A note on direct product of complex intuitionistic fuzzy subfield

1 Introduction

Human life is mostly shrouded in mystery. Accurate calculations and hypotheses do not apply in our reality. For human intellect, this assessment inaccuracy is quite troubling. Therefore, complex fuzzy sets and fuzzy sets are among the mathematical ideas that have been developed as practical solutions to this problem. Complex fuzzy logic was developed using a group of ambiguous information. Moreover, complex intuitionistic fuzzy sets have an elastic property that can withstand unreliability. This occurrence is considered extraordinarily significant for the humanistic and undoubtedly a fundamental component of traditional complex fuzzy sets. On the other hand, fuzzy mathematics is the area of mathematics that deals with fuzzy set theory. Zadeh [1] explored the properties of fuzzy sets, which are an extension of the concept of classical sets. Rosenfeld [2] created a fuzzy subgroup by combining group theory with a fuzzy set. Consequently, Jana and Pal [3] established a bridge to connect a soft set, and the union operations on sets then applied it to BCK/BCI-algebras. They also introduced the notion of the (α, β)-Union-Soft ((α, β)-US) set and discussed the soft BCK/BCI-algebras. Al-Masarwah and Ahmad [4] introduced subalgebras of type (α, β) based on m-polar fuzzy points in BCK/BCI-algebras. Rasuli [5] discussed fuzzy subgroups on the direct product of groups over a t-norm and characterized some basic properties of the T-fuzzy direct product of groups. Meanwhile, Jana and Pal [6] studied the application of (α, β)-soft intersectional sets on BCK/BCI-algebras, and some properties of (α, β)-soft intersectional BCK/BCI-algebras were investigated. Ejegwa [7] established a generalized direct product of fuzzy multigroup. Then the concept of fuzzy subgroups was generalized by many researchers [8–10].

Moreover, Kausar and Waqar [11] introduced a direct product of finite fuzzy normal subrings over non-associative rings and investigated some basic characteristics of the direct product of fuzzy normal subrings. Correspondingly, Kausar et al. [12] defined the direct product of finite anti-fuzzy normal subrings over non-associative and noncommutative rings. Furthermore, Jana et al. [13] introduced the notion of (∈ , ∈ ∨  _q )-bipolar fuzzy subalgebras and ideals of BCK/BCI-algebras and their related properties were investigated. Al-Masarwah et al. [14] presented the idea of m-polar fuzzy positive implicative ideals of BCK-algebras. Subsequently, Malik and Mordeson [15] identified several fuzzy subfield properties. Meanwhile, Rasuli [16] demonstrated the notions of anti-Q-fuzzy subring and Q-fuzzy subring and some of their properties were verified under norms. Giri and Ananth [17] investigated the anti S-fuzzy subfield and proved some theorems related to it, while Hussain [18] introduced the Q-fuzzy field and gave some related theorems.

In addition, Atanassov [19] proposed the idea of intuitionistic fuzzy sets. Hur et al. [20] investigated the properties of intuitionistic fuzzy subrings and intuitionistic fuzzy subgroups. Meanwhile, Jana and Pal [21] presented the generalized intuitionistic fuzzy ideals of BCK/BCI-algebras based on 3-valued logic and its computational study. Moreover, Joshi et al. [22] discussed intuitionistic fuzzy parameterized fuzzy soft set theory and its application. After that, Yilmaz and Çuvalcioğlu [23] defined (T,S)-intuitionistic fuzzy algebras, and some properties were proved. Castillo et al. [24] also presented intuitionistic fuzzy control of twin rotor multiple input multiple output systems. On the other hand, Abed Alhaleem and Ahmad [25] extended the normed rings to intuitionistic fuzzy normed rings. Nguyen and Thu [26] proved fixed point results for intuitionistic fuzzy mappings and their application. Later, Gulistan et al. [27] defined the direct product of (∈ , ∈ ∨ q_k)-intuitionistic fuzzy sets and direct product of (∈ , ∈ ∨ q_k)-intuitionistic fuzzy soft sets of subtraction algebras and investigated some related properties. Traneva and Tranev [28] developed an intuitionistic fuzzy two-factor variance analysis of movie ticket sales. Subsequently, Abed Alhaleem and Ahmad [29, 30] discussed the intuitionistic fuzzy and anti-fuzzy normal subrings over normed rings. Zhang et al. [31] also introduced the intuitionistic fuzzy subfield and investigated its characteristics. Correspondingly, Anandh and Giri [32] proposed the concept of intuitionistic fuzzy subfields in terms of (T,S)-norm. Vasu [33] defined the properties of an intuitionistic multi-fuzzy subfield.

Furthermore, the idea of a complex fuzzy set was developed by Ramot et al. [34]. They expanded the fuzzy set’s range to include the unit disc in the plane of complex instead of the interval [0,1] to present their innovative notion of complex fuzzy sets. Al-Qudah and Hassan [35] defined operations on complex multi-fuzzy sets. Then, the concept of a complex intuitionistic fuzzy set was proposed by Alkouri and Salleh [36] as an expansion of a complex fuzzy set. Later, Alsarahead and Ahmad [37] presented the complex fuzzy subring and established some new concepts such as the π-fuzzy subring and π-fuzzy ideal. The study of complex intuitionistic fuzzy groups was continued by Al-Husban et al. [38] who, introduced the concept of a complex intuitionistic fuzzy normal subgroup. Many comparisons and applications have been made between the results of complex intuitionistic fuzzy sets and situations in real life. Recent studies have discussed the applications of complex intuitionistic fuzzy sets to solve one of the famous real-life problems, which is the decision-making problem. Garg and Kumar [39] studied the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision-making. Then, Garg and Rani [40] examined a robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making. After that, Azam et al. [41] demonstrated a decision-making approach for evaluating information security management under a complex intuitionistic fuzzy set environment. Moreover, Al-Sharqi et al. [42] established similarity measures on interval-complex neutrosophic soft sets with applications to decision-making and medical diagnosis under uncertainty. Alsarahead and Ahmad [43] developed the complex intuitionistic fuzzy subring and proposed several ideas, including homogeneous complex intuitionistic fuzzy sets and intuitionistic π-fuzzy sets. Jana et al. [44] studied the emerging applications of fuzzy algebraic structures. On the other hand, Gulzar et al. [45] investigated the homomorphic image and inverse image of complex intuitionistic fuzzy subgroups under group homomorphism. Gulzar et al. [46] presented the concept of the direct product of a complex intuitionistic fuzzy subring and showed its algebraic properties. Rifayathali [47] introduced the notion of complex anti-Q-fuzzy subring and discussed its different algebraic properties. They proved that every generated complex anti-Q-fuzzy subring had two anti-Q-fuzzy subrings. Recently, Khamis and Ahmad [48] presented the concept of a complex intuitionistic Q-fuzzy subfield, and they defined the necessity and possibility operators on a complex intuitionistic Q-fuzzy subfield. Al-Sharqi et al. [49–52] also proposed the application of the interval-valued complex neutrosophic soft set in decision-making.

Furthermore, Gulzar et al. [53] introduced the concept of a complex fuzzy subfield as a new structure. Nevertheless, they did not use the idea of intuitionistic fuzzy sets for a complex fuzzy subfield. Hence, in this research, we add the idea of intuitionistic fuzzy sets to a complex fuzzy subfield. We expand the complex fuzzy subfield to CIFSF and present the direct product of CIFSF, and many new results are discussed.

The contributions of this paper lie in introducing the concepts of CIFSF and the direct product of CIFSF. We extend the idea of a complex fuzzy subfield to CIFSF by combining the concept of intuitionistic fuzzy sets with a complex fuzzy subfield. Moreover, we demonstrate that the direct product of two CIFSFs is CIFSF. Basic properties of the direct product of CIFSF are also presented. In addition, the definition of necessity and possibility operators on the direct product of CIFSF are defined, and some related theorems are proven. Furthermore, we establish the level subsets of the direct product of two CIFSFs of a field. We also indicate that the level subset of the direct product of two CIFSFs is a subfield and prove some related results.

The motivation for undertaking this research is that the idea of research is new and has not been studied before. Also, in this research, new concepts are presented which are CIFSF, and the direct product of CIFSF. Furthermore, some basic properties are studied, and related theorems are proven. These new results will be a new scientific addition to the field of mathematics.

This work is divided into the following sections. Section 2 presents fundamental concepts and preliminary findings that will be utilized in this study. In Section 3, we propose the concepts of CIFSF and the direct product of CIFSF and describe the notions of the necessity and possibility operators on a direct product of CIFSF. Also in this section, we introduce the level subsets of the direct product of two complex intuitionistic fuzzy subsets of a field, and some fundamental theorems are investigated. Finally, Section 4 provides a summary of this research.

2 Preliminaries

We review some fundamental definitions, theorems, and findings of fuzzy sets and fuzzy subfields in this part that will be utilized throughout the study. A fuzzy set is a generalization of a classical set that enables any element to be included in the closed interval [0, 1] by a membership function. In intuitionistic fuzzy sets, we deal with both membership and non-membership degrees with the added condition that the total of these degrees should not be more than one. By Ramot et al. [34] the idea of a complex fuzzy set was created. Instead of using the interval  [0, 1], they expanded the range of fuzzy sets to the unit disc in the complex plane.

Definition 2.1. [1] In a universal set X, a fuzzy set ψ is a membership function ψ : X → [0, 1] that accepts values from X to the range [0, 1] and symbolized by μ _ψ (k). A fuzzy set ψ is represented by: ψ = { 〈 k , μ ψ ( k ) : k 〉 ∈ X } , where for every k ∈ X, 0 ⩽ μ _ψ (k) ⩽ 1.

If ψ is a set in the ordinary sense, then its membership function can only accept the values 0 and 1, with μ _ψ (k) = 1 or depending on whether or not k belongs to ψ. As a result, in this situation, μ _ψ (k) simplifies to the well-known characteristic function of a set.

The classical set theory views the world as either black or white. Let X be the universe of discourse and k be its elements. According to the classical set theory, crisp set ψ of X is defined by the characteristic function f _ψ (k) of set ψ as: f ψ ( k ) = { 1 , if k ∈ ψ 0 , if k ∉ ψ .

Definition 2.2. [1] let ψ ={ 〈 k, μ _ψ (k) 〉 : k ∈ X } in a fixed universe X be a fuzzy set. Then the complement set of the fuzzy set ψ is the fuzzy set in X that denoted by ψ ¯ , and the following formula is used to determine the membership function of the complement set: ψ ¯ ( k ) = 1 - μ ψ ( k ) , ∀ k ∈ X .

Definition 2.3. [1] Let ψ₁ ={ 〈 k, μ _ψ1 (k) 〉 : k ∈ X } and ψ₂ ={ 〈 k, μ _ψ2 (k) 〉 : k ∈ X } be two fuzzy sets. It is said to be that ψ₁ is contained in ψ₂ (or ψ₁ is smaller than or equal to ψ₂, or ψ₁ is a subset of ψ₂, or equivalently) if and only if μ _ψ1 (k) ⩽ μ _ψ2 (k). It is written as: ψ 1 ⊂ ψ 2 ⇔ μ ψ 1 ( k ) ⩽ μ ψ 2 ( k ) .

Many of the fundamental identities that hold for ordinary sets were extended to fuzzy sets with the operations of union, intersection, and complementation. Consider De Morgan’s laws as an illustration: ψ 1 ∪ ψ 2 ¯ = ψ 1 ¯ ∩ ψ 2 ¯ , ψ 1 ∩ ψ 2 ¯ = ψ 1 ¯ ∪ ψ 2 ¯ .

It is possible to establish such equality by demonstrating that the relevant relations for the membership functions of ψ₁ and ψ₂ are identities. It will be shown by the following proposition.

Proposition 2.1. [1] Let ψ₁ ={ 〈 k, μ _ψ1 (k) 〉 :   k ∈ X }, ψ₂ ={ 〈 k, μ _ψ2 (k) 〉 : k ∈ X } two fuzzy sets. Then for every k ∈ X:

1 - max { μ ψ 1 ( k ) , μ ψ 2 ( k ) } = min { 1 - μ ψ 1 ( k ) , 1 - μ ψ 2 ( k ) } ,

Malik and Moreson (1990) studied the notion of fuzzy subfields and developed some basic properties of fuzzy subfields. The fuzzy subfield is combined between the fuzzy set and subfield and defined as the following definition.

Definition 2.4. [15] Let (F, + , ·) is a field. A fuzzy subset ψ ={ 〈 k, μ _ψ (k) 〉 : k ∈ F } of a field F is called a fuzzy subfield if:

μ _ψ (k - m)⩾ min { μ _ψ (k) , μ _ψ (m) },

Theorem 2.1 [15] The intersection of any two fuzzy subfields is a fuzzy subfield.

Atanassov [19] defined the concept of an intuitionistic fuzzy set. He extended the concept of a fuzzy set to an intuitionistic fuzzy set. He presented an intuitionistic fuzzy set as: ψ = { 〈 k , μ ψ ( k ) , γ ψ ( k ) 〉 : k ∈ X } , where 0 ⩽ μ _ψ (k) + γ _ψ (k) ⩽ 1.

Definition 2.5. [19] Let ψ be a set in a nonempty set of X. It is said to be that ψ is an intuitionistic fuzzy set if it is described as the following form: ψ = { 〈 k , μ ψ ( k ) , γ ψ ( k ) 〉 : k ∈ X } , where its membership and non-membership functions are μ _ψ (k) : X → [0, 1], γ _ψ (k) : X → [0, 1], respectively, with the following condition: 0 ⩽ μ ψ ( k ) + γ ψ ( k ) ⩽ 1 .

Definition 2.6. [19] An intuitionistic fuzzy set ψ is defined as ψ ={ 〈 k, μ _ψ (k) , γ _ψ (k) 〉 : k ∈ X }. The following is the definition of the complement of ψ: ψ ¯ = { 〈 k , γ ψ ( k ) , μ ψ ( k ) 〉 : k ∈ X } .

Definition 2.7. [54] Let X is nonempty set. The necessity and possibility operators on an intuitionistic fuzzy set ψ ={ 〈 k, μ _ψ (k) , γ _ψ (k) 〉 : k ∈ X } are given as:

□ψ ={ 〈 k, μ _ψ (k) , 1 - μ _ψ (k) 〉 : k ∈ X },

The novel idea of complex fuzzy sets was examined by Ramot et al. [34]. In contrast to a usual fuzzy membership function, this range is expanded to the complex plane’s unit circle rather than being constrained to the range [0,1]. Consequently, the complex fuzzy set offers a mathematical foundation for representing membership in a set in terms of a complex number.

Definition 2.8. [34] A complex fuzzy set (CFS) ψ, defined on a universe of discourse X, is characterized by a membership functionμ _ψ (k), that assigns to any elementk ∈ X a complex-valued grade of membership in ψ. By definition, the values of μ _ψ (k), may receive all lying within the unit circle in the complex plane, and its form is μ _ψ (k) = r _ψ (k) e^{iω _ψ(k)}, where i = - 1 , each of r _ψ (k) and ω _ψ (k) are both real-valued such that the value of r _ψ (k) ∈ [0, 1] and ω _ψ (k) ∈ [0, 2π]. The complex fuzzy set ψ may be represented as the set of ordered pairs: ψ = { 〈 k , μ ψ ( k ) 〉 : k ∈ X } .

Note that a fuzzy set with a complex-valued membership function is referred to as a complex fuzzy set, whereas a typical fuzzy set with a real-valued membership function is referred to as a fuzzy set.

Definition 2.9. [34] Let ψ be complex fuzzy sets of setX for all k ∈ X . The complement of complex fuzzy set ψ is specified by a function: μ ψ ¯ ( x ) = r ψ ¯ ( x ) e i ω ψ ¯ ( x ) = { 1 - r ψ ( x ) } e i { 2 π - ω ψ ( x ) } .

The complex intuitionistic fuzzy set represents the problems with uncertainty and periodicity simultaneously and it is described as the following definition.

Definition 2.10. [36] A complex intuitionistic fuzzy set (CIFS) ψ, defined on a universe of discourse X, is characterized by membership function: μ ψ ( k ) = r ψ ( k ) e i ω ψ ( k ) , and non-membership function: γ ψ ( k ) = r ˆ ψ ( k ) e i ω ˆ ψ ( k ) , that assign any element k ∈ X a complex-valued grade of both membership and non-membership inψ. By definition, ψ ={ 〈 k, μ _ψ (k) , γ _ψ (k) 〉 : k ∈ X } where r ψ ( k ) + r ˆ ψ ( k ) ⩽ 1 .

It is important to note that one can obtain the traditional intuitionistic fuzzy set by choosing the value ω ψ ( k ) = ω ˆ ψ ( k ) = 0 in definition (2.10). The term CIFS refers to an intuitionistic fuzzy set with complex-valued membership and non-membership functions, while the term intuitionistic fuzzy set refers to a conventional intuitionistic fuzzy set with real-valued membership and non-membership functions.

Definition 2.11. [36] Let X nonempty set. Let ψ on X where ψ ={ 〈 k, μ _ψ (k) , γ _ψ (k) 〉 : k ∈ X } is a complex intuitionistic fuzzy set. The complement of ψ is denoted by ψ ¯ and defined as: ψ ¯ = { 〈 k , γ ψ ( k ) , μ ψ ( k ) 〉 : k ∈ X } = { 〈 k , r ˆ ψ ( k ) e i ω ˆ ψ ¯ ( k ) , r ψ ( k ) e i ω ψ ¯ ( k ) 〉 : k ∈ X } , where ω ψ ¯ ( k ) = ω ψ ( k ) , 2 π - ω ψ ( k ) , or ω ψ ( k ) + π .

Definition 2.12. [43] Let ψ be an intuitionistic fuzzy set where ψ ={ 〈 k, μ _ψ (k) , γ _ψ (k) 〉 : k ∈ X }. Then ψ _π (k) ={ 〈 k, ν _{ψ π} (k) , ρ _{ψ π} (k) 〉 : k ∈ X } is said to be an intuitionistic π-fuzzy set whereν _{ψ π} (k) = 2πμ _ψ (k) and ρ _{ψ π} (k) = 2πγ _ψ (k). Note that the conditionν _{ψ π} (k)+ρ _{ψ π} (k) ⩽ 2π is already satisfied.

Definition 2.13. [53] Let (F, + , ·) be a field. A complex fuzzy set ψ ={ 〈 k, μ _ψ (k) 〉 : k ∈ F } of a field F is said to be a complex fuzzy subfield of a field F if it satisfies the following for all k, m ∈ F:

μ _ψ (k - m) ⩾ min { μ _ψ (k) , μ _ψ (m) },

Furthermore, the level subset of a complex fuzzy set ψ is denoted by ψ_(δ,σ) where the symbol δ ∈ [0, 1] and σ ∈ [0, 2π] and described as the following definition.

Definition 2.14. [53] Let ψ ={ 〈 k, μ _ψ (k) 〉 : k ∈ X } be a complex fuzzy subset of the universe of discourseX with membership function μ _ψ (k) = r _ψ (k) e^{iω _ψ(k)}. For δ ∈ [0, 1] andσ ∈ [0, 2π] the level subset of a complex fuzzy set ψ is denoted by ψ_(δ,σ) and defined as: ψ ( δ , σ ) = { k ∈ X : r ψ ( k ) ⩾ δ , ω ψ ( k ) ⩾ σ } .

For σ = 0, we obtain the lower-level subset: ψ δ = { k ∈ X : r ψ ( k ) ⩾ δ } , and for δ = 0, then we obtain the lower-level subset: ψ σ = { k ∈ X : ω ψ ( k ) ⩾ σ } .

Theorem 2.2. [53] Let ψ ={ 〈 k, μ _ψ (k) 〉 : k ∈ F } be a complex fuzzy set of a field F. Then ψ is a complex fuzzy subfield of a field F if and only if ψ_(δ,σ) is a subfield of a field F, for all δ ∈ [0, 1] and σ ∈ [0, 2π], where r _ψ (0) ⩾ δ ⩽ r _ψ (1) , ω _ψ (0) ⩾ σ ⩽ ω _ψ (1).

In mathematics, the cartesian product of sets ψ₁ and ψ₂ is defined as the set of all ordered pairs (k, m)such that k belongs toψ₁and m belongs to ψ₂. The following definitions show the cartesian product of complex fuzzy sets and complex intuitionistic fuzzy subsets.

Definition 2.15. [55] Let ψ₁ andψ₂ two complex fuzzy sets of universe of discourse X. The cartesian product of complex fuzzy setsψ₁ and ψ₂ is represented by a function: μ ψ 1 × ψ 2 ( k , m ) = r ψ 1 × ψ 2 ( k , m ) e i ω ψ 1 × ψ 2 ( k , m ) = min { r ψ 1 ( k ) , r ψ 2 ( m ) } e imin { ω ψ 1 ( k ) , ω ψ 2 ( m ) } .

Theorem 2.3. [53] Let ψ₁ and ψ₂ be two complex fuzzy subfields of fields F₁ and F₂, respectively. Then ψ₁ × ψ₂ is complex fuzzy subfield of a field F₁ × F₂.

Definition 2. 16. [56] Let ψ₁ and ψ₂ be two complex intuitionistic fuzzy subsets of X₁ and X₂, respectively, such that ψ₁ ={ 〈 k, μ _ψ1 (k) , γ _ψ1 (k) 〉 : k ∈ X₁ } and ψ₂ ={ 〈 k, μ _ψ2 (k) , γ _ψ2 (k) 〉 : k ∈ X₂ }. Then their direct product is symbolized by ψ₁ × ψ₂ and represented as: ( ψ 1 × ψ 2 ) = { 〈 ( k , m ) , μ ψ 1 × ψ 2 ( k , m ) , γ ψ 1 × ψ 2 ( k , m ) 〉 : ( k , m ) ∈ X 1 × X 2 }

where for all (k, m) ∈ X₁ × X₂: μ ψ 1 × ψ 2 ( k , m ) = r ψ 1 × ψ 2 ( k , m ) e i ω ψ 1 × ψ 2 ( k , m ) = min { r ψ 1 ( k ) , r ψ 2 ( m ) } e imin { ω ψ 1 ( k ) , ω ψ 2 ( m ) } , γ ψ 1 × ψ 2 ( k , m ) = r ˆ ψ 1 × ψ 2 ( k , m ) e i ω ˆ ψ 1 × ψ 2 ( k , m ) = max { r ˆ ψ 1 ( k ) , r ˆ ψ 2 ( m ) } e imax { ω ˆ ψ 1 ( k ) , ω ˆ ψ 2 ( m ) } .

3 Direct product of complex intuitionistic fuzzy subfield

In this part, we begin by introducing the definitions of complex intuitionistic fuzzy subfields and the direct product of complex intuitionistic fuzzy subfields. We study some basic properties of direct product complex intuitionistic fuzzy subfields and prove some related theorems.

Definition 3.1. Let (F, + , ·) be a field. If a homogeneous complex intuitionistic fuzzy subset ψ ={ 〈 k, μ _ψ (k) , γ _ψ (k) 〉 : k ∈ F } of a field F holds the following conditions for any k, m ∈ F, it is said to be a complex intuitionistic fuzzy subfield (CIFSF) of that field:

μ _ψ (k - m)⩾ min { μ _ψ (k) , μ _ψ (m) },

We can define the complex intuitionistic anti-fuzzy subfield (CIAFSF) by the following definition.

Definition 3.2. Let (F, + , ·) be a field. If a homogeneous complex intuitionistic fuzzy subset ψ ={ 〈 k, μ _ψ (k) , γ _ψ (k) 〉 : k ∈ F } of a field F holds the following conditions for any k, m ∈ F, it is said to be a complex intuitionistic anti-fuzzy subfield (CIAFSF) of that field:

μ _ψ (k - m)⩽ max { μ _ψ (k) , μ _ψ (m) },

If we have two π-intuitionistic fuzzy sets ψ_1π and ψ_2π of fields F₁ and F₂, respectively, that are defined in the definition (2.12), then the direct product of ψ_1π and ψ_2π is given by the following definition.

Definition 3.3. Let F₁ and F₂ be two fields where ψ₁ in F₁, ψ₂ in F₂ are two π-intuitionistic fuzzy sets, such that ψ_1π ={ 〈 k, ν _ψ1π (k) , ρ _ψ1π (k) 〉 : k ∈ F₁ } and ψ_2π ={ 〈 k, ν _ψ2π (k) , ρ _ψ2π (k) 〉 : k ∈ F₂ }. Then their direct product of ψ_1π and ψ_2π is represented by ψ_1π × ψ_2π and is described as: ψ 1 π × ψ 2 π = { 〈 ( k , m ) , ν ψ 1 π × ψ 2 π ( k , m ) , ρ ψ 1 π × ψ 2 π ( k , m ) 〉 : ( k , m ) ∈ F 1 × F 2 }

The direct product of two CIFSFs ψ₁ and ψ₂ of fields F₁ and F₂, respectively can be presented by the membership function: μ ψ 1 × ψ 2 ( k , m ) = r ψ 1 × ψ 2 ( k , m ) e i ω ψ 1 × ψ 2 ( k , m ) , and non-membership function: γ ψ 1 × ψ 2 ( k , m ) = r ˆ ψ 1 × ψ 2 ( k , m ) e i ω ˆ ψ 1 × ψ 2 ( k , m ) .

For all k ∈ F₁, m ∈ F₂. The following definition shows this concept.

Definition 3.4. Let fields F₁ and F₂, have two CIFSFs ψ₁ ={ 〈 k, μ _ψ1 (k) , γ _ψ1 (k) 〉 : k ∈ F₁ } and the other set ψ₂ ={ 〈 k, μ _ψ2 (k) , γ _ψ2 (k) 〉 : k ∈ F₂ }. Then their direct product is denoted by ψ₁ × ψ₂ and for all (k, m) ∈ F₁ × F₂ defined as: ψ 1 × ψ 2 = { 〈 ( k , m ) , μ ψ 1 × ψ 2 ( k , m ) , γ ψ 1 × ψ 2 ( k , m ) 〉 : ( k , m ) ∈ F 1 × F 2 } ,

where μ ψ 1 × ψ 2 ( k , m ) = r ψ 1 × ψ 2 ( k , m ) e i ω ψ 1 × ψ 2 ( k , m ) = min { r ψ 1 ( k ) , r ψ 2 ( m ) } e imin { ω ψ 1 ( k ) , ω ψ 2 ( m ) } , γ ψ 1 × ψ 2 ( k , m ) = r ˆ ψ 1 × ψ 2 ( k , m ) e i ω ˆ ψ 1 × ψ 2 ( k , m ) = max { r ˆ ψ 1 ( k ) , r ˆ ψ 2 ( m ) } e imax { ω ˆ ψ 1 ( k ) , ω ˆ ψ 2 ( m ) } .

By the following theorem, we can prove that the direct product of two complex intuitionistic fuzzy subfields will be a complex intuitionistic fuzzy subfield.

Theorem 3.1. Let fields F₁ and F₂, have two CIFSFs ψ₁ and ψ₂. Then ψ₁ × ψ₂ is CIFSF of a field F₁ × F₂.

Proof. Let ψ₁ ={ 〈 k, μ _ψ1 (k) , γ _ψ1 (k) 〉 : k ∈ F₁ } and ψ₂ ={ 〈 k, μ _ψ2 (k) , γ _ψ2 (k) 〉 : k ∈ F₂ } be two complex intuitionistic fuzzy subfields of fields F₁ and F₂, respectively. We must demonstrate that ψ₁ × ψ₂ satisfies all conditions of a complex intuitionistic fuzzy subfield to establish that ψ₁ × ψ₂ is a complex intuitionistic fuzzy subfield.

For each (k, m) , (a, b) ∈ F₁ × F₂, we have μ ψ 1 × ψ 2 ( ( k , m ) - ( a , b ) ) = μ ψ 1 × ψ 2 ( k - a , m - b ) = r ψ 1 × ψ 2 ( k - a , m - b ) e i ω ψ 1 × ψ 2 ( k - a , m - b ) = min { r ψ 1 ( k - a ) , r ψ 2 ( m - b ) } e imin { ω ψ 1 ( k - a ) , ω ψ 2 ( m - b ) } = min { r ψ 1 ( k - a ) e i ω ψ 1 ( k - a ) , r ψ 2 ( m - b ) e i ω ψ 2 ( m - b ) } = min { μ ψ 1 ( k - a ) , μ ψ 2 ( m - b ) } ⩾ min { min { μ ψ 1 ( k ) , μ ψ 1 ( a ) } , min { μ ψ 2 ( m ) , μ ψ 2 ( b ) } } = min { min { μ ψ 1 ( k ) , μ ψ 2 ( m ) } , min { μ ψ 1 ( a ) , μ ψ 2 ( b ) } } = min { μ ψ 1 × ψ 2 ( k , m ) , μ ψ 1 × ψ 2 ( a , b ) } .

Moreover,

μ _ψ1×ψ2 ((k, m) (a, b))

= μ _ψ1×ψ2 (ka, mb)

= r _ψ1×ψ2 (ka, mb) e^{iω _ψ1×ψ2(ka,bm)}

= min { r _ψ1 (ka) , r _ψ2 (mb) } e^{imin{ω _ψ1(ka),ω _ψ2(mb)}}

= min{ r _ψ1 (ka) e^{iω _ψ1(ka)}, r _ψ2 (yb) e^{iω _ψ2(mb)} }

= min{ μ _ψ1 (ka) , μ _ψ2 (mb) }

⩾min { min { μ _ψ1 (k) , μ _ψ1 (a) } ,

    min{ μ _ψ2 (m) , μ _ψ2 (b) }}

= min { min { μ _ψ1 (k) , μ _ψ2 (m) } ,

   min { μ _ψ1 (a) , μ _ψ2 (b) }}

= min{ μ _ψ1×ψ2 (k, m) , μ _ψ1×ψ2 (a, b) }.

Furthermore,

μ _ψ1×ψ2 (k^-1, m^-1)

= r _ψ1×ψ2 (k^-1, m^-1) e^{iω _ψ1×ψ2(k-1,m-1)}

= min{ r _ψ1 (k^-1) , r _ψ2 (m^-1) }

   e^{imin{ω _ψ1(k-1),ω _ψ2(m-1)}}

= min { r _ψ1 (k^-1) e^{iω _ψ1(k-1)},

r _ψ2 (m^-1) e^{iω _ψ2(m-1)}}

= min{ μ _ψ1 (k^-1) , μ _ψ2 (m^-1) }

⩾min{ μ _ψ1 (k) , μ _ψ2 (m) }.

= μ _ψ1×ψ2 (k, m).

On the other hand,

γ _ψ1×ψ2 ((k, m) - (a, b))

= γ _ψ1×ψ2 (k - a, m - b)

= r ˆ ψ 1 × ψ 2 ( k - a , m - b ) e i ω ˆ ψ 1 × ψ 2 ( k - a , m - b )

= max { r ˆ ψ 1 ( k - a ) , r ˆ ψ 2 ( m - b ) }

e imax { ω ˆ ψ 1 ( k - a ) , ω ˆ ψ 2 ( m - b ) }

= max { r ˆ ψ 1 ( k - a ) e i ω ˆ ψ 1 ( k - a ) ,

r ˆ ψ 2 ( m - b ) e i ω ˆ ψ 2 ( m - b ) }

= max{ γ _ψ1 (k - a) , γ _ψ2 (m - b) }

⩽max { max { γ _ψ1 (k) , γ _ψ1 (a) } ,

   max { γ _ψ2 (m) , γ _ψ2 (b) }}

= max { max { γ _ψ1 (k) , γ _ψ2 (m) } ,

   max { γ _ψ1 (a) , γ _ψ2 (b) }}

= max{ γ _ψ1×ψ2 (k, m) , γ _ψ1×ψ2 (a, b) }.

Also,

γ _ψ1×ψ2 ((k, m) (a, b))

= γ _ψ1×ψ2 (ka, mb)

= r ˆ ψ 1 × ψ 2 ( ka , mb ) e i ω ˆ ψ 1 × ψ 2 ( ka , mb )

= max { r ˆ ψ 1 ( ka ) , r ˆ ψ 2 ( mb ) } e imax { ω ˆ ψ 1 ( ka ) , ω ˆ ψ 2 ( mb ) }

= max { r ˆ ψ 1 ( ka ) e i ω ˆ ψ 1 ( ka ) , r ˆ ψ 2 ( mb ) e i ω ˆ ψ 2 ( mb ) }

= max{ γ _ψ1 (ka) , γ _ψ2 (mb) }

⩽max { max { γ _ψ1 (k) , γ _ψ1 (a) } ,

   max { γ _ψ2 (m) , γ _ψ2 (b) }}

= max { max { γ _ψ1 (k) , γ _ψ2 (m) } ,

   max { γ _ψ1 (a) , γ _ψ2 (b) }}

= max{ γ _ψ1×ψ2 (k, m) , γ _ψ1×ψ2 (a, b) }.

Moreover,

γ _ψ1×ψ2 (k^-1, m^-1)

= r ˆ ψ 1 × ψ 2 ( k - 1 , m - 1 ) e i ω ˆ ψ 1 × ψ 2 ( k - 1 , m - 1 )

= max { r ˆ ψ 1 ( k - 1 ) , r ˆ ψ 2 ( m - 1 ) }

e imax { ω ˆ ψ 1 ( k - 1 ) , ω ˆ ψ 2 ( m - 1 ) }

= max { r ˆ ψ 1 ( k - 1 ) e i ω ˆ ψ 1 ( k - 1 ) ,

r ˆ ψ 2 ( m - 1 ) e i ω ˆ ψ 2 ( m - 1 ) }

= max{ γ _ψ1 (k^-1) , γ _ψ2 (m^-1) }

⩽max{ γ _ψ1 (k) , γ _ψ2 (m) }.

= γ _ψ1×ψ2 (k, m).

Therefore, ψ₁ × ψ₂ is CIFSF of a field F₁ × F₂ .

Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in the philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logic extends other systems by adding unary operators ◊ and □, representing possibility and necessity respectively. For instance, the modal formula ♦ψ can be read as “possibly ψ” while  □ ψ can be read as “necessarily ψ”. Modal logic can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. The following definition presents the concepts of necessity and possibility operators on the direct product of complex intuitionistic fuzzy subfields and some related theorems are proven.

Definition 3.5. Let ψ₁ × ψ₂ is CIFSF of a field F₁ × F₂ such that: ψ 1 × ψ 2 = { 〈 ( k , m ) , μ ψ 1 × ψ 2 ( k , m ) , γ ψ 1 × ψ 2 ( k , m ) 〉 : ( k , m ) ∈ F 1 × F 2 } .

Then the necessity and possibility operators on the direct product of CIFSF are defined as: ( i ) □ ( ψ 1 × ψ 2 ) = { 〈 ( k , m ) , μ ψ 1 × ψ 2 ( k , m ) , 1 - μ ψ 1 × ψ 2 ( k , m ) 〉 : ( k , m ) ∈ F 1 × F 2 } , ( ii ) ♦ ( ψ 1 × ψ 2 ) = { 〈 ( k , m ) , 1 - γ ψ 1 × ψ 2 ( k , m ) , γ ψ 1 × ψ 2 ( k , m ) 〉 : ( k , m ) ∈ F 1 × F 2 } .

We can study the relation between the direct product of complex intuitionistic fuzzy subfields and the necessity and possibility operators on the direct product of complex intuitionistic fuzzy subfields where if we have the direct product of complex intuitionistic fuzzy subfields then the necessity and possibility operators on the direct product of complex intuitionistic fuzzy subfields are also complex intuitionistic fuzzy subfield. The following theorems explain these results in detail.

Theorem 3.2. If ψ₁ × ψ₂ is CIFSF of a field F₁ × F₂, then □ (ψ₁ × ψ₂) is CIFSF of a field F₁ × F₂.

Proof. Let ψ₁ × ψ₂ be CIFSF of a field F₁ × F₂. Let □ ( ψ 1 × ψ 2 ) = { 〈 ( k , m ) , μ ψ 1 × ψ 2 ( k , m ) , μ ψ 1 × ψ 2 ¯ ( k , m ) 〉 : (k, m)  ∈ F₁ × F₂}. We will prove that □ (ψ₁ × ψ₂) is CIFSF of a field F₁ × F₂.

Since the direct product ψ₁ × ψ₂ is CIFSF of a field F₁ × F₂. So, we have

μ _ψ1×ψ2 ((k, m) - (a, b)) ⩾min{ μ _ψ1×ψ2   (k, m) , μ _ψ1×ψ2 (a, b) },

We must demonstrate the following to establish that □ (ψ₁ × ψ₂) is a complex intuitionistic fuzzy subfield:

μ _ψ1×ψ2 ((k, m) - (a, b)) ⩾min{ μ _ψ1×ψ2   (k, m) , μ _ψ1×ψ2 (a, b) },

We observe that i), ii), and iii) are provided. Therefore, we just need to demonstrate iv), v), and vi).

Now, μ ψ 1 × ψ 2 ¯ ( ( k , m ) - ( a , b ) ) = μ ψ 1 × ψ 2 ¯ ( ( k - a ) , ( m - b ) ) = r ψ 1 × ψ 2 ¯ ( k - a , m - b ) e i ω ψ 1 × ψ 2 ¯ ( k - a , m - b ) = { 1 - r ψ 1 × ψ 2 ( k - a , m - b ) } e i { 2 π - ω ψ 1 × ψ 2 ( k - a , m - b ) } = { 1 - min { r ψ 1 ( k - a ) , r ψ 2 ( m - b ) } } e i { 2 π - min { ω ψ 1 ( k - a ) , ω ψ 2 ( m - b ) } } = max { 1 - r ψ 1 ( k - a ) , 1 - r ψ 2 ( m - b ) } e i { max { 2 π - ω ψ 1 ( k - a ) , 2 π - ω ψ 2 ( m - b ) } } = max { r ψ 1 ¯ ( k - a ) , r ψ 2 ¯ ( m - b ) } e i { max { ω ψ 1 ¯ ( k - a ) , ω ψ 2 ¯ c ( m - b ) } } = max { r ψ 1 ¯ ( k - a ) e i ω ψ 1 ¯ ( k - a ) , r ψ 2 ¯ ( m - b ) e i ω ψ 2 ¯ ( m - b ) } = max { μ ψ 1 ¯ ( k - a ) , μ ψ 2 ¯ ( m - b ) } ⩽ max { max { μ ψ 1 ¯ ( k ) , μ ψ 1 ¯ ( a ) } , max { μ ψ 2 ¯ ( m ) , μ ψ 2 ¯ ( b ) } } = max { max { μ ψ 1 ¯ ( k ) , μ ψ 2 ¯ ( m ) } , max { μ ψ 1 ¯ ( a ) , μ ψ 2 ¯ ( b ) } } = max { μ ψ 1 × ψ 2 ¯ ( k , m ) , μ ψ 1 × ψ 2 ¯ ( a , b ) } .

Moreover, μ ψ 1 × ψ 2 ¯ ( ( k , m ) ( a , b ) ) = μ ψ 1 × ψ 2 ¯ ( ( ka ) , ( mb ) ) = r ψ 1 × ψ 2 ¯ ( ka , mb ) e i ω ψ 1 × ψ 2 ¯ ( ka , mb ) = { 1 - r ψ 1 × ψ 2 ( ka , mb ) } e i { 2 π - ω ψ 1 × ψ 2 ( ka , mb ) } = { 1 - min { r ψ 1 ( ka ) , r ψ 2 ( mb ) } } e i { 2 π - min { ω ψ 1 ( ka ) , ω ψ 2 ( mb ) } }

= max{ 1 - r _ψ1 (ka) , 1 - r _ψ2 (mb) }

e ^{i{max{2π-ω _ψ1(ka),2π-ω _ψ2(mb)}}}

= max { r ψ 1 ¯ ( ka ) , r ψ 2 ¯ ( mb ) }

e i { max { ω ψ 1 ¯ ( ka ) , ω ψ 2 ¯ ( mb ) } }

= max { r ψ 1 ¯ ( ka ) e i ω ψ 1 ¯ ( ka ) , r ψ 2 ¯ ( mb ) e i ω ψ 2 ¯ ( mb ) }

= max { μ ψ 1 ¯ ( ka ) , μ ψ 2 ¯ ( mb ) }

⩽ max { max { μ ψ 1 ¯ ( k ) , μ ψ 1 ¯ ( a ) } ,

max { μ ψ 2 ¯ ( m ) , μ ψ 2 ¯ ( b ) } }

= max { max { μ ψ 1 ¯ ( k ) , μ ψ 2 ¯ ( m ) } ,

max { μ ψ 1 ¯ ( a ) , μ ψ 2 ¯ ( b ) } }

= max { μ ψ 1 × ψ 2 ¯ ( x , y ) , μ ψ 1 × ψ 2 ¯ ( a , b ) } .

Also, μ ψ 1 × ψ 2 ¯ ( k - 1 , m - 1 ) = r ψ 1 × ψ 2 ¯ ( k - 1 , m - 1 ) e i ω ψ 1 × ψ 2 ¯ ( k - 1 , m - 1 ) = { 1 - r ψ 1 × ψ 2 ( k - 1 , m - 1 ) } e i { 2 π - ω ψ 1 × ψ 2 ( k - 1 , m - 1 ) } = { 1 - min { r ψ 1 ( k - 1 ) , r ψ 2 ( m - 1 ) } } e i { 2 π - min { ω ψ 1 ( k - 1 ) , ω ψ 2 ( m - 1 ) } } = max { 1 - r ψ 1 ( k - 1 ) , 1 - r ψ 2 ( m - 1 ) } e i { max { 2 π - ω ψ 1 ( k - 1 ) , 2 π - ω ψ 2 ( m - 1 ) } } = max { r ψ 1 ¯ ( k - 1 ) , r ψ 2 ¯ ( m - 1 ) } e i { max { ω ψ 1 ¯ ( k - 1 ) , ω ψ 2 ¯ ( m - 1 ) } } = max { r ψ 1 ¯ ( k - 1 ) e i ω ψ 1 ¯ ( k - 1 ) , r ψ 2 ¯ ( m - 1 ) e i ω ψ 2 ¯ ( m - 1 ) } = max { μ ψ 1 ¯ ( k - 1 ) , μ ψ 2 ¯ ( m - 1 ) } ⩽ max { μ ψ 1 ¯ ( k ) , μ ψ 2 ¯ ( m ) } = μ ψ 1 × ψ 2 ¯ ( k , m ) .

So, it is given that: μ ψ 1 × ψ 2 ( ( k , m ) - ( a , b ) ) ⩾ min { μ ψ 1 × ψ 2 ( k , m ) , μ ψ 1 × ψ 2 ( a , b ) } , μ ψ 1 × ψ 2 ( ( k , m ) ( a , b ) ) ⩾ min { μ ψ 1 × ψ 2 ( k , m ) , μ ψ 1 × ψ 2 ( a , b ) } , μ ψ 1 × ψ 2 ( k - 1 , m - 1 ) ⩾ μ ψ 1 × ψ 2 ( k , m ) .

Also, we proved that: μ ψ 1 × ψ 2 ¯ ( ( k , m ) - ( a , b ) ) ⩽ max { μ ψ 1 × ψ 2 ¯ ( k , m ) , μ ψ 1 × ψ 2 ¯ ( a , b ) } , μ ψ 1 × ψ 2 ¯ ( ( k , m ) ( a , b ) ) ⩽ max { μ ψ 1 × ψ 2 ¯ ( k , m ) , μ ψ 1 × ψ 2 ¯ ( a , b ) } , μ ψ 1 × ψ 2 ¯ ( k - 1 , m - 1 ) ⩽ μ ψ 1 × ψ 2 ¯ ( k , m ) .

Therefore, □ ( ψ 1 × ψ 2 ) = { 〈 ( k , m ) , μ ψ 1 × ψ 2 ( k , m ) , μ ψ 1 × ψ 2 ¯ ( k , m ) 〉 :

(k, m)  ∈ F₁ × F₂} is a complex intuitionistic fuzzy subfield of a field F₁ × F₂.

Theorem 3.3. If ψ₁ × ψ₂ is CIFSF of a field F₁ × F₂, then ♦ (ψ₁ × ψ₂) is CIFSF of a field F₁ × F₂.

Proof. Let ψ₁ × ψ₂ be CIFSF of a field F₁ × F₂. Let ♦ ( ψ 1 × ψ 2 ) = { 〈 ( k , m ) , γ ψ 1 × ψ 2 ¯ ( k , m ) , γ ψ 1 × ψ 2 ( k , m ) 〉 :

(k, m)  ∈ F₁ × F₂}. We will prove that ♦ (ψ₁ × ψ₂) is CIFSF of a field F₁ × F₂.

Since the direct product ψ₁ × ψ₂ is CIFSF of a field F₁ × F₂. So, we have

μ _ψ1×ψ2 ((k, m) - (a, b))

⩾min{ μ _ψ1×ψ2 (k, m) , μ _ψ1×ψ2 (a, b) },