Image development in the framework of ξ

Abstract

The study of complex fuzzy sets defined over the meet operator (ξ – CFS) is a useful mathematical tool in which range of degrees is extended from [0, 1] to complex plane with unit disk. These particular complex fuzzy sets plays a significant role in solving various decision making problems as these particular sets are powerful extensions of classical fuzzy sets. In this paper, we define ξ – CFS and propose the notion of complex fuzzy subgroups defined over ξ – CFS (ξ – CFSG) along with their various fundamental algebraic characteristics. We extend the study of this idea by defining the concepts of ξ – complex fuzzy homomorphism and ξ – complex fuzzy isomorphism between any two ξ – complex fuzzy subgroups and establish fundamental theorems of ξ – complex fuzzy morphisms. In addition, we effectively apply the idea of ξ – complex fuzzy homomorphism to refine the corrupted homomorphic image by eliminating its distortions in order to obtain its original form. Moreover, to view the true advantage of ξ – complex fuzzy homomorphism, we present a comparative analysis with the existing knowledge of complex fuzzy homomorphism which enables us to choose this particular approach to solve many decision-making problems.

Keywords

08A72 20N25 03E72

1 Introduction

A homomorphism is a way of translating one object into another that reflects its original structure. It covers a number of topics in interdisciplinary studies in mathematics like mechanics, camera calibration, quantum theory, telephone network design and non-linear network problems. In real life, we come across a lot of problems which can easily be countered through this phenomenon. For instance, one can use the idea of homomorphism in national security as a monitoring plate form to prevent the attacks. It also plays a key role to protect cyber security systems from hackers. It is indeed an analytical tool to study communication in various information systems. This particular notion is highly suitable for genomics data sharing in the field of medicine. Uncertainty, imprecision and vagueness are the main characteristics of the problems occurring in many real-world phenomena which cannot be effectively handled by using traditional mathematical tools. Although, the theory of fuzzy sets is used to deal with these limitations, however, this particular setup does not handle the periodicity or seasonality that exists in many physical situations. This leads to the introduction of complex fuzzy logic, which is a linear augmentation of the classic fuzzy logic. These sets also allow a natural reduction in difficulty based on the fuzzy logic. The complex fuzzy sets have a great significance in numerous executions, specifically, advanced control systems and predicting of the periodic events where multiple fuzzy variables are interlinked in a complex way that cannot be properly identified by conventional fuzzy operations. Zadeh [1] introduced the concept of fuzzy sets in 1965. Rosenfeld [2] innovated the idea of fuzzy subgroups based on fuzzy sets in 1971. Das [3] defined level subgroups of fuzzy groups in 1981. A comprehensive study about the development of elementary concepts of fuzzy subgroups can be viewed in [4 –8]. The notion of fuzzy homomorphism was proposed by Chakraborty and Khare in [9]. Fang [10] characterized the decomposition theorems of fuzzy isomorphisms in 1994. Choudhury et al. [11] described the effects of fuzzy homomorphism on fuzzy subgroups. Chengyi and Qingan [12] investigated various key features of this phenomenon. One can study an effective research regarding applications of fuzzy homomorphism in [13 –18]. Changzhong et al. [19] used the concept of fuzzy homomorphism as the main tool for studying the relationship between fuzzy information systems in 2014. Buckley [20] initiated the study of complex fuzzy numbers in 1989. Later on, he established a useful analysis of these numbers in the framework of derivation and integration in [21, 22]. In [23], Ramote et al. launched the study of complex fuzzy sets (CFS) in 2002 whereas two novel operations, namely reflection and rotations of these newly defined sets were explored in [24]. The authors defined complex fuzzy hyper structure [25], complex fuzzy subgroups [26] and complex fuzzy normal subgroups [27] over a complex fuzzy space in recent times. Alsarahead and Ahmad [28] extended the ideology of complex fuzzy sets to present the idea of complex fuzzy subgroups in 2017, whereas, the concept of complex intuitionistic fuzzy sets was originated in the same year [29]. Moreover, Yousaf et al. [30] gave the solution of decision-making problems on the basis of complex multi-fuzzy relations in 2018. More useful applications of complex fuzzy sets can be viewed in [31, 32]. Akram [33] presented a new technique of combining the complex pythagorean fuzzy information with competition graph in 2020. A comprehensive study of the significance of complex fuzzy sets in the frame work of complex fuzzy graphs can be viewed in [34, 35]. The competency of complex fuzzy set has played an effective role to solve many physical problems. It provides us the meaningful representations of measuring uncertainty and periodicity. Despite all of these advantages, we still face vast complications to counter various physical situations based on a complex valued membership function. This motivates us to define the notion of ξ –complex fuzzy set through which one can have multiple options to investigate a specific real-world situation in much efficient way by choosing appropriate value of the parameter ξ. The main theme of this paper is to initiate the study of ξ –CFS as a powerful extension of classical fuzzy sets and to prove various fundamental characteristics of the concept of support of these newly defined sets. We extend this ideology to innovate the notion of ξ –CFSG and prove many important algebraic aspects of this phenomenon. Moreover, we characterize the notions of ξ –complex fuzzy homomorphism and ξ –complex fuzzy isomorphism between these particular fuzzy subgroups and explore the significance of these ideas by proving their fundamental theorems along with the practical applications of these notions related to the real-world situations. After a brief debate about the historical background and importance of CFS, the rest of the work in this article is organized as: the second section contains basic preliminaries which are quite essential to understand the main results of this article. In section three, we commence the study of ξ –CFS and ξ –CFSG along with their many important algebraic characteristics. In section four, we innovate the ideas of ξ –complex fuzzy homomorphism, ξ –complex fuzzy isomorphism and prove three fundamental theorems of ξ –complex fuzzy isomorphism. Section five contains a useful description of potential applications of ξ –complex fuzzy homomorphism related to many important physical situations. We also elucidate the importance of ξ –complex fuzzy homomorphism by presenting a comprehensive comparative analysis of this particular notion with the existing concept of complex fuzzy homomorphism.

2 Preliminaries

This section contains a brief review of the notion of CFS and related ideas, which are important to master the technicalities of the work presented in this article.

Definition 2.1 [28]. A CFS A defined on a universe of discourse U is characterized by a membership function μ_A (m) that allocates each element of U to a unit circle C^* in complex plane and is written as r_A (m) e^iω_A(m), where r_A (m) denotes the real-valued function from U to the closed unit interval and e^iω_A(m) represents a periodic function ∀m, n ∈ U whose periodic law and principal period are 2π and 0 < arg_A (m) ⩽ 2π, respectively. Note that ω_A (m) = arg_A (m) + 2kπ, where arg_A (m) is the principal argument and k ∈ Z.

Definition 2.2 [28]. Let A and B be any two CFS of a universe U, then

i. A is homogeneous CFS if r_A (m) ≤ r_A (n) implies ω_A (m) ≤ ω_A (n) and vice versa for all m, n ∈ U. ∀m, n ∈ U

ii. A is homogeneous CFS with B if r_A (m) ⩽ r_B (n) implies ω_A (m) ⩽ ω_B (n) and vice versa for all m, n ∈ U. ∀m, n ∈ U.

Definition 2.3 [28]. A homogeneous CFS A of a group G is said to be a CFSG if μ_A (mn)⩾ min { μ_A (m) , μ_A (n) } and μ_A (m^-1) ⩾ μ_A (m), ∀m, n ∈ G.

Definition 2.4 [28]. Let A be a CFSG(G) and m ∈ G. The complex fuzzy left coset of A in G is represented by mA and is given by: mA (g) ={ μ_A (m^-1g) : g ∈ G }. Similarly, one can define the complex fuzzy right coset of A in G.

Definition 2.5 [28]. A CFSG A of a group G is CFNSG(G) if mA = Am, ∀ m ∈ G.

3 Fundamental algebraic attributes of ξ –CFSG

In this section our main focus is to propose the concepts of ξ –CFS, ξ –CFSG and to investigate various fundamental algebraic aspects of these notions.

Definition 3.1. Let A be a CFS of a universe U and ξ = αe^iβ be an element of a unit circle with 0 ⩽ α ⩽ 1 and 0 ⩽ β ⩽ 2π. The CFS A^ξ is called the ξ –CFS with respect to CFS A and is represented by the following membership function: μ_{A
^ξ} (m) = min(r_A (m) e^iω_A(m), αe^iβ) = min {r_A (m) , α} e^{imin(ω_A(m),β)} = r_{A
^ξ} (m) e^{iω_{A
^ξ}(m)}, ∀ m ∈ U, where r_{A
^ξ} (m) is a real valued function from U to closed unit interval and e^{iω_{A
^ξ}(m)} represents a periodic function whose periodic law and principal period are 2π and 0 < arg_{A
^ξ} (m) ⩽ 2π, respectively.

The family of all ξ –CFS defined on the universe U is denoted by F^ξ (U).

Example 3.2. Consider the complex fuzzy set A defined on set of integers as: $μ_{A} (m) = {\begin{matrix} 1 e^{i 2 π} & if m \in {0, \pm 2, \pm \pm 4 \dots} \\ 0.7 e^{i 1.1 π} & if m \in {\pm 3, \pm 9, \pm 15, \dots .} \end{matrix}$

The ξ –CFSG A^ξ corresponding to the value ξ = 0.9e^i1.9π is defined as: $μ_{A^{ξ}} (m) = {\begin{matrix} 0.9 e^{i 1.9 π} & if m \in {0, \pm 2, \pm \pm 4 \dots} \\ 0.7 e^{i 1.1 π} & if m \in {\pm 3, \pm 9, \pm 15, \dots .} \end{matrix} .$

Definition 3.3. Let A^ξ and B^ξ ∈ F^ξ (U).

i. The union of ξ –CFS A^ξ and B^ξ is denoted by A^ξ ∪ B^ξ and is described by the following membership function:

μ_A^ξ∪B^ξ (m) = r_A^ξ∪B^ξ (m) e^{iω_A^ξ∪B^ξ(m)} = max(r_{A
^ξ} (m) , r_{B
^ξ} (m)) e^{imax(ω_{A
^ξ}(m),ω_{B
^ξ}(m))}.

ii. The intersection of ξ –CFS A^ξ and B^ξ is denoted by A^ξ ∩ B^ξ and is expressed by the following membership function:

μ_A^ξ∩B^ξ (m) = r_A^ξ∩B^ξ (m) e^{iω_A^ξ∩B^ξ(m)} = min(r_{A
^ξ} (m) , r_{B
^ξ} (m)) e^{imin(ω_{A
^ξ}(m),ω_{B
^ξ}(m))}.

iii. The complement of ξ –CFS A^ξ is denoted by A^ξ′ and is depicted as follows:

μ_{A
^ξ′} (m) = r_{A
^ξ′} (m) e^{iω_{A
^ξ′}(m)} = (1 - r_{A
^ξ} (m)) e^{i(2π-ω_{A
^ξ}(m))}, ∀m ∈ U.

iv. The product of any two ξ –CFS A^ξ and B^ξ is denoted by A^ξ ∘ B^ξ and is defined as: $(A^{ξ} \circ B^{ξ}) (m) = \underset{x . y = m}{Sup} {\min (μ_{A^{ξ}} (x), μ_{B^{ξ}} (y)) : m \in U}$ .

Remark 3.4. The union and intersection of two ξ –CFS is also an ξ –CFS.

Definition 3.5. For any A^ξ ∈ F^ξ (U), the support of A^ξ is defined as: $A_{*}^{ξ} = {m \in U : r_{A^{ξ}} (m) > 0, ω_{A^{ξ}} (m) > 0}$ .

Example 3.6. Consider the complex fuzzy set A defined on set of integers as: $μ_{A} (m) = {\begin{matrix} 1 e^{i 2 π} & if m \in {0, \pm 2, \pm \pm 4 \dots} \\ 0.7 e^{i 0.9 π} & if m \in {\pm 5, \pm 15, \pm 25, \dots .} \\ 0 & otherwise \end{matrix}$

The ξ –CFSG A^ξ corresponding to the value ξ = 0.8e^i1.1π is defined as: $μ_{A^{ξ}} (m) = {\begin{matrix} 0.8 e^{i 1.1 π} & if m \in {0, \pm 2, \pm \pm 4 \dots} \\ 0.7 e^{i 0.9 π} & if m \in {\pm 5, \pm 15, \pm 25, \dots .} \\ 0 & otherwise \end{matrix}$

Then the support of A^ξ is $A_{*}^{ξ} =$ ${\begin{matrix} 0.8 e^{i 1.1 π} & if m \in {0, \pm 2, \pm \pm 4 \dots} \\ 0.7 e^{i 0.9 π} & if m \in {\pm 5, \pm 15, \pm 25, \dots .} \end{matrix}$ .

Definition 3.7. Let A^ξ and B^ξ be any two ξ –CFS of a universe U, then

i. A^ξ is homogeneous ξ –CFS if r_{A
^ξ} (m) ⩽ r_{A
^ξ} (n) implies ω_{A
^ξ} (m) ⩽ ω_{A
^ξ} (n) and vice versa ∀m, n ∈ U.

ii. A^ξ is homogeneous ξ –CFS with B^ξ if r_{A
^ξ} (m) ⩽ r_{B
^ξ} (n) implies ω_{A
^ξ} (m) ⩽ ω_{B
^ξ} (n) and vice versa ∀m, n ∈ U.

Note: Throughout this article, we consider ξ –CFS as a homogeneous ξ –CFS.

Definition 3.8. An ξ –CFS A^ξ of a group G is called ξ –complex fuzzy subgroup (ξ –CFSG) of G if A^ξ admits the following conditions:

i. μ_{A
^ξ} (mn)⩾ min { μ_{A
^ξ} (m) , μ_{A
^ξ} (n) }.

ii. μ_{A
^ξ} (m^-1) ⩾ μ_{A
^ξ} (m) , ∀ m, n ∈ G. The family of all ξ –CFSG defined on the group G is denoted by F^ξ (G).

Example 3.9. The CFS A of group S₃ is defined as: $μ_{A} (m) = {\begin{matrix} 1 e^{1.9 π} & if m = 1 \\ 0.5 e^{1.1 π} & if m \in {(123), (132)} \\ 0.3 e^{0.5 π} & otherwise \end{matrix}$

Then the ξ –CFSG A^ξ of G corresponding to the value ξ = 0.6e^i1.2π is defined as: $μ_{A^{ξ}} (m) = {\begin{matrix} 0.6 e^{1.2 π} & if m = 1 \\ 0.5 e^{1.1 π} & if m \in {(123), (132)} \\ 0.3 e^{0.5 π} & otherwise \end{matrix}$

Note that A^ξ is ξ –CFSG because each of its level set is a group of S₃. For instance the cut set Ω_{A
^ξ} corresponding to the value α = 0.5e^1.1π is {1, (123) , (132)].

Definition 3.10. Let A^ξ be an ξ –CFSG(G) and m ∈ G. The ξ –complex fuzzy left coset of A^ξ in G is represented by mA^ξ and is defined as: mA^ξ (g) = μ_{A
^ξ} (m^-1g) ={ min { μ_A (m^-1g) , ξ } : g ∈ G }.

Similarly, one can define the ξ –complex fuzzy right coset of A^ξ in G.

Definition 3.11. An ξ –CFSG A^ξ of a group G is ξ –complex fuzzy normal subgroup (ξ –CFNSG) of G if mA^ξ = A^ξm, ∀ m ∈ G.

The family of all ξ –CFNSG defined on the group G is denoted by F^ξN (G).

Definition 3.12. For any ξ –CFNSG A^ξ of G, the set of all ξ –complex fuzzy cosets of A^ξ in G forms a group under multiplication of ξ –complex fuzzy cosets. This group is known as quotient group of G relative to ξ –CFNSG A^ξ.

The following result defines an important property of the product of support of any two ξ –CFSG of a group G.

Theorem 3.13. Let A^ξ and B^ξ be the ξ –CFSG, then ${(A^{ξ} \circ B^{ξ})}_{*} = A_{*}^{ξ} \circ B_{*}^{ξ}$ .

Proof. By using the Definition 3.5, we have μ_A^ξ∘B^ξ (m) > 0 for all m ∈ (A^ξ ∘ B^ξ) _*. This implies that $\underset{m = x . y}{Sup} \min {μ_{A^{ξ}} (x), μ_{B^{ξ}} (y)} > 0$ . Then clearly μ_{A
^ξ} (x) > 0 and μ_{B
^ξ} (y) > 0.

$\begin{matrix} Therefore m \in A_{*}^{ξ} \circ B_{*}^{ξ} and ultimately \\ {(A^{ξ} \circ B^{ξ})}_{*} \subseteq A_{*}^{ξ} \circ B_{*}^{ξ} \end{matrix}$ (3.1)

Let $m \in A_{*}^{ξ} \circ B_{*}^{ξ}$ where m = x . y, $x \in A_{*}^{ξ}$ and $y \in B_{*}^{ξ}$ . By applying the Definition 3.5 in the above relation, we get μ_{A
^ξ} (x) > 0 and μ_{B
^ξ} (y) > 0. This shows that $\underset{m = x . y}{Sup} \min {μ_{A^{ξ}} (x), μ_{B^{ξ}} (y)} > 0$ , therefore, m ∈ (A^ξ ∘ B^ξ) _*. Consequently, $A_{*}^{ξ} \circ B_{*}^{ξ} \subseteq {(A^{ξ} \circ B^{ξ})}_{*}$ (3.2)

By comparing (3.1) and (3.2), we obtain the required equality. In the subsequent result, we prove the normality of an ξ –CFSG of G.

Theorem 3.14. Let A^ξ be an ξ –CFNSG(G), then $A_{*}^{ξ}$ is a normal subgroup of G.

Proof. By applying the Definition 3.5 and using the fact that A^ξ is ξ –CFSG, we have μ_{A
^ξ} (mn^-1) > 0 for all $m, n \in A_{*}^{ξ}$ . This implies that ${mn}^{- 1} \in A_{*}^{ξ}$ , therefore $A_{*}^{ξ} \subseteq G$ . Moreover, by applying Definition 3.5 and using the given condition on elements m ∈ G and $n \in A_{*}^{ξ}$ , we have μ_{A
^ξ} (mnm^-1) > 0. Thus, ${mnm}^{- 1} \in A_{*}^{ξ}$ , which establishes the condition of normality for $A_{*}^{ξ}$ .

In the following result, we show that support of any two ξ –CFSG admits the intersection property.

Theorem 3.15. For any A^ξ and B^ξ ∈ F^ξ (G), ${(A^{ξ} \cap B^{ξ})}_{*} = A_{*}^{ξ} \cap B_{*}^{ξ}$ .

Proof. In view of the Definition 3.5, we get μ_A^ξ∩B^ξ (m) > 0, m ∈ (A^ξ ∩ B^ξ) _*. This implies that min(μ_{A
^ξ} (m) , μ_{B
^ξ} (m)) > 0. Then μ_{A
^ξ} (m) > 0 and μ_{B
^ξ} (m) > 0, therefore, $m \in A_{*}^{ξ} \cap B_{*}^{ξ}$ . $Consequently {(A^{ξ} \cap B^{ξ})}_{*} \subseteq A_{*}^{ξ} \cap B_{*}^{ξ}$ (3.3)

Moreover, by applying Definition 3.5 on an element $m \in A_{*}^{ξ} \cap B_{*}^{ξ}$ , we obtain μ_{A
^ξ} (m) > 0 and μ_{B
^ξ} (m) > 0. This shows that min(μ_{A
^ξ} (m) , μ_{B
^ξ} (m)) > 0, therefore m ∈ (A^ξ ∩ B^ξ) _*. Consequently, $A_{*}^{ξ} \cap B_{*}^{ξ} \subseteq {(A^{ξ} \cap B^{ξ})}_{*}$ (3.4)

By using of the relations (3.3) and (3.4), we get the required equality.

Definition 3.16. Let A^ξ and B^ξ ∈ F^ξ (G) such that A^ξ ⊆ B^ξ. Then A^ξ is an ξ –CFNSG of B^ξ if μ_{A
^ξ} (mnm^-1)⩾ min { μ_{A
^ξ} (n) , μ_{B
^ξ} (m) }, ∀n ∈ A^ξ and m ∈ B^ξ.

The following result is about the relation of normality between the support of any two ξ –CFSG.

Theorem 3.17. If A^ξ, B^ξ ∈ F^ξ (G) such that A^ξ is an ξ –CFNSG of B^ξ, then $A_{*}^{ξ}) B_{*}^{ξ}$ .

Proof. The application of Definition 3.5 on the elements $n \in A_{*}^{ξ}$ and $m \in B_{*}^{ξ}$ yields that μ_{A
^ξ} (n) > 0 and μ_{B
^ξ} (m) > 0. This implies that min { μ_{A
^ξ} (n) , μ_{B
^ξ} (m) } > 0. In view of Definition 3.16 and using the fact that A^ξ is an ξ –CFNSG of B^ξ, we have μ_{A
^ξ} (mnm^-1) > 0. This means that ${mnm}^{- 1} \in A_{*}^{ξ}$ . Hence $A_{*}^{ξ}) B_{*}^{ξ}$ .

Theorem 3.18. Let A^ξ be an ξ –CFNSG and B^ξ be an ξ –CFSG of G, then A^ξ ∩ B^ξ is an ξ –CFNSG of B^ξ.

Proof. In view of the Remark 3.4 and using the given condition A^ξ ∩ B^ξ ⊆ B^ξ, there exist elements n ∈ A^ξ ∩ B^ξ and m ∈ B^ξ in G such that $\begin{matrix} μ_{A^{ξ} \cap B^{ξ}} ({mnm}^{- 1}) = min {μ_{A^{ξ} \cap B^{ξ}} ({mnm}^{- 1}), \\ μ_{B^{ξ}} ({mnm}^{- 1})} ⩾ \\ min {\begin{matrix} min {μ_{A^{ξ}} ({mnm}^{- 1}), μ_{B^{ξ}} ({mnm}^{- 1})}, \\ min {μ_{B^{ξ}} (mn), μ_{B^{ξ}} (n^{- 1})} \end{matrix}} \end{matrix}$ $\begin{matrix} ⩾ min {\begin{matrix} min {μ_{A^{ξ}} (n), μ_{B^{ξ}} ({mnm}^{- 1})}, \\ min {μ_{B^{ξ}} (m), μ_{B^{ξ}} (n)} \end{matrix}} = \\ min {\begin{matrix} min {μ_{A^{ξ}} (n), μ_{B^{ξ}} (n)}, \\ min {μ_{B^{ξ}} (m), μ_{B^{ξ}} ({mnm}^{- 1})} \end{matrix}} . \end{matrix}$

Thus, μ_A^ξ∩B^ξ (mnm^-1)⩾ min {μ_A^ξ∩B^ξ (n) , μ_{B
^ξ} (m)}. This concludes the proof.

4 Fundamental theorems of ξ –complex fuzzy isomorphisms of ξ –CFSG

In this section, we commence the study of ξ –complex fuzzy homomorphism and ξ –complex fuzzy isomorphism between any two ξ –CFSG and develop the significance of these notions by establishing their fundamental theorems.

Definition 4.1. Suppose A^ξ and B^ξ are any two ξ –complex fuzzy subgroups of G and G′ respectively. A group homomorphism f from G to G′ is called a weak ξ –complex fuzzy homomorphism from A^ξ to B^ξ if f (A^ξ) ⊆ B^ξ. In this situation A^ξ is said to be a weak ξ –complex fuzzy homomorphic to B^ξ and is represented as A^ξ ∼ B^ξ. Moreover, f is called an ξ –complex fuzzy homomorphism from A^ξ to B^ξ if f (A^ξ) = B^ξ. We say that A^ξ is an ξ –complex fuzzy homomorphic to B^ξ and write it as A^ξ ≈ B^ξ. A group isomorphism f : G → G′ is called a weak complex fuzzy isomorphism from A^ξ to B^ξ if f (A^ξ) ⊆ B^ξ. In this case A^ξ is a weak ξ –complex fuzzy isomorphism to B^ξ and is represented as A^ξ ≃ B^ξ. Moreover, f is called an ξ –complex fuzzy isomorphism from A^ξ to B^ξ if f (A^ξ) = B^ξ. We say that A^ξ is an ξ –complex fuzzy isomorphism to B^ξ and is expressed as A^ξ ≅ B^ξ.

Definition 4.2. Let A^ξ and B^ξ be any ξ –CFSG of G and G′ respectively and f be a group homomorphism from G to G′. The images of A^ξ and B^ξ under ξ –complex fuzzy homomorphism f is defined as follows: $\begin{matrix} μ_{f (A^{ξ})} (n) \\ = {\begin{matrix} max {μ_{A^{ξ}} (m) : f (m) = n} & if f^{- 1} (n) \neq \emptyset \\ 0 & otherwise \end{matrix} \end{matrix}$ where n ∈ G′ and μ_f^-1(B^ξ) (m) = μ_{B
^ξ} (f (m)) for all m ∈ G.

We observe the above stated algebraic facts in the following example.

Example 4.3. Consider the group homomorphism f from a multiplicative group G = R -{ 0 } to a multiplicative group G′ ={ 1, - 1 } in the following way: $f (m) = {\begin{matrix} 1 & if m is rational \\ - 1 & if m is irrational \end{matrix} .$

The CFSG A and B of G and G′ are given by: $μ_{A} (m) = {\begin{matrix} 1 e^{i 2 π} & if m is rational \\ 0.6 e^{i 0.9 π} & if m is irrationl \end{matrix}$ $and μ_{B} (m) = {\begin{matrix} 0.8 e^{i 1.9 π} & if m = 1 \\ 0.6 e^{i 0.9 π} & if m = - 1 \end{matrix} .$

The ξ –CFSG A^ξ and B^ξ of G and G′ corresponding to the value ξ = 0.7e^i1.2π are defined as: $μ_{A^{ξ}} (m) = {\begin{matrix} 0.7 e^{i 1.2 π} & if m is rational \\ 0.6 e^{i 0.9 π} & if m is irrational \end{matrix}$ $and μ_{B^{ξ}} (m) = {\begin{matrix} 0.7 e^{i 1.2 π} & if m = 1 \\ 0.6 e^{i 0.9 π} & if m = - 1 \end{matrix} .$

The required ξ –complex fuzzy homomorphism is given by: f (μ_{A
^ξ}) (1) = 0.7e^i1.2π and (μ_{A
^ξ}) (- 1) = 0.6e^i0.9π.

An important feature of the homomorphic image of the support of a given group under a ξ –complex fuzzy epimorphism has been investigated in the subsequence result.

Theorem 4.4. For any A^ξ ∈ F^ξ (G), B^ξ ∈ F^ξ (G′) and a ξ –complex fuzzy epimorphism f : A^ξ → B^ξ then $f (A_{*}^{ξ}) = B_{*}^{ξ}$ .

Proof. For any element $n \in f (A_{*}^{ξ})$ , we have $n = f (m), m \in A_{*}^{ξ}$ . Consider, μ_f(A^ξ) (n) = max { μ_{A
^ξ} (m) , m ∈ f^-1 (n) } ⩾ μ_{A
^ξ} (m) > 0. Therefore, $n \in B_{*}^{ξ}$ . $Thus f (A_{*}^{ξ}) \subseteq B_{*}^{ξ}$ (4.1)

Moreover, the application of Definition 4.2 and the given condition establishes that $B_{*}^{ξ} \subseteq f (A_{*}^{ξ})$ (4.2)

In the light of relations (4.1) and (4.2), we obtain $f (A_{*}^{ξ}) = B_{*}^{ξ}$ .

The following result indicates that the image of ξ –CFSG is always ξ –CFSG.

Theorem 4.5. Let A^ξ be a ξ –CFSG of G and f be a bijective homomorphism from G to G′ then f (A^ξ) is a ξ –CFSG of G′.

Proof. In view of the given condition, for any two elements n₁, n₂ ∈ G′ there exist m₁, m₂ ∈ G such that f (m₁) = n₁ and f (m₂) = n₂. Consider $\begin{matrix} μ_{{(f (A))}^{ξ}} (n_{1} n_{2}) = min {μ_{f (A)} (n_{1} n_{2}), ξ} \\ = \min {μ_{f (A)} ({fm}_{1}) f (m_{2}), ξ} \\ = min {μ_{f (A)} (f (m_{1} m_{2})), ξ} = min {μ_{A} (m_{1} m_{2}), ξ} \\ = μ_{A^{ξ}} (m_{1} m_{2}) \end{matrix}$

Thus, μ_(f(A))^ξ (n₁n₂) ⩾ min {μ_{A
^ξ} (m₁) , μ_{A
^ξ} (m₂)}.

This shows that μ_(f(A))^ξ (n₁n₂) ⩾ min {μ_f(A^ξ) (n₁) , μ_f(A^ξ) (n₂)}.

Further, μ_f(A^ξ) (n^-1) = μ_(f(A))^ξ (n).

The following result illustrates that every image of ξ –CFNSG is always ξ –CFNSG.

Theorem 4.6. Let A^ξ be a ξ –CFNSG and f be bijective homomorphism from G to G′, then f (A^ξ) is a ξ –CFNSG of G′.

Proof. In view of the given condition, for any elements n₁, n₂ ∈ G′ there exist elements m₁, m₂ ∈ G such that f (m₁) = n₁ and f (m₂) = n₂. Consider $\begin{matrix} μ_{{(f (A))}^{ξ}} (n_{1} n_{2}) = \min {μ_{f (A)} (n_{1} n_{2}), ξ} \\ = \min {μ_{f (A)} (f (m_{1}) f (m_{2})), ξ} \\ = \min {μ_{f (A)} (f (m_{1} m_{2})), ξ} \\ = \min {μ_{A} (m_{1} m_{2}), ξ} = μ_{A^{ξ}} (m_{1} m_{2}) \\ = μ_{A^{ξ}} (m_{2} m_{1}) = \min {μ_{A} (m_{2} m_{1}), ξ} \\ = \min {μ_{f (A)} (f (m_{2} m_{1})), ξ} \\ = \min {μ_{f (A)} (f (m_{2}) f (m_{1})), ξ} . \end{matrix}$

Therefore, μ_(f(A))^ξ (n₁n₂) = min {μ_f(A) (n₂n₁) , ξ}. Consequently, μ_(f(A))^ξ (n₁n₂) = μ_(f(A))^ξ (n₂n₁).

In the following result, we establish that inverse image of a ξ –CFSG is always ξ –CFSG.

Theorem 4.7. Let B^ξ be ξ –CFSG of G′ and f be a group homomorphism from group G to G′. Then f^-1 (B^ξ) is ξ –CFSG of G.

Proof. In the light of Definition 4.2 and using the given conditions, there exist elements m and n in G such that $\begin{matrix} μ_{f^{- 1} (B^{ξ})} (mn) = μ_{B^{ξ}} (f (mn)) = μ_{B^{ξ}} (f (m) f (n)) \\ ⩾ \min {μ_{B^{ξ}} (f (m)), μ_{B^{ξ}} (f (n))} . \end{matrix}$

Thus, μ_f^-1(B^ξ) (mn) ⩾ min {μ_f^-1(B^ξ) (m), μ_f^-1(B^ξ) (n)}. Moreover, $\begin{matrix} μ_{f^{- 1} (B^{ξ})} (m^{- 1}) = μ_{B^{ξ}} (f (m^{- 1})) = μ_{B^{ξ}} (f (m)) \\ = μ_{f^{- 1} (B^{ξ})} (m) . \end{matrix}$

The following result explains that every inverse image of an ξ –CFNSG is an ξ –CFNSG.

Theorem 4.8. Let B^ξ be an ξ –CFNSG of G′ and f be a group homomorphism from G to G′, then f^-1 (B^ξ) is an ξ –CFNSG of G.

Proof. By applying the Definition 4.2 and the given conditions, there exist m,n ∈ G such that μ_f^-1(B^ξ) (mn) = μ_{B
^ξ} (f (mn)). This means that μ_f^-1(B^ξ) (mn) = μ_{B
^ξ} (f (nm)). Hence f^-1 (B^ξ) is an ξ –CFNSG of G.

In the following result, we obtain ξ –complex fuzzy homomorphism between A^ξ ∈ F^ξ (G) and $A_{ρ}^{ξ} \in F^{ξ} (G / N)$ .

Theorem 4.9. Let A^ξ ∈ F^ξ (G) and f be a natural homomorphism from G onto G/N, then A^ξ is an ξ –complex fuzzy homomorphic to $A_{ρ}^{ξ}$ , where N ) G and $A_{ρ}^{ξ} \in F^{ξ} (G / N)$ .

Proof. Let us define the natural homomorphism f : G → G/N in the following way: f (g) = gN, g ∈ G.

For gN ∈ G/N, we have $\begin{matrix} f (A^{ξ}) (gN) = μ_{f (A^{ξ})} (gN) = \max {μ_{A^{ξ}} (z) : \\ z \in f^{- 1} (gN)} = \max {μ_{A^{ξ}} (z) : f (z) = gN} \\ = \max {μ_{A^{ξ}} (z) : zN = gN} = \max {μ_{A^{ξ}} (z) : \\ z = gn, n \in N} = \max {μ_{A^{ξ}} (gn) : n \in N} . \end{matrix}$

Thus, $μ_{f (A^{ξ})} (gN) = μ_{A_{ρ}^{ξ}} (gN)$ . This means that $f (A^{ξ}) = A_{ρ}^{ξ}$ . Hence $A^{ξ} \approx A_{ρ}^{ξ}$ .

The subsequent illustration supports the fact discussed in the above result.

Example 4.10. Consider a quotient group G/N ={ N, iN }, where G ={ ± 1, ± i } and N ={ ± 1 }. The CFSG A of G is defined as: $μ_{A} (m) = {\begin{matrix} 1 e^{i 1.7 π} & if m \in {\pm 1} \\ 0.5 e^{i π} & otherwise \end{matrix} .$

The ξ –CFSG A^ξ of G for the value of ξ = 0.6e^i1.2π is given as: $μ_{A^{ξ}} (m) = {\begin{matrix} 0.6 e^{i 1.2 π} & if m \in {\pm 1} \\ 0.5 e^{i π} & otherwise \end{matrix} .$

The natural homomorphism f from G onto G/N is defined as: f (g) = gN, ∀ g ∈ G.

Define CFSG A_ρ of G/N as follows: $μ_{A_{ρ}} (n) = {\begin{matrix} 1 e^{i 1.7 π} & if n \in N \\ 0.5 e^{i π} & if n \in iN \end{matrix} .$

The ξ –CFSG $A_{ρ}^{ξ}$ of G/N for the value of ξ = 0.6e^i1.2π is given as: $μ_{A_{ρ}^{ξ}} (n) = {\begin{matrix} 0.6 e^{i 1.2 π} & if n \in N \\ 0.5 e^{i π} & if n \in iN \end{matrix} .$

From the above discussion, it is quite clear that $A^{ξ} \approx A_{ρ}^{ξ}$ .

The following result establishes an ξ –complex fuzzy homomorphism between $A_{ρ}^{ξ} \in F^{ξ} (G / N)$ and B^ξ ∈ F^ξ (G′).

Theorem 4.11. Fundamental theorem of ξ –complex fuzzy homomorphism. Let A^ξ ∈ F^ξ (G), B^ξ ∈ F^ξ (G′) and f : A^ξ → B^ξ be an ξ –complex fuzzy epimorphism. Then a function φ : G/N → G′ is ξ –complex fuzzy epimorphism from $A_{ρ}^{ξ}$ to B^ξ, where $A_{ρ}^{ξ} \in F^{ξ} (G / N)$ .

Proof. By using the fact that f is an ξ –complex fuzzy homomorphism from A^ξ onto B^ξ, we have f (A^ξ) = B^ξ. Moreover, the given mapping φ : G/N → G′ is defined as: φ (mN) = f (m) = n, ∀ m ∈ G. Clearly, φ is a group homomorphism from G/N onto G′. Consider

$φ (A_{ρ}^{ξ}) (n) = μ_{φ (A_{ρ}^{ξ})} (n), n \in G^{'} = max {μ_{A_{ρ}^{ξ}} (mN) : mN \in φ^{- 1} (n)} =$

$max {μ_{A_{ρ}^{ξ}} (mN) : φ (mN) = n} = max {μ A^{ξ} (mx) : f (m) = n, x \in N} = \max {μ A^{ξ} (z) : z \in f^{- 1} (n)} = μ_{f (A^{ξ})} (n)$ .

Thus, $μ_{φ (A_{ρ}^{ξ})} (n) = μ_{B^{ξ}} (n)$ . Consequently, $φ (A_{ρ}^{ξ}) = B^{ξ}$ , which is the required ξ - complex fuzzy homomorphism from $A_{ρ}^{ξ}$ to B^ξ.

The above algebraic fact can be visualized in the following example.

Example 4.12. Let us define a group homomorphism f from G ={ 1, a, b, ab } to G′ ={ 1, - 1 } as; $f (m) = {\begin{matrix} 1 & if m \in {1, a} \\ - 1 & otherwise \end{matrix}$ .

The CFSG A and B of groups G and G′ are $μ_{A} (m) = {\begin{matrix} 1 e^{i 1.7 π} & if m \in {1, a} \\ 0.5 e^{i π} & otherwise \end{matrix}$ $and μ_{B} (m) = {\begin{matrix} 0.9 e^{i 1.7 π} & if m = 1 \\ 0.5 e^{i π} & if m = - 1 \end{matrix} .$

For ξ = 0.6e^i1.2π, the ξ –CFSG A^ξ of G and B^ξ of G′ are given by $μ_{A^{ξ}} (m) = {\begin{matrix} 0.6 e^{i 1.2 π} & if m \in {1, a} \\ 0.5 e^{i π} & otherwise \end{matrix}$ $and μ_{B^{ξ}} (m) = {\begin{matrix} 0.6 e^{i 1.2 π} & if m = 1 \\ 0.5 e^{i π} & if m = - 1 \end{matrix} .$

Note that, f is an ξ –complex fuzzy homomorphism from A^ξ to B^ξ, that is, $μ_{f (A^{ξ})} (1) = 0.6 e^{i 1.2 π} and μ_{f (A^{ξ})} (- 1) = 0.5 e^{i π} .$

The quotient group of G with respect to its normal subgroup N ={ 1, a } is G/N ={ N, bN }.

Define CFSG A_ρ on G/N as follows; $μ_{A_{ρ}} (z) = {\begin{matrix} 1 e^{i 2 π} & if z \in N \\ 0.5 e^{i 1.1 π} & if z \in bN \end{matrix} .$

The ξ –CFSG $A_{ρ}^{ξ}$ of G/N for the value of ξ = 0.6e^i1.2π is given as; $μ_{A_{ρ}^{ξ}} (z) = {\begin{matrix} 0.6 e^{i 1.2 π} & if z \in N \\ 0.5 e^{i 1.1 π} & if z \in bN \end{matrix} .$

The required ξ –complex fuzzy homomorphism from $A_{ρ}^{ξ}$ to B^ξ is obtained in the following way:

$μ_{φ (A_{ρ}^{ξ})} (1) = 0.6 e^{i 1.2 π}$ and $μ_{φ (A_{ρ}^{ξ})} (- 1) = 0.5 e^{i π}$ . Consequently, $A_{ρ}^{ξ} \approx B^{ξ}$ .

Remark 4.13. Let A^ξ ∈ F^ξ (G), B^ξ ∈ F^ξ (G′) and f : A^ξ → B^ξ be an ξ –complex fuzzy epimorphism with kerf = K. Then the function φ defined in the above theorem describes an ξ –complex fuzzy epimorphism from $A_{K}^{ξ}$ to B^ξ, where $A_{K}^{ξ} \in F^{ξ} (G / K)$ .

Theorem 4.14. Let A^ξ ∈ F^ξ (G), B^ξ ∈ F^ξ (G′), N ) G, f (N) = N′ and f : A^ξ → B^ξ be an ξ –complex fuzzy epimorphism. Then a group epimorphism η : G′ → G′/N′ establishes an ξ –complex fuzzy epimorphism from B^ξ to $A_{ρ^{'}}^{ξ}$ , Moreover, the map ψ = η ∘ f depicts an ξ –complex fuzzy epimorphism from A^ξ to $A_{ρ^{'}}^{ξ}$ , where $A_{ρ^{'}}^{ξ} \in F^{ξ} (G^{'} / N^{'})$ .

Proof. By using the fact that η and f are natural homomorphisms, we have (η ∘ f) (A^ξ) (g′N′) = μ_{(η∘f)(A^ξ)} (g′N′) = max {μ_{A
^ξ} (z) : z ∈ (η ∘ f) ^-1 (g′N′)} = max {μ_{A
^ξ} (z) : z ∈ f^-1 (η^-1 (g′N′))}, where, g′N′ ∈ G′/N′.

Therefore, μ_{(η∘f)(A^ξ)} (g′N′) = μ_f(A^ξ) (η^-1 (g′N′)).

Since f is an ξ –complex fuzzy homomorphism from A^ξ to B^ξ, therefore μ_{(η∘f)(A^ξ)} (g′N′) = μ_{B
^ξ} (η^-1 (g′N′)).

This shows that η is an ξ - complex fuzzy epimorphism from B^ξ to $A_{ρ^{'}}^{ξ}$ . Moreover, μ_{(η∘f)(A^ξ)} (g′N′) = μ_η(B^ξ) (g′N′). Consequently, $μ_{(η \circ f) (A^{ξ})} (g^{'} N^{'}) = μ_{A_{ρ^{'}}^{ξ}} (g^{'} N^{'})$ . Hence $ψ (A^{ξ}) = A_{ρ^{'}}^{ξ}$ , which is the required ξ –complex fuzzy homomorphism from A^ξ to $A_{ρ^{'}}^{ξ}$ . We prove the existence of ξ –complex fuzzy homomorphism between ξ –CFSG $A_{ρ}^{ξ}$ and $A_{ρ^{'}}^{ξ}$ in the following result.

Theorem 4.15. Let f be an epimorphism from G to G′ and let A^ξ and B^ξ be ξ –CFSG of G and G′ respectively such that f (A^ξ) = B^ξ. Let η be the natural epimorphism from G′ to G′/N′ (where N′ ) G′) such that N ={ m ∈ G : f (m) ∈ N′ }. Then a group epimorphism φ : G/N → G′/N′ is an ξ –complex fuzzy epimorphism from $A_{ρ}^{ξ}$ to $A_{ρ^{'}}^{ξ}$ , where $A_{ρ}^{ξ} \in F^{ξ} (G / N)$ and $A_{ρ^{'}}^{ξ} \in F^{ξ} (G^{'} / N^{'})$ .

Proof. The given epimorphism f may be defined as f (g) = g′, g′ ∈ G′. Moreover, the natural epimorphism η may be defined as η (g′) = g′N′. Then the epimorphism ψ = η ∘ f from G to G′/N′ is defined as ψ (g) = (η ∘ f) (g) = η (f (g)) = η (g′) = g′N′, ∀ g ∈ Gandg′ ∈ G′. Moreover, ψ (N) = (η ∘ f) (N) = η (f (N)) = η (N′) = N′. This means that ker(ψ) = N. It follows that ψ is an epimorphism from G to G′/N′ with N as its kernel. Hence there is an epimorphism φ from G/N to G′/N′ and is defined as: φ (gN) = g′N′.

Consider

$\begin{matrix} φ (A_{ρ}^{ξ}) (g^{'} N^{'}) = μ_{φ (A_{ρ}^{ξ})} (g^{'} N^{'}) \\ = max {μ A_{ρ}^{ξ} (gN) : gN \in φ^{- 1} (g^{'} N^{'})} = \\ max {max {μ_{A^{ξ}} (gn) : f (g) = g^{'}, n \in N, g \in G} : \\ φ (gN) = g^{'} N^{'}} = max {μ_{A^{ξ}} (gn) : f (g) = g^{'}, \\ n \in N, g \in G} = max {μ_{A^{ξ}} (z) : z \in f^{- 1} (g^{'})} \\ = μ_{B^{ξ}} (g^{'}) = μ_{B^{ξ}} (η^{- 1} (g^{'} N^{'})) = max {μ_{B^{ξ}} (g^{'}) : \\ η (g^{'}) = g^{'} N^{'}} = max {μ_{B^{ξ}} (g^{'}) : g^{'} \in η^{- 1} (g^{'} N^{'})} \\ = μ_{η (B^{ξ})} (g^{'} N^{'}) \end{matrix}$ .

Therefore, $μ_{φ (A_{ρ}^{ξ})} (g^{'} N^{'}) = μ_{A_{ρ^{'}}^{ξ}} (g^{'} N^{'})$ . Consequently, $φ (A_{ρ}^{ξ}) = A_{ρ^{'}}^{ξ}$ , which is the required ξ –complex fuzzy homomorphism from $A_{ρ}^{ξ}$ to $A_{ρ^{'}}^{ξ}$ .

Theorem 4.16. First fundamental theorem of ξ –complex fuzzy isomorphism. Let A^ξ ∈ F^ξ (G), B^ξ ∈ F^ξ (G′) and f : A^ξ → B^ξ be an ξ –complex fuzzy epimorphism with kerf = K. Then A^ξ/C^ξ ≅ B^ξ, where C^ξ is an ξ –CFNSG of A^ξ.

Proof. Consider an ξ –CFSG C^ξ as follows: $μ_{C^{ξ}} (m) = {\begin{matrix} μ_{A^{ξ}} (m) & if m \in K \\ 0 & if m \notin K \end{matrix} .$

Moreover, for any n ∈ C^ξ, the given ξ –CFNSG C^ξ of A^ξ is defined as $μ_{C^{ξ}} ({mnm}^{- 1}) = {\begin{matrix} μ_{A^{ξ}} ({mnm}^{- 1}) & if m \in K \\ 0 & if m \notin K \end{matrix} .$

By applying Theorem 4.4 and using the fact that A^ξ ≈ B^ξ, we have $f (A_{*}^{ξ})$ = $B_{*}^{ξ}$ . Consider an epimorphism $χ : A_{*}^{ξ} \to B_{*}^{ξ}$ with $\ker χ = C_{*}^{ξ}$ . Then there exists an isomorphism $ζ : (A_{*}^{ξ} / C_{*}^{ξ}) \to B_{*}^{ξ}$ defined as follows: $ζ ({mC}_{*}^{ξ}) = χ (m) = f (m) = x, \forall m \in A_{*}^{ξ}$ .

Suppose $x \in B_{*}^{ξ}$ , then $\begin{matrix} ζ (A^{ξ} / C^{ξ}) (x) = μ_{ζ (A^{ξ} / C^{ξ})} (x) = \max {μ_{A^{ξ} / C^{ξ}} ({mC}_{*}^{ξ}) : \\ m \in A_{*}^{ξ}, {mC}_{*}^{ξ} \in ζ^{- 1} (x)} \end{matrix}$ $\begin{matrix} = \max {μ_{A^{ξ} / C^{ξ}} ({mC}_{*}^{ξ}) : m \in A_{*}^{ξ}, ζ ({mC}_{*}^{ξ}) = x} \\ = \max {max {μ_{A^{ξ}} (y) : y \in {mC}_{*}^{ξ}} : m \in A_{*}^{ξ}, \\ χ (m) = x} = \max {μ_{A^{ξ}} (y) : y \in A_{*}^{ξ}, χ (y) = x} \\ = \max {μ_{A^{ξ}} (y) : y \in G, f (y) = x} = μ_{f (A^{ξ})} (x) \end{matrix}$

Therefore, μ_{ζ(A^ξ/C^ξ)} (x) = μ_{B
^ξ} (x). Hence ζ (A^ξ/C^ξ) = B^ξ. Consequently, A^ξ/C^ξ ≅ B^ξ.

Theorem (4.17) Second fundamental theorem of ξ –complex fuzzy isomorphism. Let A^ξ ∈ F^ξN (G) and B^ξ ∈ F^ξ (G), such that μ_{A
^ξ} (e) = μ_{B
^ξ} (e). Then there is a weak ξ –complex fuzzy isomorphism between B^ξ/A^ξ ∩ B^ξ and A^ξ ∘ B^ξ/A^ξ.

Proof. In view of Theorem (3.14), we have $A_{*}^{ξ}) G$ . In addition, by using the second fundamental theorem of group isomorphism, we have $B_{*}^{ξ} / (A^{ξ} \cap B^{ξ})_{*} ≅ (A^{ξ} \circ B^{ξ})_{*} / A_{*}^{ξ}$ , where the corresponding group isomorphism f is defined as follows:

$f (m {(A^{ξ} \cap B^{ξ})}_{*}) = {mA}_{*}^{ξ}, \forall m \in B_{*}^{ξ}$ .

Consider $\begin{matrix} f (B^{ξ} / (A^{ξ} \cap B^{ξ})) ({mA}_{*}^{ξ}) = μ_{f} (B^{ξ} / (A^{ξ} \cap B^{ξ})) ({mA}_{*}^{ξ}) \\ = max {μ_{B^{ξ} / (A^{ξ} \cap B^{ξ})} (m (A^{ξ} \cap B^{ξ})_{*}) : m \in B_{*}^{ξ}, \\ m (A^{ξ} \cap B^{ξ})_{*} \in f^{- 1} ({mA}_{*}^{ξ})} = max {μ_{B^{ξ} / (A^{ξ} \cap B^{ξ})} \\ (m (A^{ξ} \cap B^{ξ})_{*}) : m \in B_{*}^{ξ}, f (m (A^{ξ} \cap B^{ξ})_{*}) = {mA}_{*}^{ξ}} \\ = max {μ_{B^{ξ}} (x) : x \in m (A^{ξ} \cap B^{ξ})_{*}} ⩽ max {μ_{A^{ξ} {oB}^{ξ}} \\ (x) : x \in m (A^{ξ} \cap B^{ξ})_{*}} ⩽ max {μ_{A^{ξ} {oB}^{ξ}} (x) : \\ x \in {mA}_{*}^{ξ}} . \end{matrix}$

Therefore, $μ_{f (B^{ξ} / (A^{ξ} \cap B^{ξ}))} ({mA}_{*}^{ξ}) ⩽ μ_{A^{ξ} \circ B^{ξ} / A^{ξ}} ({mA}_{*}^{ξ})$ . Hence μ_{f(B^ξ/(A^ξ∩B^ξ))} ⩽ μ_{A^ξ∘B^ξ/A^ξ}. Thus, f (B^ξ/(A^ξ ∩ B^ξ)) ⊆ A^ξ ∘ B^ξ/A^ξ. Consequently, we obtain the required week ξ –complex fuzzy isomorphism between B^ξ/(A^ξ ∩ B^ξ) and A^ξ ∘ B^ξ/A^ξ.

Theorem 4.18. Third fundamental theorem of ξ –complex fuzzy isomorphism. Let A^ξ, B^ξ,C^ξ ∈ F^ξ (G) such that A^ξ and B^ξ are ξ –CFNSG of C^ξ and A^ξ ⊆ B^ξ. Then (C^ξ/A^ξ)/(B^ξ/A^ξ) ≅ C^ξ/B^ξ.

Proof. In view of Theorem (3.17), $A_{*}^{ξ}) C_{*}^{ξ}$ , $B_{*}^{ξ}) C_{*}^{ξ}$ and $A_{*}^{ξ}) B_{*}^{ξ}$ . The algebraic facts lead to note that the quotient groups $C_{*}^{ξ} / A_{*}^{ξ}$ , $C_{*}^{ξ} / B_{*}^{ξ}$ and $B_{*}^{ξ} / A_{*}^{ξ}$ are well defined. Moreover, we obtain the following isomorphism by means of third theorem of group isomorphism:

$(C_{*}^{ξ} / A_{*}^{ξ}) / (B_{*}^{ξ} / A_{*}^{ξ}) ≅ C_{*}^{ξ} / B_{*}^{ξ}$ , where the corresponding group isomorphism f is defined as follows: $f ({mA}_{*}^{ξ} (B_{*}^{ξ} / A_{*}^{ξ})) = {mB}_{*}^{ξ}, \forall m \in C_{*}^{ξ}$ Consider $\begin{matrix} f ((C^{ξ} / A^{ξ}) / (B^{ξ} / A^{ξ})) ({mB}_{*}^{ξ}) = μ_{f ((C^{ξ} / A^{ξ}) / (B^{ξ} / A^{ξ}))} ({mB}_{*}^{ξ}) \\ = μ_{(C^{ξ} / A^{ξ}) / (B^{ξ} / A^{ξ})} ({mA}_{*}^{ξ} (B_{*}^{ξ} / A_{*}^{ξ})), f is injective \\ = \max {μ_{C^{ξ} / A^{ξ}} ({nA}_{*}^{ξ}) : n \in C_{*}^{ξ}, {nA}_{*}^{ξ} \in {mA}_{*}^{ξ} (B_{*}^{ξ} / A_{*}^{ξ}) \\ = \max {\begin{matrix} \max {μ_{C^{ξ}} (x) : x \in {nA}_{*}^{ξ}} : \\ n \in C_{*}^{ξ}, {nA}_{*}^{ξ} \in {mA}_{*}^{ξ} (B_{*}^{ξ} / A_{*}^{ξ})} \end{matrix}} \\ = \max {μ_{C^{ξ}} (x) : x \in C_{*}^{ξ}, {xA}_{*}^{ξ} \in {mA}_{*}^{ξ} (B_{*}^{ξ} / A_{*}^{ξ})} \\ = \max {μ_{C^{ξ}} (x) : x \in {mA}_{*}^{ξ} (B_{*}^{ξ} / A_{*}^{ξ})} = \max {μ_{C^{ξ}} (x) : \\ x \in C_{*}^{ξ}, f (x) \in {mB}_{*}^{ξ}} = μ_{C^{ξ} / B^{ξ}} ({mB}_{*}^{ξ}), \forall m \in C_{*}^{ξ} \end{matrix} .$

Therefore, f ((C^ξ/A^ξ)/(B^ξ/A^ξ)) = C^ξ/B^ξ. Consequently, (C^ξ/A^ξ)/(B^ξ/A^ξ) ≅ C^ξ/B^ξ.

5 Solutions of decision-making problems under the environment of ξ –complex fuzzy homomorphism

Science and technology questioned the concept of similarity in art and that the rhythm of scientific and technological discoveries at the turn of the 20th century was parallel to the transition from the concept of similarity to the concept of homomorphism in the conceptualization of artistic representation. Homomorphism refers to a representation that allocates with a point correspondence between depicted objects and perceptual data. Logically, a fuzzy homomorphism is such a mathematical tool between two fuzzy varieties which preserves their properties and relations. Despite of all advantages of fuzzy homomorphism, we still come across immense complications to counter various physical challenges based on a real valued membership function. It is, therefore, very essential to present the notion of ξ –complex fuzzy homomorphism as a powerful extension of classical fuzzy homomorphism which is quite capable of counter such situations due to its structural flexibilities. This section summarizes the significance of ξ –complex fuzzy homomorphism by virtue of some important applications of this phenomenon to various decision-making problems.

The Fig. 1 indicates some of the physical challenges that can easily be countered under the environment of ξ –complex fuzzy homomorphism.

Fig. 1

Applications of ξ –complex fuzzy homomorphism.

For instance, consider the application of ξ –complex fuzzy homomorphism in positioning of the image. A photograph of a person is in fact his homomorphic image that interprets his many biological features like tall or short, heavy or thin, old or young and male or female. Sometime the homomorphic image becomes corrupted due to many distortions (like scale and pincushion distortion) in the lenses. A scale distortion enlarges or reduces an object horizontally, vertically or both. This kind of distortion can be viewed in the following Fig. 2.

Fig. 2

Scale distortion.

A distortion in which magnification increases with distance from the axis is called pincushion distortion. The Fig. 3 describes the existence of pincushion distortion in the image of an object:

Fig. 3

Pincushion distortion.

We apply ξ –complex fuzzy homomorphism on a corrupted photograph to remove scale and pincushion distortion in order to get its original form.

Example 5.1. The homomorphic image of a ten sided polygone in the context of complex fuzzy homomorphism contains two types of distortions (scale and pincushion). The following table indicates the scale and pincushion distortion of the corrupted homomorphic image by means of membership function of a complex fuzzy set A where r_A and ω_A represent the degrees of involment of the scale and pincushion distortions respectively.

Table 1

Corrupted homomorphic image in the framework of CFS

Corrupted image in terms of CFS	Distortions in terms of Membership Grades	Corrupted image in terms of CFS	Distortions in terms of Membership Grades
r_A (a) e^iω_A(a)	0.42e^i0.4π	r_A (f) e^iω_A(f)	0.5e^i0.3π
r_A (b) e^iω_A(b)	0.7e^i0.5π	r_A (g) e^iω_A(g)	0.423e^i0.5π
r_A (c) e^iω_A(c)	0.91e^i1.8π	r_A (h) e^iω_A(h)	0.72e^i1.4π
r_A (d) e^iω_A(d)	0.82e^i1.5π	r_A (i) e^iω_A(i)	1.0e^i1.7π
r_A (e) e^iω_A(e)	0.6e^i0.6π	r_A (j) e^iω_A(j)	0.85e^i0.5π

The scale and pincushion distortions in the corrupted homomorphic image can be viewed in the Figs. 4 and 5.

Fig. 4

Diagrammatic view of scale distortion.

Fig. 5

Diagrammatic view of pincushion distortion.

Let ξ = αe^iβ be a parameter through which the corrupted homomorphic image is refined using ξ –complex fuzzy homomorphism, where α and β denote the degrees of refinement in scale and pincushion distortions respectively. Initially, we obtain the refined homomorphic image after removing the scale distortion by setting α = 0.4. This can be viewed in the following diagram.

Now we eliminate the effect of pincushion distortion from the image by setting β = 0.3π. In this way, the original form of the homomorphic image looks like as:

Comparative Analysisof the application of ξ –complex fuzzy homomorphism: When we apply complex fuzzy homomorphism on a ten sided polygon, the homomorphic image contains two types of distortions (scale and pincushion) as shown in Figs. 4 and 5 respectively. These distortions have been removed from the homomorphic image in the frame work of ξ –complex fuzzy homomorphism by setting an appropriate value of the parameter ξ. This process has graphically been shown in Figs. 6 and 7. From the above discussion, it is quite evident that the notion of ξ –complex fuzzy homomorphism is more effective tool than the concept of complex fuzzy homomorphism in order to obtain the solutions of various real-world problems based on complex valued functions.

Fig. 6

Homomorphic image after removing scale distortion.

Fig. 7

Purified homomorphic image.

Note: Many useful applications of fundamental theorems of ξ –complex fuzzy isomorphism to solve many real-world situations will be explored in our next paper.

6 Conclusion

In the present work, we have introduced the concept of ξ –CFSG over ξ –CFS and have proved their numerous algebraic postulates. Moreover, the notions of ξ –complex fuzzy homomorphism and ξ –complex fuzzy isomorphism between any two ξ –CFSG have also been proposed in this article. We have also explored the importance of these newly defined ideas by presenting their fundamental theorems. In addition, the significance of ξ –complex fuzzy homomorphism has also been highlighted by presenting some important applications of this phenomenon to the solutions of various decision-making problems.

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Image development in the framework of ξ – complex fuzzy morphisms

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Fundamental algebraic attributes of ξ –CFSG

References