1. Introduction

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10.12688/f1000research.161430.1

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Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras

[version 1; peer review: 1 approved, 1 approved with reservations]

Tilahun

Gebrie Yeshiwas

Writing – Review & Editing https://orcid.org/0000-0002-7010-4847 a 1 1Department of Mathematics, Assosa University, Asosa, Benishangul-Gumuz, Ethiopia

a gebrieyeshiwas0@gmail.com

No competing interests were disclosed.

10 2 2025

2025

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30 1 2025

2025

This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

autometrized algebra hesitant fuzzy subalgebra hesitant fuzzy ideal hesitant fuzzy congruence

The author(s) declared that no grants were involved in supporting this work.

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 , γ ∈ Γ ; α ≼ β ⇒ α + γ ≼ β + γ .

(iii)

⋆ : Γ × Γ → Γ is autometric on Γ , that is, ⋆ satisfies metric operation axioms:

( M1) ∀ α , β ∈ Γ ; α ⋆ β ≽ 0 and, α ⋆ β = 0 ⇔ α = β ,

( M2) ∀ α , β ∈ Γ ; α ⋆ β = β ⋆ α ,

( M3) ∀ α , β , γ ∈ Γ ; α ⋆ γ ≼ α ⋆ β + β ⋆ γ .

Definition 2.2

¹⁸ Γ is called normal if and only if (i)

α ≼ α ⋆ 0 ∀ α ∈ Γ .

(ii)

( α + γ ) ⋆ ( β + δ ) ≼ ( α ⋆ β ) + ( γ ⋆ δ ) ∀ α , β , γ , δ ∈ Γ .

(iii)

( α ⋆ γ ) ⋆ ( β ⋆ δ ) ≼ ( α ⋆ β ) + ( γ ⋆ δ ) ∀ α , β , γ , δ ∈ Γ .

(iv)

For any α and β in Γ , α ≼ β ⇒ ∃ γ ≽ 0 such that α + γ = β .

Definition 2.3

³⁰ Let ϒ ⊆ Γ . 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, α ≼ β ⇒ α + γ ≼ β + γ for any α , β , γ ∈ ϒ .

(iii)

⋆ | ϒ : ϒ × ϒ → ϒ is metric.

Definition 2.4

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

α , β ∈ I imply α + β ∈ I .

(ii)

α ∈ I , β ∈ Γ and β ⋆ 0 ≼ α ⋆ 0 imply β ∈ I .

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 ) ∈ Ψ ∀ α , β , γ , δ ∈ Γ ,

(ii)

( α , β ) , ( γ , δ ) ∈ Ψ ⇒ ( α ⋆ γ , β ⋆ δ ) ∈ Ψ ∀ α , β , γ , δ ∈ Γ ,

(iii)

( α , β ) ∈ Ψ and γ ⋆ δ ≼ α ⋆ β ⇒ ( γ , δ ) ∈ Ψ ∀ α , β , γ , δ ∈ Γ .

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 χ ε = { α ∈ Γ | 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 : Γ → P ( [ 0 , 1 ] ) , where P ( [ 0 , 1 ] ) means the power set of [ 0 , 1 ] .

Definition 2.9

² Let Γ be a reference set. If H ⊆ Γ , the characteristic hesitant fuzzy set χ H on Γ is a function of Γ into P ( [ 0 , 1 ] ) defined as for all α ∈ Γ : χ H ( α ) = { [ 0 , 1 ] , if α ∈ H . ∅ , 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 α ∈ Γ 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 , β ∈ Γ ; (i)

χ ( α + β ) ⊇ χ ( α ) ∩ χ ( β ) .

(ii)

χ ( α ⋆ β ) ⊇ χ ( α ) ∩ χ ( β ) .

Example 3.2

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

⋆ 0 α β γ

0 0 α β γ

α α 0 γ β

β β γ 0 α

γ γ β α 0

+ 0 α β γ

0 0 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 : Γ → P ( [ 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 α ∈ Γ ; χ ( 0 ) ⊇ χ ( α ) .

Proof.

Assume that χ is a hesitant fuzzy subalgebra of Γ . Then, χ ( α ⋆ α ) ⊇ 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 . 0 ∈ H if and only if χ H ( 0 ) ⊇ χ H ( α ) for all a ∈ Γ .

Proof.

Assume that 0 ∈ H . So, χ H ( 0 ) = [ 0 , 1 ] . Therefore, χ H ( 0 ) ⊇ χ H ( α ) for all α ∈ Γ .

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

Theorem 3.5

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

Proof.

Assume that H is subalgebra of Γ . Let χ H ( α ) , β ∈ Γ . (i)

Here we will consider three cases. (a)

Let α , β ∈ H . Clearly, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = [ 0 , 1 ] . So, χ H ( α ) ∩ χ H ( β ) = [ 0 , 1 ] . Since H is a subalgebra of Γ ; α ⋆ β ∈ H . As a result, χ H ( α ⋆ β ) = [ 0 , 1 ] . Therefore, χ H ( α ⋆ β ) ⊇ χ H ( α ) ∩ χ H ( β ) .

(b)

Let α ∈ H and β ∉ H . Then, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = ∅ . So, χ H ( α ) ∩ χ H ( β ) = ∅ . Clearly, χ H ( α ⋆ β ) ⊇ ∅ . Therefore, χ H ( α ⋆ β ) ⊇ χ H ( α ) ∩ χ H ( β ) .

(c)

Let α ∉ H and β ∉ H . Then, χ H ( α ) = ∅ and χ H ( β ) = ∅ . So, χ H ( α ) ∩ χ H ( β ) = ∅ . Clearly, χ H ( α ⋆ β ) ⊇ ∅ . Therefore, χ H ( α ⋆ β ) ⊇ χ H ( α ) ∩ χ H ( β ) .

(ii)

Here we will consider three cases. (a)

Let α , β ∈ H . Clearly, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = [ 0 , 1 ] . So, χ H ( α ) ∩ χ H ( β ) = [ 0 , 1 ] . Since H is a subalgebra of Γ ; α + β ∈ H . As a result, χ H ( α + β ) = [ 0 , 1 ] . Therefore, χ H ( α + β ) ⊇ χ H ( α ) ∩ χ H ( β ) .

(b)

Let α ∈ H and β ∉ H . Clearly, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = ∅ . So, χ H ( α ) ∩ χ H ( β ) = ∅ . Clearly, χ H ( α + β ) ⊇ ∅ . Therefore, χ H ( α + β ) ⊇ χ H ( α ) ∩ χ H ( β ) .

(c)

Let α ∉ H and β ∉ H . Clearly, χ H ( α ) = ∅ and χ H ( β ) = ∅ . So, χ H ( α ) ∩ χ H ( β ) = ∅ . Clearly, χ H ( α + β ) ⊇ ∅ . Therefore, χ H ( α + β ) ⊇ χ H ( α ) ∩ χ H ( β ) .

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

To show that 0 ∈ H . Since χ H ( 0 ) ⊇ χ H ( α ) for all α ∈ Γ ; by theorem (3.4), 0 ∈ H .

(ii)

Let α , β ∈ H . Then, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = [ 0 , 1 ] . Since χ H is a hesitant fuzzy subalgebra of Γ ; χ H ( α ⋆ β ) ⊇ χ H ( α ) ∩ χ H ( β ) = [ 0 , 1 ] . So, χ H ( α ⋆ β ) = [ 0 , 1 ] . Hence, α ⋆ β ∈ H .

(iii)

Let α , β ∈ H . Then, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = [ 0 , 1 ] . Since χ H is a hesitant fuzzy subalgebra of Γ ; χ H ( α + β ) ⊇ χ H ( α ) ∩ χ H ( β ) = [ 0 , 1 ] . So, χ H ( α + β ) = [ 0 , 1 ] . Hence, α + β ∈ H . Hence Γ is a subalgebra of Γ .

Let χ be a hesitant fuzzy set of Γ . For all ε ∈ P ( [ 0 , 1 ] ) , define the level subsets of χ as F ⊇ ( χ , ε ) = { α ∈ Γ | χ ( a ) ⊇ ε } , F ⊃ ( χ , ε ) = { α ∈ Γ | χ ( a ) ⊃ ε } , F ⊆ ( χ , ε ) = { α ∈ Γ | χ ( a ) ⊆ ε } , and F ⊂ ( χ , ε ) = { α ∈ Γ | 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 ε ∈ P ( [ 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 ε ∈ P ( [ 0 , 1 ] ) be such that F ⊇ ( χ , ε ) ≠ ∅ . (i)

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

(ii)

Let α , β ∈ F ⊇ ( 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 ⋆ β ∈ F ⊇ ( χ , ε ) .

(iii)

Let α , β ∈ F ⊇ ( 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 + β ∈ F ⊇ ( 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 , β ∈ Γ . Take ε = χ ( α ) ∩ χ ( β ) . Therefore, χ ( α ) ⊇ ε and χ ( β ) ⊇ ε . As a result, α , β ∈ F ⊇ ( χ , ε ) ≠ ∅ . Since F ⊇ ( χ , ε ) is a subalgebra of Γ ; α ⋆ β ∈ F ⊇ ( χ , ε ) . Hence, χ ( α ⋆ β ) ⊇ ε = χ ( α ) ∩ χ ( β ) .

(ii)

Let α , β ∈ Γ . Take ε = χ ( α ) ∩ χ ( β ) . Therefore, χ ( α ) ⊇ ε and χ ( β ) ⊇ ε . As a result, α , β ∈ F ⊇ ( χ , ε ) ≠ ∅ . Since F ⊇ ( χ , ε ) is a subalgebra of Γ ; α + β ∈ F ⊇ ( 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 ε ∈ P ( [ 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 ε ∈ P ( [ 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 , β ∈ Γ . 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 , β ∈ F ⊃ ( χ , ε ) ≠ ∅ . Since F ⊃ ( χ , ε ) is a subalgebra of Γ ; α + β ∈ F ⊃ ( 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 , β ∈ Γ .

(ii)

Let α , β ∈ Γ . 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 , β ∈ F ⊃ ( χ , ε ) ≠ ∅ . Since F ⊃ ( χ , ε ) is a subalgebra of Γ ; α ⋆ β ∈ F ⊃ ( 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 , β ∈ Γ .

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 ε ∈ P ( [ 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 ε ∈ P ( [ 0 , 1 ] ) be such that F ⊆ ( χ , ε ) ≠ ∅ . (i)

Let α ∈ F ⊆ ( 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, 0 ∈ F ⊆ ( χ , ε ) .

(ii)

Let α , β ∈ F ⊆ ( 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 + β ∈ F ⊆ ( χ , ε ) .

(iii)

Let α , β ∈ F ⊆ ( 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 ⋆ β ∈ F ⊆ ( 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 , β ∈ Γ . Take ε = χ ( α ) ∪ χ ( β ) . Therefore, χ ( α ) ⊆ ε and χ ( β ) ⊆ ε . As a result, α , β ∈ F ⊆ ( χ , ε ) ≠ ∅ . Since F ⊆ ( χ , ε ) is a subalgebra of Γ ; α ⋆ β ∈ F ⊆ ( 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 , β ∈ Γ . Take ε = χ ( α ) ∪ χ ( β ) . Therefore, χ ( α ) ⊆ ε and χ ( β ) ⊆ ε . As a result, α , β ∈ F ⊆ ( χ , ε ) ≠ ∅ . Since F ⊆ ( χ , ε ) is a subalgebra of Γ ; α + β ∈ F ⊆ ( 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 ε ∈ P ( [ 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 ε ∈ P ( [ 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 , β ∈ Γ . 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 , β ∈ F ⊂ ( χ , ε ) ≠ ∅ . Since F ⊂ ( χ , ε ) is a subalgebra of Γ ; α + β ∈ F ⊂ ( 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 , β ∈ Γ .

(ii)

Let α , β ∈ Γ . 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 , β ∈ F ⊂ ( χ , ε ) ≠ ∅ . Since F ⊂ ( χ , ε ) is a subalgebra of Γ ; α ⋆ β ∈ F ⊂ ( 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 , β ∈ Γ . 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 , β ∈ Γ ; (i)

χ ( α + β ) ⊇ χ ( α ) ∩ χ ( β ) .

(ii)

if α ⋆ 0 ≼ β ⋆ 0 , then χ ( α ) ⊇ χ ( β ) .

Example 4.2

Then, Γ is an autometrized algebra. Define a hesitant fuzzy subset χ : Γ → P ( [ 0 , 1 ] ) of Γ by: χ ( δ ) = { { 0.2 , 0.5 } , if δ ∈ { 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 α ∈ Γ ; χ ( 0 ) ⊇ χ ( α ) .

Proof.

Assume that χ is a hesitant fuzzy ideal of Γ . Since 0 ⋆ 0 ≼ α ⋆ 0 ; 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 α ∈ Γ ; χ ( α ⋆ 0 ) = χ ( 0 ) .

Proof.

Assume that χ is a hesitant fuzzy ideal of Γ . Since Γ is normal; implies that ( α ⋆ 0 ) ⋆ 0 = α ⋆ 0 . This implies that ( α ⋆ 0 ) ⋆ 0 ≼ α ⋆ 0 and α ⋆ 0 ≼ ( α ⋆ 0 ) ⋆ 0 ; implies that χ ( α ⋆ 0 ) ⊇ χ ( α ) and χ ( α ) ⊇ χ ( α ⋆ 0 ) . Therefore, χ ( α ⋆ 0 ) = 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 , β ∈ Γ . (i)

By the definition of ideal, χ ( α + β ) ⊇ χ ( α ) ∩ χ ( β ) .

(ii)

Now, to show that χ ( α ⋆ β ) ⊇ χ ( α ) ∩ χ ( β ) . Since Γ is normal; ( α ⋆ β ) ⋆ 0 ≼ α ⋆ 0 + β ⋆ 0 . Therefore, χ ( α ⋆ β ) ⊇ χ ( α ⋆ 0 + β ⋆ 0 ) ⊇ χ ( α ⋆ 0 ) ∩ χ ( β ⋆ 0 ) ⊇ χ ( α ) ∩ χ ( β ) [ ( 0 ) ]

Therefore, χ ( α ⋆ β ) ⊇ 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 χ H is a hesitant fuzzy ideal of Γ .

Proof.

Assume that H is an ideal of Γ . Let α , β ∈ Γ . (i)

Here we will consider three cases.

(a)

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

(b)

Let α ∈ H and β ∉ H . Then, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = ∅ . So, χ H ( α ) ∩ χ H ( β ) = ∅ . Clearly, χ H ( α + β ) ⊇ ∅ . Therefore, χ H ( α + β ) ⊇ χ H ( α ) ∩ χ H ( β ) .

(c)

Let α ∉ H and β ∉ H . Then, χ H ( α ) = ∅ and χ H ( β ) = ∅ . So, χ H ( α ) ∩ χ H ( β ) = ∅ . Clearly, χ H ( α + β ) ⊇ ∅ . Therefore, χ H ( α + β ) ⊇ χ H ( α ) ∩ χ H ( β ) .

(ii)

Suppose α ⋆ 0 ≼ β ⋆ 0 . To show that χ H ( α ) ⊇ χ H ( β ) . Here we will consider three cases.

(a)

Let β ∈ H . Clearly, χ H ( β ) = [ 0 , 1 ] . Since H is an ideal; α ∈ H . As a result, χ H ( α ) = [ 0 , 1 ] . Hence, χ H ( α ) ⊇ χ H ( β ) .

(b)

Let α ∈ H and β ∉ H . Clearly, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = ∅ . Therefore, χ H ( α ) ⊇ χ H ( β ) .

(c)

Let α ∉ H and β ∉ H . Clearly, χ H ( α ) = ∅ and χ H ( β ) = ∅ . Therefore, χ H ( α ) ⊇ χ H ( β ) = ∅ .

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

Let α , β ∈ H . Then, χ H ( α ) = [ 0 , 1 ] and χ H ( β ) = [ 0 , 1 ] . Since χ H is a hesitant fuzzy ideal of Γ ; χ H ( α + β ) ⊇ χ H ( α ) ∩ χ H ( β ) = [ 0 , 1 ] . So, χ H ( α + β ) = [ 0 , 1 ] . Hence, α + β ∈ H .

(ii)

Let α , β ∈ Γ . Suppose α ⋆ 0 ≼ β ⋆ 0 . Let β ∈ H . Clearly, χ H ( β ) = [ 0 , 1 ] . To show that α ∈ H . Since χ H is a hesitant fuzzy ideal of Γ , χ H ( α ) ⊇ χ H ( β ) = [ 0 , 1 ] . Therefore, χ H ( α ) = [ 0 , 1 ] . Thus, α ∈ H . 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 ε ∈ P ( [ 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 ε ∈ P ( [ 0 , 1 ] ) be such that F ⊇ ( χ , ε ) ≠ ∅ . (i)

Let α , β ∈ F ⊇ ( 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 + β ∈ F ⊇ ( χ , ε ) .

(ii)

Let α , β ∈ Γ . Suppose α ⋆ 0 ≼ β ⋆ 0 . Let β ∈ F ⊇ ( χ , ε ) . Clearly, χ ( β ) ⊇ ε . To show that α ∈ F ⊇ ( χ , ε ) . Since χ is a hesitant fuzzy ideal of Γ , χ ( α ) ⊇ χ ( β ) ⊇ ε . Therefore, χ ( α ) ⊇ ε . Thus, α ∈ F ⊇ ( 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 , β ∈ Γ . Take ε = χ ( α ) ∩ χ ( β ) . Therefore, χ ( α ) ⊇ ε and χ ( β ) ⊇ ε . As a result, α , β ∈ F ⊇ ( χ , ε ) ≠ ∅ . Since F ⊇ ( χ , ε ) is an ideal of Γ ; α + β ∈ F ⊇ ( χ , ε ) . Hence, χ ( α + β ) ⊇ ε = χ ( α ) ∩ χ ( β ) .

(ii)

Let α , β ∈ Γ . Suppose α ⋆ 0 ≼ β ⋆ 0 . Take ε = χ ( β ) . So, β ∈ F ⊇ ( χ , ε ) . Since F ⊇ ( χ , ε ) is an ideal; α ∈ F ⊇ ( 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 ε ∈ P ( [ 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 ε ∈ P ( [ 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 , β ∈ Γ . 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 , β ∈ F ⊃ ( χ , ε ) ≠ ∅ . Since F ⊃ ( χ , ε ) is an ideal of Γ ; α + β ∈ F ⊃ ( 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 , β ∈ Γ .

(ii)

Let α , β ∈ Γ . Suppose α ⋆ 0 ≼ β ⋆ 0 . 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, β ∈ F ⊃ ( χ , ε ) . Since F ⊃ ( χ , ε ) is an ideal of Γ ; α ∈ F ⊃ ( χ , ε ) . So, χ ( α ) ⊃ ε = χ ( α ) . This is a contradiction. Therefore, χ ( α ) ⊇ ( β ) for all α , β ∈ Γ .

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 ε ∈ P ( [ 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 ε ∈ P ( [ 0 , 1 ] ) be such that F ⊆ ( χ , ε ) ≠ ∅ . (i)

Let α ∈ F ⊆ ( 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, 0 ∈ F ⊆ ( χ , ε ) .

(ii)

Let α , β ∈ F ⊆ ( 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 + β ∈ F ⊆ ( χ , ε ) .

(iii)

Let α , β ∈ Γ . Suppose α ⋆ 0 ≼ β ⋆ 0 . Let β ∈ F ⊆ ( χ , ε ) . Clearly, χ ( β ) ⊆ ε . To show that α ∈ F ⊆ ( 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, α ∈ F ⊆ ( 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 , β ∈ Γ . Take ε = χ ( α ) ∪ χ ( β ) . Therefore, χ ( α ) ⊆ ε and χ ( β ) ⊆ ε . As a result, α , β ∈ F ⊆ ( χ , ε ) ≠ ∅ . Since F ⊆ ( χ , ε ) is an ideal of Γ ; α + β ∈ F ⊆ ( 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 , β ∈ Γ . Suppose α ⋆ 0 ≼ β ⋆ 0 . Take ε = χ ( β ) . So, β ∈ F ⊆ ( χ , ε ) . Since F ⊆ ( χ , ε ) is an ideal; α ∈ F ⊆ ( 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 ε ∈ P ( [ 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 ε ∈ P ( [ 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 , β ∈ Γ . 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 , β ∈ F ⊂ ( χ , ε ) ≠ ∅ . Since F ⊂ ( χ , ε ) is an ideal of Γ ; α + β ∈ F ⊂ ( 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 , β ∈ Γ .

(ii)

Let α , β ∈ Γ . Suppose α ⋆ 0 ≼ β ⋆ 0 . 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, β ∈ F ⊂ ( χ , ε ) . Since F ⊂ ( χ , ε ) is an ideal of Γ ; α ∈ F ⊂ ( χ , ε ) . So, χ ( α ) ⊂ ε = χ ( α ) . This is a contradiction. Therefore, χ ¯ ( α ) ⊇ χ ¯ ( β ) for all α , β ∈ Γ . 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 × Γ → P ( [ 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 , b ∈ Γ 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 , γ ∈ Γ } .(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 , δ ∈ Γ ,

(b)

Ψ ( α ⋆ γ , β ⋆ δ ) ⊇ Ψ ( α , β ) ∩ Ψ ( γ , δ ) ∀ α , β , γ , δ ∈ Γ ,

(c)

For any α , β , γ , δ ∈ Γ , if γ ⋆ δ ≼ α ⋆ β , 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 × Γ → P ( [ 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 χ Ψ on Γ is a function of Γ × Γ into P ( [ 0 , 1 ] ) defined as for all ( α , β ) ∈ Γ × Γ : χ Ψ ( 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 χ Ψ is a hesitant fuzzy congruence relation on Γ .

Proof.

Assume that Ψ is a congruence relation on Γ . To show that χ Ψ is a hesitant fuzzy congruence relation on Γ . Let α , β , γ , δ ∈ Γ . (i)

To show that χ Ψ ( α + γ , β + δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) . Here we will consider three cases.

(a)

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

(b)

Let ( α , β ) ∈ Ψ and ( γ , δ ) ∉ Ψ . Clearly, χ Ψ ( α , β ) = [ 0 , 1 ] and χ Ψ ( γ , δ ) = ∅ . So, χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) = ∅ . Therefore, χ Ψ ( α + γ , β + δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) .

(c)

Let ( α , β ) ∉ Ψ and ( γ , δ ) ∉ Ψ . Clearly, χ Ψ ( α , β ) = ∅ and χ Ψ ( γ , δ ) = ∅ . So, χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) = ∅ . Therefore, χ Ψ ( α + γ , β + δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) .

(ii)

To show that χ Ψ ( α ⋆ γ , β ⋆ δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) . Here we will consider three cases.

(a)

Let ( α , β ) , ( γ , δ ) ∈ Ψ . Clearly, χ Ψ ( α , β ) = [ 0 , 1 ] and χ Ψ ( γ , δ ) = [ 0 , 1 ] . So, χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) = [ 0 , 1 ] . Since Ψ is a congruence relation on Γ ; ( α ⋆ γ , β ⋆ δ ) ∈ Ψ . As a result, χ Ψ ( α ⋆ γ , β ⋆ δ ) = [ 0 , 1 ] . Therefore, χ Ψ ( α ⋆ γ , β ⋆ δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) .

(b)

Let ( α , β ) ∈ Ψ and ( γ , δ ) ∉ Ψ . Clearly, χ Ψ ( α , β ) = [ 0 , 1 ] and χ Ψ ( γ , δ ) = ∅ . So, χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) = ∅ . Therefore, χ Ψ ( α ⋆ γ , β ⋆ δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) .

(c)

Let ( α , β ) ∉ Ψ and ( γ , δ ) ∉ Ψ . Clearly, χ Ψ ( α , β ) = ∅ and χ Ψ ( γ , δ ) = ∅ . So, χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) = ∅ . Therefore, χ Ψ ( α ⋆ γ , β ⋆ δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) .

(iii)

Let α , β , γ , δ ∈ Γ and suppose that γ ⋆ δ ≼ α ⋆ β . To show that χ Ψ ( γ , δ ) ⊇ χ Ψ ( α , β ) . Here we will consider two cases. (a)

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

(b)

Let ( α , β ) ∉ Ψ . Clearly, χ Ψ ( α , β ) = ∅ . Therefore, χ Ψ ( γ , δ ) ⊇ χ Ψ ( α , β ) . Hence, χ Ψ is a hesitant fuzzy congruence relation on Γ .

Conversely, assume that χ Ψ 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, χ Ψ ( α , β ) = [ 0 , 1 ] and χ Ψ ( γ , δ ) = [ 0 , 1 ] . Since χ Ψ is a hesitant fuzzy congruence relation on Γ ; χ Ψ ( α + γ , β + δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) = [ 0 , 1 ] . So, χ Ψ ( α + γ , β + δ ) = [ 0 , 1 ] . Hence, ( α + γ , β + δ ) ∈ Ψ .

(ii)

Let ( α , β ) , ( γ , δ ) ∈ Ψ . Then, χ Ψ ( α , β ) = [ 0 , 1 ] and χ Ψ ( γ , δ ) = [ 0 , 1 ] . Since χ Ψ is a hesitant fuzzy congruence relation on Γ ; χ Ψ ( α ⋆ γ , β ⋆ δ ) ⊇ χ Ψ ( α , β ) ∩ χ Ψ ( γ , δ ) = [ 0 , 1 ] . So, χ Ψ ( α ⋆ γ , β ⋆ δ ) = [ 0 , 1 ] . Hence, ( α ⋆ γ , β ⋆ δ ) ∈ Ψ .

(iii)

Let ( α , β ) ∈ Ψ and γ ⋆ δ ≼ α ⋆ β . Clearly, Clearly, χ Ψ ( α , β ) = [ 0 , 1 ] . Since Ψ is a hesitant fuzzy congruence relation on Γ ; then χ Ψ ( γ , δ ) ⊇ χ Ψ ( α , β ) = [ 0 , 1 ] . Therefore, χ Ψ ( 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 ε ∈ P ( [ 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 t ∈ P ( [ 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 , γ ∈ Γ } ⊇ 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 ( α ⋆ γ , β ⋆ δ ) ⊇ Ψ ( α , β ) ∩ Ψ ( γ , δ ) ⊇ ε . Therefore, ( α ⋆ γ , β ⋆ δ ) ∈ F ⊇ ( Ψ , ε ) .

(c)

Let ( α , β ) ∈ F ⊇ ( Ψ , ε ) and γ ⋆ δ ≼ α ⋆ β . 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 , γ ∈ Γ , γ ⋆ γ = 0 ≼ α ⋆ β . 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 , β ∈ Γ } .

(ii)

Let α , β ∈ Γ . 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 , γ ∈ Γ . 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 , δ ∈ Γ . 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 , δ ∈ Γ . 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 ) .

(c)

Let α , β , γ , δ ∈ Γ and suppose that γ ⋆ δ ≼ α ⋆ β . 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 ε ∈ P ( [ 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 ε ∈ P ( [ 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 , δ ∈ Γ . 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 , δ ∈ Γ .

(ii)

Let α , β , γ , δ ∈ Γ . 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 , δ ∈ Γ .

(iii)

Let α , β , γ , δ ∈ Γ and suppose that γ ⋆ δ ≼ α ⋆ β . 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 ε ∈ P ( [ 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 ε ∈ P ( [ 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 , γ ∈ Γ } . Then, [ 0 , 1 ] \ Ψ ( α , α ) = [ 0 , 1 ] \ ∩ { Ψ ( β , γ ) | β , γ ∈ Γ } . As a result, Ψ ( α , α ) = ∩ { Ψ ( β , γ ) | β , γ ∈ Γ } ⊆ 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 ¯ ( 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 ) .

(c)

Let ( α , β ) ∈ F ⊆ ( Ψ , ε ) and γ ⋆ δ ≼ α ⋆ β . 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 , γ ∈ Γ , α ⋆ α = 0 ≼ β ⋆ γ . 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 , γ ∈ Γ } .

(ii)

Let α , β ∈ Γ . 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 , γ ∈ Γ . 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 , δ ∈ Γ . 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 , δ ∈ Γ . 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 ) .

(c)

Let α , β , γ , δ ∈ Γ and suppose that γ ⋆ δ ≼ α ⋆ β . 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 ε ∈ P ( [ 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 ε ∈ P ( [ 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 , δ ∈ Γ . 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 , δ ∈ Γ . 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 ) .

(iii)

Let α , β , γ , δ ∈ Γ and suppose that γ ⋆ δ ≼ α ⋆ β . 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 χ H 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 ε ∈ P ( [ 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 ε ∈ P ( [ 0 , 1 ] ) , F ⊆ ( Ψ , ε ) is either empty or congruence relation on Γ .

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: Equivalent conditions of normal autometrized l-algebra. Res. Math. 2023;10(1):2215037. 10.1080/27684830.2023.2215037

Tilahun

Parimi

Melesse

: Convex subalgebras and convex spectral topology on autometrized algebras. Res. Math. 2023;10(1):2283261.

Tilahun

Parimi

Melesse

: Direct product, subdirect product, and representability in autometrized algebras. Korean J. Math. 2023;31(4):445–463.

Tilahun

: Congruency, homomorphism and isomorphism on autometrized algebras. F1000Res. 2025;14:22. 10.12688/f1000research.159591.1

10.5256/f1000research.177460.r391499

Reviewer response for version 1

Das

Ajoy Kanti

1 Referee https://orcid.org/0000-0002-9326-1677 1Tripura University, Suryamani Nagar, Tripura, India

Competing interests: No competing interests were disclosed.

20 6 2025

2025

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

approve-with-reservations

REVIEW REPORT

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 hesitant fuzzy congruences. It provides formal definitions, examples, and characterizations through theorems. The approach combines hesitant fuzzy set theory with abstract algebraic structures to fill gaps in the literature on fuzzy algebraic generalizations. The paper addresses a novel and underexplored intersection between hesitant fuzzy sets and autometrized algebra. The theoretical framework is built systematically, starting from definitions to illustrative examples and theorems. The inclusion of level sets and their algebraic properties adds depth to the analysis. The work appears to be original and free of plagiarism, and no ethical concerns are noted. However, there are several areas where the manuscript could benefit from improvement, both in terms of structure and content. Some comments are suggested to improve the quality of this paper.

Suggestions for Improvement:

The introduction lacks a clear motivation for the study, and no real-world relevance or potential applications are discussed. Include a deeper comparison with related works to position your model within the existing body of research. Emphasize the specific limitations in existing models that your work addresses.

Examples are presented without sufficient explanation, particularly in verifying algebraic closure in the operation tables.

Key definitions (e.g., hesitant fuzzy set operations, level set interpretation) are missing.

Some of the mathematical formulas use different styles or symbols that might be confusing to readers. It would be helpful to use consistent symbols throughout the paper and to explain each symbol clearly when it first appears. For example, symbols like “⊇”, “∩”, and “⊂” are used without explaining their meaning in the context of hesitant fuzzy sets. Often appears as “χ(α⋆β) ⊇ χ(α) ∩ χ(β)” without defining what the intersection of hesitant fuzzy sets means. Define all operations (intersection, union) explicitly in the hesitant fuzzy context.

The theoretical contributions in the paper are clear, but it would be helpful to explain why hesitant fuzzy subalgebras, ideals, and congruences are important or needed compared to traditional approaches. Including examples of real-world problems or areas where hesitant fuzzy set theory works better than classical methods would make the paper stronger. Consider adding a short section that discusses practical situations or fields where these concepts are especially useful. This would help justify the value of the theory.

The focus is primarily theoretical; real-world applications or potential use cases are not discussed. Expand the literature review by including more recent research on Fuzzy and Hesitant Fuzzy set theories and their applications in various domains. You may consider citing the following relevant and recent studies:

(refer to 1, 2, 3 4, 5 and 6)

Include a future research section that outlines the next steps for further research to emphasize the study’s broader relevance and discuss any limitations of the current study.

The paper makes a significant theoretical contribution and has strong potential for real-world applications but requires minor revision before it can be accepted for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Fuzzy set theory and its applications, Soft set and Decision-making, Soft Computing, Sequence space, Topology

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

References 1

Weighted hesitant bipolar-valued fuzzy soft set in decision-making. Songklanakarin Journal of Science and Technology, 45(6): 681-690 https://sjst.psu.ac.th/journal/45-6/10.pdf.

An efficient water quality evaluation model using weighted hesitant fuzzy soft sets for water pollution rating. https://doi.org/10.1201/9781003494478-10.

: An efficient water quality evaluation model using weighted hesitant fuzzy soft sets for water pollution rating.2025; 10.1201/9781003494478-10 179-195

10.1201/9781003494478-10

: An Advanced Approach to Fuzzy Soft Group Decision-Making Using Weighted Average Ratings. SN Computer Science .2021;2(6) : 10.1007/s42979-021-00873-5

10.1007/s42979-021-00873-5

A new fuzzy parameterized intuitionistic fuzzy soft multiset theory and group decision-making. Journal of Current Science and Technology, 12(3), 547-567. https://doi.org/10.14456/jcst.2022.42.

IFP-intuitionistic multi fuzzy N-soft set and its induced IFP-hesitant N-soft set in decision-making, Journal of Ambient Intelligence and Humanized Computing, 14: 10143–10152. https://doi.org/10.1007/s12652-021-03677-w.

Tilahun

Gebrie Yeshiwas

Mathematics, Assosa University, Asosa, Benishangul-Gumuz, Ethiopia

Competing interests: No competing interests were disclosed.

22 8 2025

We incorporated all the comments as the reviewer's suggestions. We add the following in the new version.

1. In the introduction section, we are incorporating a clear motivation, real-world relevance and a deeper comparison of hesitant fuzzy set with related works.

2. Examples are presented with sufficient explanation.

3. The definitions of hesitant fuzzy set operations and level sets are discussed.

4. We add a discussion section that explain the significance of hesitant fuzzy subalgebras, ideals, and congruences compared to traditional fuzzy sets or crisp sets. We also discuss practical situations or fields where these concepts are especially beneficial.

5. The future scope of the research is also discussed.

6. Recent reference materials suggested by you are incorporated.

10.5256/f1000research.177460.r366481

Reviewer response for version 1

Korma

Sileshe Gone

1 Referee https://orcid.org/0000-0001-7426-9027 1Arba Minch University, Hawassa, Ethiopia

Competing interests: No competing interests were disclosed.

11 3 2025

2025

recommendation

approve

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 the concept of hesitant fuzzy ideals and examines their properties. Additionally, it defines hesitant fuzzy congruences and investigates their characteristics, including conditions under which a hesitant fuzzy equivalence relation qualifies as a hesitant fuzzy congruence.

The author should review the points I have highlighted in my comments and address them accordingly to improve clarity and accuracy.

Definition 2.3; condition (ii) is already satisfied in autometrized algebra Γ . Is there a possibility this condition fails for a subposet?

Definition 2.8, 2.9; what do you mean reference set?

Definition 2.3; it is better to write set complement χ α = 0,1 -χ(α)

Theorem 3.5 Proof(i) (b) and (c) it is better to combine both cases as α∉H or β∉H . The same is true for converse part (ii) (b) and (c)

Page 5 line 3 in the definition of F ⊇ χ, ϵ ={α∈Γ | χ ( α )⊇ ϵ } (in the brace both variables needs to be α and check for the remaining)

If it is possible try to merge proof (ii) and (iii) as one for Theorem 3.6

Theorem 3.7 Proof(i) line 2 χ(α+β)∈Ρ[0,1] is trivial please exclude it. The same is for Proof(ii) line 2 .

Lemma 4.3 proof line 1 there is repetition please check it

Lemma 4.3 line 2 typing error χ α*0 =χ(α)

Theorem 4.5 line 2 please add one line χ( α*β *0)⊇χ(α*0+β*0) and remove [ 0 ] in line 4

Theorem 4.6 Proof(i) (b) and (c) it is better to combine both cases as α∉H or β∉H . The same is true for converse part (ii) (b) and (c)

Definition 5.2 better to use as Ψ a,b =Ρ0,1 -Ψ ( a , b )

Definition 5.2 Please write your definition as a hesitant fuzzy relation Ψ on Γ is said to be hesitant fuzzy equivalence relation if . . .

Definition 5.6 line 1make it “ If Ψ is a relation on Γ … “

Theorem 4.6 Proof(i) (b) and (c) it is better to combine both cases as α∉H or β∉H . The same is true for converse part (ii) (b) and (c)

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Algebra, lattice theory, universal algebra, fuzzy algebra

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Tilahun

Gebrie Yeshiwas

Mathematics, Assosa University, Asosa, Benishangul-Gumuz, Ethiopia

Competing interests: No competing interests were disclosed.

22 8 2025

We have incorporated all the suggested comments and submitted the revised version.

Answers for questions

1. There is no possibility for the condition to fail as a subposet. This is simply an alternative definition of autometrized algebra intended to encompass all conditions. If we discard this definition, there is no difference.

2. In this context, the term "reference set" simply refers to a universal set.