Abstract
The intuitionistic fuzzification in algebras about the concepts of ideals and subalgebras given with several related characterizations is considered. Some new concepts like intuitionistic fuzzy ideal (), intuitionistic fuzzy subalgebra (), homomorphism, and intuitionistic fuzzy ideal () are introduced and some of their descriptions are given in this work. Further, we show some applications on the family of all intuitionistic fuzzy subalgebras in algebra like the binary relations , and on . Also, their equivalence classes are given and studied.
1. Introduction
The fuzzy set (FS) as suggested by Zadeh [1] in 1965 is a regulation to vagueness and encounter uncertainty. A FS maps each element of the universe of discourse to the interval After the introduction of fuzzy sets theory by him, many mathematicians were conducted on the generalizations of the this concept and studied in the groups, algebras, and soft spaces (see [2–5]). By including a fuzzy set the degree of nonmembership, Atanassov [6] in 1986 suggested the intuitionistic fuzzy set (IFS), which seems more precise for provides opportunities and uncertainty quantification to accurately model a problem based on existing knowledge and monitoring. Also, this notion is discussed in different fields (see [7–11])
algebra, class of algebra of logic, was investigated by Imai and Iseki [12]. After that, the notion of algebras was investigated by Neggers and Kim [13]. In 2017, the concepts of algebra, ideal, ideal, subalgebra, and permutation topological algebra were first proposed by Mahmood and Abud Alradha [14]. Next, they showed the notion of the soft algebra and soft edge algebra [15].
In the present work, the notions of intuitionistic fuzzy ideal (), intuitionistic fuzzy subalgebra (), homomorphism, and intuitionistic fuzzy ideal () are introduced. Further, we show some applications on the family of all intuitionistic fuzzy subalgebras in algebra like the binary relations , and on . Also, their equivalence classes are given and studied.
2. Preliminaries and Notations
We will recall basic definitions and results to obtain properties developed in this work.
Definition 1 (see [16]). An intuitionistic fuzzy set (IFS, in short) over the universe is defined by , where : [0; 1], : [0; 1] with 0 , . and are real numbers and their values represent the degree of membership and nonmembership of to , respectively.
Definition 2 (see [6]). The IF whole and empty sets of are defined by and , respectively.
2.1. Basic Relations and Operations on Intuitionistic Fuzzy Sets [7]
Assume and are two IF sets of . We deduced the following relations:(1)[inclusion] iff and , ,(2)[equality] iff and ,(3)[intersection] , (4)[union] , ,(5)[complement] .
Definition 3 (see [14]). We say is algebra if () is a binary operation on with a constant and such that(1),(2),(3) imply that ,(4)For all imply that .
Definition 4 (see [14]). Assume is a algebra and . We say is a subalgebra of if , .
Definition 5 (see [14]). Assume is algebra and . We say is ideal of if (1) imply ,(2) and imply , .
Definition 6 (see [14]). Assume is a algebra and subset of . We say is a ideal of if (1),(2) and , .
Definition 7 (see [11]). Assume that is an IFS in and The set (resp., is said to be level cut (resp., level cut) of .
3. Intuitionistic Fuzzy Subalgebras in Algebras
In this section, we introduce some new concepts, such as (), (), (), and homomorphism which are introduced and discussed. Further, some binary relations , and on are given, and some basic properties are shown.
Definition 8. Assume is a algebra and is IFS of . We say is an () of if and , .
Example 9. Let be algebra with Table 1.
Then, is an () of .
Definition 10. Assume is algebra and is IFS of . We say is () of if (1) and ,(2) and , .
Example 11. Let be algebra in Example 9 and let be IFS of . Then, is () of .
Definition 12. Assume is algebra and is IFS of . We say is () of if (1) and ,(2) and , .
Example 13. Let be algebra in Example 9 and let be IFS of . Then, is () of .
Remark 14. (1)If is () of , then is ().(2)If is () of , then is ().(3)If is () of and satisfies (2) in Definition 10, then is ().(4)If is () of and satisfies (1) in Definition 12, then is ().
Lemma 15. If is () of , then and , .
Proof. Let Then and .
Theorem 16. If is any family of () of , then is () of , where .
Proof. Let . Thus we consider that
. Also .
Thus satisfies condition (2) in Definition 12. Also, let Hence, we consider that Furthermore, . Then (1) in Definition 12 is held and hence is () of .
Theorem 17. If is () of , then is () of .
Proof. We need only to show that satisfies the first and second condition in Definition 10. Assume . Then Furthermore, . Hence is () of .
Theorem 18. If is () of , then the sets and are subalgebras of .
Proof. Let Hence , and By using Lemma 15, we consider that or equivalently . Now, let This implies that and, by applying Lemma 15, we conclude that . Therefore .
Definition 19. Assume is () of . We say has finite image, if each image of and is with finite cardinality (i.e., and such that and ).
Definition 20. Assume that is () of and The set (resp., is said to be level cut (resp., level cut) of .
Theorem 21. If is () of , then and of are subalgebras of .
Proof. Let . Hence and . This implies that so that . Thus is subalgebra of Now let . Thus and . Therefore is subalgebra of .
Theorem 22. If is IFS of such that the sets and are subalgebras of , then is an () of .
Proof. Suppose that there are two members and in with . Let . Hence , and so , but This is a contradiction, and therefore , . Now assume that for some . Taking , then we consider that It follows that and This is a contradiction. Therefore, we consider that , . Then is () of .
Theorem 23. If is subalgebra of , then there exists () of , where satisfies both level subalgebra and level subalgebra of in .
Proof. Assume is subalgebra of and let and be fuzzy sets in defined byand, where are fixed real numbers with . Assume . Then whenever . This implies that and . If at least one of or does not belong to , then either or and hence either or . It follows that , Hence is () of . Obviously, .
Definition 24. Assume is a mapping of algebras. We say is homomorphism if , . And is IFS in algebra for any IFS of algebra . Also, if is IFS in algebra , then is IFS in and defined by
, whereand
Theorem 25. Let be homomorphism of algebra into algebra and be () of . Then is () of .
Proof. Assuming , we have = and ≤ . Thus is () of .
Theorem 26. Assume is homomorphism of algebra into algebra and is () of . Then is () of .
Proof. Let be () of and let Noticing that , we have ≥ = = . Also, we consider that = ≤ ≤ = = Hence is () of
Theorem 27. Assume is homomorphism of algebra into algebra and is () of . Then is () of .
Proof. Since is () of , then by Theorem 26 and Remark 14 we have as () of . Hence condition (1) in Definition 10 is held. Since is surjective, then for any , such that and Also, . Further, noticing that and , for any , we have = ≥ = = . Also, = . Thus we consider that is () of .
Theorem 28. Assume is homomorphism of algebra into algebra and is () of . Then is () of .
Proof. Since is () of . Then by Theorem 26 and Remark 14 we have as () of . Hence condition (2) in Definition 12 is held. Assume that and are constants of and , respectively. Since is () of , hence and , . Since is homomorphism of algebras, then , where and are constants for algebras and , respectively. Noticing that and for any , then we have = . Also, . Hence is () of
4. Some Applications on
In this section, some applications on are shown like the binary relations , and on . Also, in this section the equivalence classes for theses binary relations are given, and some of their basic properties are studied.
4.1. Equivalence Classes Modulo (/)
Denote the collection of all () of by and let . Define binary relations and on as follows.
and , respectively, for and in . Moreover, it is clear that and are equivalence relations on . If , then we refer to the equivalence class of modulo (resp., ) by (resp., ), and we refer to the family of all equivalence classes of modulo (resp., by (resp., ); i.e., (resp., ). Moreover, denote the collection of all ideals of by and let . Let and be maps from to by and , respectively, . In other words, and are well-defined.
Theorem 29. Let and be the maps from to . Then and are surjective, for each .
Proof. Let . Then is in , where each one of and is (FS) in defined by and , . Furthermore, Let . , let , and ; thus Now, we want to prove that Since , then by condition (1) in Definition 5 we have as subalgebra of and this implies that and are subalgebras of . By Theorem 22 we consider Therefore, we consider and for some This completes the proof.
Theorem 30. Let and be quotient sets. Then they are equipotent to , .
Proof. Assume and let be a map from (resp., ) to and they are defined by , . Hence, and , and in , if and . Then and . This implies the maps and are injective. Moreover, let and , let, and thus We consider that , and Finally, for we have and Therefore and are surjective, and we are done.
4.2. Equivalence Class Modulo
Another relation on is defined by , and, , . Moreover, the relation is also an equivalence relation on . Let denote the equivalence class of modulo , .
Theorem 31. For any , the map defined by , is surjective.
Proof. Let . For , we get For any , there exists , whereand such that This completes the proof.
Theorem 32. For any , the quotient set is equipotent to .
Proof. Assume and is a map defined by , .
Suppose that for any . We consider that , i.e., Hence , and so . Therefore is injective. Furthermore, for we get = = = = Let , , be the same () of that is defined in the proof of Theorem 22. Then we have . Hence is surjective. This completes the proof.
5. Conclusion
In this work, we introduce the notions of (), (), (), and others; then we proved that for any of can be considered as both level subalgebra and level subalgebra of some () of . At the same time, we proved that intersection of any family of () of is () of . Also, we show that if IFS of such that the sets and are subalgebras of . Then is () of . Further, some interesting theorems about homomorphism are given. Finally, some binary relations , and on are obtained, and some of their basic properties are discussed. In future work, we will investigate IF in new types of algebras like algebras, subalgebras, ideals and others. Next, we will study their characteristics.
Data Availability
Data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.