Fermatean fuzzy semi-prime ideals of ordered semigroups

Amal Kumar Adak; Nilkamal; Navendu Barman

doi:10.1515/taa-2023-0102

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Fermatean fuzzy semi-prime ideals of ordered semigroups

Amal Kumar Adak , Nilkamal and Navendu Barman

Published/Copyright: October 5, 2023

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From the journal Topological Algebra and its Applications Volume 11 Issue 1

Abstract

Fermatean fuzzy sets (FFSs) are invented to resolve the underlying limitations of intuitionistic fuzzy sets and Pythagorean fuzzy sets. The major goal of this study is to introduce Fermatean fuzzy semi-prime ideals of ordered semigroups. Fermatean fuzzy semi-prime ideals and Fermatean fuzzy prime ideals are introduced. Also, we illustrate some novel concepts to construct Fermatean fuzzy intra-regular and regular ideals. Using the conception of function of the characteristic of ordered semigroups of FFSs, we show certain fundamental facts. Several relations are given for the family of Fermatean fuzzy regular ideals of ordered semigroups.

Keywords: Pythagorean fuzzy sets; intuitionistic fuzzy sets; Fermatean fuzzy set; Fermatean fuzzy ideals; Fermatean fuzzy regular ideals; Fermatean fuzzy semi-prime ideals

MSC 2010: 08A72; 16Y30; 03E72; 20N25

1 Introduction

In our daily life, uncertainty is unavoidable. The physical world was not produced exact measurements or presumptions. Forward decision making is not always possible. There are several significant problems in dealing with crisp data in decision making. In 1965, Zadeh [23] conceived fuzzy sets (FSs) to deal with unclear uncertain data in real-world problems. FSs represent the grade to each element under study, and every component of the conceptual universe is renumbered from of the unit range [ 0 , 1 ] . We designate a value from the unit range [ 0 , 1 ] to each element of the discursive multiverse to signify the degree of sense of esteem and self-actualization to the set under study. FSs are a subclass of set theory that allows for states that are midway between completeness and nothingness. The membership value is assigned from 0 to 1. The membership value 0 indicates that the element is not a member of the set, whereby 1 reflecting that it is, and additional considerations revealing the level of participation.

Assigning only membership value is not always adequate for decision making. To handle hesitation in human mind, FSs do not have the ability to solve this situation. Atanassov [7] developed the notions of intuitionistic fuzzy sets (IFSs) to define hesitations more clearly, which were important generalizations of FSs. Membership and non-membership levels are assigned using this method to model vagueness and impression, while maintaining that this sum is less than or equal to 1. The main contribution of IFSs is their capacity to manage uncertainty that may arise from incomplete information. It has succeeded in a variety of fields due to its capacity to handle uncertainty [2,5,9,16].

IFSs cannot demonstrate the situations when it does not fulfill restrictions. Pythagorean fuzzy sets (PFSs) were proposed to address the limitations of IFSs. Yager and Abbasov [22] and Yager [21] pioneered the PFSs that relaxed the space of IFSs, introducing the restriction that the sum of square of its grade should lie between 0 and 1. When IFSs and PFss are compared, it is noted that PFSs provide power to express the uncertainty and are flexible since the space of PFS membership degree is larger than that of the space of IFS membership degree.

Although, PFSs generalize the IFSs, it cannot describe the following decision information. If we consider the degree of membership as 0.7 and that of non-membership as 0.8, then it is obvious that 0.7 + 0.8 > 1 and 0 . 7 2 + 0 . 8 2 > 1 , and in this situation, they are not represented by IFSs and PFSs. To demonstrate such information, Senapati and Yager [19] developed Fermatean fuzzy sets (FFSs). FFSs have degree membership and non-membership satisfying the requirements that the sum of cube of grades must be in between 0 and 1. In the aforementioned situations, 0 . 7 3 + 0 . 8 3 < 1 . The membership space of FFSs is bigger than that of PFSs and also IFSs (Figure 1).

Figure 1

Spaces for IFSs, PFSs, and FFSs.

Algebraic patterns are used in theoretical physics, information science, computer science, control engineering, and along with many other domains. These patterns provide sufficient impetus for scholars to review many real abstract algebra topics and discoveries in the context of a broader fuzzy setting. Biswas [8] defined fuzzy subgroups and anti-fuzzy subgroups. Jun et al. [10] gave some deductions on intuitionistic fuzzy ideals of near rings. The concept of intuitionistic fuzzy interior ideals was introduced by Kim and Jun [11,12]. Kim and Lee [13] developed the implication of intuitionistic fuzzy bi-ideals of semigroups. Kuroki [14,15] invented important results on fuzzy bi-ideals. Sardar et al. [17] introduced semi-prime ideals, prime ideals of Γ -semigroup with intuitionistic fuzzy information. Adak and Salokokolaei [3] introduced the concept of rough PFSs and cut-sets of PFSs. Adak and Salokolaei [1] and Adak [4] discussed some properties of Pythagorean fuzzy ideals. Adak and Kumar [6] invented the important characterizations of Pythagorean fuzzy ideals of Γ -near rings. Senapati and Yager [19] defined some new operations over Fermatean fuzzy numbers. In [18,20] some works were found in the field of FFSs.

In this study, the algebraic structures of FFSs were investigated. The phrases Fermatean fuzzy semi-prime ideals and ordered semigroups are introduced . The notions Fermatean fuzzy right regular, left regular, and intra-regular ideals of ordered semigroups are presented. Some important theorems on Fermatean fuzzy interior ideals are described.

The following is how the rest of this article is organized. Section 2 reviews preliminaries and definitions for terms such as ordered set, ordered subgroups, IFSs and PFSs. We looked at how Fermatean fuzzy semi-prime ideals and prime ideals of ordered semigroups are defined, as well as some of the key features of Fermatean fuzzy regular and intra-regular ordered semigroup, in Section 3. At the end, conclusion is laid out in Section 4.

2 Preliminaries and definition

Connections of fuzzy sets, IFSs and PFSs, are discussed in this section. The etymology of ordered set, ordered semigroup, semi-prime ideal, and prime ideal is mentioned.

Definition 2.1

M ( ≠ ϕ ) is said to be an ordered semigroup if it is both an ordered set and a semigroup that meets the requirement:

a ≤ b implies m a ≤ m b and a m ≤ b m , ∀ a , b , m ∈ M .

In this study, ( M , . , ≤ ) be the ordered semigroup and it is simply denoted as M .

Definition 2.2

Consider ( M , . , ≤ ) the ordered semigroup. If S 2 ⊆ S , then S is called the subsemigroup of M , where S ≠ ϕ and S ⊆ M .

Definition 2.3

Consider A ≠ ϕ and A ⊆ M . Then, A is claimed as a left ideal (respectively, right ideal) of M if:

M A ⊆ A (respectively, A M ⊆ A ),
( ∀ p ∈ A ) ( ∀ q ∈ M ), ( q ≤ p ⇒ q ∈ A ) .

If both the conditions of left ideals and right ideals are satisfied by A , then A is identified as ideal of M .

Definition 2.4

Let A ( ≠ ϕ ) and A ⊆ M .

If p q ∈ A ⇒ p ∈ A or, q ∈ A for all p , q ∈ M , then A is stated as prime.

If A is prime and also ideal of M , then A is referred to as prime ideal.

Definition 2.5

A ≠ ϕ subset of M is defined semi-prime if it satisfies the following criteria:

p 2 ∈ A ⇒ p ∈ A for all p ∈ M .

A is defined as semi-prime ideal if it is semi-prime and ideal of M .

Definition 2.6

Fuzzy set F on X , universal set, is laid out as:

F = { ⟨ x , μ F ( x ) ⟩ : x ∈ X } ,

where μ F ( x ) indicates the fuzzy membership levels that assign have the values in between 0 and 1.

Supplement of μ is stipulated as μ ¯ ( x ) = 1 − μ ( x ) for each x ∈ X and denoted by μ ¯ .

Definition 2.7

Let μ ( ≠ ϕ ) and μ ⊆ M , then μ is defined as fuzzy ideal of M , if:

p ≤ q , then μ ( p ) ≥ μ ( q ) ,
μ ( p q ) ≥ max { μ ( p ) , μ ( q ) } , ∀ ( p , q ) ∈ M .

Definition 2.8

In X , an IFS I is outlined as:

I = { ⟨ x , α I ( x ) , β I ( x ) ⟩ : x ∈ X } ,

where α A ( x ) indicates the membership value and β A ( x ) indicates the non-membership value of x ∈ X , respectively.

Also, α I and β I are assigned the values in between 0 and 1. These fulfills

0 ≤ α I ( x ) + β I ( x ) ≤ 1 ,

for every x ∈ X .

h I ( x ) = 1 − α I ( x ) − β I ( x ) reflects the grade of hesitate.

In some circumstances, for whatever reason, 0 ≤ α ( x ) + β ( x ) ≤ 1 , this may not be true. We take some situations, where α = 0.7 and β = 0.5 such that 0.7 + 0.5 = 1.2 > 1 , but 0.7 2 + 0.5 2 < 1 . Again, if α = 0.6 and β = 0.6 where 0.6 + 0.6 = 1.2 > 1 , but 0.6 2 + 0.6 2 < 1 . To deal with this situation, Yager and Abbasov [21], Yager [22] proposed the perspective assumptions of PFS in 2013.

Definition 2.9

In universe of discourse X , A PFS P is specified as:

P = { ⟨ x , u P ( x ) , v P ( x ) ⟩ : x ∈ X } ,

u P ( x ) : X → [ 0 , 1 ] refers to the membership value and v P ( x ) : X → [ 0 , 1 ] refers to the value to which the element x ∈ X is not a member of the set P , fulfills that

0 ≤ ( u P ( x ) ) 2 + ( v P ( x ) ) 2 ≤ 1 , ∀ x ∈ X .

h P ( x ) = 1 − ( u P ( x ) ) 2 − ( v P ( x ) ) 2 indicates indeterminacy.

In practice, the condition 0 ≤ μ 2 ( x ) + ν 2 ( x ) ≤ 1 may not be true for any reason. For example, if we consider u = 0.9 v = 0.5 , where 0 . 9 2 + 0 . 5 2 = 1.06 > 1 , but 0.9 3 + 0.5 3 = 0.854 < 1 . Again, 0.8 2 + 0.7 2 = 1.13 > 1 , but 0.8 3 + 0.7 3 = 0.855 < 1 . To address this issue, Senapati and Yager [18] proposed FFS in 2021.

Definition 2.10

An FFS A in X is defined as:

A = { ⟨ x , u A ( x ) , v A ( x ) ⟩ ∣ x ∈ X } ,

u A ( x ) : X → [ 0 , 1 ] signifies the membership value and v A ( x ) : X → [ 0 , 1 ] stands for the non-membership value to which the element x ∈ X is not a member of the set A , with the condition that:

0 ≤ ( u A ( x ) ) 3 + ( v A ( x ) ) 3 ≤ 1 ,

for every x ∈ X .

h A ( x ) = 1 − ( u A ( x ) ) 3 − ( v A ( x ) ) 3 3 corresponds to the value of hesitate.

Definition 2.11

Let A = ( u A , v A ) be an FFS of ordered semigroup M . Then, A = ( u A , v A ) is designed as Fermatean fuzzy subsemigroup of M if:

u A ( α β ) ≥ min { u A ( α ) , u A ( β ) } ,
v A ( α β ) ≤ max { v A ( α ) , v A ( β ) } , ∀ α , β ∈ M .

Definition 2.12

An FFS A = ( u A , v A ) in M is said to be the Fermatean fuzzy left ideal of M if it fulfills:

α ≤ β ⇒ u A ( α ) ≥ u A ( β ) ; u A ( α β ) ≥ u A ( β ) ,
α ≤ β ⇒ v A ( α ) ≤ v A ( β ) ; v A ( α β ) ≤ v A ( β ) , ∀ α , β ∈ M .

Definition 2.13

An FFS A = ( u A , v A ) of M , is indicated to be Fermatean fuzzy right ideal of M if comply with:

α ≤ β ⇒ u A ( α ) ≥ u A ( β ) ; u A ( α β ) ≥ u A ( α ) ,
α ≤ β ⇒ v A ( α ) ≤ v A ( β ) ; v A ( α β ) ≤ v A ( α ) , ∀ α , β ∈ M .

A = ( u A , v A ) is referred to as the Fermatean fuzzy ideal of ordered semigroup M if it is both right ideal and left ideal.

3 Main results

This section sets for Fermatean fuzzy semi-prime ideals, Fermatean fuzzy prime ideals, Fermatean fuzzy intra-regular ideals, and Fermatean fuzzy regular ideals of ordered semigroups. Also, this section brings about numerous important structures on non-empty subset of ordered semigroups using the characteristic function.

Definition 3.1

Let u ˜ be the fuzzy subset of M , then u ˜ is known as prime, if:

u ˜ ( a b ) = max { u ˜ ( a ) , u ˜ ( b ) } , ∀ a , b ∈ M .

Let u ˜ be a fuzzy ideal as well as a prime fuzzy subset of M , then it is recognized as a fuzzy prime ideal of M .

Definition 3.2

Let A ˜ = ( u A , v A ) be an FFS in M and A ˜ is indicated as Fermatean fuzzy prime of M if:

u A ( a b ) = max { u A ( a ) , u A ( b ) } ,
v A ( a b ) = min { v A ( a ) , v A ( b ) } , ∀ a , b ∈ M .

χ S is depicted as a characteristic function of a nonempty subset S of M .

Theorem 3.1

Consider a prime ideal S, then S ˜ = ( χ S , χ ˜ S ) is a Fermatean fuzzy prime ideal of M.

Proof

Let us consider a , b ∈ M . If a b ∈ S , then a ∈ S or, b ∈ S . Thus, χ S ( a ) = 1 or, χ S ( b ) = 1 .

Thus, we have

χ S ( a b ) = 1 = max { χ S ( a ) , χ S ( b ) }

and

χ ˜ S ( a b ) = 1 − χ S ( a b ) = 0 = min { χ ˜ S ( a ) , χ ˜ S ( b ) } .

If a b ∉ S , then a ∉ S and b ∉ S .

Thus, χ S ( a ) = 0 and χ S ( b ) = 0 .

Therefore,

χ S ( a b ) = 0 = max { χ S ( a ) , χ S ( b ) }

and

χ ˜ S ( a b ) = 1 − χ S ( a b ) = 1 = min { χ ˜ S ( a ) , χ ˜ S ( b ) } .

Therefore, S ˜ = ( χ S , χ ˜ S ) is a Fermatean fuzzy prime of M .□

Theorem 3.2

Suppose S ( ≠ ϕ ) and S ⊆ M . If S ˜ = ( χ S , χ ˜ S ) is the prime ideal of M, then S is the prime ideal of M.

Proof

Suppose that S ˜ = ( χ S , χ ˜ S ) is the prime ideal of M and a b ∈ S .

In this case, s = a b for some s ∈ S .

Therefore,

1 = χ S ( s ) = χ S ( a b ) = 1 = max { χ S ( a ) , χ S ( b ) } .

Hence, χ S ( a ) = 1 or χ S ( b ) = 1 , i.e., a ∈ S or b ∈ S .

Thus, S is the prime ideal.

Now, assume that S ˜ = ( χ S , χ ˜ S ) is a prime ideal of M and a ′ b ′ ∈ S .

Then, s ′ = a ′ b ′ for some s ′ ∈ S .

Now, from the property of prime, we obtain

χ ˜ S ( s ′ ) = 1 − χ S ( s ′ ) = 0 = χ ˜ S ( a ′ b ′ ) = min { χ ˜ S ( a ′ ) , χ ˜ S ( b ′ ) } = min { 1 − χ S ( a ′ ) , 1 − χ S ( b ′ ) }

and so 1 − χ S ( a ′ ) = 0 , or 1 − χ S ( b ′ ) = 0 .

Therefore, χ S ( a ′ ) = 1 , or χ S ( b ′ ) = 1 , i.e., a ′ ∈ S or b ′ ∈ S .

Therefore, S is a prime ideal of M .□

Definition 3.3

Let u be a fuzzy subset of an ordered semigroup M . If u ( a ) ≥ u ( a 2 ) , for all a ∈ M , then u is called the fuzzy semi-prime.

A fuzzy ideal u of M is called a fuzzy semi-prime ideal of M if u is a fuzzy semi-prime subset of M .

Definition 3.4

Let A = ( u A , v A ) be an FFS in M . Then, A = ( u A , v A ) is called the Fermatean fuzzy semi-prime of M if the following criteria are satisfied:

u A ( a ) ≥ u A ( a 2 ) ,
v A ( a ) ≤ v A ( a 2 ) , for all a ∈ M .

Theorem 3.3

If S is the semi-prime of M, then S ˜ = ( χ S , χ ˜ S ) is a Fermatean fuzzy semi-prime of M.

Proof

If s 2 ∈ S , then since S is semi-prime, we have s ∈ S .

Thus,

χ S ( s ) = 1 ≥ χ S ( s 2 ) and χ ¯ S ( s ) = 1 − χ S ( s ) = 0 ≤ χ ¯ S ( s 2 ) .

If s 2 ∉ S , then we have χ S ( s 2 ) = 0 .

Therefore,

χ S ( s ) ≥ 0 = χ S ( s 2 ) and χ ¯ S ( s 2 ) = 1 − χ S ( s 2 ) = 1 ≥ χ ¯ S ( s ) .

Hence, the result is obtained.□

Theorem 3.4

Let S be a non-empty subset of ordered semigroup M. If S ˜ = ( χ S , χ S ˜ ) is the Fermatean fuzzy semi-prime of M, then S is the Fermatean semi-prime.

Proof

Suppose that S ˜ = ( χ S , χ ˜ S ) is a Fermatean fuzzy semi-prime of M and a 2 ∈ S .

In this case, s = a 2 for some s ∈ S . It follows that:

1 = χ S ( s ) = χ S ( a 2 ) ≤ χ S ( a ) .

Hence, χ S ( a ) = 1 , i.e., a ∈ S .

Thus, S is the Fermatean fuzzy semi-prime.

Now, assume that S ˜ = ( χ S , χ ˜ S ) is a Fermatean fuzzy semi-prime of M and a 0 2 ∈ S . Then, s 0 = a 0 2 for some s 0 ∈ S .

Therefore,

χ ˜ S ( a 0 ) ≤ χ ˜ S ( a 0 2 ) = 1 − χ S ( a 0 2 ) = 1 − 1 = 0 ,

i.e., χ ˜ S ( a 0 ) = 1 − χ S ( a 0 ) = 0 .

Hence, χ S ( a 0 ) = 1 , and so, a 0 ∈ S .

This completes the proof.□

Theorem 3.5

Let A = ( u A , v A ) be the Fermatean fuzzy sub-semigroup of M, if A is the Fermatean fuzzy semi-prime, then A ( a ) = A ( a 2 ) holds.

Proof

Since u A is a fuzzy subsemigroup of M , then

u A ( a ) ≥ u A ( a 2 ) = min { u A ( a ) , u A ( a ) } = u A ( a ) , a ∈ M ,

and so we have u A ( a ) = u A ( a 2 ) .

Again,

v A ( a ) ≤ v A ( a 2 ) ≡ max { v A ( a ) , v A ( a ) } = v A ( a ) .

Therefore, v A ( a ) = v A ( a 2 ) .

Combining the aforementioned two results, it is concluded that A ( a ) = A ( a 2 ) .□

Definition 3.5

If there is an element x in an ordered semigroup S such that α ≤ x α 2 (resp. α ≤ α 2 x ), then S is said to be left (resp. right) regular ideal, where α ∈ M .

Theorem 3.6

Let F = ( u F , v F ) Fermatean fuzzy left ideal of S, then F ( α ) = F ( α 2 ) holds for all α ∈ S , where S is left regular ideal.

Proof

Let α ∈ S and F = ( u F , v F ) be a Fermatean fuzzy left ideal. Therefore, ∃ an element x ∈ S such that α ≤ x α 2 , since S is left regular ideal.

Consequently,

u F ( α ) ≥ u F ( x α 2 ) ≥ u F ( α 2 ) ≥ u F ( α ) .

This gives u F ( α ) = u F ( α 2 ) .

Again,

v F ( α ) ≤ v F ( x α 2 ) ≤ v F ( α 2 ) ≤ v F ( α ) .

This means that v F ( α ) = v F ( x α 2 ) .

Combining these gives A ( α ) = A ( α 2 ) .□

Theorem 3.7

Fermatean fuzzy left ideal of the Fermatean fuzzy left regular ideal is Fermatean fuzzy semi-prime ideal.

Proof

Let F = ( u F , v F ) be a Fermatean fuzzy left ideal of S and let α ∈ S .

Then, ∃ x ∈ S such that α ≤ x α 2 , as S is the left regular ideal.

Therefore,

u F ( α ) ≥ u F ( x α 2 ) ≥ u F ( α 2 ) and v F ( α ) ≤ v F ( x α 2 ) ≤ v F ( α 2 ) .

Hence, F = ( u F , v F ) is an Fermatean fuzzy semi-prime ideal.□

Definition 3.6

An ordered semigroup S is called intra-regular ideal if, for each element α of S , there exist elements x and y in S such that α ≤ x α 2 y .

Definition 3.7

Let F = ( u F , v F ) be an FSS in intra-regular ideal S . Then, F is referred to as the Fermatean fuzzy interior ideal of S if it adheres to:

x ≥ y ⇒ u F ( x ) ≥ u F ( y ) ; u F ( x s y ) ≥ u F ( s ) ,
x ≤ y ⇒ v F ( x ) ≤ v F ( y ) ; v F ( x s y ) ≤ v F ( s ) , ∀ x , y ∈ F , s ∈ S .

Theorem 3.8

A Fermatean fuzzy intra-regular ideal is a Fermatean fuzzy interior ideal if and only if it is a Fermatean fuzzy ideal.

Proof

Let F = ( u F , v F ) be a Fermatean fuzzy interior ideal of S . Then, since S is intra-regular, there exist elements x , y , c , and d in S such that x 1 ≤ x x 1 2 y and x 2 ≤ c x 2 2 d , x 1 , x 2 ∈ S .

Since F is a fuzzy interior ideal of S , so

u F ( x 1 x 2 ) ≥ u F ( ( x x 1 2 y ) x 2 ) = u F ( ( x x 1 ) x 1 ( y x 2 ) ) ≥ u F ( x 1 )

and

u F ( x 1 x 2 ) ≥ u F ( x 1 ( c x 2 2 d ) ) = u F ( ( x 1 c ) x 2 ( x 2 d ) ) ≥ u F ( x 2 ) .

Also, we have

v F ( x 1 x 2 ) ≤ v F ( ( x x 1 2 y ) x 2 ) = v F ( ( x x 1 ) x 1 ( y x 2 ) ) ≤ v F ( x 1 )

and

v F ( x 1 x 2 ) ≤ v F ( x 1 ( c x 2 2 d ) ) = v F ( ( x 1 c ) x 2 ( x 2 d ) ) ≤ v F ( x 2 ) .

On the other hand, let F = ( u F , v F ) be a Fermatean fuzzy ideal of S . Then, since u F is a fuzzy ideal of S , we have

u F ( x x 1 y ) = u F ( x ( x 1 y ) ) ≥ u F ( x 1 y ) ≥ u F ( x 1 ) and v F ( x x 1 y ) = v F ( x ( x 1 y ) ) ≤ v F ( x 1 y ) ≤ v F ( x 1 )

for all x , x 1 and y ∈ S . This completes the proof.□

Theorem 3.9

Let F = ( u F , v F ) be a Fermatean fuzzy ideal of S . If S is intra-regular, then F is the Fermatean fuzzy semi-prime ideal.

Proof

Let x 1 be any element of S . Then, since S is intra-regular, there exist x and y in M such that x 1 ≤ x x 1 2 y . So, we have

u F ( x 1 ) ≥ u F ( x x 1 2 y ) ≥ u F ( x 1 2 y ) ≥ u F ( x 1 2 )

and

v F ( x 1 ) ≤ v F ( x x 1 2 y ) ≤ v F ( x 1 2 y ) ≤ v F ( x 1 2 ) .

Hence, the result is obtained.□

Theorem 3.10

Let S be the intra-regular ideal and F = ( u F , v F ) the Fermatean fuzzy interior ideal, F ( x 1 ) = F ( x 1 2 ) holds for all x 1 ∈ S .

Proof

Since S is intra-regular ideal, ∃ x , y in S such that x 1 ≤ x x 1 2 y ∀ x 1 ∈ S . Therefore,

u F ( x 1 ) ≥ u F ( x x 1 2 y ) ≥ u F ( x 1 2 ) ≥ u F ( ( x x 1 2 y ) ( x x 1 2 y ) ) = u F ( ( x x 1 ) x 1 ( y x x 1 2 y ) ) ≥ u F ( x 1 ) and v F ( x 1 ) ≤ v F ( x x 1 2 y ) ≤ v F ( x 1 2 ) ≤ v F ( ( x x 1 2 y ) ( x x 1 2 y ) ) = v F ( ( x x 1 ) x 1 ( y x x 1 2 y ) ) ≤ v F ( x 1 ) .

Thus, it is concluded that F ( x 1 ) = F ( x 1 2 ) .□

Theorem 3.11

Let S be the intra-regular ideal and F = ( u F , v F ) the Fermatean fuzzy interior ideal; if x 1 , x 2 ∈ S , then F ( x 1 x 2 ) = F ( x 2 x 1 ) holds.

Proof

Since S is intra-regular, there exist α , β ∈ S such that x 1 ≤ α x 1 2 β , where x 1 ∈ S . Therefore,

u F ( x 1 x 2 ) = u F ( ( x 1 x 2 ) 2 ) = u F ( x 1 ( x 2 x 1 ) x 2 ) ≥ u F ( x 2 x 1 ) = u F ( ( x 2 x 1 ) 2 ) = u F ( x 2 ( x 1 x 2 ) x 1 ) ≥ u F ( x 1 x 2 )

and

v F ( x 1 x 2 ) = v F ( ( x 1 x 2 ) 2 ) = v F ( x 1 ( x 2 x 1 ) x 2 ) ≤ v F ( x 2 x 1 ) = v F ( ( x 2 x 1 ) 2 ) = v F ( x 2 ( x 1 x 2 ) x 1 ) ≤ v F ( x 1 x 2 ) .

Hence, F ( x 1 x 2 ) = F ( x 2 x 1 ) .□

Definition 3.8

If there would be a positive integer n that has the property of a n ∈ M b M for any element a , b ∈ M , then an ordered semigroup M is called Archimedean.

Theorem 3.12

Each Fermatean fuzzy semi-prime ideal of ordered Archimedean semigroup S is a constant function.

Proof

Consider F = ( u F , v F ) any Fermatean fuzzy semi-prime ideal of S , while x 1 , x 2 ∈ S be a Fermatean fuzzy semi-prime fuzzy ideal. Whenever S is Archimedean, α and β exist in S so that x 1 n = α y 1 β for some integer n . Then, there is

u F ( x 1 ) = u F ( x 1 n ) = u F ( α x 2 β ) ≥ u F ( x 2 ) and , u F ( x 2 ) = u F ( x 2 n ) = u F ( α x 1 β ) ≥ u F ( x 1 ) .

Therefore, u F ( x 1 ) = u F ( x 2 ) .

Again,

v F ( x 1 ) = v F ( x 1 n ) = v F ( α x 2 β ) ≥ v F ( x 2 ) and v F ( x 2 ) = v F ( x 2 n ) = v F ( α x 1 β ) ≥ v F ( x 1 ) .

Hence, F ( x 1 ) = F ( x 2 ) for all x 1 , x 2 ∈ S .□

4 Conclusion

With respect to cognitive uncertainty, FFS is an effective generalization of such fuzzy set theory. The ideas of Fermatean fuzzy semi-prime ideals and prime ideals of ordered semigroups are introduced. Several attractive qualities are investigated. Also, FF regular ideals by exploring various results on Fermatean fuzzy regular ideals and intra-regular ideals of ordered semigroups are implemented. Later, we try to solve some problems on decision-making process using the interval-valued FSS uncertain data.

Acknowledgments

The authors are very grateful to the anonymous referees and editor for their valuable comments to improve the quality of this manuscript.

Funding information: No external fund was received for this research work.
Conflict of interest: The authors declare that there is no competing of interest.

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Received: 2022-08-03

Revised: 2023-08-25

Accepted: 2023-09-01

Published Online: 2023-10-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/taa-2023-0102

Keywords for this article

Pythagorean fuzzy sets; intuitionistic fuzzy sets; Fermatean fuzzy set; Fermatean fuzzy ideals; Fermatean fuzzy regular ideals; Fermatean fuzzy semi-prime ideals

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