Investigation of Cyber-Security and Cyber-Crimes in Oil and Gas Sectors Using the Innovative Structures of Complex Intuitionistic Fuzzy Relations

Recently, there has been enormous development due to advancements in technology. Industries and enterprises are moving towards a digital system, and the oil and gas industries are no exception. There are several threats and risks in digital systems, which are controlled through cyber-security. For the first time in the theory of fuzzy sets, this research analyzes the relationships between cyber-security and cyber-crimes in the oil and gas sectors. The novel concepts of complex intuitionistic fuzzy relations (CIFRs) are introduced. Moreover, the types of CIFRs are defined and their properties are discussed. In addition, an application is presented that uses the Hasse diagram to make a decision regarding the most suitable cyber-security techniques to implement in an industry. Furthermore, the omnipotence of the proposed methods is explained by a comparative study.


Introduction
The techniques and methods used for reasoning, modeling and computing are mostly of precise, deterministic and crisp nature. The term crisp refers to the concept of a dichotomy, i.e., yes or no rather than more or less. In conventional dual logic, a statement is either true or false-there are no other possibilities. In general, precision implies that the models are unambiguous and clear. Crisp knowledge can be modeled using the crisp/classical set theory, also known as Cantor's set theory. Meanwhile, in mathematics, the uncertainty is modeled through the theory of fuzzy sets (FSs) and fuzzy logic (FL). In practice, uncertainty cannot be avoided. There have been numerous structures, techniques and formulations introduced to model uncertainty in the theory of FSs and FL. Each of these methods have their advantages, accompanied by some limitations, leaving some gaps. Thus, this paper focuses on the formulation of some novel structures and methods that aim to solve certain cyber-security and hacking issues faced by the oil and gas sectors.
The complex intuitionistic fuzzy set (CIFS) is a powerful tool in FS theory that is used to model ambiguity and uncertainty, but the concepts of relations have not yet been defined for CIFSs. This article introduces the concepts of relations in the theory of CIFSs. Using the Cartesian product of two CIFSs, the current study presents the definition of complex intuitionistic fuzzy relations (CIFRs). The formation of CIFSs is based on a pair of complex valued functions whose values and their sum are contained within the unit disc of a complex plane. These functions are called the membership grade and non-membership grade. The real portion of each of the complex valued functions is called the amplitude The world is developing and every organization is being digitalized. This ensures that the operations of organizations are smooth, time-efficient, systematized and secure as compared to non-digital infrastructures. In recent years, hackers and other criminals have targeted the digital industries. There are certain threats and risks to the digital systems that need to be countered by implementing specific techniques. The oil and gas sectors are not an exception and have been targeted recently. Scientists and engineers have carried out research to make these industries secure. Lamba [40] described measures to protect the 'cybersecurity and resiliency' of the oil, energy and gas infrastructures of a nation; Line et al. [41] discussed the cyber-security challenges in smart grids; Lu et al. [42] and Lakhanpal and Samuel [43] reviewed the opportunities, applications, risks and challenges of implementing block-chain technology in oil and gas sectors. Based on factor state space, Yang et al. [44] presented a new cyber-security risk evaluation method; Stergiopoulos et al. [45] surveyed attack patterns and carried out incident assessment in the oil and gas industries to evaluate the cyber-attacks in the oil and gas sector; Wanasinghe et al. [46] carried out a systematic review of internet of things in the oil and gas sector, and Bjerga and Aven [47] used some new risk perspectives for adaptive risk management in the oil and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed, such as fuzzy sets (FSs), complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (CIFSs), Cartesian product (CP) of two CFSs and complex fuzzy relations (CFRs). In addition, the definitions are supported by examples. Definition 1 ([2]). An FSḮ is characterized by a function m :Ḯ → [0, 1] that assigns to each u ∈Ḯ a fuzzy number (FN) m(u) ∈ [0, 1]. The function m is called the membership grade. Henceforth, an FSḮ on a universal set terms and phase terms. Thus, they can model uncertain events with time periods and phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] studied the complex relations and applied them in cellular networks and economic relationships. Ngan et al. [38] represented CIFS by quaternion numbers and applied them in decision-making, while Kumar and Bajaj [39] worked on the distance measures and entropies in CIF soft sets.
The world is developing and every organization is being digitalized. This ensures that the operations of organizations are smooth, time-efficient, systematized and secure as compared to non-digital infrastructures. In recent years, hackers and other criminals have targeted the digital industries. There are certain threats and risks to the digital systems that need to be countered by implementing specific techniques. The oil and gas sectors are not an exception and have been targeted recently. Scientists and engineers have carried out research to make these industries secure. Lamba [40] described measures to protect the 'cybersecurity and resiliency' of the oil, energy and gas infrastructures of a nation; Line et al. [41] discussed the cyber-security challenges in smart grids; Lu et al. [42] and Lakhanpal and Samuel [43] reviewed the opportunities, applications, risks and challenges of implementing block-chain technology in oil and gas sectors. Based on factor state space, Yang et al. [44] presented a new cyber-security risk evaluation method; Stergiopoulos et al. [45] surveyed attack patterns and carried out incident assessment in the oil and gas industries to evaluate the cyber-attacks in the oil and gas sector; Wanasinghe et al. [46] carried out a systematic review of internet of things in the oil and gas sector, and Bjerga and Aven [47] used some new risk perspectives for adaptive risk management in the oil and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed, such as fuzzy sets (FSs), complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (CIFSs), Cartesian product (CP) of two CFSs and complex fuzzy relations (CFRs). In addition, the definitions are supported by examples. : ∈ Ḯ, ∈ ʝ is of the following form: Ï = (u, m(u)) : u ∈ Entropy 2021, 23, x FOR PEER REVIEW terms and phase terms. Thus, they can model uncertain events with tim phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] studied the com and applied them in cellular networks and economic relationships. Ngan e sented CIFS by quaternion numbers and applied them in decision-making and Bajaj [39] worked on the distance measures and entropies in CIF soft s The world is developing and every organization is being digitalized that the operations of organizations are smooth, time-efficient, systematiz as compared to non-digital infrastructures. In recent years, hackers and o have targeted the digital industries. There are certain threats and risks to t tems that need to be countered by implementing specific techniques. The o tors are not an exception and have been targeted recently. Scientists and e carried out research to make these industries secure. Lamba [40] describe protect the 'cybersecurity and resiliency' of the oil, energy and gas infra nation; Line et al. [41] discussed the cyber-security challenges in smart grid and Lakhanpal and Samuel [43] reviewed the opportunities, applications, lenges of implementing block-chain technology in oil and gas sectors. Based space, Yang et al. [44] presented a new cyber-security risk evaluation metho los et al. [45] surveyed attack patterns and carried out incident assessment gas industries to evaluate the cyber-attacks in the oil and gas sector; Wanasi carried out a systematic review of internet of things in the oil and gas sect and Aven [47] used some new risk perspectives for adaptive risk managem and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed, such as fu complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (CIFSs), Car (CP) of two CFSs and complex fuzzy relations (CFRs). In addition, the defin ported by examples. terms and phase terms. Thus, they can model uncertain events with time periods and phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] studied the complex relations and applied them in cellular networks and economic relationships. Ngan et al. [38] represented CIFS by quaternion numbers and applied them in decision-making, while Kumar and Bajaj [39] worked on the distance measures and entropies in CIF soft sets. The world is developing and every organization is being digitalized. This ensures that the operations of organizations are smooth, time-efficient, systematized and secure as compared to non-digital infrastructures. In recent years, hackers and other criminals have targeted the digital industries. There are certain threats and risks to the digital systems that need to be countered by implementing specific techniques. The oil and gas sectors are not an exception and have been targeted recently. Scientists and engineers have carried out research to make these industries secure. Lamba [40] described measures to protect the 'cybersecurity and resiliency' of the oil, energy and gas infrastructures of a nation; Line et al. [41] discussed the cyber-security challenges in smart grids; Lu et al. [42] and Lakhanpal and Samuel [43] reviewed the opportunities, applications, risks and challenges of implementing block-chain technology in oil and gas sectors. Based on factor state space, Yang et al. [44] presented a new cyber-security risk evaluation method; Stergiopoulos et al. [45] surveyed attack patterns and carried out incident assessment in the oil and gas industries to evaluate the cyber-attacks in the oil and gas sector; Wanasinghe et al. [46] carried out a systematic review of internet of things in the oil and gas sector, and Bjerga and Aven [47] used some new risk perspectives for adaptive risk management in the oil and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed, such as fuzzy sets (FSs), complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (CIFSs), Cartesian product (CP) of two CFSs and complex fuzzy relations (CFRs). In addition, the definitions are supported by examples. : ∈ Ḯ, ∈ ʝ is of the following form: Entropy 2021, 23, x FOR PEER REVIEW terms and phase terms. Thus, they can model uncertain events with phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] studied the and applied them in cellular networks and economic relationships. Nga sented CIFS by quaternion numbers and applied them in decision-mak and Bajaj [39] worked on the distance measures and entropies in CIF so The world is developing and every organization is being digital that the operations of organizations are smooth, time-efficient, system as compared to non-digital infrastructures. In recent years, hackers an have targeted the digital industries. There are certain threats and risks tems that need to be countered by implementing specific techniques. T tors are not an exception and have been targeted recently. Scientists an carried out research to make these industries secure. Lamba [40] desc protect the 'cybersecurity and resiliency' of the oil, energy and gas in nation; Line et al. [41] discussed the cyber-security challenges in smart g and Lakhanpal and Samuel [43] reviewed the opportunities, applicatio lenges of implementing block-chain technology in oil and gas sectors. Ba space, Yang et al. [44] presented a new cyber-security risk evaluation me los et al. [45] surveyed attack patterns and carried out incident assessm gas industries to evaluate the cyber-attacks in the oil and gas sector; Wa carried out a systematic review of internet of things in the oil and gas and Aven [47] used some new risk perspectives for adaptive risk man and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed, such a complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (CIFSs), (CP) of two CFSs and complex fuzzy relations (CFRs). In addition, the d ported by examples.   [36,37] stud and applied them in cellular networks and economic relationsh sented CIFS by quaternion numbers and applied them in decis and Bajaj [39] worked on the distance measures and entropies i The world is developing and every organization is being that the operations of organizations are smooth, time-efficient as compared to non-digital infrastructures. In recent years, ha have targeted the digital industries. There are certain threats a tems that need to be countered by implementing specific techn tors are not an exception and have been targeted recently. Scie carried out research to make these industries secure. Lamba [4 protect the 'cybersecurity and resiliency' of the oil, energy an nation; Line et al. [41] discussed the cyber-security challenges in and Lakhanpal and Samuel [43] reviewed the opportunities, ap lenges of implementing block-chain technology in oil and gas se space, Yang et al. [44] presented a new cyber-security risk evalu los et al. [45] surveyed attack patterns and carried out incident gas industries to evaluate the cyber-attacks in the oil and gas sec carried out a systematic review of internet of things in the oil a and Aven [47] used some new risk perspectives for adaptive r and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (C (CP) of two CFSs and complex fuzzy relations (CFRs). In additio ported by examples. terms and phase terms. Thus, they can model uncertain events with time periods and phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] studied the complex relations and applied them in cellular networks and economic relationships. Ngan et al. [38] represented CIFS by quaternion numbers and applied them in decision-making, while Kumar and Bajaj [39] worked on the distance measures and entropies in CIF soft sets. The world is developing and every organization is being digitalized. This ensures that the operations of organizations are smooth, time-efficient, systematized and secure as compared to non-digital infrastructures. In recent years, hackers and other criminals have targeted the digital industries. There are certain threats and risks to the digital systems that need to be countered by implementing specific techniques. The oil and gas sectors are not an exception and have been targeted recently. Scientists and engineers have carried out research to make these industries secure. Lamba [40] described measures to protect the 'cybersecurity and resiliency' of the oil, energy and gas infrastructures of a nation; Line et al. [41] discussed the cyber-security challenges in smart grids; Lu et al. [42] and Lakhanpal and Samuel [43] reviewed the opportunities, applications, risks and challenges of implementing block-chain technology in oil and gas sectors. Based on factor state space, Yang et al. [44] presented a new cyber-security risk evaluation method; Stergiopoulos et al. [45] surveyed attack patterns and carried out incident assessment in the oil and gas industries to evaluate the cyber-attacks in the oil and gas sector; Wanasinghe et al. [46] carried out a systematic review of internet of things in the oil and gas sector, and Bjerga and Aven [47] used some new risk perspectives for adaptive risk management in the oil and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed, such as fuzzy sets (FSs), complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (CIFSs), Cartesian product (CP) of two CFSs and complex fuzzy relations (CFRs). In addition, the definitions are supported by examples.   Definition 4 ([12]). Any subset of the CP of two CFSs is known as a complex fuzzy relation (CFR), which is symbolized by R.

Definition 4. [12]
Any subset of the CP of two CFSs is known as a complex fu which is symbolized by .

Definition 5 ([24]
). An IFSḮ is characterized by a pair of functions m, n :Ḯ → [0, 1] that are each assigned u ∈Ḯ a pair of fuzzy numbers (FN) m(u), n(u) ∈ [0, 1], provided that the sum m(u) + n(u) ≤ 1. The function m is called the membership grade and the function n is called the non-membership grade. Henceforth, an IFSḮ on a universal set Entropy 2021, 23, x FOR PEER REVIEW terms and phase terms. Thus, they can model uncertain events with phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] studied the and applied them in cellular networks and economic relationships. Ng sented CIFS by quaternion numbers and applied them in decision-ma and Bajaj [39] worked on the distance measures and entropies in CIF s The world is developing and every organization is being digita that the operations of organizations are smooth, time-efficient, system as compared to non-digital infrastructures. In recent years, hackers a have targeted the digital industries. There are certain threats and risk tems that need to be countered by implementing specific techniques. T tors are not an exception and have been targeted recently. Scientists a carried out research to make these industries secure. Lamba [40] des protect the 'cybersecurity and resiliency' of the oil, energy and gas i nation; Line et al. [41] discussed the cyber-security challenges in smart and Lakhanpal and Samuel [43] reviewed the opportunities, applicati lenges of implementing block-chain technology in oil and gas sectors. B space, Yang et al. [44] presented a new cyber-security risk evaluation m los et al. [45] surveyed attack patterns and carried out incident assess gas industries to evaluate the cyber-attacks in the oil and gas sector; Wa carried out a systematic review of internet of things in the oil and gas and Aven [47] used some new risk perspectives for adaptive risk man and gas sector.

Definition 1 ([2]). An FS Ḯ is characterized by a function
Entropy 2021, 23, x FOR PEER REVIEW terms and phase terms. Thus, they can model uncertain events with phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] studied the and applied them in cellular networks and economic relationships. Nga sented CIFS by quaternion numbers and applied them in decision-mak and Bajaj [39] worked on the distance measures and entropies in CIF so The world is developing and every organization is being digitali that the operations of organizations are smooth, time-efficient, system as compared to non-digital infrastructures. In recent years, hackers an have targeted the digital industries. There are certain threats and risks tems that need to be countered by implementing specific techniques. Th tors are not an exception and have been targeted recently. Scientists an carried out research to make these industries secure. Lamba [40] descr protect the 'cybersecurity and resiliency' of the oil, energy and gas in nation; Line et al. [41] discussed the cyber-security challenges in smart g and Lakhanpal and Samuel [43] reviewed the opportunities, application lenges of implementing block-chain technology in oil and gas sectors. Ba space, Yang et al. [44] presented a new cyber-security risk evaluation me los et al. [45] surveyed attack patterns and carried out incident assessm gas industries to evaluate the cyber-attacks in the oil and gas sector; Wan carried out a systematic review of internet of things in the oil and gas s and Aven [47] used some new risk perspectives for adaptive risk mana and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed, such as complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (CIFSs), C (CP) of two CFSs and complex fuzzy relations (CFRs). In addition, the de ported by examples.

Definition 1 ([2]). An FS Ḯ is characterized by a function
The function m C (u) is called the membership grade and the function n C (u) is called the non-membership grade, which are defined as m C (u) = α m (u)e ρ m (u)2πi and n C (u) = α n (u)e ρ n (u)2πi , where i = √ −1, α m (u), α n (u) ∈ [0, 1] are called the amplitude terms of the membership and non-membership grades, respectively, and ρ m (u), ρ n (u) ∈ [0, 1] are called the phase terms of the membership and non-membership grades, respectively. Henceforth, a CIFSḮ on a universal set Entropy 2021, 23, x FOR PEER REVIEW terms and phase terms. Thus, they can model uncertain events with time per phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] studied the complex and applied them in cellular networks and economic relationships. Ngan et al. [3 sented CIFS by quaternion numbers and applied them in decision-making, whi and Bajaj [39] worked on the distance measures and entropies in CIF soft sets.
The world is developing and every organization is being digitalized. Thi that the operations of organizations are smooth, time-efficient, systematized an as compared to non-digital infrastructures. In recent years, hackers and other have targeted the digital industries. There are certain threats and risks to the di tems that need to be countered by implementing specific techniques. The oil and tors are not an exception and have been targeted recently. Scientists and engine carried out research to make these industries secure. Lamba [40] described me protect the 'cybersecurity and resiliency' of the oil, energy and gas infrastruct nation; Line et al. [41] discussed the cyber-security challenges in smart grids; Lu and Lakhanpal and Samuel [43] reviewed the opportunities, applications, risks lenges of implementing block-chain technology in oil and gas sectors. Based on fa space, Yang et al. [44] presented a new cyber-security risk evaluation method; Ste los et al. [45] surveyed attack patterns and carried out incident assessment in th gas industries to evaluate the cyber-attacks in the oil and gas sector; Wanasinghe carried out a systematic review of internet of things in the oil and gas sector, an and Aven [47] used some new risk perspectives for adaptive risk management and gas sector.

Preliminaries
In this section, some fundamental concepts are reviewed, such as fuzzy s complex FSs (CFSs), intuitionistic FSs (CFSs), complex IFSs (CIFSs), Cartesian (CP) of two CFSs and complex fuzzy relations (CFRs). In addition, the definitions ported by examples.
is of the following form: terms and phase terms. Thus, they can model uncertain eve phase alterations. Yaqoob et al. [35] and Nasir et al. [36,37] stu and applied them in cellular networks and economic relationsh sented CIFS by quaternion numbers and applied them in deci and Bajaj [39] worked on the distance measures and entropies The world is developing and every organization is bein that the operations of organizations are smooth, time-efficien as compared to non-digital infrastructures. In recent years, h have targeted the digital industries. There are certain threats a tems that need to be countered by implementing specific techn tors are not an exception and have been targeted recently. Sci carried out research to make these industries secure. Lamba [ protect the 'cybersecurity and resiliency' of the oil, energy an nation; Line et al. [41] discussed the cyber-security challenges i and Lakhanpal and Samuel [43] reviewed the opportunities, a lenges of implementing block-chain technology in oil and gas se space, Yang et al. [44] presented a new cyber-security risk evalu los et al. [45] surveyed attack patterns and carried out inciden gas industries to evaluate the cyber-attacks in the oil and gas se carried out a systematic review of internet of things in the oil and Aven [47] used some new risk perspectives for adaptive r and gas sector.

Complex Intuitionistic Fuzzy Relations and Their Properties
This section introduces the novel concepts of CP of two CIFSs, complex intuitionistic fuzzy relations (CIFRs) and their types. Every definition is supported by a suitable example. Moreover, some interesting results for CIFRs are provided. Additionally, the Hasse diagram for the complex intuitionistic partial order relations is presented. The notions of maximum, minimum, maximal, minimal, supremum, infimum, upper and lower bounds are defined as well.
tors are not an exception and have been targeted carried out research to make these industries secu protect the 'cybersecurity and resiliency' of the o nation; Line et al. [41] discussed the cyber-security and Lakhanpal and Samuel [43] reviewed the opp lenges of implementing block-chain technology in o space, Yang et al. [44] presented a new cyber-secur los et al. [45] surveyed attack patterns and carried gas industries to evaluate the cyber-attacks in the o carried out a systematic review of internet of thin and Aven [47] used some new risk perspectives fo and gas sector.

Preliminaries
In this section, some fundamental concepts a complex FSs (CFSs), intuitionistic FSs (CFSs), com (CP) of two CFSs and complex fuzzy relations (CFR ported by examples. and have targeted the digital industries. There are certain threats and risks to the digital sy tems that need to be countered by implementing specific techniques. The oil and gas se tors are not an exception and have been targeted recently. Scientists and engineers ha carried out research to make these industries secure. Lamba [40] described measures protect the 'cybersecurity and resiliency' of the oil, energy and gas infrastructures of nation; Line et al. [41] discussed the cyber-security challenges in smart grids; Lu et al. [4 and Lakhanpal and Samuel [43] reviewed the opportunities, applications, risks and cha lenges of implementing block-chain technology in oil and gas sectors. Based on factor sta space, Yang et al. [44] presented a new cyber-security risk evaluation method; Stergiopo los et al. [45] surveyed attack patterns and carried out incident assessment in the oil an gas industries to evaluate the cyber-attacks in the oil and gas sector; Wanasinghe et al. [4 carried out a systematic review of internet of things in the oil and gas sector, and Bjer and Aven [47] used some new risk perspectives for adaptive risk management in the o and gas sector.
The CIFR between Ḯ and ʝ is given as follows (

NOTE:
For convenience, (u, v) will be used to denote throughout this paper, unless otherwise stated.

Definition 9.
Let R be a CIFR on a CIFSḮ. Then,

4.
A complex intuitionistic equivalence FR R onḮ possesses the following properties: a. Complex intuitionistic reflexive; b.

5.
A complex intuitionistic preorder FR R onḮ possesses the following properties: Complex intuitionistic transitive.

7.
A complex intuitionistic partial order FR R onḮ possesses the following properties: a.
R is also called the complex intuitionistic order FR.

9.
A complex intuitionistic linear order FR R onḮ possesses the following properties: a. Complex intuitionistic reflexive; b.
Complex intuitionistic complete.
It is also called the complex intuitionistic total order FR. 10. R is complex intuitionistic irreflexive FR if (u, u) / ∈ R, ∀u ∈Ḯ. 11. A complex intuitionistic strict order FR R onḮ possesses the following properties: a.

1.
The complex intuitionistic equivalence fuzzy relation R 1 onḮ ×Ḯ is The complex intuitionistic partial order fuzzy relation R 2 onḮ ×Ḯ is The complex intuitionistic linear order fuzzy relation R 3 onḮ ×Ḯ is Then, the converse of a CIFR R is given as The complex intuitionistic equivalence fuzzy relations give rise to the concept of complex intuitionistic fuzzy equivalence classes, which are defined as follows.
Definition 11. Let R be a complex intuitionistic equivalence fuzzy relation on a CIFSḮ. For u ∈Ḯ, a complex intuitionistic fuzzy equivalence class of u mod R is defined and symbolized as The complex intuitionistic equivalence fuzzy relation onḮ is Then, the complex intuitionistic fuzzy equivalence class of 1.
x mod R is

Example 10.
Let R 1 and R 2 be two CIFRs on some CIFSḮ , Then, the complex intuitionistic composite fuzzy relation R 1 • R 2 is given by Proof. Let R = R c , then Hence, R is a complex intuitionistic symmetric FR on a CIFSḮ. Conversely, suppose that R is a complex intuitionistic symmetric FR on a CIFSḮ, then Proof. Let R is a complex intuitionistic transitive FR on a CIFSḮ. Assume that (u, w) ∈ R • R then, by the transitivity of R, Conversely, suppose that R • R ⊆ R; then, for Hence, R is complex intuitionistic transitive FR onḮ.

Theorem 3. If R is a complex intuitionistic equivalence FR on a CIFSḮ, then R • R = R.
Proof. Let (u, v) ∈ R Then, by the symmetry of a complex intuitionistic equivalence FR R, (v, u) ∈ R Now, by using the transitive property of a complex intuitionistic equivalence FR R, However, by the definition of complex intuitionistic composite FR, Conversely, suppose that (u, v) ∈ R • R, then ∃ w ∈ U (u, w) ∈ R and (w, v) ∈ R However, since R is a complex intuitionistic equivalence FR onḮ, R is also a complex intuitionistic transitive FR. Thus, Hence, by (1) and (2), Theorem 4. The converse of a complex intuitionistic partial order FR R on a CIFSḮ is also a complex intuitionistic partial order FR onḮ.
Proof. In order to prove the assertion, it is sufficient to show that the converse of a complex intuitionistic partial order FR R c satisfies the three properties of a complex intuitionistic partial order FR.
i Since R is a complex intuitionistic reflexive FR. Thus, for some u ∈ U, However, R is a complex intuitionistic anti-symmetric FR. Thus, Therefore, R c is also a complex intuitionistic anti-symmetric FR.
iii Let (u, v) ∈ R c and (v, w) ∈ R c Then (w, v) ∈ R and (v, u) ∈ R However, since R is a complex intuitionistic transitive FR. Thus, Thus, R c is also a complex intuitionistic transitive FR. From i, ii and iii, it is proven that R c is also a complex intuitionistic partial order FR.

Theorem 5. If R is a complex intuitionistic equivalence FR on a CIFSḮ, then
Proof. Let (u, v) ∈ R and w ∈ R[u] ⇒ (w, u) ∈ R . Now, by the transitive property of R, Since (u, v) ∈ R, by using the symmetric property of R Hence, (3) and (4) Now, by the transitive property of R, (u, w) ∈ R and (w, v) ∈ R ⇒ (u, v) ∈ R , which completes the proof.  The CP is found to bé

1.
That succeeds all the other elements is known as the maximum or greatest element.

2.
That precedes all the other elements is known as the minimum or least element.

3.
That is not related to any other element is known as the maximal element. The topmost elements of the Hasse diagram are the maximal elements.

4.
To whom any other element(s) is(are) not related is(are) known as the minimal element(s). In other words, the element(s) that is(are) related to every other element is(are) the minimal element(s). The bottommost elements of the Hasse diagram are the minimal elements.
Example 12. Let {p, q, r, s, t, u, v, w, x, y, z} be the elements of a complex intuitionistic partial order fuzzy setḮ. For convenience, we ignore the membership and non-membership grades. Figure 4 shows the Hasse diagram of setḮ.   In the above diagram, is the maximum and maximal element, while is the minimum imal element. In the above diagram, z is the maximum and maximal element, while p is the minimum and minimal element.

Definition 15.
For a subset J ofḮ, an element u ∈ R ⊆Ḯ ×Ḯ is known as the Definition 16. Let J be a subset of a CIFSḮ, then the least upper bound and the greatest lower bound of J are called the supremum and infimum of J, respectively.
Example 13. Let {p, q, r, s, t, u, v, w, x, y, z} be the elements of a complex intuitionistic partial order fuzzy setḮ and J = {q, r, s, t, u, y} be a subset ofḮ. For convenience, we ignore the membership and non-membership grades. Figure 5 shows the Hasse diagram of set J. The elements of setḮ are colored blue.

Definition 16.
Let ʝ be a subset of a CIFS Ḯ, then the least upper bound and the greatest lower bound of ʝ are called the supremum and infimum of ʝ, respectively.
Example 13. Let , , , , , , , , , , be the elements of a complex intuitionistic partial order fuzzy set Ḯ and ʝ = , , , , , be a subset of Ḯ. For convenience, we ignore the membership and non-membership grades. Figure 5 shows the Hasse diagram of set Ḯ. The elements of set ʝ are colored blue. In the above diagram, and are the upper bounds of ʝ, whereas, is the supremum of ʝ. On the other hand, and are the lower bound of ʝ, while is the infimum of ʝ.

Application
This section presents a couple of applications of the proposed concepts in the fields of information technology; more specifically, we consider cyber-security and cyber-crimes in the oil and gas industries.

Cyber-Security in Oil and Gas Industries
Huge development and modernization has taken place as a result of digitalization among various industries, and the oil and gas sector is no exception. Although advanced technological solutions such as IIOT (Industrial Internet of Things) have improved efficiency and reduced industrial expenditures, they have also exposed the oil and gas industries to the risks of cyber-crimes. These threats can have severely negative impacts on the company, resulting in massive losses of money and reputation and leading to environmental disasters. Below are some threats and the methods used for security purposes by an oil and gas company. Figure 6 presents the flowchart for the process followed in the application. In the above diagram, y and z are the upper bounds of J, whereas, y is the supremum of J. On the other hand, p and r are the lower bound of J, while r is the infimum of J.

Application
This section presents a couple of applications of the proposed concepts in the fields of information technology; more specifically, we consider cyber-security and cyber-crimes in the oil and gas industries.

Cyber-Security in Oil and Gas Industries
Huge development and modernization has taken place as a result of digitalization among various industries, and the oil and gas sector is no exception. Although advanced technological solutions such as IIOT (Industrial Internet of Things) have improved efficiency and reduced industrial expenditures, they have also exposed the oil and gas industries to the risks of cyber-crimes. These threats can have severely negative impacts on the company, resulting in massive losses of money and reputation and leading to environmental disasters. Below are some threats and the methods used for security purposes by an oil and gas company. Figure 6 presents the flowchart for the process followed in the application.
Entropy 2021, 23, x FOR PEER REVIEW 15 of 26 Figure 6. Flowchart for the process being followed.

Threats
Some of the threats that an oil and gas company are vulnerable to are explained below. Moreover, each threat and malware has been assigned the membership and nonmembership grades. These membership grades are set by professionals according to their performance and operation. The membership grade for a threat indicates its weakness, while the non-membership grade shows the strength or the severity of the threat. Since the grades range between 0 and 1, in the case of the membership grade, values closer to 1 represent greater effectiveness or success in accomplishing the target, while lower values i.e., close to zero, indicate less effectiveness and success in accomplishing the target. On Read the Information Catesian Product Figure 6. Flowchart for the process being followed.

Threats
Some of the threats that an oil and gas company are vulnerable to are explained below. Moreover, each threat and malware has been assigned the membership and nonmembership grades. These membership grades are set by professionals according to their performance and operation. The membership grade for a threat indicates its weakness, while the non-membership grade shows the strength or the severity of the threat. Since the grades range between 0 and 1, in the case of the membership grade, values closer to 1 represent greater effectiveness or success in accomplishing the target, while lower values i.e., close to zero, indicate less effectiveness and success in accomplishing the target. On the other hand, higher values of the non-membership grade indicate a higher likelihood of failure in achieving the goals and vice versa. Table 1 summarizes this section. The amplitude term represents the level of strength or weakness, while the phase term refers to the timeframe. In addition, higher values of the phase terms reflect a greater time period and the lower values refer to a shorter timespan.
Espionage and data theft (E&DT) are very serious concerns. Industries and companies are highly dependent on unique information that keeps them ahead of their competitors. In the oil and gas sector, data such as experimental results, boring procedures, new oil reserves and the chemistry of top products are extremely valuable. Thus, such data carry the greatest risk. Some tactics used for such attacks include DNS hijacking, phishing emails and corporate VPN servers or even scraping information that is openly available to obtain data.
E&DT, 0.5e (0.6)πi , 0.5e (0.2)πi c. Ever-changing malware (ECM): Usually, there are different malwares that are executed to fulfil different purposes, such as intrusion, data theft, propagation, etc. A cyber-criminal wishes to maintain their access to the targeted system in order to steal critical information by continuously and constantly updating the malware codes. Some malwares used by cyber-criminals to infect targets, maintain persistence and communicate are web-shells, DNS tunneling, email and cloud services.
Ransomware (RW) is a malware that is used to steal or encode the valuable information of a company. These malwares hugely impact the regular operations of an industry. In order to recover lost data, the industries are more likely to pay the ransom. RW, 0.6e (0.7)πi , 0.4e (0.2)πi e. Insider threat (InT) is a serious threat to an industry that comes from the employees, former employees, contractors or business associates of the industry, i.e., people with confidential information about security practices, data and the digital system.
Henceforth, the CIFS J summarizing the security threats is given below:

Security Methods
The methods, techniques and practices adopted by the oil and gas industries to protect against cyber-crimes are discussed below. Each security method is assigned a pair of functions in the form of membership and non-membership grades. The membership grade represents the security level of a method, while the non-membership grade represents the risk levels that can result from implementing these techniques. Table 2 recaps this section. The amplitude term refers to the level or grade of security or risk, while the phase term refers to time. The assignation of the values the phase terms and amplitude term of membership and non-membership grades is similar to the assignment of values to the threats. Obviously, greater values of amplitude terms of membership grades are preferable with regard to security because they indicate greater security. However, higher values of phase terms of membership grades are also important, as they indicate long-term security. On the contrary, smaller values of amplitude terms of non-membership grades refer to a lower risk or insecurity. Thus, better security is indicated by a greater amplitude term value as well as a greater phase term value of its membership grade and lower values of the amplitude term and phase term of non-membership grades. Some of the most important security methods for the oil and gas industry are defined below with their fuzzy grades. a. Deep Armor Industrial (DMI) is an AI-based technology that identifies and reports new devices or irregular activities such as insider threats and digital-physical attacks. Due to its predictive analysis, the execution of malicious codes is prevented. It is highly effective and can stop codes that are not yet present in threat intelligence packages. DMI provides unique, innovative and exceptional security to the oil and gas industries, even against new threats that emerge between updates, or cyberattacks that arrive at isolated sites before patches can be deployed. It intends to provide the latest antivirus software, detection of threats, control of application and zero-day attack prevention to the industry. Henceforth, the following CIFSḮ summarizing the security methods is constructed:

Calculations
Here, in order to study the relationships among the effectiveness and incompetency of each cyber-security technique against every cyber-crime, we carry out the following mathematics.
We have the following two CIFSsḮ and J representing the set of securities and the set of threats, respectively.
With the intention of determining the effectiveness of the security methods against each threat, we calculate the CPḮ × J. By using Definition 8, we have Each element ofḮ × J is in the form of an order pair, which represents the relationship among the pair, i.e., the effects and impacts of the first term on the second term in the pair. The membership grades indicate the effectiveness of a security technique to eliminate a particular threat with respect to some time unit. On the other hand, the non-membership grades reflect the uselessness or ineffectiveness of a certain security method against a specific threat. For example, the element (EM, RW), 0.6e (0.6)πi , 0.4e (0.3)πi conveys that the event monitoring solution can effectively resolve the ransomware and the level of ineffectiveness is low. The numbers translate as follows: the level of security of event monitoring solution against the ransomware is 0.6 with respect to 0.6 time units and the level of risk due to ransomware after applying the event monitoring solution is 0.4 with respect to 0.3 time units. Regarding the security, a greater timeframe in the membership grade is better, while a shorter time in the non-membership grade is better. The diagrams in Figure 7 demonstrate the above relationships.

Selection of the Cyber-Security Techniques
Suppose that an enterprise or company wishes to implement some cyber-security techniques in order to control and counter the potential threats and reduce the risks of cyber-attacks. There are certain possible techniques, which are given in the following Ta-

Selection of the Cyber-Security Techniques
Suppose that an enterprise or company wishes to implement some cyber-security techniques in order to control and counter the potential threats and reduce the risks of cyberattacks. There are certain possible techniques, which are given in the following Table 3. However, the company needs to select the best possible techniques that would resolve the issues that are being faced. The complete process of this application is shown in Figure 8. ble 3. However, the company needs to select the best possible techniques that would resolve the issues that are being faced. The complete process of this application is shown in Figure 8.   Let us assign the supposed membership and non-membership grades to each of the security measures and construct a CIFSḮ: Based on the CPḮ ×Ḯ (using Definition 8), a complex intuitionistic partial order FR R (Definition 9) is obtained: Using the Definition 13, the Hasse diagram for the above complex intuitionistic partial order FR is constructed, which is given in Figure 9. For convenience, the membership and non-membership grades are not listed in the diagram. According to R, block chain is the best security technique among the available competitors because it is the maximum as well as the maximal element (according to Definition 14), while security software such as antivirus and antimalware software provides the least security because SSW is the minimum as well as the minimal element of the diagram (according to Definition 14).
Assume that the company has certain priorities, and the following security techniques have been separated from a larger set of techniques. These security measures are listed in the subset J, which is indicated in blue in the diagram. The objective is to choose the best security technique among the members of set J. Thus, one may be interested in the upper bounds and supremum. In this case, according to Definition 15, the upper bounds are {DMI, EM, NNS, BC}. By Definition 16, the supremum is the lowest upper bound, which is NNS. Hence, the Nozomi network solution is the most suitable cyber-security measure among the shortlisted ones that will eliminate the threats or reduce the risks of cyber-attacks. .

Comparative Analysis
In this section, the omnipotence of the proposed framework of CIFRs through a comparison of CIFRs with the existing structures such as CFRs or IF The major advantage of a CIFR over FR and IFR is the complex-valued m and non-membership grades. The structure of a CIFR is composed of amplitud phase term, which enable it to model the situations with phase alteration and On the other hand, the FRs and IFRs lack the phase term; thus, they are lim single dimensional models. Meanwhile, the structure of CFRs is based on a complex number and thus con plitude and phase terms. A detailed comparison between CIFRs and other s given in the following subsections.

CIFRs vs. CFRs
Let us study the relationships between the set of cyber-security techniques crimes using the CFRs. As CFRs are superior to FRs, the comparison is carried o CIFRs and CFRs. Consider the following two CFSs Ḯ and ʝ representing the rity measures and the set of threats, respectively. The details of the abbreviati sets Ḯ and ʝ are given in Tables 4 and 5.

Comparative Analysis
In this section, the omnipotence of the proposed framework of CIFRs is verified through a comparison of CIFRs with the existing structures such as CFRs or IFRs.
The major advantage of a CIFR over FR and IFR is the complex-valued membership and non-membership grades. The structure of a CIFR is composed of amplitude term and phase term, which enable it to model the situations with phase alteration and periodicity. On the other hand, the FRs and IFRs lack the phase term; thus, they are limited to only single dimensional models.
Meanwhile, the structure of CFRs is based on a complex number and thus consists of amplitude and phase terms. A detailed comparison between CIFRs and other structures is given in the following subsections.

CIFRs vs. CFRs
Let us study the relationships between the set of cyber-security techniques and cybercrimes using the CFRs. As CFRs are superior to FRs, the comparison is carried out between CIFRs and CFRs. Consider the following two CFSsḮ and J representing the set of security measures and the set of threats, respectively. The details of the abbreviations used in setsḮ and J are given in Tables 4 and 5  By applying the procedure detailed in Figure 6, the following CFR R is found, which discusses the effectiveness of security methods against each threat: Each of the elements in the above CFR demonstrates the connection between a pair of elements in an ordered pair. The effects of the first element (appearing first in the ordered pair) on the second element (appearing latter in the ordered pair) are described by the membership grades. Since the Cartesian product is carried out from the set of security measures to the set of threats, the found relation indicates the effects of the security measures on the threats. Since it is a CFR, its membership grades only display the effectiveness of a cyber-security measure against a particular threat. It also fails to provide valuable information about the ineffectiveness and failure levels of each cyber-security measure against certain threats. Meanwhile, the CIFRs provide the complete information. Hence, this shows the dominance of CIFRs over CFRs and FRs.

CIFRs vs. IFRs
In this section, the IFRs and IFSs are used to investigate the matter discussed in the proposed applications, i.e., the relationships between the set of cyber-security techniques and the set of cyber-crimes. Consider the following two IFSsḮ and J representing the set of security measures and the set of threats, respectively. The details of the abbreviations used in setsḮ and J are given in Tables 4 and 5 Following the process discussed in Figure 6, the following IFR R is obtained, which discusses the usefulness of the security methods against each threat: Based on the information in the above IFR, each element shows the relationship between a pair of elements. The approach to the interpretation of the information and determination of the results is similar to the previous examples. Thus, being an IFR, it only contains the amplitude terms, and the phase terms are missing. It produces incomplete results because the duration is missing. Therefore, it only defines the effectiveness and ineffectiveness of the cyber-security measures against the threats through the membership grade and non-membership grade, respectively. This structure is unsuccessful in providing the required results. The CIFRs produce satisfactory results that are required to obtain detailed information. Hence, this example illustrates the supremacy of CIFRs over IFRs. For this reason, this article chose the structure of CIFR to analyze the matter of cyber-security and cyber-crimes in the oil and gas industries.

Conclusions
This article introduces the novel concepts of complex intuitionistic fuzzy relation (CIFR) and the Cartesian product of two complex intuitionistic fuzzy sets (CIFSs). In addition, the types of CIFRs are defined, such as equivalence, pre-order, partial order, total order, strict order relations, equivalence class and the composition of two CIFRs. Moreover, the Hasse diagram of complex intuitionistic partial order fuzzy relations is introduced. The notions of maximum, minimum, maximal, minimal, supremum and infimum, etc., are defined for a Hasse diagram. The development of these innovative frameworks and novel modeling techniques aims to address the cyber-security concerns in the oil and gas industries. These industries have recently been targeted by hackers and cyber-criminals. Thus, the current study analyzes the relationships among the effectiveness of cyber-security measures and the most serious and common risks to the mentioned industries. Then, the CIFRs are applied for the security analysis of the oil and gas industries to explore the effects of certain cyber-security measures on the threats. Moreover, the complex intuitionistic partial order fuzzy relation and the Hasse diagram are used to determine the most appropriate cyber-security method for an industry. Lastly, the proposed methods are compared with the other methods in the literature. The weaknesses of the proposed methods include the absence of a neutral grade as well as the limitations and the constraints on the sum of grades. In future, these concepts can be extended to the other generalizations of fuzzy sets [48][49][50][51], which will give rise to many interesting structures with a vast range of applications.