An algebraic framework for fuzzy soft subrings, cosets, and ring homomorphisms

1. Introduction

Fuzzy sets (FSs), proposed by Zadeh (Zadeh, 1965), have many applications in fields such as data mining, commerce and economics. They are represented by the notation $\Gamma =\left(\frac{{\mu }_{\Gamma }\left(x\right)}{x}:x\in X\right)$, where ${\mu }_{\Gamma }\left(x\right):X\to [0,1]$ is the membership function for all $x\in X$. As an extension of crisp set theory, FSs are a useful analytical framework. They generalize the characteristic function of classical sets. To overcome the ambiguity and uncertainty, new concepts and methods have been developed from fuzzy set theory. Data representation limitations are offset with the introduction of non-membership value in addition to membership values. Atanassov (Atanassov, 1986) extended FSs to intuitionistic FSs (IFSs), offering greater freedom and flexibility in uncertainty representation while dealing with decision-making contexts (Garg and Kumar, 2018; Garg and Kumar, 2019; Ejegwa and Onyeke, 2021; Ejegwa and Agbetayo, 2023). The introduction of Pythagorean Fuzzy Set (PFS) $\Gamma =\left(\frac{{\mu }_{\Gamma }\left(x\right),{\nu }_{\Gamma }\left(x\right)}{x}:x\in X \right)$, where ${\mu }_{\Gamma }:X\to \left(0,1\right)$ and ${\nu }_{\Gamma }:X\to [0,1]$, are membership and non-membership functions satisfying ${\left({\mu }_{\Gamma }\left(x\right)\right)}^2+{\left({\nu }_{\Gamma }\left(x\right)\right)}^2\le 1$ for all $x\in X$, by Yager (Yager, 2013) extended the applicability of IFSs. This approach seeks to convert an uncertain and confusing setting into mathematical frameworks that reveal more effective strategies for addressing such issues in the physical domain (Naz et al., 2018; Li et al., 2019; Zhou et al., 2020; Hussain et al., 2022). The theory of FSs and its subsequent developments serve as mathematical instruments for addressing uncertainty. However, all these theories possess intrinsic challenges, as noted by Molodtsov (Molodtsov, 1999). The theoretical framework of soft set theory was developed in (Molodtsov, 1999), and the notion of fuzzy soft sets was introduced in (Maji et al., 2001). Additionally, (Roy and Maji, 2007)showed several applications of this concept to decision-making issue.

In classical fuzzy ring theory, a fuzzy coset is generated from a single fuzzy ideal by translating its membership function by a fixed ring element. Although this extends ordinary cosets to uncertain settings, it is non-parameterized and cannot capture parameter-dependent uncertainty. On the other hand, the fuzzy soft cosets presented in this paper are based on fuzzy soft subrings or ideals, in which each parameter generates a fuzzy coset. As a result, fuzzy soft cosets are a parameter-indexed family of fuzzy cosets and become the classical fuzzy coset construction when the parameter set has a single element.

2. Basic Definitions

This section provides the necessary background and definitions for understanding the major results proved in this paper.

Definition 2.1 (Maji et al., 2001) Let $X$ represent the universal set, $I^X$ denote the set of all fuzzy subsets over$X,$ and $\psi$ be the set of parameters. A mapping $\omega :\psi \to I^X$ is referred to as a fuzzy soft set (FSS), denoted by $(\omega ,\psi ,X)$, over the universal set $X$ with respect to the parameters $\psi$. This is formally represented as follows:

$\left(\omega ,\psi ,X\right)=\left(\left(\alpha , \omega \left(\alpha \right)={\Gamma }_{\alpha }\right):\alpha \in \psi \right)=\left(\left(\alpha ,\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(x\right)}{x} :x\in X\right)\right): \alpha \in \psi \right).$

Definition 2.2 (Maji et al., 2001) Let $(\omega ,\psi ,X)=\left\{\right(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha })\text{:}\forall \alpha \in \psi \}$ and $(\phi ,{\psi }^{\prime },X)=\left\{\right(\beta ,\phi \left(\beta \right)=K_{\beta }):\forall \beta \in {\psi }^{\prime }\}$ be two be two ft set meters then Ri FSSs over $X$. Suppose that the sets of parameters $\psi$ and ${\psi }^{\prime }$ have a non-empty intersection, denoted as $W=\psi \cap {\psi }^{\prime }$. The intersection of these two fuzzy soft sets is denoted as $(\delta ,W,X)=\left\{\right(c,\left\{\delta \right(c)={\Gamma }_c\cap K_c\}):c\in W\}$, where $\delta$ is a new mapping over W, and ${\Gamma }_c$ and $K_c$ are fuzzy subsets of $X$ for each $c\in W$.

Formally, the intersection of these fuzzy soft sets is expressed as:

$\begin{array}{c}\left(\delta ,W,X\right)=\left(\left(c,\left(\frac{{\mu }_{{\Gamma }_c\cap K_c}\left(x\right)}{x}:x\in X\right):c\in W\right)\right)\\ =\left(\left(c,\left(\frac{min\left({\mu }_{{\Gamma }_c}\left(x\right),{\mu }_{K_c}\left(x\right)\right)}{x}:x\in X\right):c\in \mathrm{W.}\right)\right)\end{array}$

Now, let’s define the union of two fuzzy soft sets.

Definition 2.3 (Maji et al., 2001) Let $(\omega ,\psi ,X)=\left\{\right(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha })\text{:}\forall \alpha \in \psi \}$ and $(\phi ,{\psi }^{\prime },X)=\left\{\right(\beta ,\phi \left(\beta \right)=K_{\beta }):\forall \beta \in {\psi }^{\prime }\}$ be two be twoft set meters then RiFSSs over $X$. Suppose that the union of sets of parameters $\psi$ and ${\psi }^{\prime }$ is denoted as $W=\psi \cup {\psi }^{\prime }$. The union of these two fuzzy soft sets is denoted as $(\delta ,W,X)=\left\{\right(c,\left\{\delta \right(c)={\Gamma }_c\cup K_c\})\text{:}c\in W\}$, where $\delta$ is a new mapping over $W$, and ${\Gamma }_c$ and $K_c$ are fuzzy subsets of $X$ for each $c\in W.$

Formally, the union of these fuzzy soft sets is expressed as:

$\begin{array}{c}\left(\delta ,W,X\right)=\left(\left(c,\delta \left(c\right)={\Gamma }_c\cup K_c\right):c\in W\right)\\ =\left(\left(c,\left(\frac{{\mu }_{{\Gamma }_c\cup K_c}\left(x\right)}{x}:x\in X\right):c\in W\right)\right),\end{array}$

where ${\mu }_{{\Gamma }_c\cup K_c}\left(x\right)=\left(\begin{array}{c}max\left({\mu }_{{\Gamma }_c}\left(x\right),{\mu }_{K_c}\left(x\right)\right) if c\in \psi \cap {\psi }^{\prime }\\ {\mu }_{{\Gamma }_c}\left(x\right), if c\in \psi -{\psi }^{\prime }\\ {\mu }_{K_c}\left(x\right), if c\in {\psi }^{\prime }-\psi \end{array}\right)$ for all $x\in X$.

Definition 2.4 (Maji et al., 2001) Let $\left(\omega ,\psi ,X\right)=\left(\left(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha }\right):\alpha \in \psi \right)$ and $\left(\phi ,{\psi }^{\prime },X\right)=\left(\left(\alpha ,\phi \left(\alpha \right)=K_{\alpha }\right):\alpha \in {\psi }^{\prime }\right)$ be two FSSs over $X$. Then, the FSS (ω, $\psi$, X) is said to be a subset of the FSS $\left(\phi , {\psi }^{\prime }, X\right)$, denoted as $\left(\omega , \psi , X\right) \subseteq \left(\phi ,{\psi }^{\prime }, X\right)$, if the following conditions hold:

• i. $\psi \subseteq {\psi }^{\prime }$

• ii. If $\alpha \in \psi ,{\psi }^{\prime }$, then ${\mu }_{{\Gamma }_{\alpha }}\left(x\right)\le {\mu }_{K_{\alpha }}\left(x\right)$ for all $x\in X$.

Definition 2.5 (İnan et al., 2012) Let ${\rm Y}$ be a ring and for each fixed parameter $\alpha \in \psi$, ${\Gamma }_{\alpha }$ is a fuzzy subset of ${\rm Y}$. A fuzzy soft set $(\omega ,\psi ,{\rm Y})=\left\{\right(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha }):\alpha \in \psi \}$ is referred to as a fuzzy soft subring (FSSR) over ${\rm Y}$ in terms of $\psi$, if for ${\kappa }_1,{\kappa }_2\in {\rm Y}$, the following two conditions are satisfied:

• i. ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1-{\kappa }_2\right)\ge min\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right)$

• ii. ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1.{\kappa }_2\right)\ge min\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right)$

In other words, the fuzzy soft set $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha }\right):\alpha \in \psi \right)$ is classified as an FSSR over the ring ${\rm Y}$ if and only if each fuzzy subset ${\Gamma }_{\alpha }$, corresponding to every parameter $\alpha \in \psi$, qualifies as a fuzzy subring of ${\rm Y}$.

Example 2.1 Consider $X=\{0,1,2,3\}$, then $2^X$ the power set of $X$, that is,

$2^X=\left(\begin{array}{c}\phi ,X,\left(0\right),\left(1\right),\left(2\right),\left(3\right),\left(0,1\right),\left(0,2\right),\left(0,3\right),\left(1,2\right),\\ \left(1,3\right),\left(2,3\right),\left(0,1,2\right),\left(0,1,3\right),\left(0,2,3\right),\left(1,2,3\right)\end{array}\right)$

Forms a ring under symmetric difference $\Delta$ and intersection $\cap$. We choose $P=\{\alpha ,\beta \}$ as a set of parameters, then we construct an FSSR $(\omega ,P,2^X)$ over $2^X$ in terms of $P$ as follows;

$\begin{array}{c}\left(\omega ,P,2^X\right)=\\ \left(\begin{array}{c}\left(\alpha ,\left(\begin{array}{c}\frac{0.90}{\phi },\frac{0.65}{X},\frac{0.75}{\left(0\right)},\frac{0.90}{\left(1\right)},\frac{0.90}{\left(2\right)},\frac{0.65}{\left(3\right)},\frac{0.75}{\left(0,1\right)},\frac{0.75}{\left(0,2\right)},\frac{0.65}{\left(0,3\right)},\\ \frac{0.90}{\left(1,2\right)},\frac{0.65}{\left(1,3\right)},\frac{0.65}{\left(2,3\right)},\frac{0.75}{\left(0,1,2\right)},\frac{0.65}{\left(0,1,3\right)},\frac{0.65}{\left(0,2,3\right)},\frac{0.65}{\left(1,2,3\right)}\end{array}\right)\right),\\ \left(\beta ,\left(\begin{array}{c}\frac{0.85}{\phi },\frac{0.55}{X},\frac{0.70}{\left(0\right)},\frac{0.85}{\left(1\right)},\frac{0.85}{\left(2\right)},\frac{0.55}{\left(3\right)},\frac{0.70}{\left(0,1\right)},\frac{0.70}{\left(0,2\right)},\frac{0.55}{\left(0,3\right)},\\ \frac{0.85}{\left(1,2\right)},\frac{0.55}{\left(1,3\right)},\frac{0.55}{\left(2,3\right)},\frac{0.70}{\left(0,1,2\right)},\frac{0.55}{\left(0,1,3\right)},\frac{0.55}{\left(0,2,3\right)},\frac{0.55}{\left(1,2,3.\right)}\end{array}\right)\right)\end{array}\right)\end{array}$

Definition 2.6 (İnan et al., 2012) An FSS $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,{\Gamma }_{\alpha }=\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa }:\kappa \in {\rm Y}\right)\right):\alpha \in \psi \right)$ is a fuzzy soft ideal (FSI) over ${\rm Y}$ if for every ${\kappa }_1,{\kappa }_2\in {\rm Y}$ and $\alpha \in \psi$, the following axioms are true:

• i. ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1-{\kappa }_2\right)\ge min\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right).$

• ii. ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1.{\kappa }_2\right)\ge max\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right).$

Theorem 2.1 (İnan et al., 2012) The intersection of two FSIs over ${\rm Y}$ is an FSI over ${\rm Y}$.

3. Characterization of Fuzzy Soft Subrings and Ideals in Rings

The section examines fuzzy soft level subsets and the developed theorems in this context provide insights into the relationships between fuzzy soft ideals and their corresponding fuzzy level sets.

Remark 3.1 Every fuzzy soft Ideal (FSI) of a ring ${\rm Y}$ is an FSSR of ${\rm Y}$. However, the converse is not always true. Example 2.1 illustrates a fuzzy soft subring $(\omega ,\psi ,2^X)$ over the power set $2^X$ that is not an FSI over $2^X$, where $X=\{0,1,2,3\}$.

Definition 3.1 Let $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,{\Gamma }_{\alpha }=\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa }:\kappa \in {\rm Y}\right)\right):\alpha \in \psi \right)$ be an FSS over a ring ${\rm Y}$. For a given $\sigma \in \left(0,1\right),$ let ${\left({\Gamma }_{\alpha }\right)}_{\left(\sigma \right)}$ represent the fuzzy level subset of the fuzzy set ${\Gamma }_{\alpha }$. Then, the set ${\left(\omega ,\psi ,{\rm Y}\right)}_{\left(\sigma \right)}=\left(\kappa \in {\rm Y}:{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)\ge \sigma \text{for every} \alpha \in \psi \right)=\left(\kappa \in {\rm Y}:\kappa \in {\left({\Gamma }_{\alpha }\right)}_{\left(\sigma \right)}\right)$ is called fuzzy soft level subset (FSLSS) of $\left(\omega ,\psi ,{\rm Y}\right)$.

Theorem 3.1 An FSSR $\left(\omega ,\psi ,{\rm Y}\right)$ is an FSI over ${\rm Y}$ if and only if for every $\sigma \in \left(0,{\mu }_{{\Gamma }_{\alpha }}\left(0\right)\right),$ the fuzzy level set ${\left({\Gamma }_{\alpha }\right)}_{\left(\sigma \right)}$ is an ideal of ${\rm Y}$.

Proof Suppose that $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,{\Gamma }_{\alpha }:\alpha \in \psi \right)\right)$ is an FSI. For a fixed $\alpha \in \psi$ and $\sigma \in \left(0,{\mu }_{{\Gamma }_{\alpha }}\left(0\right)\right)$, let ${\kappa }_1,{\kappa }_2\in {\left({\Gamma }_{\alpha }\right)}_{\left(\sigma \right)}.$ Then ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\ge \sigma$. By Definition 2.6 of an FSI:

$\begin{array}{c}{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1-{\kappa }_2\right)\ge min\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right)\ge \sigma ,\\ {\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1.{\kappa }_2\right)\ge max\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right)\ge \sigma .\end{array}$

Hence, ${\kappa }_1-{\kappa }_2\in {\left({\Gamma }_{\alpha }\right)}_{\left(\sigma \right)},$ ${\kappa }_1.{\kappa }_2\in {\left({\Gamma }_{\alpha }\right)}_{\left(\sigma \right)}$, proving ${\left({\Gamma }_{\alpha }\right)}_{\left(\sigma \right)}$ is an ideal of ${\rm Y}$.

Conversely, let ${\left({\Gamma }_{\alpha }\right)}_{\left(\sigma \right)}$ be an ideal of ${\rm Y}$ for every $\sigma \in \left(0,{\mu }_{{\Gamma }_{\alpha }}\left(0\right)\right)$. Suppose that ${\kappa }_1,{\kappa }_2\in {\rm Y}$ and ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right)={\sigma }_1$, ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)={\sigma }_2$. Then

• i.

  ${\kappa }_1,{\kappa }_2\in {\left({\Gamma }_{\alpha }\right)}_{\left(min\left({\sigma }_1,{\sigma }_2\right)\right)}$. Since ${\left({\Gamma }_{\alpha }\right)}_{\left(min\left({\sigma }_1,{\sigma }_2\right)\right)}$ is an ideal of ${\rm Y}$, therefore ${\kappa }_1-{\kappa }_2\in {\left({\Gamma }_{\alpha }\right)}_{\left(min\left({\sigma }_1,{\sigma }_2\right)\right)}$, which yields ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1-{\kappa }_2\right)\ge min\left({\sigma }_1,{\sigma }_2\right)=min\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right)$.

• 1.

  ii. either ${\kappa }_1\in {\Gamma }_{\alpha }{}_{\left(max\left({\sigma }_1,{\sigma }_2\right)\right)}$ or ${\kappa }_2\in {\Gamma }_{\alpha }{}_{\left(max\left({\sigma }_1,{\sigma }_2\right)\right)}$. Since ${\Gamma }_{\alpha }{}_{\left(max\left({\sigma }_1,{\sigma }_2\right)\right)}$ is an ideal of ${\rm Y}$, therefore, we yield ${\kappa }_1{\kappa }_2\in {\Gamma }_{\alpha }{}_{\left(max\left({\sigma }_1,{\sigma }_2\right)\right)}$. Thus, ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1.{\kappa }_2\right)\ge max\left({\sigma }_1,{\sigma }_2\right)=\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right)$.

Therefore, $\left(\omega ,\psi ,{\rm Y}\right)$ is an FSI.

Theorem 3.2 If $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa } :\kappa \in {\rm Y}\right)\right):\alpha \in \psi \right)$ is an FSI over ${\rm Y}$, then the fuzzy soft level subset ${\left(\omega ,\psi ,{\rm Y}\right)}_{\sigma }$ of $\left(\omega ,\psi ,{\rm Y}\right)$ is an ideal of ${\rm Y}$, where $\sigma \le {\mu }_{{\Gamma }_{\alpha }}\left(0\right)$ for every $\alpha \in \psi$.

Proof: Let $\left(\omega ,\psi ,{\rm Y}\right)$ be a fuzzy soft ideal of ${\rm Y}$. Then ${\Gamma }_{\alpha }$ is a fuzzy ideal of ${\rm Y},$ implying that ${\left({\Gamma }_{\alpha }\right)}_{\sigma }$ is an ideal of ${\rm Y}$ for every $\alpha \in \psi$. Since ${\left(\omega ,\psi ,{\rm Y}\right)}_{\sigma }=\underset{\alpha \in \psi }{\cap } {\left({\Gamma }_{\alpha }\right)}_{\sigma }$, therefore ${\left(\omega ,\psi ,{\rm Y}\right)}_{\sigma }$ is an ideal of ${\rm Y}$.

Theorem 3.3 Let ${\rm Y}$ be a division ring. An FSS $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa }:\kappa \in {\rm Y}\right):\alpha \in \psi \right)\right)$ is an FSI of ${\rm Y}$ if and only if the following condition holds for every nonzero element $\kappa \in {\rm Y}$ and $\alpha \in \psi$:

${\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)={\mu }_{{\Gamma }_{\alpha }}\left(1\right)\le {\mu }_{{\Gamma }_{\alpha }}\left(0\right).$

Proof Assume that $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa }:\kappa \in {\rm Y}\right):\alpha \in \psi \right)\right)$ is an FSI over ${\rm Y}$. Let $\alpha \in \psi$ such that $\omega \left(\alpha \right)={\Gamma }_{\alpha }$. Since $\left(\omega ,\psi ,{\rm Y}\right)$ is an FSI of ${\rm Y}$, therefore ${\Gamma }_{\alpha }$ is an FI of ${\rm Y}$.

Now, for $0\ne \kappa \in {\rm Y}$, consider

$\begin{array}{c}{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa .1\right)\ge max\left({\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right),{\mu }_{{\Gamma }_{\alpha }}\left(1\right)\right)\ge {\mu }_{{\Gamma }_{\alpha }}\left(1\right)={\mu }_{{\Gamma }_{\alpha }}\left({\kappa }^{-1}\kappa \right)\\ \ge max\left({\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }^{-1}\right)\right)\ge {\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)\\ \Rightarrow {\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)={\mu }_{{\Gamma }_{\alpha }}\left(1\right)\end{array}$

Furthermore, it is well known that if ${\Gamma }_{\alpha }$ is fuzzy ideal of ${\rm Y}$, then ${\mu }_{{\Gamma }_{\alpha }}\left(0\right)\ge {\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)$ for every $\kappa \in {\rm Y}$.

Conversely, suppose that ${\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)={\mu }_{{\Gamma }_{\alpha }}\left(1\right)\le {\mu }_{{\Gamma }_{\alpha }}\left(0\right)$ for every $\kappa \in {\rm Y}\setminus \left\{0\right\}$. Let ${\kappa }_1,{\kappa }_2\in {\rm Y}$.

• ii.

  1. If ${\kappa }_1\ne {\kappa }_2$, then ${\kappa }_1-{\kappa }_2\ne 0,$ so ${\mu }_{{\Gamma }_{\alpha }}({\kappa }_1-{\kappa }_2)={\mu }_{{\Gamma }_{\alpha }}\left(1\right)$$\ge \min \left\{{\mu }_{{\Gamma }_{\alpha }}\right({\kappa }_1),{\mu }_{{\Gamma }_{\alpha }}({\kappa }_2\left)\right\}$JK1210_179 - Copy.eps]. If ${\kappa }_1={\kappa }_2$, then${\kappa }_1-{\kappa }_2=0,$ and ${\mu }_{{\Gamma }_{\alpha }}\left(0\right)\ge \min \left\{{\mu }_{{\Gamma }_{\alpha }}\right({\kappa }_1),{\mu }_{{\Gamma }_{\alpha }}({\kappa }_2\left)\right\}$.

• 2.

  If either ${\kappa }_1=0$ or ${\kappa }_2=0$, then ${\kappa }_1{\kappa }_2=0$, and ${\mu }_{{\Gamma }_{\alpha }}\left(0\right)\ge max\left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right).$ If both are non-zero, ${\kappa }_1{\kappa }_2\ne 0$, and by assumption, ${\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1.{\kappa }_2\right)={\mu }_{{\Gamma }_{\alpha }}\left(1\right)=\max \left({\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right)\right).$

Hence, $\left(\omega ,\psi ,{\rm Y}\right)$ is an FSI.

Definition 3.2 Let $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,{\Gamma }_{\alpha }=\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa }:\kappa \in {\rm Y}\right)\right):\alpha \in \psi \right)$ be an FSS over a ring ${\rm Y}$. Then, the subset ${\left(\omega ,\psi ,{\rm Y}\right)}_{*}$ of ${\rm Y}$ is defined as follows:

${\left(\omega ,\psi ,{\rm Y}\right)}_{\text{*}}=\left(\kappa \in {\rm Y}:{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)={\mu }_{{\Gamma }_{\alpha }}\left(0\right)\text{ for every }\alpha \in \mathrm{\psi .}\right)$

Theorem 3.4 If $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa } :\kappa \in {\rm Y}\right)\right):\alpha \in \psi \right)$ is an FSI over ${\rm Y}$, then the subset ${\left(\omega ,\psi ,{\rm Y}\right)}_{*}$ of ${\rm Y}$ is an ideal of ${\rm Y}$.

Proof Since $\left(\omega ,\psi ,{\rm Y}\right)$ is an FSI, Theorem 3.2 shows that every fuzzy soft level subset ${\left(\omega ,\psi ,{\rm Y}\right)}_{\sigma }$ is an ideal of ${\rm Y}$ for $\sigma \le {\mu }_{{\Gamma }_{\alpha }}\left(0\right)$. In particular, choosing $\sigma$=${\mu }_{{\Gamma }_{\alpha }}\left(0\right)$ yields ${\left(\omega ,\psi ,{\rm Y}\right)}_{*}$=${\left(\omega ,\psi ,{\rm Y}\right)}_{{\mu }_{{\Gamma }_{\alpha }}\left(0\right) }$, which is therefore an ideal of ${\rm Y}$.

Theorem 3.5 Let $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\left(\frac{{\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa } :\kappa \in {\rm Y}\right)\right):\alpha \in \psi \right)$ be an FSI over ${\rm Y}$. Consider the quotient ring $\frac{{\rm Y}}{{\left(\omega ,\psi ,{\rm Y}\right)}_{*}}$, and let

$\begin{array}{c}\left(\phi ,\psi ,\frac{{\rm Y}}{{\left(\omega ,\psi ,{\rm Y}\right)}_{*}}\right)\\ =\left(\left(\alpha ,\phi \left(\alpha \right)={\text{Π}}_{\alpha }=\left(\begin{array}{c}\frac{{\mu }_{{\text{Π}}_{\alpha }}\left(\kappa +{\left(\omega ,\psi ,{\rm Y}\right)}_{*}\right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)}{\kappa +{\left(\omega ,\psi ,{\rm Y}\right)}_{*}}\\ :\kappa +{\left(\omega ,\psi ,{\rm Y}\right)}_{*}\in \frac{{\rm Y}}{{\left(\omega ,\psi ,{\rm Y}\right)}_{*}}\end{array}\right)\right):\alpha \in \psi \right)\end{array}$

be an FSS over $\frac{{\rm Y}}{{[\omega ,\psi ,{\rm Y}]}_{*}}$. Then, $\left(\phi ,\psi ,\frac{{\rm Y}}{{[\omega ,\psi ,{\rm Y}]}_{*}}\right)$ is an FSI over $\frac{{\rm Y}}{{\left(\omega ,\psi ,{\rm Y}\right)}_{*}}$.

Proof Since $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha }\right):\alpha \in \psi \right)$ is an FSI, Theorem 3.4 ensures ${\left(\omega ,\psi ,{\rm Y}\right)}_{*}$ is an ideal of ${\rm Y},$ so the quotient ring $\frac{{\rm Y}}{{\left(\omega ,\psi ,{\rm Y}\right)}_{*}}$ is well defined. To prove the required results, we have to show that, for every $\alpha \in \psi$, ${\Pi }_{\alpha }=\left(\frac{{\mu }_{{\Pi }_{\alpha }}\left(\kappa +{\left(\omega ,\psi ,{\rm Y}\right)}_{*}\right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right), }{\kappa +{\left(\omega ,\psi ,{\rm Y}\right)}_{*}}:\kappa +{\left(\omega ,\psi ,{\rm Y}\right)}_{*}\in \frac{{\rm Y}}{{\left(\omega ,\psi ,{\rm Y}\right)}_{*}}\right)$ is an FI of $\frac{{\rm Y}}{{[\omega ,\psi ,{\rm Y}]}_{*}}$. For this, firstly we will prove that the mapping ${\mu }_{{\Pi }_{\alpha }}: \frac{{\rm Y}}{{[\omega ,\psi ,{\rm Y}]}_{*}}\to [0,1]$ defined by ${\mu }_{{\Pi }_{\alpha }}(\kappa +{[\omega ,\psi ,{\rm Y}]}_{*})={\mu }_{{\Gamma }_{\alpha }}\left(\kappa \right)$ is well-defined.

Let

$\begin{array}{c}{\kappa }_1+{\left(\omega ,\psi ,{\rm Y}\right)}_{\text{*}}={\kappa }_2+{\left(\omega ,\psi ,{\rm Y}\right)}_{\text{*}}\\ \Rightarrow {\kappa }_1-{\kappa }_2\in {\left(\omega ,\psi ,{\rm Y}\right)}_{\text{*}}\Rightarrow {\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1-{\kappa }_2\right)={\mu }_{{\Gamma }_{\alpha }}\left(0\right)\\ \Rightarrow {\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1\right)={\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2\right).\end{array}$

4. Fuzzy Soft Cosets of Fuzzy Soft Subrings/Ideals

This section delves into fuzzy soft cosets, defining and characterizing them in the context of fuzzy soft rings. We explore the properties of these cosets, leading to the establishment of a new ring formed by fuzzy soft cosets, under specific binary operations. Moreover, it presents the concept of fuzzy soft quotient ideal, associated with the ring formed by fuzzy soft cosets. Finally, the classical ring homomorphism theorem is proved in the context of fuzzy soft rings.

Definition 4.1 Let $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha }\right):\alpha \in \psi \right)$ be an FSSR over ${\rm Y}$ and $\kappa \in {\rm Y}$. Then fuzzy soft coset (FSC) of $\left(\omega ,\psi ,{\rm Y}\right)$ associated with $\kappa$ is denoted by ${\left(\omega ,\psi ,{\rm Y}\right)}_{\kappa }$ and is defined as

$\left(\omega ,\psi ,{\rm Y}\right){ }_{\kappa }=\left(\left(\alpha ,\left({\left({\Gamma }_{\alpha }\right)}_{\kappa }=\left(\frac{{\mu }_{{\left({\Gamma }_{\alpha }\right)}_{\kappa }}\left(x\right)}{x}\right)=\left(\frac{{\mu }_{{\Gamma }_{\alpha }}(x-\kappa ),}{x} :x\in {\rm Y}\right)\right):\alpha \in \psi \right)\right).$

Theorem 4.1 Let $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha }\right):\alpha \in \psi \right)$ be an FSSR over ${\rm Y}$. Then ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}\iff {\kappa }_1+{\left({\Gamma }_{\alpha }\right)}_{*}={\kappa }_2+{\left({\Gamma }_{\alpha }\right)}_{*}$ for every $\alpha \in \psi$ and ${\kappa }_1,{\kappa }_2\in {\rm Y}$.

Proof Consider an FSSR, $\left(\omega ,\psi ,{\rm Y}\right)=\left(\left(\alpha ,\omega \left(\alpha \right)={\Gamma }_{\alpha }\right):\alpha \in \psi \right)$, over ${\rm Y}$. Assume that ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}$.

For each ${\kappa }_1,{\kappa }_2\in {\rm Y}$ and $\alpha \in \psi$, we have

$\begin{array}{c}{\mu }_{\left({\Gamma }_{\alpha }\right)}{}_{{\kappa }_1}\left(\kappa \right)={\mu }_{\left({\Gamma }_{\alpha }\right)}{}_{{\kappa }_2}\left(\kappa \right)\Rightarrow {\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1\right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2\right)\\ \Rightarrow {\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2-{\kappa }_1\right)={\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2-{\kappa }_2\right)={\mu }_{{\Gamma }_{\alpha }}\left(0\right)(Taking\kappa ={\kappa }_2)\\ \Rightarrow {\kappa }_2-{\kappa }_1\in {\left({\Gamma }_{\alpha }\right)}_{\text{*}}\\ \Rightarrow \left({\kappa }_2-{\kappa }_1\right)+{\left({\Gamma }_{\alpha }\right)}_{\text{*}}={\left({\Gamma }_{\alpha }\right)}_{\text{*}}\\ \Rightarrow {\kappa }_1+{\left({\Gamma }_{\alpha }\right)}_{\text{*}}={\kappa }_2+{\left({\Gamma }_{\alpha }\right)}_{\text{*}}\end{array}$

Conversely, suppose that ${\kappa }_1+{\left({\Gamma }_{\alpha }\right)}_{*}={\kappa }_2+{\left({\Gamma }_{\alpha }\right)}_{*}$ for every ${\kappa }_1,{\kappa }_2\in {\rm Y}$, then $\left({\kappa }_1-{\kappa }_2\right)+{\left({\Gamma }_{\alpha }\right)}_{*}={\left({\Gamma }_{\alpha }\right)}_{*}$ and $\left({\kappa }_2-{\kappa }_1\right)+{\left({\Gamma }_{\alpha }\right)}_{*}={\left({\Gamma }_{\alpha }\right)}_{*}$. Therefore, ${\kappa }_1-{\kappa }_2\in {\left({\Gamma }_{\alpha }\right)}_{*}$ and ${\kappa }_2-{\kappa }_1\in {\left({\Gamma }_{\alpha }\right)}_{*}$.

Consider,

$\begin{array}{c}{\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1\right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2+{\kappa }_2-{\kappa }_1\right)foreach\kappa \in {\rm Y}\\ \ge min\left({\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_2-{\kappa }_1\right)\right)\\ =min\left({\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2\right),{\mu }_{{\Gamma }_{\alpha }}\left(0\right)\right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2\right)\\ \Rightarrow {\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1\right)\ge {\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2\right)\end{array}$

Also,

$\begin{array}{c}{\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2\right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1+{\kappa }_1-{\kappa }_2\right)foreach\kappa \in {\rm Y}\\ \ge min\left({\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1-{\kappa }_2\right)\right)\\ =min\left({\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1\right),{\mu }_{{\Gamma }_{\alpha }}\left(0\right)\right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1\right)\\ \Rightarrow {\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2\right)\ge {\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1\right)\end{array}$

Therefore, ${\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_1\right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa -{\kappa }_2\right)$ for each $\kappa \in {\rm Y}$.

$\Rightarrow {\mu }_{\left({\Gamma }_{\alpha }\right)}{}_{{\kappa }_1}\left(\kappa \right)={\mu }_{\left({\Gamma }_{\alpha }\right)}{}_{{\kappa }_2}\left(\kappa \right)$ for every $\alpha \in \psi$. Thus, ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}$ for every ${\kappa }_1,{\kappa }_2\in {\rm Y}$.

Theorem 4.2 Let $\left(\omega ,\psi ,{\rm Y}\right)$ be an FSI of ${\rm Y}$. Then $S{\left(\omega ,\psi ,{\rm Y}\right)}_{\kappa }$ the set of all FSCs of $\left(\omega ,\psi ,{\rm Y}\right)$ over ${\rm Y}$, forms a ring with the following binary operations;

${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1}+{\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1+{\kappa }_2}$ and ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1}{\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1{\kappa }_2}$, for every ${\kappa }_1,{\kappa }_2\in {\rm Y}$.

Proof. First, we show that addition and multiplication defined on $S{\left(\omega ,\psi ,{\rm Y}\right)}_{\kappa }$ are well-defined. For this suppose that ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}$ and ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_3}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_4}$. Then for every Eqs. (4.1-4.7) $\kappa \in {\rm Y}$

So,

Putting $\kappa ={\kappa }_1+{\kappa }_3-{\kappa }_4$ in Eq. (4.3) $\kappa ={\kappa }_3$ in Eq. (4.4) and $\kappa ={\kappa }_1$ in Eq. (4.3), we have

Now

$\begin{array}{c}{\mu }_{{\left({\Gamma }_{\alpha }\right)}_{{\kappa }_1}}\left(\kappa \right)+{\mu }_{{\left({\Gamma }_{\alpha }\right)}_{{\kappa }_3}}\left(\kappa \right)={\mu }_{{\left({\Gamma }_{\alpha }\right)}_{{\kappa }_1+{\kappa }_3}}\left(\kappa \right)={\mu }_{{\Gamma }_{\alpha }}\left(\kappa -\left({\kappa }_1+{\kappa }_3\right)\right)\\ ={\mu }_{{\Gamma }_{\alpha }}\left(\left(\kappa -{\kappa }_2-{\kappa }_4\right)-\left({\kappa }_1-{\kappa }_2+{\kappa }_3-{\kappa }_4\right)\right)\\ \ge min\left({\mu }_{{\Gamma }_{\alpha }}\left(\kappa -\left({\kappa }_2+{\kappa }_4\right)\right),{\mu }_{{\Gamma }_{\alpha }}\left({\kappa }_1-{\kappa }_2+{\kappa }_3-{\kappa }_4\right)\right)\end{array}$

$\ge min\left({\mu }_{{\Gamma }_{\alpha }}\left(\kappa -\left({\kappa }_2+{\kappa }_4\right)\right),{\mu }_{{\Gamma }_{\alpha }}\left(0\right)\right)$ by using Eqs. (4.6) and (4.7)

$={\mu }_{{\Gamma }_{\alpha }}\left(\kappa -\left({\kappa }_2+{\kappa }_4\right)\right)={\mu }_{{\left({\Gamma }_{\alpha }\right)}_{{\kappa }_2+{\kappa }_4}}\left(\kappa \right)={\mu }_{{\left({\Gamma }_{\alpha }\right)}_{{\kappa }_2}}\left(\kappa \right)+{\mu }_{{\left({\Gamma }_{\alpha }\right)}_{{\kappa }_4}}\left(\kappa \right)$

So,

Similarly, we can prove that

The inequalities in Eqs. (4.8) and (4.9) yield the following:

Furthermore, the application of the similar approach results in

The Eqs. (4.10) and (4.11) yield that ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1}+{\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1+{\kappa }_2}$ and ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1}{\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}={\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_1{\kappa }_2}$ holds for every $\alpha \in \psi$.

Hence, both addition and multiplication defined on ${\left(\omega ,\psi ,{\rm Y}\right)}_{\kappa }$ are well-defined. Next, ${\left(\omega ,\psi ,{\rm Y}\right)}_0=\left(\omega ,\psi ,{\rm Y}\right)$ acts as zero of ${\left(\omega ,\psi ,{\rm Y}\right)}_{\kappa }$ and for each ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}\in {\left(\omega ,\psi ,{\rm Y}\right)}_{\kappa }$ there exists ${\left(\omega ,\psi ,{\rm Y}\right)}_{-{\kappa }_2}\in {\left(\omega ,\psi ,{\rm Y}\right)}_{\kappa }$ such that ${\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}+{\left(\omega ,\psi ,{\rm Y}\right)}_{-{\kappa }_2}={\left(\omega ,\psi ,{\rm Y}\right)}_0={\left(\omega ,\psi ,{\rm Y}\right)}_{-{\kappa }_2}+{\left(\omega ,\psi ,{\rm Y}\right)}_{{\kappa }_2}$. Associativity and distributivity follow directly from the corresponding properties of the ring ${\rm Y}$.