Keywords
autometrized algebra, hesitant fuzzy subalgebra, hesitant fuzzy ideal, hesitant fuzzy congruence
This paper introduces the study of hesitant fuzzy subalgebras of autometrized algebras, obtains some of their properties, and gives some examples. Next, we introduce the concept of the hesitant fuzzy ideal and examine some of its properties. Finally, we introduce a hesitant fuzzy congruence on autometrized algebras and discuss some of its properties. We also introduce the characteristic equations and level subsets of hesitant fuzzy sets and relations on autometrized algebras.
autometrized algebra, hesitant fuzzy subalgebra, hesitant fuzzy ideal, hesitant fuzzy congruence
The 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, denotes an autometrized algebra .
This section examines essential concepts, definitions, and theorems important in other sections.
12 A system is called an autometrized algebra if
18 is called normal if and only if
30 Let . Then is said to be a subalgebra of if;
18 An equivalence relation on is called a congruence relation if and only if
1 Let be a nonempty set, a fuzzy subset of is a mapping .
1 Let be a nonempty set and be a fuzzy subset of , for , the set is called a level subset of .
2 Let be a reference set. A hesitant fuzzy set on is a mapping , where means the power set of .
2 Let be a reference set. If , the characteristic hesitant fuzzy set on is a function of into defined as for all :
2 Let be a hesitant fuzzy set on a nonempty set . Then, for all which is said to be the complement of on .
In this section, we will introduce the hesitant fuzzy subalgebras of autometrized algebras and explore several fundamental properties related to these subalgebras.
A hesitant fuzzy subset of is called a hesitant fuzzy subalgebra of if for all ;
Let with and elements are incomparable. Define and by the following tables.
Then, is an autometrized algebra. Define a hesitant fuzzy subset of by: and .
Then is hesitant fuzzy subalgebra of .
If is a hesitant fuzzy subalgebra of , then for ; .
Assume that is a hesitant fuzzy subalgebra of . Then, . This implies that . Therefore, .
Let is a nonempty subset of . if and only if for all .
Assume that . So, . Therefore, for all .
Conversely, assume that for all . Since is nonempty subset of , we have for . Therefore, . As a result, . Hence, .
A nonempty subset of is a subalgebra of if and only if the characteristic hesitant fuzzy set is a hesitant fuzzy subalgebra of .
Assume that is subalgebra of . Let .
Conversely, assume that is a hesitant fuzzy subalgebra of Γ. To show that is a subalgebra of .
(i) To show that . Since for all ; by theorem (3.4), .
(ii) Let . Then, and . Since is a hesitant fuzzy subalgebra of ; . So, . Hence, .
(iii) Let . Then, and . Since is a hesitant fuzzy subalgebra of ; . So, . Hence, . Hence is a subalgebra of .
Let be a hesitant fuzzy set of . For all , define the level subsets of as , , , and .
Let be a hesitant fuzzy set of . Then is a hesitant fuzzy subalgebra of if and only if for all , is a subalgebra of .
Assume that is a hesitant fuzzy subalgebra of . To show that is a subalgebra of . Let be such that .
(i) Let . Then, . Since is a hesitant fuzzy subalgebra of , implies that . Therefore, .
(ii) Let . Then, and . Since is a hesitant fuzzy subalgebra of ; . This implies that .
(iii) Let . Then, and . Since is a hesitant fuzzy subalgebra of ; . This implies that .
Conversely, assume that is a subalgebra of . To show that is a hesitant fuzzy subalgebra of .
(i) Let . Take . Therefore, and . As a result, . Since is a subalgebra of ; . Hence, .
(ii) Let . Take . Therefore, and . As a result, . Since is a subalgebra of ; . Hence, .
Hence, is a hesitant fuzzy subalgebra of .
Let be a hesitant fuzzy set of . If is a chain and for all , is a subalgebra of , then is a hesitant fuzzy subalgebra of .
Assume that is a chain and for all , a nonempty subset is a subalgebra of . To show that is a hesitant fuzzy subalgebra of .
(i) Let . To show that . Suppose that . Since is a chain; implies that . Then, . Take . Therefore, and . As a result, . Since is a subalgebra of ; . So, . This is a contradiction. Thus, for all .
(ii) Let . To show that . Suppose that . Since is a chain; implies that . Then, . Take . Therefore, and . As a result, . Since is a subalgebra of ; . So, . This is a contradiction. Thus, for all .
Hence, is a hesitant fuzzy subalgebra of .
Let be a hesitant fuzzy set of . Then is a hesitant fuzzy subalgebra of if and only if for all , is a subalgebra of .
Assume that is a hesitant fuzzy subalgebra of . To show that is a subalgebra of . Let be such that .
(i) Let . Then, . Since is a hesitant fuzzy subalgebra of , implies that . Clearly, . Therefore, . Therefore, .
(ii) Let . Then, and . Since is a hesitant fuzzy subalgebra of ; . So, . Therefore, . Hence, .
(iii) Let . Then, and . Since is a hesitant fuzzy subalgebra of ; . So, . Therefore, . Hence, . Therefore, is a subalgebra of .
Conversely, assume that is a subalgebra of . To show that is a hesitant fuzzy subalgebra of .
(i) Let . Take . Therefore, and . As a result, . Since is a subalgebra of ; . Clearly, . As a result, . Therefore, .
(ii) Let . Take . Therefore, and . As a result, . Since is a subalgebra of ; . Clearly, . As a result, . Therefore, .
Hence, is a hesitant fuzzy subalgebra of .
Let be a hesitant fuzzy set of . If is a chain and for all , is a subalgebra of , then is a hesitant fuzzy subalgebra of .
Assume that is a chain and for all , a nonempty subset is a subalgebra of . To show that is a hesitant fuzzy subalgebra of .
(i) Let . To show that . Suppose that . Since is a chain; implies that . Clearly, . So, . Take . Therefore, and . As a result, . Since is a subalgebra of ; . So, . This is a contradiction. Thus, for all .
(ii) Let . To show that . Suppose that . Since is a chain; implies that . Clearly, . So, . Take . Therefore, and . As a result, . Since is a subalgebra of ; . So, . This is a contradiction. Thus, for all . Hence, is a hesitant fuzzy subalgebra of .
This section presents the concept of hesitant fuzzy ideals in autometrized algebras and explores several fundamental properties related to these ideals.
A hesitant fuzzy subset of is called a hesitant fuzzy ideal of if for all ;
Let with and elements are incomparable. Define and by the following tables.
Then, is an autometrized algebra. Define a hesitant fuzzy subset of by:
It is clear that is a hesitant fuzzy ideal of .
If is a hesitant fuzzy ideal of , then for ; .
Assume that is a hesitant fuzzy ideal of . Since ; implies that that . Therefore, .
Let be a normal autometrized algebra. If is a hesitant fuzzy ideal of , then for ; .
Assume that is a hesitant fuzzy ideal of . Since is normal; implies that . This implies that and ; implies that and . Therefore, .
Let be a normal autometrized algebra. Every hesitant fuzzy ideal of is a hesitant fuzzy subalgebra of .
Assume that is a hesitant fuzzy ideal of . Let .
Therefore, . Hence, is a hesitant fuzzy subalgebra of .
A nonempty subset of is an ideal of if and only if the characteristic hesitant fuzzy set is a hesitant fuzzy ideal of .
Assume that is an ideal of . Let .
(i) Here we will consider three cases.
(ii) Suppose . To show that . Here we will consider three cases.
Conversely, assume that is a hesitant fuzzy ideal of . To show that is an ideal of .
Let be a hesitant fuzzy set of . Then is a hesitant fuzzy ideal of if and only if for all , a nonempty subset is an ideal of .
Assume that is a hesitant fuzzy ideal of . To show that is an ideal of . Let be such that .
(i) Let . Then, and . Since is a hesitant fuzzy ideal of ; . This implies that .
(ii) Let . Suppose . Let . Clearly, . To show that . Since is a hesitant fuzzy ideal of , . Therefore, . Thus, . Hence, is an ideal of .
Conversely, assume that is an ideal of . To show that is a hesitant fuzzy ideal of .
(i) Let . Take . Therefore, and . As a result, . Since is an ideal of ; . Hence, .
(ii) Let . Suppose . Take . So, . Since is an ideal; . As a result, .
Hence, is a hesitant fuzzy ideal of .
Let be a hesitant fuzzy set of . If is a chain and for all , a nonempty subset is an ideal of , then is a hesitant fuzzy ideal of .
Assume that is a chain and for all , a nonempty subset is an ideal of . To show that is a hesitant fuzzy ideal of .
(i) Let . To show that . Suppose that . Since is a chain; implies that . Then, . Take . Therefore, and . As a result, . Since is an ideal of ; . So, . This is a contradiction. Thus, for all .
(ii) Let . Suppose . To show that . Suppose that . Since is a chain; implies that . Then, . Take . Therefore, . As a result, . Since is an ideal of ; . So, . This is a contradiction. Therefore, for all .
Hence, is a hesitant fuzzy ideal of .
Let be a hesitant fuzzy set of . Then is a hesitant fuzzy ideal of if and only if for all , a nonempty subset is an ideal of .
Assume that is a hesitant fuzzy ideal of . To show that is an ideal of . Let be such that .
(i) Let . Then, . Since is a hesitant fuzzy ideal of , implies that . Clearly, . Therefore, . Therefore, .
(ii) Let . Then, and . Since is a hesitant fuzzy ideal of ; . So, . Therefore, . Hence, .
(iii) Let . Suppose . Let . Clearly, . To show that . Since is a hesitant fuzzy ideal of , . So, . Therefore, . Thus, . Hence, is an ideal of .
Conversely, assume that is an ideal of . To show that is a hesitant fuzzy ideal of .
Let be a hesitant fuzzy set of . If is a chain and for all , a nonempty subset is an ideal of , then is a hesitant fuzzy ideal of .
Assume that is a chain and for all , a nonempty subset is an ideal of . To show that is a hesitant fuzzy ideal of .
(i) Let . To show that . Suppose that . Since is a chain; implies that . Clearly, . So, . Take . Therefore, and . As a result, . Since is an ideal of ; . So, . This is a contradiction. Thus, for all .
(ii) Let . Suppose . To show that . Suppose that . Since is a chain; implies that . Then, . Clearly, . Take . Therefore, . As a result, . Since is an ideal of ; . So, . This is a contradiction. Therefore, for all . Hence, is a hesitant fuzzy ideal of .
In this section, we introduce the hesitant fuzzy congruence relation of autometrized algebras and examine some related properties.
A hesitant fuzzy relation on is a mapping , where means the power set of .
Let be a hesitant fuzzy relation on . Then, for all which is said to be the complement of on .
If is a hesitant fuzzy equivalent relation on , then
A hesitant fuzzy equivalence relation on is called a hesitant fuzzy congruence relation on if
In Example 3.2, becomes an autometrized algebra. Define a hesitant fuzzy relation on by: and . Therefore, is a hesitant fuzzy congruence relation on .
If is a hesitant fuzzy relation on , then characteristic hesitant fuzzy relation on is a function of into defined as for all :
A nonempty equivalent relation on is a congruence relation on if and only if the characteristic hesitant fuzzy equivalent relation is a hesitant fuzzy congruence relation on .
Assume that is a congruence relation on . To show that is a hesitant fuzzy congruence relation on . Let .
(i) To show that . Here we will consider three cases.
(ii) To show that . Here we will consider three cases.
(iii) Let and suppose that . To show that . Here we will consider two cases.
Conversely, assume that is a hesitant fuzzy congruence relation on . To show that is a congruence relation on .
(i) Let . Then, and . Since is a hesitant fuzzy congruence relation on ; . So, . Hence, .
(ii) Let . Then, and . Since is a hesitant fuzzy congruence relation on ; . So, . Hence, .
(iii) Let and . Clearly, Clearly, . Since is a hesitant fuzzy congruence relation on ; then . Therefore, . Hence, . Hence, is a congruence relation on .
Let be a hesitant fuzzy relation on . For all , the sets , , , and are called level subset of .
Let be a hesitant fuzzy relation on . is a hesitant fuzzy congruence relation on if and only if for all , is either empty or a congruence relation on .
Assume that is a hesitant fuzzy congruence relation on .
(i) Since is a nonempty; let . So, . But . So, . Therefore, .
(ii) Let . So, . Clearly, . So, . Therefore, .
(iii) Let . Then, and . Clearly, . Since is a hesitant fuzzy congruence; . Therefore, . Therefore, is an equivalence relation.
(a) Let . So, and . Since is a hesitant fuzzy congruence relation on ; . Therefore, .
(b) Let . So, and . Since is a hesitant fuzzy congruence relation on ; . Therefore, .
(c) Let and . Clearly, . Since is a hesitant fuzzy congruence relation on ; then . Therefore, . Hence, is a congruence relation on .
Conversely, is a congruence relation on . To show that is a hesitant fuzzy congruence relation on .
(i) We know that for any , . Take . So, . Since is a congruence relation; . Then, . Therefore, .
(ii) Let . Take . So, . Since is a congruence; . So, . Clearly, . Again, take . So, . Since is a congruence; . So, . Clearly, . Consequently, .
(iii) Let . Take . Clearly, and . This implies that . Since is a congruence relation; . Therefore, . Thus, . Therefore, is a hesitant equivalence relation.
(a) Let . Take . Therefore, and . Since is a congruence relation on ; . As a result, .
(b) Let . Take . Therefore, and . Since is a congruence relation on ; . As a result, .
(c) Let and suppose that . Take . Therefore, . Since is a congruence relation on ; . Therefore, . Hence, is a hesitant fuzzy congruence relation on .
Let be a hesitant fuzzy equivalent relation on . If is a chain and for all , a nonempty subset is a congruence relation on , then is a hesitant fuzzy congruence relation on .
Assume that is a chain and for all , a nonempty subset is a congruence relation on . To show that is a hesitant fuzzy congruence relation on .
(i) Let . To show that . Suppose that . Since is a chain; implies that . Then, . Take . Therefore, and . As a result, . Since is a congruence relation on ; . So, . This is a contradiction. Thus, for all .
(ii) Let . To show that . Suppose that . Since is a chain; implies that . Then, . Take . Therefore, and . As a result, . Since is a congruence relation on ; . So, . This is a contradiction. Thus, for all .
(iii) Let and suppose that . To show that . Suppose that Suppose that . Since is a chain; implies that . Take . Therefore, . Since is a congruence relation on relation on ; . Therefore, is contradiction. Therefore, . Hence, is a hesitant fuzzy congruence relation on .
Let be a hesitant fuzzy equivalent relation on . Then is a hesitant fuzzy congruence relation on if and only if for all , is either empty or congruence relation on .
Assume that is a hesitant fuzzy congruence relation on . To show that is a congruence relation on . Let be such that .
(i) Since is a nonempty; let . So, . Since is a hesitant fuzzy congruence relation on ; . Then, . As a result, . Therefore, .
(ii) Let . So, . Since is a hesitant fuzzy congruence relation on ; . Clearly, . So, . Therefore, .
(iii) Let . Then, and . Since is a hesitant fuzzy congruence relation on ; . Then, . Consequently, . Therefore, . Therefore, is an equivalence relation.
(a) Let . Then, and . Since is a hesitant fuzzy congruence relation on ; . So, . Therefore, . Hence, .
(b) Let . Then, and . Since is a hesitant fuzzy congruence relation on ; . So, . Therefore, . Hence, .
(c) Let and . Clearly, . To show that . Since is a hesitant fuzzy congruence relation on , . So, . Therefore, . Thus, . Hence, is congruence relation on .
Conversely, assume that is congruence relation on . To show that is a hesitant fuzzy congruence relation on .
(i) We know that for any , . Take . So, . Since is a congruence relation; . Then, . So, . Clearly, . Therefore, .
(ii) Let . Take . So, . Since is a congruence; . So, . Therefore, . Clearly, . Therefore, . Again, take . So, . Since is a congruence; . So, . Clearly, . Clearly, . Therefore, . Hence, .
(iii) Let . Take . Clearly, and . This implies that . Since is a congruence relation; . Therefore, . Clearly, . Now, consider . As a result, . Therefore, is a hesitant equivalence relation.
(a) Let . Take . Therefore, and . As a result, . Since is a congruence relation on ; . Clearly, . As a result, . Therefore, .
(b) Let . Take . Therefore, and . As a result, . Since is a congruence relation on ; . Clearly, . As a result, . Therefore, .
(c) Let and suppose that . Take . So, . Since is a congruence relation on ; . As a result, . Clearly, . Therefore, . Hence, is a hesitant fuzzy congruence relation on .
Let be a hesitant fuzzy equivalent relation on . If is a chain and for all , a nonempty subset is a congruence relation on , then is a hesitant fuzzy congruence relation on .
Assume that is a chain and for all , a nonempty subset is a congruence relation on . To show that is a hesitant fuzzy congruence relation on .
(i) Let . To show that . Suppose that . Since is a chain; implies that . Then, . Clearly, . So, . Take . Therefore, and . As a result, . Since is a congruence relation on ; . So, . This is a contradiction. Thus, .
(ii) Let . To show that . Suppose that . Since is a chain; implies that . Then, . Clearly, . So, . Take . Therefore, and . As a result, . Since is a congruence relation on ; . So, . This is a contradiction. Thus, .
(iii) Let and suppose that . To show that . Suppose that . Since is a chain; implies that . Then, . Clearly, . Take . Therefore, . As a result, . Since is a congruence relation on ; . So, . This is a contradiction. Therefore, . Hence, is a hesitant fuzzy congruence relation on .
This paper introduced the study of hesitant fuzzy subalgebras of autometrized algebras. Also, we proved that a nonempty subset of an autometrized algebra is a subalgebra of if and only if the characteristic hesitant fuzzy set is a hesitant fuzzy subalgebra of . Further, we introduced the concept of the hesitant fuzzy ideal and examined some of its properties. We showed that hesitant fuzzy set on an autometrized algebra is a hesitant fuzzy ideal of if and only if for all , a nonempty subset is an ideal of . 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 on an autometrized algebra is a hesitant fuzzy congruence relation on if and only if for all , is either empty or congruence relation on .
All authors made equal contributions to this manuscript and have approved the final version.
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Is the study design appropriate and is the work technically sound?
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Are sufficient details of methods and analysis provided to allow replication by others?
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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.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
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
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