t-Intuitionistic Fuzzy Structures on PMS-Ideals of a PMS-Algebra

Derseh, Beza Lamesgin; Alaba, Berhanu Assaye; Wondifraw, Yohannes Gedamu

doi:https://doi.org/10.1155/2022/5101293

International Journal of Mathematics and Mathematical Sciences

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5101293 | https://doi.org/10.1155/2022/5101293

t-Intuitionistic Fuzzy Structures on PMS-Ideals of a PMS-Algebra

Beza Lamesgin Derseh,¹Berhanu Assaye Alaba,¹and Yohannes Gedamu Wondifraw¹

Academic Editor: Sergejs Solovjovs

Received03 Apr 2022

Revised06 Jul 2022

Accepted11 Jul 2022

Published23 Sept 2022

Abstract

In this article, we apply the concept of a -intuitionistic fuzzy set to PMS-ideals in PMS-algebras. The notion of the -intuitionistic fuzzy PMS-ideal of PMS-algebra is introduced, and several related properties are studied. The relationships between a -intuitionistic fuzzy PMS-ideal and a -intuitionistic fuzzy PMS-subalgebra of a PMS-algebra, as well as the relationships between an intuitionistic fuzzy PMS-ideal and a -intuitionistic fuzzy PMS-ideal are discussed in detail. A condition for an intuitionistic fuzzy set to be a -intuitionistic fuzzy PMS-ideal is provided. The -intuitionistic fuzzy PMS-ideals of PMS-algebra are described using their level cuts. The homomorphism of a -intuitionistic fuzzy PMS-ideal of a PMS-algebra is studied, and its homomorphic image and inverse image are explored. The Cartesian product of any two -intuitionistic fuzzy PMS-ideals is discussed, and some related results are derived. The Cartesian product of the -intuitionistic fuzzy PMS-ideals is also characterized using its level cuts. The strongest -intuitionistic fuzzy PMS-relation in a PMS-algebra is defined. Finally, the relationships between the strongest -intuitionistic fuzzy PMS-relation and -intuitionistic fuzzy PMS-ideal are studied.

1. Introduction

In 1965, Zadeh[1] introduced the concept of a fuzzy set for dealing with uncertainty and vagueness in real-world problems. Since then, several researchers have applied it to a wide range of algebraic structures, such as BCI-algebras, BCK-algebras, BG-algebras, KU-algebras, etc. Akram and Dar [2] introduced the notions of T-fuzzy subalgebras and T-fuzzy H-ideals in BCI-algebras using a t-norm T and investigated some of their properties. Akram and Zhan [3] introduced the notion of sensible fuzzy ideals of BCK-algebras with respect to a t-conorm and investigated some of their properties. Senapati et al. [4] introduced the notion of T-fuzzy subalgebras and T-fuzzy closed ideals of BG-algebras and investigated their related results. The notion of T-fuzzy KU-ideals of KU-algebras are introduced using t-norm T and their related results are investigated by Senapati [5]. He further investigated images and pre-images of KU-ideals under homomorphism and the Cartesian product and T-product of T-fuzzy KU-ideals of KU-algebras. After the introduction of fuzzy sets by Zadeh, many mathematicians have worked to extend this fundamental concept in a variety of ways. In 1975, Zadeh [6] developed a type-2 fuzzy set as an extension of the fuzzy set with a membership grade of fuzzy set in the unit interval [0, 1] rather than a point in [0, 1]. Torra and Narukawa [7, 8] developed the concept of a hesitant fuzzy set as one of the extensions of the fuzzy set to express hesitant information more thoroughly than other extensions of the fuzzy set, as it permits several possible values for the membership degree of an element.

Atanassov [9, 10] introduced the idea of an intuitionistic fuzzy set as a generalization of the fuzzy set. An intuitionistic fuzzy set is more effective than a fuzzy set in dealing with ambiguity and uncertainty since it assigns a membership and nonmembership degree to each element of a set. Since its appearance, mathematicians have applied this fundamental concept to a number of algebraic structures. Jun et al. [11] introduced the notion of an intuitionistic fuzzy quasi-associative ideal of a BCI-algebra and investigated some related properties. Many fundamental characteristics of intuitionistic fuzzy subgroups were also explored by Sharma [12, 13]. Panigrahi and Nanda [14] studied the idea of intuitionistic fuzzy relations over intuitionistic fuzzy subsets and found several interesting properties of intuitionistic fuzzy relations in intuitionistic fuzzy subsets. Peng [15] introduced the notion of intuitionistic fuzzy B-algebras in B-algebra and studied some properties of the homomorphic image and inverse image of intuitionistic fuzzy B-algebras. Jana et al. [16] introduced the concept of intuitionistic fuzzy set to G-subalgebras of G-algebras and investigated several properties. Sharma [17, 18] developed the concept of the -intuitionistic fuzzy set as an extension of the intuitionistic fuzzy set to deal with uncertainty and vagueness within the context of some reference points in the unit interval [0,1] and then introduced the concepts of -intuitionistic fuzzy subgroups and -intuitionistic fuzzy subrings. Shuaib et al. [19] introduced the notion of -intuitionistic fuzzy subgroup over -intuitionistic fuzzy subset and studied some algebraic aspects of -fuzzy subgroups. Barbhuiya [20] introduced the notion of -intuitionistic fuzzy subalgebra and -intuitionistic fuzzy normal subalgebra of BG-algebra and studied their properties. He also investigated the homomorphic image and inverse image of both -intuitionistic fuzzy subalgebra and -intuitionistic fuzzy normal subalgebra of a BG-algebra.

Iseki and Tanaka [21] introduced a class of abstract algebras called BCK-algebras. Iseki [22] introduced another class of abstract algebra called BCI-algebra as a generalization of BCK-algebra. In 2016, Sithar Selvam and Nagalakshmi [23] introduced a new algebraic structure, known as PMS-algebra, as a generalization of BCKBCITMPS-algebras and investigated various related results. In the same year, Sithar Selvam and Nagalakshmi [24] also fuzzified a PMS-ideal in a PMS-algebra and investigated its basic properties. The study of intuitionistic fuzzification of PMS-subalgebras and PMS-ideals of PMS-algebras was done by Derseh et al. [25]. The notion of -intuitionistic fuzzy subalgebra has been studied in several algebraic structures (see [17–20, 26, 27]. However, to the best of our knowledge, no studies on -intuitionistic fuzzy ideals of any algebraic structure, including PMS-algebra, are available. This motivated us to develop -intuitionistic fuzzy PMS-ideals in PMS-algebra.

In this manuscript, we use the concept of a -intuitionistic fuzzy set to PMS-ideals in PMS-algebras. We introduce the notion of a -intuitionistic fuzzy PMS-ideal of PMS-algebra and study its properties. We consider the relationships between a -intuitionistic fuzzy PMS-ideal and a -intuitionistic fuzzy PMS-subalgebra of a PMS-algebra as well as the relationships between an intuitionistic fuzzy PMS-ideal and a -intuitionistic fuzzy PMS-ideal of a PMS-algebra. We establish a condition for an intuitionistic fuzzy set in a PMS-algebra to be a -intuitionistic fuzzy PMS-ideal of a PMS-algebra. We describe the -intuitionistic fuzzy PMS-ideals of PMS-algebra using their level cuts. We consider a -intuitionistic fuzzy PMS-ideal in a PMS-algebra under homomorphism and explore its homomorphic image and inverse image in a PMS-algebra. Furthermore, we study the Cartesian product of any two -intuitionistic fuzzy PMS-ideals of PMS-algebra and find some interesting results. We also characterize the Cartesian product of the -intuitionistic fuzzy PMS-ideals using their level cuts. We finally define the strongest -intuitionistic fuzzy PMS-relation in a PMS-algebra and study the relationship between the strongest -intuitionistic fuzzy PMS-relation and a -intuitionistic fuzzy PMS-ideal.

2. Preliminaries

In this section, we consider some basic definitions, results, and some important concepts in PMS-algebras that are needed for our work.

Definition 1. (see [23]). A PMS-algebra is a nonempty set X with a constant 0 and a binary operation of type (2, 0) satisfying the following axioms:(i)(ii), for all We can define a binary relation in by if and only if .

Definition 2. (see [23]). A nonempty subset S of a PMS-algebra is called a PMS-subalgebra of X if , for all .

Definition 3. (see [23]). A nonempty subset I of a PMS-algebra is said to be a PMS-ideal of if it satisfies the following conditions:(i) I(ii) I I, for all I.

Proposition 1 (see [23]). Let be a PMS-algebra. Then the following properties hold for all ,(i)(ii)(iii)(iv)(v)

Definition 4. (see [1]). A fuzzy subset in a nonempty set is defined as , where the mapping defines the degree of membership

Definition 5. (see [9, 10]). An intuitionistic fuzzy set A in a nonempty set is an object having the form , where the functions and define the degree of membership and the degree of nonmembership of each element to the set , respectively, with the condition , .

Definition 6. (see [9, 10]). Let A and B be two intuitionistic fuzzy subsets of the set , where and , then(i)(ii)(iii)(iv)(v)

Definition 7. (see [25]). An intuitionistic fuzzy subset of a PMS-algebra is called an intuitionistic fuzzy PMS-subalgebra of if and , for all .

Definition 8. (see [28]). An intuitionistic fuzzy set in is called an intuitionistic fuzzy PMS-ideal of if it satisfies the following conditions for all .(i) and ,(ii),(iii)

Definition 9. (see [29]). Let A be a fuzzy set in a nonempty set with membership function and let . Then the fuzzy set in is called the -fuzzy subset of whose membership function is (w.r.t fuzzy set ) and is defined by , for all .

Definition 10. (see [18]). Let be an intuitionistic fuzzy set in a nonempty set and . Then the t-intuitionistic fuzzy set (t-IFS) in a nonempty set is an object having the form , where the function and denote the degree of membership and degree of nonmembership, respectively, such that and satisfying the condition , for all .
Note: For the sake of simplicity, we shall use the symbol , for t-IFS .

Remark 1. Let be a -IFSs of the set . Then
and

Remark 2. (see [18, 27]). Let and be any two -intuitionistic fuzzy subsets of any nonempty set , then and .

Definition 11. (see [23]). Let and be PMS-algebras. The mapping is called a homomorphism of PMS-algebras if , for all . A homomorphism is called an epimorphism of PMS-algebras if .
Note: If is a homomorphism of PMS-algebras, then .

Remark 3. (see [18]). Let be a mapping and A such that B are any two -IFSs of X and Y, respectively, then and , for all .

Definition 12. (see [26]). Let and be two -intuitionistic fuzzy subsets of and , respectively. Then their Cartesian product of and denoted by is defined as , where and , for all and .

Remark 4. Let and be any two PMS-algebras, for every , we define on by . Clearly, is a PMS-algebra.

Definition 13. (see [26]). Let be -IFS of X w.r.t IFS A. Then the -cut of is a crisp subset of and is given by , where with .

3. -Intuitionistic Fuzzy PMS-Ideals of a PMS-Algebra

In this section, we study the notion of a -intuitionistic fuzzy PMS-ideal in a PMS-algebra and investigate several interesting results. In what follows, let X and Y denote PMS-algebra unless otherwise specified.

Definition 14. Let . A -IFS of is called the -intuitionistic fuzzy (t-IF) PMS-ideal of a PMS-algebra if(i) and ,(ii),(iii), for all .

Example 1. Consider such that is a PMS-algebra with Table 1.
Define the intuitionistic fuzzy set in by
and for all .
If we take , then we have and , for all .
Then by routine calculation, we can see that is a t-IF PMS-ideal of .

Theorem 1. Every t-IF PMS-ideal of X is a t-IF PMS-subalgebra of X

Proof. Let be a t-IF PMS-ideal of and . Then by Definition 14 and Definition 1, we have
andTherefore, is a t-IF PMS-subalgebra of .

Theorem 2. Let be a t-IF PMS-ideal of X. If , then and , for all

Proof. Let such that . Then . By Definition 14, Proposition 1, Definition 14, and Theorem 1, we have and, therefore, and , for all .

Theorem 3. If is an IF PMS-ideal of , then is also a -IF PMS-ideal of .

Proof. Let be an IF PMS-ideal of and . Then by the definition of t-IFS and definition of intuitionistic fuzzy PMS-ideal, we have
and
Also,andHence, and , for all . Therefore, is a -IF PMS-ideal of .

Remark 5. The converse of above theorem need not be necessarily true. This fact is shown by the following example:

Example 2. Let be a set with Table 1 as in Example 1. Define the intuitionistic fuzzy set in by
and for all .
Since and , it follows that is not an intuitionistic fuzzy PMS-ideal of as it does not satisfy Definition 14. If we take , then and , for all . Therefore, by routine calculations, we get that(i) and (ii), and(iii).Therefore, by Definition 14, is a t-IF PMS-ideal of .
The following theorem provides a condition for an intuitionistic fuzzy set in a PMS-algebra to be a -intuitionistic fuzzy PMS-ideal.

Theorem 4. Let be an IFS in and such that , where and . Then is a -IF PMS-ideal of .

Proof. Since , we have and
and , and , and , for all .
So, and , and , for all .
Therefore, and . This satisfies the condition and , for all .
Also, and
, for all .
Hence, and , for all .

Example 3. Let be the set of all integers. Let be a binary operation on defined by for all , where is the usual subtraction of integers. Then is a PMS-algebra since Definition 1 is satisfied as shown below.(1)(2).Define an intuitionistic fuzzy set bySince and , then . So by Theorem 4 for , is a -IF PMS-ideal of .
The subsequent result shows that the intersection of any two -IF PMS-ideal is a -IF PMS-ideal.

Theorem 5. The intersection of any two t-IF PMS-ideals of X is also a -IF PMS-ideal of X.

Proof. Let and be any two t-IF PMS-ideals of and . Then by Definition 14 and Definition 10, we have and thus, and and andTherefore, is a t-IF PMS-ideals of .
The above theorem can also be generalized to any family of t-IF PMS-ideals in PMS-algebra as given in the next corollary.

Corollary 1. The intersection of a family of t-IF PMS-ideals of X is again a t-IF PMS-ideal of X.

Remark 6. The union of any two t-IF PMS-ideals of X may not be a t-IF PMS-ideal of X. This is shown by the next example.

Example 4. Let be the set of all integers and is a binary operation on defined as in Example 3. Clearly, is a PMS-algebra. Define the intuitionistic fuzzy sets and in respectively byandClearly, A and B are IF PMS-ideals of . Thus, by Theorem 3 and are t-IF PMS-ideals of for . If we take t = 0.6, then and are given byandNow and . Therefore,Take , and , then , and .
Now and .Similarly, and .As a result of (16) and(17), we arrive at a contradiction with Definition 14.
Therefore, is not an IF-PMS-ideal of .

Theorem 6. If is a -IF PMS-ideal of X, then is also a -IF PMS-ideal of X.

Proof. Suppose is a -IF PMS-ideal of X. Then by Definition 14, we have and for all . So, we need to show that and for all . Now,
Also,Hence, is a -IF PMS-ideal of .

Theorem 7. If is a -IF PMS-ideal of X, then is also a -IF PMS-ideal of X.

Proof. Suppose is a -IF PMS-ideal of X. Then by Definition 14, we have
and , for all . So, we have to show that and , for all .
Now, .
And,Hence, is a -IF PMS-ideal of .

Theorem 8. Let be a -IFS of . Then is a -IF PMS-ideal of if and only if the nonempty subset of X is a PMS-ideal of X for all with .

Proof. Since , there exist such that . Then, and . Since is a -IF PMS-ideal of , and for all . Thus, and . Therefore, .
Let such that . Then and . Since is a -IF PMS-ideal of , we have and . Thus, . Therefore, is a PMS-ideal of .
Conversely, suppose is a PMS-ideal of . Let such that and . Since is a PMS-ideal of , we have that .
This implies and .
Hence, and for every .
Also, let such that and . Then and . Thus, . Since is a PMS-ideal of , it follows that . So that we have and . Hence, and , for all . Therefore, is a -IF PMS-ideal of .

4. Homomorphism of -Intuitionistic Fuzzy PMS-Ideals

In this section, we discuss -intuitionistic fuzzy PMS-ideals under homomorphism. We study the homomorphic images and inverse images of -intuitionistic fuzzy PMS-ideals and find some related results.

Definition 15. Let X and Y be two nonempty sets and be a mapping. Let and be t-IFSs of and , respectively. Then the image of under is denoted by and is defined as , whereandAlso, the inverse image of under is denoted by and is defined as
, where and for all and .
Note: For any , we have and .

Theorem 9. Let f: be an epimorphism of PMS-algebras and . If is a t-IF PMS-ideal of , then is a t-IF PMS-ideal of .

Proof. Since is an epimorphism of PMS-algebras, for each , there exists , such that . Then and .
Hence, and .
Also, let . Since is an epimorphism of PMS-algebras, there exist such that and . So, using Definitions 14 and 15, we have,and, hence, is a t-IF PMS-ideal of .

Theorem 10. Let be a homomorphism of PMS-algebras and . If is a t-IF PMS-ideal of , then is a -IF PMS-ideal of .

Proof. Let be a -IF PMS-ideal of Y for and let . Then
and
and .
Let . Then by Definition 14 and Definition 15, we haveHence, is a t-IF PMS-ideal of .
The converse of the above theorem is true if is an epimorphism of PMS-algebras.

Theorem 11. Let be an epimorphism of PMS-algebras and is a t-IFS in Y. If is a t-IF PMS-ideal of , then is a t-IF PMS-ideal of for .

Proof. Assume that is an epimorphism of PMS-algebras and is a t-IF PMS-ideal of . Since is an epimorphism of PMS-algebras for any , there exist such that . Then,
andAlso, let . Then there exist such that , and .
Now,andHence, is a -IF PMS-ideal of .

Theorem 12. Let be a -IFS of and be an epimorphism of PMS-algebras. Then the homomorphic image of the nonempty subset of X is a PMS-ideal of Y.

Proof. Let be a -IF PMS-ideal of and let . Since is an epimorphism of PMS-algebras, there exist such that , and . By Theorem 9, is a -IF PMS-ideal of , and by Theorem6, is a PMS-ideal of , that is and .
Then, andHence, .
Also, assume that , that is .
So, we haveandThus, andTherefore, . Hence, is a PMS-ideal of .

Theorem 13. Let be a -IFS of a PMS-algebra and be an epimorphism of PMS-algebras. Then the homomorphic inverse image of the nonempty subset of Y is a PMS-ideal of X.

Proof. Let be a -IF PMS-ideal of and let . Since is an epimorphism of PMS-algebras, there exist such that , and . By Theorem 10, is a -IF PMS-ideal of , and by Theorem 8, is a PMS-ideal of , that is and .
and .
Hence, .
Now let . So, we have
andHence, and .
Therefore, is a PMS-ideal of .

5. Cartesian Product of -Intuitionistic Fuzzy PMS-Ideals

In this section, we consider the Cartesian product of a -intuitionistic fuzzy PMS-ideal and investigate its related properties. We define the strongest -intuitionistic fuzzy PMS-relation and study its relationship with a -intuitionistic fuzzy PMS-ideal.

Theorem 14. Let and be two -IF PMS-ideals of and , respectively. Then is a t-IF PMS-ideal of .

Proof. Let . Then by Definition 14, we have
andThus, and .
Also, let . Again by Definition 14, we haveandTherefore, is an intuitionistic fuzzy PMS-ideal of .

Theorem 15. Let and be any two -IF sets in and , respectively. If is a -IF PMS-ideal of , then either is a -IF PMS-ideal of or is a -IF PMS-ideal of .

Proof. Let and be t-IFSs of and , respectively, such that is a t-IF PMS-ideal of . Then , .
Assume and for some . Then,Similarly, or .
Let . Since is a t-IF PMS-ideal of , we haveAs , we haveIf we put (or resp. ), we have either
or .
In similar way, we can show that either or .
Therefore, is a t-IF PMS-ideal of or is a t-IF PMS-ideal of .

Definition 16. Let and be -IFSs of and w.r.t IFSs A and B. Then, -cut of is a crisp subset of is given by
, where with .

Theorem 16. Let and be t-IFSs of and respectively. Then is a -IF PMS-ideals of if and only if the nonempty subset of is a PMS-ideal of for all with .

Proof. Let and be -IFSs of and , respectively. Since , there exist such that . Then, and . Since is a -IF PMS-ideal of , and for all . Thus, it follows that and . Therefore,
Let such that for .Then, , and . Since is a -IF PMS- ideal of , we have and .
and .
Therefore, .
Hence, is a PMS-ideal of .
Conversely, suppose is a PMS-ideal of for all with . Assume that is not a -IF PMS-ideal of . Then there exist such that and . Then by taking
and , and .
Hence, but and . This implies is not a PMS-ideal of , which is a contradiction.
Therefore, and . for all .
Hence, is an intuitionistic fuzzy PMS-ideal of .

Definition 17. Let be a -intuitionistic fuzzy set in and be a -intuitionistic fuzzy relation on . Then the strongest -intuitionistic fuzzy PMS-relation on X, that is, a -intuitionistic fuzzy PMS-relation on whose membership function and whose nonmembership function is given by and , for all .

Theorem 17. Let be a -intuitionistic fuzzy subset of PMS-algebra and let be the strongest -intuitionistic fuzzy PMS-relation on , then is a -intuitionistic fuzzy PMS-ideal of if and only if is a -intuitionistic fuzzy PMS-ideal of .

Proof. Assume that is a -intuitionistic fuzzy PMS-ideal of . Let . Then
and
, for all , and , for all .
Also, let . ThenandTherefore, is a t-intuitionistic fuzzy PMS-ideal of .
Conversely, assume that is a t-intuitionistic fuzzy PMS-ideal of . Let . Then we have(i) or and or (ii) If we put (or resp. ), then we get (or resp. )(iii)If we put (or resp. ), then we get
(or resp. ).
Hence, is a -intuitionistic fuzzy PMS-ideal of .

6. Conclusion

In this article, we used the concept of a -intuitionistic fuzzy set to PMS-ideals in a PMS-algebra. We studied the notion of a -intuitionistic fuzzy PMS-ideal of a PMS-algebra and explored some related properties. We provided the relationships between a -intuitionistic fuzzy PMS-ideal and a -intuitionistic fuzzy PMS-subalgebra of a PMS-algebra, as well as the relationships between an intuitionistic fuzzy PMS-ideal and a -intuitionistic fuzzy PMS-ideal of a PMS-algebra. We established a condition for an intuitionistic fuzzy set in a PMS-algebra to be a -intuitionistic fuzzy PMS-ideal of a PMS-algebra. We described the -intuitionistic fuzzy PMS-ideals of PMS-algebra by their level cuts. We studied a -intuitionistic fuzzy PMS-ideal of a PMS-algebra under homomorphism and explored the homomorphic image and inverse image of the -intuitionistic fuzzy PMS-ideal. Furthermore, we discussed the Cartesian product of any two -intuitionistic fuzzy PMS-ideals of PMS-algebra and obtained some interesting results. We characterized the Cartesian product of the -intuitionistic fuzzy PMS-ideals by their level cuts. Finally, we defined the strongest -intuitionistic fuzzy PMS-relation on a PMS-algebra and studied the relationship between the strongest -intuitionistic fuzzy PMS-relation and a -intuitionistic fuzzy PMS-ideal. We hope that the findings of this work will add other dimensions to the structures of -intuitionistic fuzzy PMS-ideals based on -intuitionistic fuzzy sets and serve as the foundation for further studies. In our future works, we will extend this concept to - intuitionistic fuzzy PMS-ideals, -intuitionistic multi-fuzzy, and anti-multi-fuzzy PMS-ideals to obtain additional novel results. Moreover, we will develop the neutro-algebraic structures with respect to the PMS-ideals of PMS-algebra.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are highly grateful to the Editors and referees for their valuable comments and suggestions helpful in improving this paper.

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Copyright © 2022 Beza Lamesgin Derseh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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