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Research Article

Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras

[version 1; peer review: 1 approved, 1 approved with reservations]
PUBLISHED 10 Feb 2025
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Abstract

This paper introduces the study of hesitant fuzzy subalgebras of autometrized algebras, obtains some of their properties, and gives some examples. Next, we introduce the concept of the hesitant fuzzy ideal and examine some of its properties. Finally, we introduce a hesitant fuzzy congruence on autometrized algebras and discuss some of its properties. We also introduce the characteristic equations and level subsets of hesitant fuzzy sets and relations on autometrized algebras.

Keywords

autometrized algebra, hesitant fuzzy subalgebra, hesitant fuzzy ideal, hesitant fuzzy congruence

1. Introduction

The concept of fuzzy sets was introduced by Refs. 1, 2 and 3 introduced the concept of hesitant fuzzy sets. Several researches were conducted on the generalizations of the notion of hesitant fuzzy sets and its applications such as, Refs. 4-11.

In his work,12 introduced the concept of autometrized algebras which includes Boolean algebras,13,14 Brouwerian algebras,15 Newman algebras,16 autometrized lattices,15 and commutative lattice-ordered groups (or l-groups).17 The study of ideals and congruences in autometrized algebras was conducted by.18 Further advancements in the theory of autometrized algebras were made by Refs. 18-26, and 27-29. Also Refs. 30-33. developed the theory of subalgebras, ideals, and congruences of autometrized algebras. Moreover Ref. 34, introduced the homomorphism, isomorphism, and correspondence theorems of autometrized algebra by using congruency.

The previous studies did not investigate hesitant fuzzy subalgebra, hesitant fuzzy ideal, and hesitant fuzzy congruence of autometrized algebra. Therefore, our motivation is to address these gaps. This paper introduces the concept of hesitant fuzzy subalgebras, hesitant fuzzy ideals, and hesitant fuzzy congruence relations on autometrized algebras.

The paper will be organized as follows: Section 2 will provide definitions and key terms. Section 3 will introduce the concept of hesitant fuzzy subalgebras of autometrized algebras. In Section 4, we will discuss hesitant fuzzy ideals of autometrized algebras. Section 5 will focus on hesitant fuzzy congruences on autometrized algebras. Finally, Section 6 will conclude the paper.

In this paper, Γ denotes an autometrized algebra (Γ,+,0,,) .

2. Preliminaries

This section examines essential concepts, definitions, and theorems important in other sections.

Definition 2.1

12 A system Γ=(Γ,+,0,,) is called an autometrized algebra if

  • (i) (Γ,+,0) is a commutative monoid.

  • (ii) (Γ,) is a partial ordered set, and is translation invariant, that is, α,β,γΓ;αβα+γβ+γ .

  • (iii) :Γ×ΓΓ is autometric on Γ , that is, satisfies metric operation axioms:

    (M1) α,βΓ;αβ0 and, αβ=0α=β ,

    (M2) α,βΓ;αβ=βα ,

    (M3) α,β,γΓ ; αγαβ+βγ .

Definition 2.2

18 Γ is called normal if and only if

  • (i) αα0αΓ .

  • (ii) (α+γ)(β+δ)(αβ)+(γδ)α,β,γ,δΓ .

  • (iii) (αγ)(βδ)(αβ)+(γδ)α,β,γ,δΓ .

  • (iv) For any α and β in Γ , αβγ0 such that α+γ=β .

Definition 2.3

30 Let ϒΓ . Then ϒ is said to be a subalgebra of Γ if;

  • (i) (ϒ,+,0) is a commutative monoid.

  • (ii) (ϒ,) is a subposet, and is translation invariant, that is, αβα+γβ+γ for any α,β,γϒ .

  • (iii) |ϒ:ϒ×ϒϒ is metric.

Definition 2.4

30 A nonempty subset I of Γ is called an ideal if and only if

  • (i) α,βI imply α+βI .

  • (ii) αI,βΓ and β0α0 imply βI .

Definition 2.5

18 An equivalence relation Ψ on Γ is called a congruence relation if and only if

  • (i) (α,β),(γ,δ)Ψ(α+γ,β+δ)Ψα,β,γ,δΓ ,

  • (ii) (α,β),(γ,δ)Ψ(αγ,βδ)Ψα,β,γ,δΓ ,

  • (iii) (α,β)Ψandγδαβ(γ,δ)Ψα,β,γ,δΓ .

Definition 2.6

1 Let Γ be a nonempty set, a fuzzy subset χ of Γ is a mapping χ:Γ[0,1] .

Definition 2.7

1 Let Γ be a nonempty set and χ be a fuzzy subset of Γ , for ε[0,1] , the set χε={αΓ|χ(α)ε} is called a level subset of χ .

Definition 2.8

2 Let Γ be a reference set. A hesitant fuzzy set on Γ is a mapping χ:ΓP([0,1]) , where P([0,1]) means the power set of [0,1] .

Definition 2.9

2 Let Γ be a reference set. If HΓ , the characteristic hesitant fuzzy set χH on Γ is a function of Γ into P([0,1]) defined as for all αΓ :

χH(α)={[0,1],ifαH.,otherwise.

Definition 2.10

2 Let χ be a hesitant fuzzy set on a nonempty set Γ . Then, χ¯(α)=[0,1]\χ(α) for all αΓ which is said to be the complement of χ on Γ .

3. Hesitant fuzzy subalgebra of autometrized algebra

In this section, we will introduce the hesitant fuzzy subalgebras of autometrized algebras and explore several fundamental properties related to these subalgebras.

Definition 3.1

A hesitant fuzzy subset χ of Γ is called a hesitant fuzzy subalgebra of Γ if for all α,βΓ ;

  • (i) χ(α+β)χ(α)χ(β) .

  • (ii) χ(αβ)χ(α)χ(β) .

Example 3.2

Let Γ={0,α,β,γ} with 0α,βγ and elements α,β are incomparable. Define and + by the following tables.

0 α β γ
00 α β γ
α α 0 γ β
β β γ 0 α
γ γ β α 0

+ 0 α β γ
00 α β γ
α α α γ γ
β β γ β γ
γ γ γ γ γ

Then, Γ is an autometrized algebra. Define a hesitant fuzzy subset χ:ΓP([0,1]) of Γ by: χ(0)={0.3,0.5} and χ(α)=χ(β)=χ(γ)={0.5} .

Then χ is hesitant fuzzy subalgebra of Γ .

Theorem 3.3

If χ is a hesitant fuzzy subalgebra of Γ , then for αΓ ; χ(0)χ(α) .

Proof.

Assume that χ is a hesitant fuzzy subalgebra of Γ . Then, χ(αα)χ(α)χ(α) . This implies that χ(0)χ(α)χ(α) . Therefore, χ(0)χ(α) .

Theorem 3.4

Let H is a nonempty subset of Γ . 0H if and only if χH(0)χH(α) for all aΓ .

Proof.

Assume that 0H . So, χH(0)=[0,1] . Therefore, χH(0)χH(α) for all αΓ .

Conversely, assume that χH(0)χH(α) for all αΓ . Since H is nonempty subset of Γ , we have βH for βΓ . Therefore, χH(0)χH(β)=[0,1] . As a result, χH(0)=[0,1] . Hence, 0H .

Theorem 3.5

A nonempty subset H of Γ is a subalgebra of Γ if and only if the characteristic hesitant fuzzy set χH is a hesitant fuzzy subalgebra of Γ .

Proof.

Assume that H is subalgebra of Γ . Let χH(α),βΓ .

  • (i) Here we will consider three cases.

    • (a) Let α,βH . Clearly, χH(α)=[0,1] and χH(β)=[0,1] . So, χH(α)χH(β)=[0,1] . Since H is a subalgebra of Γ ; αβH . As a result, χH(αβ)=[0,1] . Therefore, χH(αβ)χH(α)χH(β) .

    • (b) Let αH and βH . Then, χH(α)=[0,1] and χH(β)= . So, χH(α)χH(β)= . Clearly, χH(αβ) . Therefore, χH(αβ)χH(α)χH(β) .

    • (c) Let αH and βH . Then, χH(α)= and χH(β)= . So, χH(α)χH(β)= . Clearly, χH(αβ) . Therefore, χH(αβ)χH(α)χH(β) .

  • (ii) Here we will consider three cases.

    • (a) Let α,βH . Clearly, χH(α)=[0,1] and χH(β)=[0,1] . So, χH(α)χH(β)=[0,1] . Since H is a subalgebra of Γ ; α+βH . As a result, χH(α+β)=[0,1] . Therefore, χH(α+β)χH(α)χH(β) .

    • (b) Let αH and βH . Clearly, χH(α)=[0,1] and χH(β)= . So, χH(α)χH(β)= . Clearly, χH(α+β) . Therefore, χH(α+β)χH(α)χH(β) .

    • (c) Let αH and βH . Clearly, χH(α)= and χH(β)= . So, χH(α)χH(β)= . Clearly, χH(α+β) . Therefore, χH(α+β)χH(α)χH(β) .

Conversely, assume that χH is a hesitant fuzzy subalgebra of Γ. To show that H is a subalgebra of Γ .

  • (i) To show that 0H . Since χH(0)χH(α) for all αΓ ; by theorem (3.4), 0H .

  • (ii) Let α,βH . Then, χH(α)=[0,1] and χH(β)=[0,1] . Since χH is a hesitant fuzzy subalgebra of Γ ; χH(αβ)χH(α)χH(β)=[0,1] . So, χH(αβ)=[0,1] . Hence, αβH .

  • (iii) Let α,βH . Then, χH(α)=[0,1] and χH(β)=[0,1] . Since χH is a hesitant fuzzy subalgebra of Γ ; χH(α+β)χH(α)χH(β)=[0,1] . So, χH(α+β)=[0,1] . Hence, α+βH . Hence Γ is a subalgebra of Γ .

Let χ be a hesitant fuzzy set of Γ . For all εP([0,1]) , define the level subsets of χ as F(χ,ε)={αΓ|χ(a)ε} , F(χ,ε)={αΓ|χ(a)ε} , F(χ,ε)={αΓ|χ(a)ε} , and F(χ,ε)={αΓ|χ(α)ε} .

Theorem 3.6

Let χ be a hesitant fuzzy set of Γ . Then χ is a hesitant fuzzy subalgebra of Γ if and only if for all εP([0,1]) , F(χ,ε) is a subalgebra of Γ .

Proof.

Assume that χ is a hesitant fuzzy subalgebra of Γ . To show that F(χ,ε) is a subalgebra of Γ . Let εP([0,1]) be such that F(χ,ε) .

  • (i) Let αF(χ,ε) . Then, χ(a)ε . Since χ is a hesitant fuzzy subalgebra of Γ , implies that χ(0)χ(α)ε . Therefore, 0F(χ,ε) .

  • (ii) Let α,βF(χ,ε) . Then, χ(α)ε and χ(β)ε . Since χ is a hesitant fuzzy subalgebra of Γ ; χ(αβ)χ(α)χ(β)ε . This implies that αβF(χ,ε) .

  • (iii) Let α,βF(χ,ε) . Then, χ(α)ε and χ(β)ε . Since χ is a hesitant fuzzy subalgebra of Γ ; χ(α+β)χ(α)χ(β)ε . This implies that α+βF(χ,ε) .

Conversely, assume that F(χ,ε) is a subalgebra of Γ . To show that χ is a hesitant fuzzy subalgebra of Γ .

  • (i) Let α,βΓ . Take ε=χ(α)χ(β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is a subalgebra of Γ ; αβF(χ,ε) . Hence, χ(αβ)ε=χ(α)χ(β) .

  • (ii) Let α,βΓ . Take ε=χ(α)χ(β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is a subalgebra of Γ ; α+βF(χ,ε) . Hence, χ(α+β)ε=χ(α)χ(β) .

Hence, χ is a hesitant fuzzy subalgebra of Γ .

Theorem 3.7

Let χ be a hesitant fuzzy set of Γ . If Im(χ) is a chain and for all εP([0,1]) , F(χ,ε) is a subalgebra of Γ , then χ is a hesitant fuzzy subalgebra of Γ .

Proof.

Assume that Im(χ) is a chain and for all εP([0,1]) , a nonempty subset F(χ,ε) is a subalgebra of Γ . To show that χ is a hesitant fuzzy subalgebra of Γ .

  • (i) Let α,βΓ . To show that χ(α+β)χ(α)χ(β) . Suppose that χ(α+β)χ(α)χ(β) . Since Im(χ) is a chain; implies that χ(α+β)χ(α)χ(β) . Then, χ(α+β)P([0,1]) . Take ε=χ(α+β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is a subalgebra of Γ ; α+βF(χ,ε) . So, χ(α+β)ε=χ(α+β) . This is a contradiction. Thus, χ(α+β)χ(α)χ(β) for all α,βΓ .

  • (ii) Let α,βΓ . To show that χ(αβ)χ(α)χ(β) . Suppose that χ(αβ)χ(α)χ(β) . Since Im(χ) is a chain; implies that χ(αβ)χ(α)χ(β) . Then, χ(αβ)P([0,1]) . Take ε=χ(αβ) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is a subalgebra of Γ ; αβF(χ,ε) . So, χ(αβ)ε=χ(αβ) . This is a contradiction. Thus, χ(αβ)χ(α)χ(β) for all α,βΓ .

Hence, χ is a hesitant fuzzy subalgebra of Γ .

Theorem 3.8

Let χ¯ be a hesitant fuzzy set of Γ . Then χ¯ is a hesitant fuzzy subalgebra of Γ if and only if for all εP([0,1]) , F(χ,ε) is a subalgebra of Γ .

Proof.

Assume that χ¯ is a hesitant fuzzy subalgebra of Γ . To show that F(χ,ε) is a subalgebra of Γ . Let εP([0,1]) be such that F(χ,ε) .

  • (i) Let αF(χ,ε) . Then, χ(α)ε . Since χ¯ is a hesitant fuzzy subalgebra of Γ , implies that χ¯(0)χ¯(α) . Clearly, [0,1]\χ(0)[0,1]\χ(α) . Therefore, χ(0)χ(α)ε . Therefore, 0F(χ,ε) .

  • (ii) Let α,βF(χ,ε) . Then, χ(α)ε and χ(β)ε . Since χ¯ is a hesitant fuzzy subalgebra of Γ ; χ¯(α+β)χ¯(α)χ¯(β) . So, [0,1]\χ(α+β)([0,1]\χ(α))([0,1]\χ(β))=([0,1]\(χ(α)χ(β)) . Therefore, χ(α+β)χ(α)χ(β)ε . Hence, α+βF(χ,ε) .

  • (iii) Let α,βF(χ,ε) . Then, χ(α)ε and χ(β)ε . Since χ¯ is a hesitant fuzzy subalgebra of Γ ; χ¯(αβ)χ¯(α)χ¯(β) . So, [0,1]\χ(αβ)([0,1]\χ(α))([0,1]\χ(β))=([0,1]\(χ(α)χ(β)) . Therefore, χ(αβ)χ(α)χ(β)ε . Hence, αβF(χ,ε) . Therefore, F(χ,ε) is a subalgebra of Γ .

Conversely, assume that F(χ,ε) is a subalgebra of Γ . To show that χ¯ is a hesitant fuzzy subalgebra of Γ .

  • (i) Let α,βΓ . Take ε=χ(α)χ(β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is a subalgebra of Γ ; αβF(χ,ε) . Clearly, χ(αβ)ε=χ(α)χ(β) . As a result, [0,1]\χ(αβ)[0,1]\(χ(α)χ(β))=([0,1]\χ(α))([0,1]\χ(β))=χ¯(α)χ¯(β) . Therefore, χ¯(αβ)χ¯(α)χ¯(β) .

  • (ii) Let α,βΓ . Take ε=χ(α)χ(β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is a subalgebra of Γ ; α+βF(χ,ε) . Clearly, χ(α+β)ε=χ(α)χ(β) . As a result, [0,1]\χ(α+β)[0,1]\(χ(α)χ(β))=([0,1]\χ(α))([0,1]\χ(β))=χ¯(α)χ¯(β) . Therefore, χ¯(α+β)χ¯(α)χ¯(β) .

Hence, χ¯ is a hesitant fuzzy subalgebra of Γ .

Theorem 3.9

Let χ be a hesitant fuzzy set of Γ . If Im(χ) is a chain and for all εP([0,1]) , F(χ,ε) is a subalgebra of Γ , then χ¯ is a hesitant fuzzy subalgebra of Γ .

Proof.

Assume that Im(χ) is a chain and for all εP([0,1]) , a nonempty subset F(χ,ε) is a subalgebra of Γ . To show that χ¯ is a hesitant fuzzy subalgebra of Γ .

  • (i) Let α,βΓ . To show that χ¯(α+β)χ¯(α)χ¯(β) . Suppose that χ¯(α+β)χ¯(α)χ¯(β) . Since Im(χ) is a chain; implies that χ¯(α+β)χ¯(α)χ¯(β) . Clearly, [0,1]\χ(α+β)([0,1]\χ(α))([0,1]\χ(β))=[0,1]\(χ(α)χ(β)) . So, χ(α+β)χ(α)χ(β) . Take ε=χ(α+β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is a subalgebra of Γ ; α+βF(χ,ε) . So, χ(α+β)ε=χ(α+β) . This is a contradiction. Thus, χ¯(α+β)χ¯(α)χ¯(β) for all α,βΓ .

  • (ii) Let α,βΓ . To show that χ¯(αβ)χ¯(α)χ¯(β) . Suppose that χ¯(αβ)χ¯(α)χ¯(β) . Since Im(χ) is a chain; implies that χ¯(αβ)χ¯(α)χ¯(β) . Clearly, [0,1]\χ(αβ)([0,1]\χ(α))([0,1]\χ(β))=[0,1]\(χ(α)χ(β)) . So, χ(αβ)χ(α)χ(β) . Take ε=χ(αβ) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is a subalgebra of Γ ; αβF(χ,ε) . So, χ(αβ)ε=χ(αβ) . This is a contradiction. Thus, χ¯(αβ)χ¯(α)χ¯(β) for all α,βΓ . Hence, χ¯ is a hesitant fuzzy subalgebra of Γ .

4. Hesitant fuzzy ideals of autometrized algebra

This section presents the concept of hesitant fuzzy ideals in autometrized algebras and explores several fundamental properties related to these ideals.

Definition 4.1

A hesitant fuzzy subset χ of Γ is called a hesitant fuzzy ideal of Γ if for all α,βΓ ;

  • (i) χ(α+β)χ(α)χ(β) .

  • (ii) if α0β0 , then χ(α)χ(β) .

Example 4.2

Let Γ={0,α,β,γ} with 0α,βγ and elements α,β are incomparable. Define and + by the following tables.

0 α β γ
00 α β γ
α α 0 γ β
β β γ 0 α
γ γ β α 0

+ 0 α β γ
00 α β γ
α α α γ γ
β β γ β γ
γ γ γ γ γ

Then, Γ is an autometrized algebra. Define a hesitant fuzzy subset χ:ΓP([0,1]) of Γ by:

χ(δ)={{0.2,0.5},ifδ{0,α}.{0.2},otherwise.

It is clear that χ is a hesitant fuzzy ideal of Γ .

Lemma 4.3

If χ is a hesitant fuzzy ideal of Γ , then for αΓ ; χ(0)χ(α) .

Proof.

Assume that χ is a hesitant fuzzy ideal of Γ . Since 00α0 ; implies that that χ(0)χ(α) . Therefore, χ(0)χ(α) .

Lemma 4.4

Let Γ be a normal autometrized algebra. If χ is a hesitant fuzzy ideal of Γ , then for αΓ ; χ(α0)=χ(0) .

Proof.

Assume that χ is a hesitant fuzzy ideal of Γ . Since Γ is normal; implies that (α0)0=α0 . This implies that (α0)0α0 and α0(α0)0 ; implies that χ(α0)χ(α) and χ(α)χ(α0) . Therefore, χ(α0)=χ(α) .

Theorem 4.5

Let Γ be a normal autometrized algebra. Every hesitant fuzzy ideal of Γ is a hesitant fuzzy subalgebra of Γ .

Proof.

Assume that χ is a hesitant fuzzy ideal of Γ . Let α,βΓ .

  • (i) By the definition of ideal, χ(α+β)χ(α)χ(β) .

  • (ii) Now, to show that χ(αβ)χ(α)χ(β) . Since Γ is normal; (αβ)0α0+β0 . Therefore,

    χ(αβ)χ(α0+β0)χ(α0)χ(β0)χ(α)χ(β)[(0)]

Therefore, χ(αβ)χ(α)χ(β) . Hence, χ is a hesitant fuzzy subalgebra of Γ .

Theorem 4.6

A nonempty subset H of Γ is an ideal of Γ if and only if the characteristic hesitant fuzzy set χH is a hesitant fuzzy ideal of Γ .

Proof.

Assume that H is an ideal of Γ . Let α,βΓ .

  • (i) Here we will consider three cases.

    • (a) Let α,βH . Clearly, χH(α)=[0,1] and χH(β)=[0,1] . So, χH(α)χH(β)=[0,1] . Since H is an ideal of Γ ; α+βH . As a result, χH(α+β)=[0,1] . Therefore, χH(α+β)χH(α)χH(β) .

    • (b) Let αH and βH . Then, χH(α)=[0,1] and χH(β)= . So, χH(α)χH(β)= . Clearly, χH(α+β) . Therefore, χH(α+β)χH(α)χH(β) .

    • (c) Let αH and βH . Then, χH(α)= and χH(β)= . So, χH(α)χH(β)= . Clearly, χH(α+β) . Therefore, χH(α+β)χH(α)χH(β) .

  • (ii) Suppose α0β0 . To show that χH(α)χH(β) . Here we will consider three cases.

    • (a) Let βH . Clearly, χH(β)=[0,1] . Since H is an ideal; αH . As a result, χH(α)=[0,1] . Hence, χH(α)χH(β) .

    • (b) Let αH and βH . Clearly, χH(α)=[0,1] and χH(β)= . Therefore, χH(α)χH(β) .

    • (c) Let αH and βH . Clearly, χH(α)= and χH(β)= . Therefore, χH(α)χH(β)= .

Conversely, assume that χH is a hesitant fuzzy ideal of Γ . To show that H is an ideal of Γ .

  • (i) Let α,βH . Then, χH(α)=[0,1] and χH(β)=[0,1] . Since χH is a hesitant fuzzy ideal of Γ ; χH(α+β)χH(α)χH(β)=[0,1] . So, χH(α+β)=[0,1] . Hence, α+βH .

  • (ii) Let α,βΓ . Suppose α0β0 . Let βH . Clearly, χH(β)=[0,1] . To show that αH . Since χH is a hesitant fuzzy ideal of Γ , χH(α)χH(β)=[0,1] . Therefore, χH(α)=[0,1] . Thus, αH . Hence, H is an ideal of Γ .

Theorem 4.7

Let χ be a hesitant fuzzy set of Γ . Then χ is a hesitant fuzzy ideal of Γ if and only if for all εP([0,1]) , a nonempty subset F(χ,ε) is an ideal of Γ .

Proof.

Assume that χ is a hesitant fuzzy ideal of Γ . To show that F(χ,ε) is an ideal of Γ . Let εP([0,1]) be such that F(χ,ε) .

  • (i) Let α,βF(χ,ε) . Then, χ(α)ε and χ(β)ε . Since χ is a hesitant fuzzy ideal of Γ ; χ(α+β)χ(α)χ(β)ε . This implies that α+βF(χ,ε) .

  • (ii) Let α,βΓ . Suppose α0β0 . Let βF(χ,ε) . Clearly, χ(β)ε . To show that αF(χ,ε) . Since χ is a hesitant fuzzy ideal of Γ , χ(α)χ(β)ε . Therefore, χ(α)ε . Thus, αF(χ,ε) . Hence, F(χ,ε) is an ideal of Γ .

Conversely, assume that F(χ,ε) is an ideal of Γ . To show that χ is a hesitant fuzzy ideal of Γ .

  • (i) Let α,βΓ . Take ε=χ(α)χ(β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is an ideal of Γ ; α+βF(χ,ε) . Hence, χ(α+β)ε=χ(α)χ(β) .

  • (ii) Let α,βΓ . Suppose α0β0 . Take ε=χ(β) . So, βF(χ,ε) . Since F(χ,ε) is an ideal; αF(χ,ε) . As a result, χ(α)ε=χ(β) .

Hence, χ is a hesitant fuzzy ideal of Γ .

Theorem 4.8

Let χ be a hesitant fuzzy set of Γ . If Im(χ) is a chain and for all εP([0,1]) , a nonempty subset F(χ,ε) is an ideal of Γ , then χ is a hesitant fuzzy ideal of Γ .

Proof.

Assume that Im(χ) is a chain and for all εP([0,1]) , a nonempty subset F(χ,ε) is an ideal of Γ . To show that χ is a hesitant fuzzy ideal of Γ .

  • (i) Let α,βΓ . To show that χ(α+β)χ(α)χ(β) . Suppose that χ(α+β)χ(α)χ(β) . Since Im(χ) is a chain; implies that χ(α+β)χ(α)χ(β) . Then, χ(α+β)P([0,1]) . Take ε=χ(α+β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is an ideal of Γ ; α+βF(χ,ε) . So, χ(α+β)ε=χ(α+β) . This is a contradiction. Thus, χ(α+β)χ(α)χ(β) for all α,βΓ .

  • (ii) Let α,βΓ . Suppose α0β0 . To show that χ(α)χ(β) . Suppose that χ(α)χ(β) . Since Im(χ) is a chain; implies that χ(α)χ(β) . Then, χ(α)P([0,1]) . Take ε=χ(α) . Therefore, χ(β)ε . As a result, βF(χ,ε) . Since F(χ,ε) is an ideal of Γ ; αF(χ,ε) . So, χ(α)ε=χ(α) . This is a contradiction. Therefore, χ(α)(β) for all α,βΓ .

Hence, χ is a hesitant fuzzy ideal of Γ .

Theorem 4.9

Let χ¯ be a hesitant fuzzy set of Γ . Then χ¯ is a hesitant fuzzy ideal of Γ if and only if for all εP([0,1]) , a nonempty subset F(χ,ε) is an ideal of Γ .

Proof.

Assume that χ¯ is a hesitant fuzzy ideal of Γ . To show that F(χ,ε) is an ideal of Γ . Let εP([0,1]) be such that F(χ,ε) .

  • (i) Let αF(χ,ε) . Then, χ(α)ε . Since χ¯ is a hesitant fuzzy ideal of Γ , implies that χ¯(0)χ¯(α) . Clearly, [0,1]\χ(0)[0,1]\χ(α) . Therefore, χ(0)χ(α)ε . Therefore, 0F(χ,ε) .

  • (ii) Let α,βF(χ,ε) . Then, χ(α)ε and χ(β)ε . Since χ¯ is a hesitant fuzzy ideal of Γ ; χ¯(α+β)χ¯(α)χ¯(β) . So, [0,1]\χ(α+β)([0,1]\χ(α))([0,1]\χ(β))=([0,1]\(χ(α)χ(β)) . Therefore, χ(α+β)χ(α)χ(β)ε . Hence, α+βF(χ,ε) .

  • (iii) Let α,βΓ . Suppose α0β0 . Let βF(χ,ε) . Clearly, χ(β)ε . To show that αF(χ,ε) . Since χ¯ is a hesitant fuzzy ideal of Γ , χ¯(α)χ¯(β) . So, [0,1]\χ(α)[0,1]\χ(β) . Therefore, χ(α)χ(β)ε . Thus, αF(χ,ε) . Hence, F(χ,ε) is an ideal of Γ .

Conversely, assume that F(χ,ε) is an ideal of Γ . To show that χ¯ is a hesitant fuzzy ideal of Γ .

  • (i) Let α,βΓ . Take ε=χ(α)χ(β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is an ideal of Γ ; α+βF(χ,ε) . Clearly, χ(α+β)ε=χ(α)χ(β) . As a result, [0,1]\χ(α+β)[0,1]\(χ(α)χ(β))=([0,1]\χ(α))([0,1]\χ(β))=χ¯(α)χ¯(β) . Therefore, χ¯(α+β)χ¯(α)χ¯(β) .

  • (ii) Let α,βΓ . Suppose α0β0 . Take ε=χ(β) . So, βF(χ,ε) . Since F(χ,ε) is an ideal; αF(χ,ε) . As a result, χ(α)ε=χ(β) . Clearly, [0,1]\χ(α)[0,1]\χ(β) . Therefore, χ¯(α)χ¯(β) . Hence, χ¯ is a hesitant fuzzy ideal of Γ .

Theorem 4.10

Let χ be a hesitant fuzzy set of Γ . If Im(χ) is a chain and for all εP([0,1]) , a nonempty subset F(χ,ε) is an ideal of Γ , then χ¯ is a hesitant fuzzy ideal of Γ .

Proof.

Assume that Im(χ) is a chain and for all εP([0,1]) , a nonempty subset F(χ,ε) is an ideal of Γ . To show that χ¯ is a hesitant fuzzy ideal of Γ .

  • (i) Let α,βΓ . To show that χ¯(α+β)χ¯(α)χ¯(β) . Suppose that χ¯(α+β)χ¯(α)χ¯(β) . Since Im(χ) is a chain; implies that χ¯(α+β)χ¯(α)χ¯(β) . Clearly, [0,1]\χ(α+β)([0,1]\χ(α))([0,1]\χ(β))=[0,1]\(χ(α)χ(β)) . So, χ(α+β)χ(α)χ(β) . Take ε=χ(α+β) . Therefore, χ(α)ε and χ(β)ε . As a result, α,βF(χ,ε) . Since F(χ,ε) is an ideal of Γ ; α+βF(χ,ε) . So, χ(α+β)ε=χ(α+β) . This is a contradiction. Thus, χ¯(α+β)χ¯(α)χ¯(β) for all α,βΓ .

  • (ii) Let α,βΓ . Suppose α0β0 . To show that χ¯(α)χ¯(β) . Suppose that χ¯(α)χ¯(β) . Since Im(χ) is a chain; implies that χ¯(α)χ¯(β) . Then, [0,1]\χ(a)[0,1]\χ(b) . Clearly, χ(α)χ(β) . Take ε=χ(α) . Therefore, χ(β)ε . As a result, βF(χ,ε) . Since F(χ,ε) is an ideal of Γ ; αF(χ,ε) . So, χ(α)ε=χ(α) . This is a contradiction. Therefore, χ¯(α)χ¯(β) for all α,βΓ . Hence, χ¯ is a hesitant fuzzy ideal of Γ .

5. Hesitant fuzzy congruence on autometrized algebra

In this section, we introduce the hesitant fuzzy congruence relation of autometrized algebras and examine some related properties.

Definition 5.1

A hesitant fuzzy relation on Γ is a mapping Ψ:Γ×ΓP([0,1]) , where P([0,1]) means the power set of [0,1] .

Definition 5.2

Let Ψ be a hesitant fuzzy relation on Γ . Then, Ψ¯(a,b)=P([0,1])\Ψ(a,b) for all a,bΓ which is said to be the complement of Ψ on Γ .

Definition 5.3

If Ψ is a hesitant fuzzy equivalent relation on Γ , then

  • (a) Ψ(α,α)={Ψ(β,γ)|β,γΓ} .(reflexive)

  • (b) Ψ(α,β)=Ψ(β,α) .(symmetric)

  • (c) Ψ(α,γ)Ψ(α,β)Ψ(β,γ) .(transitive)

Definition 5.4

A hesitant fuzzy equivalence relation Ψ on Γ is called a hesitant fuzzy congruence relation on Γ if

  • (a) Ψ(α+γ,β+δ)Ψ(α,β)Ψ(γ,δ)α,β,γ,δΓ ,

  • (b) Ψ(αγ,βδ)Ψ(α,β)Ψ(γ,δ)α,β,γ,δΓ ,

  • (c) For any α,β,γ,δΓ , if γδαβ , then Ψ(γ,δ)Ψ(α,β) .

Example 5.5

In Example 3.2, Γ becomes an autometrized algebra. Define a hesitant fuzzy relation Ψ:Γ×ΓP([0,1]) on Γ by: Ψ(0,0)=Ψ(α,α)=Ψ(β,β)=Ψ(γ,γ)={0.2,0.8} and Ψ(0,α)=Ψ(α,0)=Ψ(β,γ)=Ψ(γ,β)={0.8} . Therefore, Ψ is a hesitant fuzzy congruence relation on Γ .

Definition 5.6

If Ψ is a hesitant fuzzy relation on Γ , then characteristic hesitant fuzzy relation χΨ on Γ is a function of Γ×Γ into P([0,1]) defined as for all (α,β)Γ×Γ :

χΨ(α,β)={[0,1],if(α,β)Ψ.,otherwise.

Theorem 5.7

A nonempty equivalent relation Ψ on Γ is a congruence relation on Γ if and only if the characteristic hesitant fuzzy equivalent relation χΨ is a hesitant fuzzy congruence relation on Γ .

Proof.

Assume that Ψ is a congruence relation on Γ . To show that χΨ is a hesitant fuzzy congruence relation on Γ . Let α,β,γ,δΓ .

  • (i) To show that χΨ(α+γ,β+δ)χΨ(α,β)χΨ(γ,δ) . Here we will consider three cases.

    • (a) Let (α,β),(γ,δ)Ψ . Clearly, χΨ(α,β)=[0,1] and χΨ(γ,δ)=[0,1] . So, χΨ(α,β)χΨ(γ,δ)=[0,1] . Since Ψ is a congruence relation on Γ ; (α+γ,β+δ)Ψ . As a result, χΨ(α+γ,β+δ)=[0,1] . Therefore, χΨ(α+γ,β+δ)χΨ(α,β)χΨ(γ,δ) .

    • (b) Let (α,β)Ψ and (γ,δ)Ψ . Clearly, χΨ(α,β)=[0,1] and χΨ(γ,δ)= . So, χΨ(α,β)χΨ(γ,δ)= . Therefore, χΨ(α+γ,β+δ)χΨ(α,β)χΨ(γ,δ) .

    • (c) Let (α,β)Ψ and (γ,δ)Ψ . Clearly, χΨ(α,β)= and χΨ(γ,δ)= . So, χΨ(α,β)χΨ(γ,δ)= . Therefore, χΨ(α+γ,β+δ)χΨ(α,β)χΨ(γ,δ) .

  • (ii) To show that χΨ(αγ,βδ)χΨ(α,β)χΨ(γ,δ) . Here we will consider three cases.

    • (a) Let (α,β),(γ,δ)Ψ . Clearly, χΨ(α,β)=[0,1] and χΨ(γ,δ)=[0,1] . So, χΨ(α,β)χΨ(γ,δ)=[0,1] . Since Ψ is a congruence relation on Γ ; (αγ,βδ)Ψ . As a result, χΨ(αγ,βδ)=[0,1] . Therefore, χΨ(αγ,βδ)χΨ(α,β)χΨ(γ,δ) .

    • (b) Let (α,β)Ψ and (γ,δ)Ψ . Clearly, χΨ(α,β)=[0,1] and χΨ(γ,δ)= . So, χΨ(α,β)χΨ(γ,δ)= . Therefore, χΨ(αγ,βδ)χΨ(α,β)χΨ(γ,δ) .

    • (c) Let (α,β)Ψ and (γ,δ)Ψ . Clearly, χΨ(α,β)= and χΨ(γ,δ)= . So, χΨ(α,β)χΨ(γ,δ)= . Therefore, χΨ(αγ,βδ)χΨ(α,β)χΨ(γ,δ) .

  • (iii) Let α,β,γ,δΓ and suppose that γδαβ . To show that χΨ(γ,δ)χΨ(α,β) . Here we will consider two cases.

    • (a) Let (α,β)Ψ . Clearly, χΨ(α,β)=[0,1] . Since Ψ is a congruence relation on Γ ; (γ,δ)Ψ . As a result, χΨ(γ,δ)=[0,1] . Therefore, χΨ(γ,δ)χΨ(α,β) .

    • (b) Let (α,β)Ψ . Clearly, χΨ(α,β)= . Therefore, χΨ(γ,δ)χΨ(α,β) . Hence, χΨ is a hesitant fuzzy congruence relation on Γ .

Conversely, assume that χΨ is a hesitant fuzzy congruence relation on Γ . To show that Ψ is a congruence relation on Γ .

  • (i) Let (α,β),(γ,δ)Ψ . Then, χΨ(α,β)=[0,1] and χΨ(γ,δ)=[0,1] . Since χΨ is a hesitant fuzzy congruence relation on Γ ; χΨ(α+γ,β+δ)χΨ(α,β)χΨ(γ,δ)=[0,1] . So, χΨ(α+γ,β+δ)=[0,1] . Hence, (α+γ,β+δ)Ψ .

  • (ii) Let (α,β),(γ,δ)Ψ . Then, χΨ(α,β)=[0,1] and χΨ(γ,δ)=[0,1] . Since χΨ is a hesitant fuzzy congruence relation on Γ ; χΨ(αγ,βδ)χΨ(α,β)χΨ(γ,δ)=[0,1] . So, χΨ(αγ,βδ)=[0,1] . Hence, (αγ,βδ)Ψ .

  • (iii) Let (α,β)Ψ and γδαβ . Clearly, Clearly, χΨ(α,β)=[0,1] . Since Ψ is a hesitant fuzzy congruence relation on Γ ; then χΨ(γ,δ)χΨ(α,β)=[0,1] . Therefore, χΨ(γ,δ)=[0,1] . Hence, (γ,δ)Ψ . Hence, Ψ is a congruence relation on Γ .

Let Ψ be a hesitant fuzzy relation on Γ . For all εP([0,1]) , the sets F(Ψ,ε)={(α,β)Γ×Γ|Ψ(α,β)ε} , F(Ψ,ε)={(α,β)Γ×Γ|Ψ(α,β)ε} , F(Ψ,ε)={(α,β)Γ×Γ|Ψ(α,β)ε} , and F(Ψ,ε)={(α,β)Γ×Γ|Ψ(α,β)ε} are called level subset of Ψ .

Theorem 5.8

Let Ψ be a hesitant fuzzy relation on Γ . Ψ is a hesitant fuzzy congruence relation on Γ if and only if for all tP([0,1]) , F(Ψ,ε) is either empty or a congruence relation on Γ .

Proof.

Assume that Ψ is a hesitant fuzzy congruence relation on Γ .

  • (i) Since F(Ψ,ε) is a nonempty; let (α,β)F(Ψ,ε) . So, Ψ(α,β)ε . But Ψ(α,α)={Ψ(β,γ)|β,γΓ}Ψ(α,β)ε . So, Ψ(α,α)ε . Therefore, (α,α)F(Ψ,ε) .

  • (ii) Let (α,β)F(Ψ,ε) . So, Ψ(α,β)ε . Clearly, Ψ(α,β)=Ψ(β,α)ε . So, Ψ(β,α)ε . Therefore, (β,α)F(Ψ,ε) .

  • (iii) Let (α,β),(β,γ)F(Ψ,ε) . Then, Ψ(α,β)ε and Ψ(β,γ)ε . Clearly, Ψ(α,β)Ψ(β,γ)ε . Since Ψ is a hesitant fuzzy congruence; Ψ(α,γ)Ψ(α,β)Ψ(β,γ)ε . Therefore, (α,γ)F(Ψ,ε) . Therefore, F(Ψ,ε) is an equivalence relation.

    • (a) Let (α,β),(γ,δ)F(Ψ,ε) . So, Ψ(α,β)ε and Ψ(γ,δ)ε . Since Ψ is a hesitant fuzzy congruence relation on Γ ; Ψ(α+γ,β+δ)Ψ(α,β)Ψ(γ,δ)ε . Therefore, (α+γ,β+δ)F(Ψ,ε) .

    • (b) Let (α,β),(γ,δ)F(Ψ,ε) . So, Ψ(α,β)ε and Ψ(γ,δ)ε . Since Ψ is a hesitant fuzzy congruence relation on Γ ; Ψ(αγ,βδ)Ψ(α,β)Ψ(γ,δ)ε . Therefore, (αγ,βδ)F(Ψ,ε) .

    • (c) Let (α,β)F(Ψ,ε) and γδαβ . Clearly, (α,β)ε . Since Ψ is a hesitant fuzzy congruence relation on Γ ; then Ψ(γ,δ)Ψ(α,β)ε . Therefore, (γ,δ)F(Ψ,ε) . Hence, F(Ψ,ε) is a congruence relation on Γ .

    Conversely, F(Ψ,ε) is a congruence relation on Γ . To show that Ψ is a hesitant fuzzy congruence relation on Γ .

  • (i) We know that for any α,β,γΓ , γγ=0αβ . Take ε=Ψ(α,β) . So, (α,β)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation; (γ,γ)F(Ψ,ε) . Then, Ψ(γ,γ)ε=Ψ(α,β) . Therefore, Ψ(γ,γ)={Ψ(α,β)|α,βΓ} .

  • (ii) Let α,βΓ . Take ε=Ψ(α,β) . So, (α,β)F(Ψ,ε) . Since F(Ψ,ε) is a congruence; (β,α)F(Ψ,ε) . So, Ψ(β,α)ε . Clearly, Ψ(β,α)Ψ(α,β) . Again, take ε=Ψ(β,α) . So, (β,α)F(Ψ,ε) . Since F(Ψ,ε) is a congruence; (α,β)F(Ψ,ε) . So, Ψ(α,β)ε . Clearly, Ψ(α,β)Ψ(β,α) . Consequently, Ψ(α,β)=Ψ(β,α) .

  • (iii) Let α,β,γΓ . Take ε=Ψ(α,β)Ψ(β,γ) . Clearly, Ψ(α,β)ε and Ψ(β,γ)ε . This implies that (α,β),(β,γ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation; (α,γ)F(Ψ,ε) . Therefore, Ψ(α,γ)ε . Thus, Ψ(α,γ)Ψ(α,β)Ψ(β,γ) . Therefore, Ψ is a hesitant equivalence relation.

    • (a) Let α,β,γ,δΓ . Take ε=Ψ(α,β)Ψ(γ,δ) . Therefore, (α,β)F(Ψ,ε) and (γ,δ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (α+γ,β+δ)F(Ψ,ε) . As a result, (α+γ,β+δ)ε=Ψ(α,β)Ψ(γ,δ) .

    • (b) Let α,β,γ,δΓ . Take ε=Ψ(α,β)Ψ(γ,δ) . Therefore, (α,β)F(Ψ,ε) and (γ,δ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (αγ,βδ)F(Ψ,ε) . As a result, (αγ,βδ)ε=Ψ(α,β)Ψ(γ,δ) .

    • (c) Let α,β,γ,δΓ and suppose that γδαβ . Take ε=Ψ(α,β) . Therefore, (α,β)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (γ,δ)F(Ψ,ε) . Therefore, Ψ(γ,δ)ε=Ψ(α,β) . Hence, Ψ is a hesitant fuzzy congruence relation on Γ .

Theorem 5.9

Let Ψ be a hesitant fuzzy equivalent relation on Γ . If Im(Ψ) is a chain and for all εP([0,1]) , a nonempty subset F(Ψ,ε) is a congruence relation on Γ , then Ψ is a hesitant fuzzy congruence relation on Γ .

Proof.

Assume that Im(Ψ) is a chain and for all εP([0,1]) , a nonempty subset F(Ψ,ε) is a congruence relation on Γ . To show that Ψ is a hesitant fuzzy congruence relation on Γ .

  • (i) Let α,β,γ,δΓ . To show that Ψ(α+γ,β+δ)Ψ(α,β)Ψ(γ,δ) . Suppose that Ψ(α+γ,β+δ)Ψ(α,β)Ψ(γ,δ) . Since Im(Ψ) is a chain; implies that Ψ(α+γ,β+δ)Ψ(α,β)Ψ(γ,δ) . Then, Ψ(α+γ,β+δ)P([0,1]) . Take ε=Ψ(α+γ,β+δ) . Therefore, Ψ(α,β)ε and Ψ(γ,δ)ε . As a result, (α,β),(γ,δ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (α+γ,β+δ)F(Ψ,ε) . So, Ψ(α+γ,β+δ)ε=Ψ(α+γ,β+δ) . This is a contradiction. Thus, Ψ(α+γ,β+δ)Ψ(α,β)Ψ(γ,δ) for all α,β,γ,δΓ .

  • (ii) Let α,β,γ,δΓ . To show that Ψ(αγ,βδ)Ψ(α,β)Ψ(γ,δ) . Suppose that Ψ(αγ,βδ)Ψ(α,β)Ψ(γ,δ) . Since Im(Ψ) is a chain; implies that Ψ(αγ,βδ)Ψ(α,β)Ψ(γ,δ) . Then, Ψ(αγ,βδ)P([0,1]) . Take ε=Ψ(αγ,βδ) . Therefore, Ψ(α,β)ε and Ψ(γ,δ)ε . As a result, (α,β),(γ,δ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (αγ,βδ)F(Ψ,ε) . So, Ψ(αγ,βδ)ε=Ψ(αγ,βδ) . This is a contradiction. Thus, Ψ(αγ,βδ)Ψ(α,β)Ψ(γ,δ) for all α,β,γ,δΓ .

  • (iii) Let α,β,γ,δΓ and suppose that γδαβ . To show that Ψ(γ,δ)Ψ(α,β) . Suppose that Suppose that Ψ(γ,δ)Ψ(α,β) . Since Im(Ψ) is a chain; implies that Ψ(γ,δ)Ψ(α,β) . Take ε=Ψ(γ,δ) . Therefore, (α,β)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on relation on Γ ; (γ,δ)F(Ψ,ε) . Therefore, Ψ(γ,δ)ε=Ψ(γ,δ) is contradiction. Therefore, Ψ(γ,δ)Ψ(α,β) . Hence, Ψ is a hesitant fuzzy congruence relation on Γ .

Theorem 5.10

Let Ψ¯ be a hesitant fuzzy equivalent relation on Γ . Then Ψ¯ is a hesitant fuzzy congruence relation on Γ if and only if for all εP([0,1]) , F(Ψ,ε) is either empty or congruence relation on Γ .

Proof.

Assume that Ψ¯ is a hesitant fuzzy congruence relation on Γ . To show that F(Ψ,ε) is a congruence relation on Γ . Let εP([0,1]) be such that F(Ψ,ε) .

  • (i) Since F(Ψ,ε) is a nonempty; let (α,β)F(Ψ,ε) . So, Ψ(α,β)ε . Since Ψ¯ is a hesitant fuzzy congruence relation on Γ ; Ψ¯(α,α)={Ψ¯(β,γ)|β,γΓ} . Then, [0,1]\Ψ(α,α)=[0,1]\{Ψ(β,γ)|β,γΓ} . As a result, Ψ(α,α)={Ψ(β,γ)|β,γΓ}Ψ(α,β)ε . Therefore, (α,α)F(Ψ,ε) .

  • (ii) Let (α,β)F(Ψ,ε) . So, Ψ(α,β)ε . Since Ψ¯ is a hesitant fuzzy congruence relation on Γ ; Ψ¯(α,β)=Ψ¯(β,α) . Clearly, Ψ(α,β)=Ψ(β,α)ε . So, Ψ(β,α)ε . Therefore, (β,α)F(Ψ,ε) .

  • (iii) Let (α,β),(β,γ)F(Ψ,ε) . Then, Ψ(α,β)ε and Ψ(β,γ)ε . Since Ψ¯ is a hesitant fuzzy congruence relation on Γ ; Ψ¯(α,γ)Ψ¯(α,β)Ψ¯(β,γ) . Then, [0,1]\Ψ(α,γ)([0,1]\Ψ(α,β))([0,1]\Ψ(β,γ))=[0,1]\Ψ(α,β)Ψ(β,γ) . Consequently, Ψ(α,γ)Ψ(α,β)Ψ(β,γ)ε . Therefore, (α,γ)F(Ψ,ε) . Therefore, F(Ψ,ε) is an equivalence relation.

    • (a) Let (α,β),(γ,δ)F(Ψ,ε) . Then, Ψ(α,β)ε and Ψ(γ,δ)ε . Since Ψ¯ is a hesitant fuzzy congruence relation on Γ ; Ψ¯(α+γ,β+δ)Ψ¯(α,β)Ψ¯(γ,δ) . So, [0,1]\Ψ(α+γ,β+δ)([0,1]\Ψ(α,β))([0,1]\Ψ(γ,δ))=([0,1]\(Ψ(α,β)Ψ(γ,δ)) . Therefore, Ψ(α+γ,β+δ)Ψ(α,β)Ψ(γ,δ)ε . Hence, (α+γ,β+δ)F(Ψ,ε) .

    • (b) Let (α,β),(γ,δ)F(Ψ,ε) . Then, Ψ(α,β)ε and Ψ(γ,δ)ε . Since Ψ¯ is a hesitant fuzzy congruence relation on Γ ; Ψ¯(αγ,βδ)Ψ¯(α,β)Ψ¯(γ,δ) . So, [0,1]\Ψ(αγ,βδ)([0,1]\Ψ(α,β))([0,1]\Ψ(γ,δ))=([0,1]\(Ψ(α,β)Ψ(γ,δ)) . Therefore, Ψ(αγ,βδ)Ψ(α,β)Ψ(γ,δ)ε . Hence, (αγ,βδ)F(Ψ,ε) .

    • (c) Let (α,β)F(Ψ,ε) and γδαβ . Clearly, Ψ(α,β)ε . To show that (γ,δ)δ)F(Ψ,ε) . Since Ψ¯ is a hesitant fuzzy congruence relation on Γ , Ψ¯(γ,δ)Ψ¯(α,β) . So, [0,1]\Ψ(γ,δ)[0,1]\Ψ(α,β) . Therefore, Ψ(γ,δ)Ψ(α,β)ε . Thus, (γ,δ)F(Ψ,ε) . Hence, F(Ψ,ε) is congruence relation on Γ .

Conversely, assume that F(Ψ,ε) is congruence relation on Γ . To show that Ψ¯ is a hesitant fuzzy congruence relation on Γ .

  • (i) We know that for any α,β,γΓ , αα=0βγ . Take ε=Ψ(β,γ) . So, (β,γ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation; (α,α)F(Ψ,ε) . Then, Ψ(α,α)ε=Ψ(β,γ) . So, [0,1]\Ψ(α,α)[0,1]\Ψ(β,γ) . Clearly, Ψ¯(α,α)Ψ¯(β,γ) . Therefore, Ψ¯(α,α)={Ψ¯(β,γ)|β,γΓ} .

  • (ii) Let α,βΓ . Take ε=Ψ(α,β) . So, (α,β)F(Ψ,ε) . Since F(Ψ,ε) is a congruence; (β,α)F(Ψ,ε) . So, Ψ(β,α)ε . Therefore, Ψ(β,α)Ψ(α,β) . Clearly, [0,1]\Ψ(β,α)[0,1]\Ψ(α,β) . Therefore, Ψ¯(β,α)Ψ¯(α,β) . Again, take ε=Ψ(β,α) . So, (β,α)F(Ψ,ε) . Since F(Ψ,ε) is a congruence; (α,β)F(Ψ,ε) . So, Ψ(α,β)ε . Clearly, Ψ(α,β)Ψ(β,α) . Clearly, [0,1]\Ψ(α,β)[0,1]\Ψ(β,α) . Therefore, Ψ¯(α,β)Ψ¯(β,α) . Hence, Ψ¯(α,β)=Ψ¯(β,α) .

  • (iii) Let α,β,γΓ . Take ε=Ψ(α,β)Ψ(β,γ) . Clearly, Ψ(α,β)ε and Ψ(β,γ)ε . This implies that (α,β),(β,γ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation; (α,γ)F(Ψ,ε) . Therefore, Ψ(α,γ)ε . Clearly, Ψ(α,γ)Ψ(α,β)Ψ(β,γ) . Now, consider [0,1]\Ψ(α,γ)([0,1]\Ψ(α,β)Ψ(β,γ))=([0,1]\Ψ(α,β))([0,1]\Ψ(β,γ)) . As a result, Ψ¯(α,γ)Ψ¯(α,β)Ψ¯(β,γ) . Therefore, Ψ¯ is a hesitant equivalence relation.

    • (a) Let α,β,γ,δΓ . Take ε=Ψ(α,β)Ψ(γ,δ) . Therefore, Ψ(α,β)ε and Ψ(γ,δ)ε . As a result, (α,β),(γ,δ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (α+γ,β+δ)F(Ψ,ε) . Clearly, Ψ(α+γ,β+δ)ε=Ψ(α,β)Ψ(γ,δ) . As a result, [0,1]\Ψ(α+γ,β+δ)[0,1]\(Ψ(α,β)Ψ(γ,δ))=([0,1]\Ψ(α,β))([0,1]\Ψ(γ,δ))=Ψ¯(α,β)Ψ¯(γ,δ) . Therefore, Ψ¯(α+γ,β+δ)Ψ¯(α,β)Ψ¯(γ,δ) .

    • (b) Let α,β,γ,δΓ . Take ε=Ψ(α,β)Ψ(γ,δ) . Therefore, Ψ(α,β)ε and Ψ(γ,δ)ε . As a result, (α,β),(γ,δ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (αγ,βδ)F(Ψ,ε) . Clearly, Ψ(αγ,βδ)ε=Ψ(α,β)Ψ(γ,δ) . As a result, [0,1]\Ψ(αγ,βδ)[0,1]\(Ψ(α,β)Ψ(γ,δ))=([0,1]\Ψ(α,β))([0,1]\Ψ(γ,δ))=Ψ¯(α,β)Ψ¯(γ,δ) . Therefore, Ψ¯(αγ,βδ)Ψ¯(α,β)Ψ¯(γ,δ) .

    • (c) Let α,β,γ,δΓ and suppose that γδαβ . Take ε=Ψ(α,β) . So, (α,β)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (γ,δ)F(Ψ,ε) . As a result, Ψ(γ,δ)ε=Ψ(α,β) . Clearly, [0,1]\Ψ(γ,δ)[0,1]\Ψ(α,β) . Therefore, Ψ¯(γ,δ)Ψ¯(α,β) . Hence, Ψ¯ is a hesitant fuzzy congruence relation on Γ .

Theorem 5.11

Let Ψ be a hesitant fuzzy equivalent relation on Γ . If Im(Ψ) is a chain and for all εP([0,1]) , a nonempty subset F(Ψ,ε) is a congruence relation on Γ , then Ψ¯ is a hesitant fuzzy congruence relation on Γ .

Proof.

Assume that Im(Ψ) is a chain and for all εP([0,1]) , a nonempty subset F(Ψ,ε) is a congruence relation on Γ . To show that Ψ¯ is a hesitant fuzzy congruence relation on Γ .

  • (i) Let α,β,γ,δΓ . To show that Ψ¯(α+γ,β+δ)Ψ¯(α,β)Ψ¯(γ,δ) . Suppose that Ψ¯(α+γ,β+δ)Ψ¯(α,β)Ψ¯(γ,δ) . Since Im(Ψ) is a chain; implies that Ψ¯(α+γ,β+δ)Ψ¯(α,β)Ψ¯(γ,δ) . Then, Ψ¯(α+γ,β+δ)P([0,1]) . Clearly, [0,1]\Ψ(α+γ,β+δ)([0,1]\Ψ(α,β))([0,1]\Ψ(γ,δ))=[0,1]\(Ψ(α,β)Ψ(γ,δ)) . So, Ψ(α+γ,β+δ)Ψ(α,β)Ψ(γ,δ) . Take ε=Ψ(α+γ,β+δ) . Therefore, Ψ(α,β)ε and Ψ(γ,δ)ε . As a result, (α,β),(γ,δ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (α+γ,β+δ)F(Ψ,ε) . So, Ψ(α+γ,β+δ)ε=Ψ(α+γ,β+δ) . This is a contradiction. Thus, Ψ¯(α+γ,β+δ)Ψ¯(α,β)Ψ¯(γ,δ) .

  • (ii) Let α,β,γ,δΓ . To show that Ψ¯(αγ,βδ)Ψ¯(α,β)Ψ¯(γ,δ) . Suppose that Ψ¯(αγ,βδ)Ψ¯(α,β)Ψ¯(γ,δ) . Since Im(Ψ) is a chain; implies that Ψ¯(αγ,βδ)Ψ¯(α,β)Ψ¯(γ,δ) . Then, Ψ¯(αγ,βδ)P([0,1]) . Clearly, [0,1]\Ψ(αγ,βδ)([0,1]\Ψ(α,β))([0,1]\Ψ(γ,δ))=[0,1]\(Ψ(α,β)Ψ(γ,δ)) . So, Ψ(αγ,βδ)Ψ(α,β)Ψ(γ,δ) . Take ε=Ψ(αγ,βδ) . Therefore, Ψ(α,β)ε and Ψ(γ,δ)ε . As a result, (α,β),(γ,δ)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (αγ,βδ)F(Ψ,ε) . So, Ψ(αγ,βδ)ε=Ψ(αγ,βδ) . This is a contradiction. Thus, Ψ¯(αγ,βδ)Ψ¯(α,β)Ψ¯(γ,δ) .

  • (iii) Let α,β,γ,δΓ and suppose that γδαβ . To show that Ψ¯(γ,δ)Ψ¯(α,β) . Suppose that Ψ¯(γ,δ)Ψ¯(α,β) . Since Im(Ψ) is a chain; implies that Ψ¯(γ,δ)Ψ¯(α,β) . Then, [0,1]\Ψ(γ,δ)[0,1]\Ψ(α,β) . Clearly, Ψ(γ,δ)Ψ(α,β) . Take ε=Ψ(γ,δ) . Therefore, Ψ(α,β)ε . As a result, (α,β)F(Ψ,ε) . Since F(Ψ,ε) is a congruence relation on Γ ; (γ,δ)F(Ψ,ε) . So, Ψ(γ,δ)ε=Ψ(γ,δ) . This is a contradiction. Therefore, Ψ¯(γ,δ)Ψ¯(α,β) . Hence, Ψ¯ is a hesitant fuzzy congruence relation on Γ .

6. Conclusion

This paper introduced the study of hesitant fuzzy subalgebras of autometrized algebras. Also, we proved that a nonempty subset H of an autometrized algebra Γ is a subalgebra of Γ if and only if the characteristic hesitant fuzzy set χH is a hesitant fuzzy subalgebra of Γ . Further, we introduced the concept of the hesitant fuzzy ideal and examined some of its properties. We showed that hesitant fuzzy set χ on an autometrized algebra Γ is a hesitant fuzzy ideal of Γ if and only if for all εP([0,1]) , a nonempty subset F(χ,ε) is an ideal of Γ . Finally, we introduced a hesitant fuzzy congruence on autometrized algebras and discussed some of its properties. In particular, we explored a hesitant fuzzy equivalent relation Ψ¯ on an autometrized algebra Γ is a hesitant fuzzy congruence relation on Γ if and only if for all εP([0,1]) , F(Ψ,ε) is either empty or congruence relation on Γ .

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Tilahun GY. Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras [version 1; peer review: 1 approved, 1 approved with reservations]. F1000Research 2025, 14:183 (https://doi.org/10.12688/f1000research.161430.1)
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Reviewer Report 20 Jun 2025
Ajoy Kanti Das, Tripura University, Suryamani Nagar, Tripura, India 
Approved with Reservations
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Title: Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras

This paper aims to extend the study of autometrized algebras by introducing the notions of hesitant fuzzy subalgebras, hesitant fuzzy ideals, and ... Continue reading
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Das AK. Reviewer Report For: Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras [version 1; peer review: 1 approved, 1 approved with reservations]. F1000Research 2025, 14:183 (https://doi.org/10.5256/f1000research.177460.r391499)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
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Reviewer Report 11 Mar 2025
Sileshe Gone Korma, Arba Minch University, Hawassa, Ethiopia 
Approved
VIEWS 7
This paper explores hesitant fuzzy subalgebras within autometrized algebras. It establishes that a nonempty subset of an autometrized algebra is a subalgebra if and only if its corresponding hesitant fuzzy set is a hesitant fuzzy subalgebra. The study also introduces ... Continue reading
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Korma SG. Reviewer Report For: Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras [version 1; peer review: 1 approved, 1 approved with reservations]. F1000Research 2025, 14:183 (https://doi.org/10.5256/f1000research.177460.r366481)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.

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Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions
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