Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras

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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." } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/14-183/v1", "name": "Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized..." } } ] } Home Browse Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized... ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article 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 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras [version 1; peer review: 1 approved, 1 approved with reservations] Gebrie Yeshiwas Tilahun https://orcid.org/0000-0002-7010-4847 Gebrie Yeshiwas Tilahun https://orcid.org/0000-0002-7010-4847 PUBLISHED 10 Feb 2025 Author details Author details Department of Mathematics, Assosa University, Asosa, Benishangul-Gumuz, Ethiopia Gebrie Yeshiwas Tilahun Roles: Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS 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. READ ALL READ LESS Keywords autometrized algebra, hesitant fuzzy subalgebra, hesitant fuzzy ideal, hesitant fuzzy congruence Corresponding Author(s) Gebrie Yeshiwas Tilahun ( [email protected] ) Close Corresponding author: Gebrie Yeshiwas Tilahun Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2025 Tilahun GY. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: 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 ) First published: 10 Feb 2025, 14 :183 ( https://doi.org/10.12688/f1000research.161430.1 ) Latest published: 09 Sep 2025, 14 :183 ( https://doi.org/10.12688/f1000research.161430.2 )  There is a newer version of this article available. Suppress this message for one day. 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: ( M 1) ∀ α , β ∈ Γ ; α ⋆ β ≽ 0 and , α ⋆ β = 0 ⇔ α = β , ( M 2) ∀ α , β ∈ Γ ; α ⋆ β = β ⋆ α , ( M 3) ∀ 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 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 ( 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 . Author contributions All authors made equal contributions to this manuscript and have approved the final version. 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Publisher Full Text 32. Tilahun GY, Parimi RK, Melesse MH: Convex subalgebras and convex spectral topology on autometrized algebras. Res. Math. 2023; 10 (1): 2283261. 33. Tilahun GY, Parimi RK, Melesse MH: Direct product, subdirect product, and representability in autometrized algebras. Korean J. Math. 2023; 31 (4): 445–463. 34. Tilahun GY: Congruency, homomorphism and isomorphism on autometrized algebras. F1000Res. 2025; 14 : 22. Publisher Full Text Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 10 Feb 2025 ADD YOUR COMMENT Comment Author details Author details Department of Mathematics, Assosa University, Asosa, Benishangul-Gumuz, Ethiopia Gebrie Yeshiwas Tilahun Roles: Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (2) version 2 Revised Published: 09 Sep 2025, 14:183 https://doi.org/10.12688/f1000research.161430.2 version 1 Published: 10 Feb 2025, 14:183 https://doi.org/10.12688/f1000research.161430.1 Copyright © 2025 Tilahun GY. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article 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 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 10 Feb 2025 Views 0 Cite How to cite this report: 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 ) The direct URL for this report is: https://f1000research.com/articles/14-183/v1#referee-response-391499 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 20 Jun 2025 Ajoy Kanti Das , Tripura University, Suryamani Nagar, Tripura, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.177460.r391499 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 ... Continue reading READ ALL 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 Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes 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. 2. An efficient water quality evaluation model using weighted hesitant fuzzy soft sets for water pollution rating. https://doi.org/10.1201/9781003494478-10. 3. Das A, Gupta N, Mahmood T, Tripathy B, et al.: An efficient water quality evaluation model using weighted hesitant fuzzy soft sets for water pollution rating. 2025. 179-195 Publisher Full Text 4. Das A, Granados C: An Advanced Approach to Fuzzy Soft Group Decision-Making Using Weighted Average Ratings. SN Computer Science . 2021; 2 (6). Publisher Full Text 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. 6. 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. Competing Interests: No competing interests were disclosed. 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT 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 ) The direct URL for this report is: https://f1000research.com/articles/14-183/v1#referee-response-391499 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 10 Sep 2025 Gebrie Yeshiwas Tilahun , Department of Mathematics, Assosa University, Asosa, Ethiopia 10 Sep 2025 Author Response 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 ... Continue reading 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. 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. Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 10 Sep 2025 Gebrie Yeshiwas Tilahun , Department of Mathematics, Assosa University, Asosa, Ethiopia 10 Sep 2025 Author Response 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 ... Continue reading 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. 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. Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: 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 ) The direct URL for this report is: https://f1000research.com/articles/14-183/v1#referee-response-366481 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 11 Mar 2025 Sileshe Gone Korma , Arba Minch University, Hawassa, Ethiopia Approved VIEWS 0 https://doi.org/10.5256/f1000research.177460.r366481 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 READ ALL 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 Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT 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 ) The direct URL for this report is: https://f1000research.com/articles/14-183/v1#referee-response-366481 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 10 Sep 2025 Gebrie Yeshiwas Tilahun , Department of Mathematics, Assosa University, Asosa, Ethiopia 10 Sep 2025 Author Response 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 ... Continue reading 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. 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. Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 10 Sep 2025 Gebrie Yeshiwas Tilahun , Department of Mathematics, Assosa University, Asosa, Ethiopia 10 Sep 2025 Author Response 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 ... Continue reading 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. 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. Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 10 Feb 2025 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 3 Version 2 (revision) 09 Sep 25 read read Version 1 10 Feb 25 read read Sileshe Gone Korma , Arba Minch University, Hawassa, Ethiopia Ajoy Kanti Das , Tripura University, Suryamani Nagar, India Jehad R Kider , University of Technology-Iraq, Baghdad, Iraq Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Das A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 17 Sep 2025 | for Version 2 Ajoy Kanti Das , Tripura University, Suryamani Nagar, Tripura, India 0 Views copyright © 2025 Das A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Thank you for sending me the revised version of the article “Hesitant Fuzzy Subalgebras, Ideals and Congruences on Autometrized Algebras.” I have carefully assessed the revisions, and the authors have addressed my earlier concerns. Overall, the revisions significantly improve the clarity and quality of the paper. I find the manuscript suitable for approval. Competing Interests No competing interests were disclosed. 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. reply Respond to this report Responses (0) Das AK. Peer Review 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.187101.r412595) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-183/v2#referee-response-412595 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Kider J. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 16 Sep 2025 | for Version 2 Jehad R Kider , University of Technology-Iraq, Baghdad, Iraq 0 Views copyright © 2025 Kider J. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions 1.Generally, the paper is well written and has a good structure. 2.The paper addresses a novel and underexplored intersection between hesitant fuzzy sets and autometrized algebra. 3.The theoretical framework is built systematically, starting from definitions to illustrative examples and theorems. 4.The work appears to be original and free of plagiarism, and no ethical concerns are noted Taking the above into consideration, I recommend the paper for indexing in your journal. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise fuzzy metric spaces, fuzzy normed spaces, fuzzy inner product spaces. 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. reply Respond to this report Responses (0) Kider JR. Peer Review 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.187101.r413664) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-183/v2#referee-response-413664 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Das A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 20 Jun 2025 | for Version 1 Ajoy Kanti Das , Tripura University, Suryamani Nagar, Tripura, India 0 Views copyright © 2025 Das A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions 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 Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes 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. 2. An efficient water quality evaluation model using weighted hesitant fuzzy soft sets for water pollution rating. https://doi.org/10.1201/9781003494478-10. 3. Das A, Gupta N, Mahmood T, Tripathy B, et al.: An efficient water quality evaluation model using weighted hesitant fuzzy soft sets for water pollution rating. 2025. 179-195 Publisher Full Text 4. Das A, Granados C: An Advanced Approach to Fuzzy Soft Group Decision-Making Using Weighted Average Ratings. SN Computer Science . 2021; 2 (6). Publisher Full Text 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. 6. 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. Competing Interests No competing interests were disclosed. 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. reply Respond to this report Responses (1) Author Response 10 Sep 2025 Gebrie Yeshiwas Tilahun, Department of Mathematics, Assosa University, Asosa, Ethiopia 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. View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Das AK. Peer Review 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 this citation. The direct URL for this report is: https://f1000research.com/articles/14-183/v1#referee-response-391499 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Korma S. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 11 Mar 2025 | for Version 1 Sileshe Gone Korma , Arba Minch University, Hawassa, Ethiopia 0 Views copyright © 2025 Korma S. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions 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 Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. 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. reply Respond to this report Responses (1) Author Response 10 Sep 2025 Gebrie Yeshiwas Tilahun, Department of Mathematics, Assosa University, Asosa, Ethiopia 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. View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Korma SG. Peer Review 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 this citation. The direct URL for this report is: https://f1000research.com/articles/14-183/v1#referee-response-366481 Alongside their report, reviewers assign a status to the article: Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. 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