteq 𝒮\},\text{(4)}$$

If $\overline{\eta }(𝒮;\varsigma )\ne \underset{\_}{\eta }(𝒮;\varsigma ),$ Then $\eta 𝑅(𝒮;\varsigma )≔\left(\overline{\eta }\right(𝒮;\varsigma ),\underset{\_}{\eta }(𝒮;\varsigma \left)\right)$ is known as a rough substructure of $𝒮$

$$in(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right).$$

Definition 2.14 [29]

Let $(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS and $\varnothing \ne 𝒮\subseteq 𝒴.\overline{\eta }(𝒮;\varsigma )$ is called a nonempty $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-UA$ of $𝒮$ in $(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ if $\overline{\eta }(𝒮;\varsigma )$ is nonempty subset of $𝒴.$

$\underset{\_}{\eta }(𝒮;\varsigma )$ is called a nonempty $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-LA$ of $𝒮$ in $(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$if $\underset{\_}{\eta }(𝒮;\varsigma )$ is nonempty

subset of $𝒴.\eta 𝑅(𝒮;\varsigma )$ is referred to as a nonempty $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$rough set in $(𝒴,𝒳,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ if $\overline{\eta }(𝒮;\varsigma )$ is a nonempty $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-UA$ and $\underset{\_}{\eta }(𝒮;\varsigma )$ is a nonempty $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-LA.$

Definition 2.15 [30]

We define FR $\eta$ on $𝒴.$ Then $\eta$ is said to be compatible if $\forall {𝔟}_1,{𝔟}_2,{𝔟}_3,{𝔟}_4\in 𝒴,$ we have

$$𝜂({𝔟}_1⊛{𝔟}_3,{𝔟}_2⊛{𝔟}_4)\ge 𝜂({𝔟}_1,{𝔟}_2)\wedge 𝜂({𝑞}_3,{𝑞}_4)\text{(5)}$$

Definition 2.16 [31]

Let $\eta$ be a FR on $𝒴.$ If $\eta$ is reflexive, transitive and compatible then $\eta$ is called a compatible preorder FR (CPFR). If $\eta$ is a CPFR then $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ is called an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS of CPFR.

Definition 2.17 [32]

Assume that a nonempty set ${\dot{\mathbb{Q}}}_{𝓉}$ is a complete lattice and that a binary operation $⊛$ satisfies the assertion that $\forall 𝔵,{𝔵}_j\in {\dot{\mathbb{Q}}}_{𝓉}:$

$$𝔵⊛\left({\bigvee }_{𝑗\in 𝐽}{𝔵}_{𝑗}\right)={\bigvee }_{𝑗\in 𝐽}(𝔵⊛{𝔵}_{𝑗}),$$

$$\left({\bigvee }_{𝑗\in 𝐽}{𝔵}_{𝑗}\right)⊛𝔵={\bigvee }_{𝑗\in 𝐽}({𝔵}_{𝑗}⊛𝔵).$$

It is then known as quantale. Suppose that ${𝒮}_j,𝒮,𝒯\subseteq {\dot{\mathbb{Q}}}_{𝓉}$ Following that, we define binary operations and arbitrary joins as

$$𝒮\vee 𝒯=\{{𝔵}_1\vee 𝔵|{𝔵}_1\in 𝒮,{𝔵}_2\in 𝒯\},\text{(6)}$$

$$𝒮⊛𝒯=\{{𝔵}_1⊛{𝔵}_2|{𝔵}_1\in 𝒮,{𝔵}_2\in 𝒯\},\text{(7)}$$

$${\bigvee }_{𝑗\in 𝐽}{𝒮}_{𝑗}=\left\{{\bigvee }_{𝑗\in 𝐽}{𝔵}_{𝑗}\right|{𝔵}_{𝑗}\in {𝒮}_{𝑗}\}.\text{(8)}$$

RESULTS

This section will cover the fundamental characteristics of JU-algebra and associated findings. The findings of JU-algebra, JU-subalgebra, JU-ideal, JU-weak ideal, and JU-strong ideal with respect to the CSCs associated with $\varsigma -$level under FRs have been examined. Furthermore, this section will cover the fundamental characteristics of JU-ideals in quantales and associated findings.

Proposition 3.1

Let $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS of CPFR. Then for all ${𝔟}_1,{𝔟}_2\in 𝒴,$ we have

$$𝒞𝒮{𝒞}_{\eta }(b_1;\varsigma )⊛𝒞𝒮{𝒞}_{\eta }(b_2;\varsigma )\subseteq 𝒞𝒮{𝒞}_{\eta }(b_1⊛b_2;\varsigma )\text{(9)}$$

Example 3.1 Let the following Table 4 define $𝕁𝕌-$subalgebra on $⊛$ as $𝒳=\{0,𝔞,𝔟,𝔠,𝔡,𝔢\}.$

For every two components in $𝒴,$ define the membership grades of their connection under $\eta$ on $𝒴$ has the following relations:

$\eta$ is obviously compatible. The SCs of each element in $𝒴$ associated with 0.6 under $\eta$ for $\varsigma =0.6$ are:

$$𝑆{𝐶}_{\eta }(0;0.6)=\left\{0\right\},$$

$$𝑆{𝐶}_{\eta }(𝔞;0.6)=\{𝔟,𝔡\},$$

$$𝑆{𝐶}_{\eta }(𝔟;0.6)=\{𝔟,𝔡\},$$

$$𝑆{𝐶}_{\eta }(𝔠;0.6)=\left\{𝔠\right\},$$

$$𝑆{𝐶}_{\eta }(𝔡;0.6)=\{𝔟,𝔡\}\text{and}$$

$$𝑆{𝐶}_{\eta }(𝔢;0.6)=\left\{𝔢\right\}.$$

Therefore, each element CSCs in $𝒴$ that is connected to the 0.6 level under $\eta$ are:

$$𝐶𝑆{𝐶}_{\eta }(0;0.6)=\left\{0\right\},$$

$$C𝑆{𝐶}_{\eta }(𝔞;0.6)=\{𝔞,𝔟,𝔡\},$$

$$C𝑆{𝐶}_{\eta }(𝔟;0.6)=\{𝔞,𝔟,𝔡\},$$

$$𝐶𝑆{𝐶}_{\eta }(𝔠;0.6)=\left\{𝔠\right\},$$

$$C𝑆{𝐶}_{\eta }(𝔡;0.6)=\{𝔞,𝔟,𝔡\}\text{and}$$

$$𝐶𝑆{𝐶}_{\eta }(𝔢;0.6)=\left\{𝔢\right\}.$$

Here, it is simple to confirm that $\forall {𝔟}_1,{𝔟}_2\in 𝒴,$

$$𝒞𝒮{𝒞}_{\eta }(b_1;\varsigma )⊛𝒞𝒮{𝒞}_{\eta }(b_2;\varsigma )\subseteq 𝒞𝒮{𝒞}_{\eta }(b_1⊛b_2;\varsigma ).$$

Note that equality in general does not apply in this case. Let's look at the example that holds equality is as follows.

Example 3.2 Using the following Table 5 and let $𝒳=$ $\{0,𝔞,𝔟,𝔠,𝔡,𝔢\}$ be $𝕁𝕌-$subalgebra defined on $⊛.$

For every two components in $𝒴,$ define the membership grades of their connection under $\eta$ on $𝒴$ has the following relations:

Here $\eta$ is compatible. For $\varsigma =0.6,$ any element in $𝒴$ that is connected to 0.6 under $\eta$ has a SCs that are:

$$𝑆{𝐶}_{\eta }(0;0.6)=\left\{𝔡\right\},$$

$$𝑆{𝐶}_{\eta }(𝔞;0.6)=\{𝔞,𝔟,𝔠\},$$

$$𝑆{𝐶}_{\eta }(𝔟;0.6)=\{𝔞,𝔟,𝔠\},$$

$$𝑆{𝐶}_{\eta }(𝔠;0.6)=\{𝔞,𝔟,𝔠\},$$

$$𝑆{𝐶}_{\eta }(𝔡;0.6)=\left\{𝔡\right\}\text{and}$$

$$𝑆{𝐶}_{\eta }(𝔢;0.6)=\left\{𝔢\right\}.$$

Accordingly, each member in $𝒴$ CSCs at the 0.6 level under $\eta$ are:

$$𝐶𝑆{𝐶}_{\eta }(0;0.6)=\{0,𝔡\},$$

$$C𝑆{𝐶}_{\eta }(𝔞;0.6)=\{𝔞,𝔟,𝔠\},$$

$$𝐶𝑆{𝐶}_{\eta }(𝔟;0.6)=\{𝔞,𝔟,𝔠\},$$

$$𝐶𝑆{𝐶}_{\eta }(𝔠;0.6)=\{𝔞,𝔟,𝔠\},$$

$$𝐶𝑆{𝐶}_{\eta }(𝔡;0.6)=\{0,𝔡\}\text{and}$$

$$𝐶𝑆{𝐶}_{\eta }(𝔢;0.6)=\left\{𝔢\right\}.$$

It is simple to confirm here that $\forall {𝔟}_1,{𝔟}_2\in 𝒴,$

$$𝒞𝒮{𝒞}_{\eta }(b_1;\varsigma )⊛𝒞𝒮{𝒞}_{\eta }(b_2;\varsigma )=𝒞𝒮{𝒞}_{\eta }(b_1⊛b_2;\varsigma ).$$

Remark 3.1

The FR matrices used in the examples are constructed mathematical examples rather than arbitrary empirical datasets. They are particularly chosen to fulfill assumptions used in the definitions and theorems of the corresponding classes and to facilitate the construction of SCs, CSCs, lower and upper approximation.

Definition 3.1

Let $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS of CPFR. If for all ${𝔟}_1,{𝔟}_2\in 𝒴,$

$$𝒞𝒮{𝒞}_{\eta }(b_1;\varsigma )⊛𝒞𝒮{𝒞}_{\eta }(b_2;\varsigma )=𝒞𝒮{𝒞}_{\eta }(b_1⊛b_2;\varsigma )\text{(10)}$$

Then the collection $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )$ is called $⊛-$Complete FR (CFR).

Let $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS of CPFR. If for all ${𝔟}_1,{𝔟}_2\in 𝒴,$

$$𝒞𝒮{𝒞}_{\eta }(b_1;\varsigma )\vee 𝒞𝒮{𝒞}_{\eta }(b_2;\varsigma )=𝒞𝒮{𝒞}_{\eta }(b_1\vee b_2;\varsigma )\text{(11)}$$

Then the collection $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )$ is called $\vee -$CFR.

If it is both $⊛-$Complete and $\vee -$Complete is called CFR.

Theorem 3.1

Let $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ and $(𝒳,𝒞𝒮{𝒞}_{\theta }(𝒳;\tau \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS and $𝒞𝒮{𝒞}_{\theta }(𝒳;\tau )-$AS respectively. These characteristics are true if $\varphi \ne 𝒮,𝒯\subseteq 𝒴:$

$$1.\underset{\_}{𝜂}(𝒮,\varsigma )\subseteq 𝒮\subseteq \overline{𝜂}(𝒮,\varsigma );$$

$$2.\underset{\_}{𝜂}(\varphi ,\varsigma )=\varphi =\overline{𝜂}(\varphi ,\varsigma );$$

$$3.\overline{𝜂}(𝒮\cup 𝒯,\varsigma )=\overline{𝜂}(𝒮,\varsigma )\cup \overline{𝜂}(𝒯,\varsigma );$$

$$4.\underset{\_}{𝜂}(𝒮\cap 𝒯,\varsigma )=\underset{\_}{𝜂}(𝒮,\varsigma )\cap \underset{\_}{𝜂}(𝒯,\varsigma );$$

$5.𝒮\subseteq 𝒯$ implies that $\underset{\_}{𝜂}(𝒮,\varsigma )\subseteq \underset{\_}{𝜂}(𝒯,\varsigma )$ and $\overline{𝜂}(𝒮,\varsigma )\subseteq \overline{𝜂}(𝒯,\varsigma ).$

$$6.\underset{\_}{𝜂}(𝒮,\varsigma )\cap \underset{\_}{𝜂}(𝒯,\varsigma )\subseteq \underset{\_}{𝜂}(𝒮\cap 𝒯,\varsigma );$$

$$7.\overline{𝜂}(𝒮,\varsigma )\cup \overline{𝜂}(𝒯,\varsigma )\subseteq \overline{𝜂}(𝒮\cup 𝒯,\varsigma );$$

$8.\eta \subseteq \theta$ and $\varsigma \subseteq \tau$ implies that $𝜃(𝒮;\tau )\subseteq \underset{\_}{𝜂}(𝒮;\varsigma )$ and $\overline{𝜂}(𝒮;\varsigma )\subseteq 𝜃(𝒮;\tau ).$

Proof

(1) If $𝔟\in \underset{\_}{𝜂}(𝒮;\varsigma ),$ then $𝔟\in 𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\subseteq 𝒮.$ This implies that $𝔟\in 𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )$ and $𝔟\in 𝒮.$ From

there we can write $𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\cap 𝒮\ne \varphi .$ This implies that $𝔟\in \overline{𝜂}(𝒮;\varsigma ).$ Hence, $𝒮\subseteq \overline{𝜂}(𝒮,\varsigma ).$ Consequently, $\underset{\_}{𝜂}(𝒮,\varsigma )\subseteq 𝒮\subseteq \overline{𝜂}(𝒮,\varsigma ).$

(2) It is straightforward.

(3) Let,

$$𝔟\in \overline{𝜂}(𝒮\cup 𝒯,\varsigma )\iff 𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\cap (𝒮\cup 𝒯)\ne \varphi$$

$$\iff \left(𝐶𝑆{𝐶}_{\eta }\right(𝔟;\varsigma )\cap 𝒮)\cup \left(𝐶𝑆{𝐶}_{\eta }\right(𝔟;\varsigma )\cap 𝒯)\ne \varphi$$

$$\iff 𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\cap 𝒮\ne 𝜙𝑜𝑟𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\cap 𝒯\ne \varphi$$

$$\iff 𝔟\in \overline{𝜂}(𝒮;\varsigma )or𝔟\in \overline{𝜂}(𝒯;\varsigma )$$

$$\iff 𝔟\in \overline{𝜂}(𝒮;\varsigma )\cup \overline{𝜂}(𝒯;\varsigma ).$$

Thus,

$$\overline{𝜂}(𝒮\cup 𝒯,\varsigma )=\overline{𝜂}(𝒮,\varsigma )\cup \overline{𝜂}(𝒯,\varsigma ).$$

(4) Let,

$$𝔟\in \underset{\_}{𝜂}(𝒮\cap 𝒯,\varsigma )\iff 𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\subseteq (𝒮\cap 𝒯)$$

$$\iff 𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\subseteq 𝒮𝑎𝑛𝑑𝐶𝑆{𝐶}_{\eta }(𝔟;\varsigma )\subseteq 𝒯$$

$$\iff 𝔟\in \underset{\_}{𝜂}(𝒮;\varsigma )\text{and}𝔟\in \underset{\_}{𝜂}(𝒯;\varsigma )$$

$$\iff 𝔟\in \underset{\_}{𝜂}(𝒮;\varsigma )\cap \underset{\_}{𝜂}(𝒯;\varsigma ).$$

Thus,

$$\underset{\_}{𝜂}(𝒮\cap 𝒯,\varsigma )=\underset{\_}{𝜂}(𝒮,\varsigma )\cap \underset{\_}{𝜂}(𝒯,\varsigma ).$$

(5) Since $𝒮\subseteq 𝒯$ if and only if $𝒮\cap 𝒯=𝒮,$ by (3) we have

$$\underset{\_}{𝜂}(𝒮,\varsigma )=\underset{\_}{𝜂}(𝒮\cap 𝒯,\varsigma )=\underset{\_}{𝜂}(𝒮,\varsigma )\cap \underset{\_}{𝜂}(𝒯,\varsigma )$$

This implies that $\underset{\_}{𝜂}(𝒮,\varsigma )\subseteq \underset{\_}{𝜂}(𝒯,\varsigma ).$ Additionally, take note that $𝒮\subseteq 𝒯$ if and only if $𝒮\cap 𝒯=𝒮.$

From (2), we obtain $\overline{𝜂}(𝒮,\varsigma )=\overline{𝜂}(𝒮\cup 𝒯,\varsigma )=\overline{𝜂}(𝒮,\varsigma )\cup \overline{𝜂}(𝒯,\varsigma ).$

This implies that $\overline{𝜂}(𝒮,\varsigma )\subseteq \overline{𝜂}(𝒯,\varsigma ).$

(6) Since $𝒮\subseteq 𝒮\cup 𝒯$ and $𝒯\subseteq 𝒮\cup 𝒯,$ by (4) we have

$$\underset{\_}{𝜂}(𝒮,\varsigma )=\underset{\_}{𝜂}(𝒮\cap 𝒯,\varsigma )\text{and}\underset{\_}{𝜂}(𝒯,\varsigma )=\underset{\_}{𝜂}(𝒮\cap 𝒯,\varsigma )$$

This implies that $\underset{\_}{𝜂}(𝒮,\varsigma )\cup \underset{\_}{𝜂}(𝒯,\varsigma )=\underset{\_}{𝜂}(𝒮\cap 𝒯,\varsigma ).$

(7) Since $𝒮\cap 𝒯\subseteq 𝒮$ and $𝒮\cap 𝒯\subseteq 𝒯,$ by (4) we have

$$\overline{𝜂}(𝒮\cap 𝒯,\varsigma )\subseteq \overline{𝜂}(𝒮,\varsigma )\text{and}\overline{𝜂}(𝒮\cap 𝒯,\varsigma )\subseteq \overline{𝜂}(𝒯,\varsigma )$$

which implies that $\overline{𝜂}(𝒮\cap 𝒯,\varsigma )\subseteq \overline{𝜂}(𝒮,\varsigma )\cap \overline{𝜂}(𝒯,\varsigma ).$

(8) Since $\varsigma \subseteq \tau .$ If $x\in 𝜃(𝒮;\tau ),$ then $𝐶𝑆{𝐶}_{\theta }(x;\tau )\subseteq 𝒮.$ But $\varsigma \subseteq \tau .$ Therefore, $𝐶𝑆{𝐶}_{\eta }(x;\varsigma )\subseteq$ $𝐶𝑆{𝐶}_{\theta }(x;\tau )\subseteq 𝒮.$

This implies that $𝐶𝑆{𝐶}_{\eta }(x;\varsigma )\subseteq 𝒮.$ Thus, $x\in \underset{\_}{𝜂}(𝒮;\varsigma ).$ Hence, $𝜃(𝒮;\tau )\subseteq \underset{\_}{𝜂}(𝒮;\varsigma ).$

Now, let $x\in \overline{𝜂}(𝒮;\varsigma ),$ then we have $𝐶𝑆{𝐶}_{\eta }(x;\varsigma )\cap 𝒮\ne 𝜙.$

Then $\exists y\in 𝐶𝑆{𝐶}_{\eta }(x;\varsigma )\cap 𝒮s.t.y\in 𝐶𝑆{𝐶}_{\eta }(x;\varsigma )$ and $y\in 𝒮.$ Hence, $𝑆{𝐶}_{\eta }(y;\varsigma )=𝑆{𝐶}_{\eta }(x;\varsigma ),$ Since $\varsigma \subseteq \tau ,$ so we have $y\in 𝐶𝑆{𝐶}_{\theta }(x;\tau ).$

Therefore, we have $y\in 𝐶𝑆{𝐶}_{\theta }(x;\tau )\cap 𝒮.$

Which means that $x\in 𝜃(𝒮;\tau ).$

Hence, $\overline{𝜂}(𝒮;\varsigma )\subseteq 𝜃(𝒮;\tau ).$

Theorem 3.2 Let $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS. If $𝒮$ are nonempty subsets of $𝒴,$ then

$$1.\underset{\_}{𝜂}\left(\underset{\_}{𝜂}\right(𝒮,\varsigma ),\varsigma )=\underset{\_}{𝜂}(𝒮,\varsigma )$$

$$2.\overline{𝜂}\left(\overline{𝜂}\right(𝒮,\varsigma ),\varsigma )=\overline{𝜂}(𝒮,\varsigma )$$

$$3.\overline{𝜂}\left(\underset{\_}{𝜂}\right(𝒮,\varsigma ),\varsigma )=\underset{\_}{𝜂}(𝒮,\varsigma )$$

$$4.\underset{\_}{𝜂}\left(\overline{𝜂}\right(𝒮,\varsigma ),\varsigma )=\overline{𝜂}(𝒮,\varsigma )$$

$$5.\underset{\_}{𝜂}(𝒮,\varsigma )=\left(\right(\overline{𝜂}({𝒮}^c,\varsigma ){)}^{𝑐}$$

$$6.\overline{𝜂}(𝒮,\varsigma )=\left(\right(\underset{\_}{𝜂}({𝒮}^c,\varsigma ){)}^{𝑐}$$

$7.\underset{\_}{𝜂}\left(𝐶𝑆{𝐶}_{\eta }\right(𝔟;\varsigma ),\varsigma )=𝒴=\underset{\_}{𝜂}\left(𝐶𝑆{𝐶}_{\eta }\right(𝔟;\varsigma ),\varsigma ),$ for all $𝔟\in 𝒮.$

Proof

Proof is simple.

Theorem 3.3

Let $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS. If $\varphi \ne 𝒮\subseteq 𝒴,$ then

$$1.\overline{𝜂}(𝒮,\varsigma )⊛\overline{𝜂}(𝒯,\varsigma )\subseteq \overline{𝜂}(𝒮⊛𝒯,\varsigma )$$

2. If $\eta$ is CFR, then $\underset{\_}{𝜂}(𝒮,\varsigma )⊛\underset{\_}{𝜂}(𝒯,\varsigma )\subseteq \underset{\_}{𝜂}(𝒮⊛𝒯,\varsigma ).$

Proof

(1) Any element of $\overline{𝜂}(𝒮,\varsigma )⊛\overline{𝜂}(𝒯,\varsigma )$ can be $𝔠.$ Consequently, $𝔠=𝔭⊛𝔮,$ where $𝔭\in \overline{𝜂}(𝒮,\varsigma )$ and $𝔮\in$ $\overline{𝜂}(𝒯,\varsigma ).$ As a result, $𝔵,𝔶$ exist in $𝒮$ such that $𝔵\in CS{C}_{\eta }(𝔭,\varsigma )\cap 𝒮$ and $𝔶\in CS{C}_{\eta }(𝔮,\varsigma )\cap 𝒯.$ It follows from this that $𝔵\in CS{C}_{\eta }(𝔭,\varsigma ),𝔵\in 𝒮$ and $𝔶\in CS{C}_{\eta }(𝔮,\varsigma ),𝔶\in 𝒯.$

Proposition 3. gives us

$$𝔵⊛𝔶\in CS{C}_{\eta }(𝔭,\varsigma )⊛CS{C}_{\eta }(𝔮,\varsigma )\subseteq CS{C}_{\eta }(𝔭⊛𝔮,\varsigma )$$

Furthermore, since we have $𝔵⊛𝔶\in 𝒮⊛𝒯,𝔵⊛𝔶\in CS{C}_{\eta }(𝔭⊛𝔮,\varsigma ).$ This suggests that $𝔵⊛𝔶\in$ $CS{C}_{\eta }(𝔭⊛𝔮,\varsigma )\cap 𝒮⊛𝒯.$ This implies that $𝔠=𝔭⊛𝔮\in \overline{𝜂}(𝒮⊛𝒯,\varsigma ).$ Thus

$$\overline{𝜂}(𝒮,\varsigma )⊛\overline{𝜂}(𝒯,\varsigma )\subseteq \overline{𝜂}(𝒮⊛𝒯,\varsigma )$$

(2) Let $𝔠$ be any element of $\underset{\_}{𝜂}(𝒮,\varsigma )⊛\underset{\_}{𝜂}(𝒯,\varsigma ).$ Then $𝔠=𝔭⊛𝔮$ with $𝔭\in \underset{\_}{𝜂}(𝒮,\varsigma )$ and $𝔮\in \underset{\_}{𝜂}(𝒯,\varsigma ).$

This follows that $CS{C}_{\eta }(𝔭,\varsigma )\subseteq 𝒮$ and $CS{C}_{\eta }(𝔮,\varsigma )\subseteq 𝒯.$ Since $\eta$ is CFR, so we have

$$CS{C}_{\eta }(𝔭,\varsigma )⊛CS{C}_{\eta }(𝔮,\varsigma )=CS{C}_{\eta }(𝔭⊛𝔮,\varsigma )\subseteq 𝒮⊛𝒯.$$

This implies that $𝔠=𝔭⊛𝔮\in \underset{\_}{𝜂}(𝒮⊛𝒯,\varsigma ).$ Thus,

$$\underset{\_}{𝜂}(𝒮,\varsigma )⊛\underset{\_}{𝜂}(𝒯,\varsigma )\subseteq \underset{\_}{𝜂}(𝒮⊛𝒯,\varsigma ).$$

Theorem 3.4

Let $𝒮$ is a $𝕁𝕌-$subalgebra of $𝒴$ and $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS of $⊛-$CFR, then

$1.\overline{𝜂}(𝒮,\varsigma )$ is a $𝕁𝕌-$subalgebra of $𝒴.$

$2.\underset{\_}{𝜂}(𝒮,\varsigma )$ is a $𝕁𝕌-$subalgebra of $𝒴.$

Proof

(1) Assume that $𝔵,𝔶are$ in $\overline{𝜂}(𝒮,\varsigma ).$ Then $CS{C}_{\eta }(𝔵,\varsigma )\cap 𝒮\ne 𝜙$ and $CS{C}_{\eta }(𝔶,\varsigma )\cap 𝒮\ne 𝜙,$ and so there exist $𝔞,𝔟\in 𝒮$ such that $𝔞\in CS{C}_{\eta }(𝔵,\varsigma )$ and $𝔟\in CS{C}_{\eta }(𝔶,\varsigma ).$ According to Proposition 3., we have

$$𝔞⊛𝔟\in CS{C}_{\eta }(𝔵,\varsigma )⊛CS{C}_{\eta }(𝔶,\varsigma )\subseteq CS{C}_{\eta }(𝔵⊛𝔶,\varsigma ).$$

It follows from this that $𝔞⊛𝔟\in CS{C}_{\eta }(𝔵⊛𝔶,\varsigma ).$ Given that $𝒮$ represent a $𝒴𝕁𝕌-$subalgebra, $𝔞⊛$ $𝔟\in 𝒮.$ Accordingly, $𝔞⊛𝔟\in CS{C}_{\eta }(𝔵⊛𝔶,\varsigma )\cap 𝒮,$ that is $CS{C}_{\eta }(𝔵⊛𝔶,\varsigma )\cap 𝒮\ne 𝜙.$ In this case, $𝔵⊛$ $𝔶\in \overline{𝜂}(𝒮,\varsigma ).$ Hence, $\overline{𝜂}(𝒮,\varsigma )$ is a $𝕁𝕌-$subalgebra of $𝒴.$

(2) Assume that $𝔵,𝔶\in \underset{\_}{𝜂}(𝒮,\varsigma ).$ Then $CS{C}_{\eta }(𝔵,\varsigma )\subseteq 𝒮$ and $CS{C}_{\eta }(𝔶,\varsigma )\subseteq 𝒮,$ respectively. Given that

$𝒮$ is a $𝕁𝕌-$subalgebra of $𝒴$ and $\eta$ is $⊛-$CFR, so

$$CS{C}_{\eta }(𝔵,\varsigma )⊛CS{C}_{\eta }(𝔶,\varsigma )=CS{C}_{\eta }(𝔵⊛𝔶,\varsigma )\subseteq 𝒮$$

Hence $𝔵⊛𝔶\in \underset{\_}{𝜂}(𝒮,\varsigma ).$ Thus, $\underset{\_}{𝜂}(𝒮,\varsigma )$ is $𝕁𝕌-$subalgebra of $𝒴.$

Theorem 3.5

If $𝒮$ is a $𝕁𝕌-$ideal of $𝒴$ and $(𝒴,𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma \left)\right)$ be an $𝒞𝒮{𝒞}_{\eta }(𝒴;\varsigma )-$AS of $⊛-$CFR, then

$1.\overline{𝜂}(𝒮,\varsigma )$ is a $𝕁𝕌-$ideal of $𝒴.$

$2.\underset{\_}{𝜂}(𝒮,\varsigma )$ is a $𝕁𝕌-$ideal of $𝒴.$

Proof

(1) Consider $𝒮$ is a $𝕁𝕌-$ideal of $𝒴.$ Indeed, $0\in \overline{𝜂}(𝒮,\varsigma ).$ For $𝔵,𝔶,𝔷\in 𝒴,$ let $𝔶\in \overline{𝜂}(𝒮,\varsigma )$ and $𝔵⊛$ $(𝔶⊛𝔷)\in \overline{𝜂}(𝒮,\varsigma ).$ After that,

$$CS{C}_{\eta }(𝔶,𝜍)\cap 𝒮\ne 𝜙𝑎𝑛𝑑CS{C}_{\eta }(𝔵⊛(𝔶⊛𝔷),𝜍)\cap 𝒮\ne 𝜙,$$

Thus, $𝔞,𝔟\in 𝒮$ such that $𝔞\in CS{C}_{\eta }(𝔶,𝜍)$ and $𝔟\in CS{C}_{\eta }(𝔵⊛(𝔶⊛z),𝜍)$ respectively. Since $S{C}_{\eta }(𝔞,𝜍)=S{C}_{\eta }(𝔶,𝜍)$ and $S{C}_{\eta }(𝔟,𝜍)=S{C}_{\eta }(𝔵⊛(𝔶⊛𝔷),𝜍),$ it follows that $𝔶⊛𝔞\in CS{C}_{\eta }(𝔶,𝜍)\subseteq 𝒮$ and $𝔵⊛(𝔶⊛𝔷)⊛𝑏\in CS{C}_{\eta }(𝔵⊛(𝔶⊛𝔷),𝜍)\subseteq 𝒮.$ Given that $𝒮$ is a $𝕁𝕌-$ideal and $𝔞,𝔟\in 𝒮,$