Abstract
In the present paper, using Lukaswize triple-valued logic, we introduce the notion of -intuitionistic fuzzy soft ideal of -algebras, where are the membership values between an intuitionistic fuzzy soft point and intuitionistic fuzzy set. Moreover, intuitionistic fuzzy soft ideals with thresholds are introduced, and their related properties are investigated.
1. Introduction
In the real world, there are several difficult problems that cannot be solved by usual mathematical methods. As a result, several theories were introduced to solve the existing problems. One of them is the theory of fuzzy sets proposed by Zadeh [1]. After the introduction of fuzzy sets by Zadeh, fuzzy set theory has become an active area of research in various fields up to now (see, for example, [2, 3]), and many generalizations of fuzzy sets have been defined and studied.
Atanassov defined a new generalization of the fuzzy set, namely, intuitionistic fuzzy set [4]. Kim considered intuitionistic fuzziness on subalgebras and ideals in -algebras [5]. All these theories have their weaknesses as pointed out in [6]; thus, Molodstov introduced the idea of soft sets which have been a useful tool for dealing with uncertainties. Maji et al. defined soft binary operations [7]. Also, Maji et al. presented the concept of fuzzy soft set [8]. Later on, Maji et al. introduced and studied intuitionistic fuzzy soft sets (see [9–11]), and more specifically, Akram et al. studied intuitionistic fuzzy soft -algebras (see [12]). In recent years, a number of research papers have been devoted to the study of soft set theory applied to different algebraic structures (see, for example, [13–18]). Jun et al. applied soft set and fuzzy soft set theories to -algebras in [19, 20], respectively, and Akram et al. applied the same theories on -algebras in [21]. Larimi and Jun introduced the concepts of -intuitionistic fuzzy h-ideals of hemiring [22]. Various attributes of -algebra are considered in [23–31].
Based on fuzzy points, Jana et al. studied different types of ideals [32, 33]. Moreover, the same authors studied generalized intuitionistic fuzzy ideals of -algebras and Lukaswize intuitionistic fuzzy -subalgebras based on 3-valued logic (see [34, 35]).
This motivated us to study intuitionistic fuzzy soft ideals of -algebra using cut sets and the degree of existence of fuzzy soft point in an intuitionistic fuzzy soft set of . Also, an -intuitionistic fuzzy soft ideal of is introduced by applying the Lukaswize triple-valued logic, where , with . Moreover, intuitionistic fuzzy soft ideals of -algebras with thresholds are investigated, and related results are obtained.
2. Preliminaries
The algebraic structures of - and -algebras were introduced by K. Iseki.
An algebra of type with 0 as the identity element is called a -algebra if for every , the following conditions are satisfied:
The partial ordering is defined as .
If -algebra X satisfies for every , then is a -algebra.
A nonempty subset of is called an ideal of if it satisfies the following: for every , , and
Unless or otherwise mentioned, denotes a -algebra.
Definition 1. (see [12]). For an initial set and a set of parameters , a pair is said to be a soft set over which is a mapping of into the set of all subsets of the set .
Definition 2. (see [14]). Let be a collection of parameters, and let denote the collection of all fuzzy sets in . Then, is called a fuzzy soft set over , where is a subset of and is a mapping given by .
It is easy to see that every classical soft set may be considered as a fuzzy soft set. In general, for every , is a fuzzy subset of , and it is called a fuzzy value set. If for every , is a crisp subset of , then is generated as a standard soft set. Let denote the degree of existence function; then, can be written as a fuzzy set such that .
Definition 3. (see [15]). Let be a collection of parameters, and let denote the set of all intuitionistic fuzzy sets in . Then, is called an intuitionistic fuzzy soft set (IFSS) over , where is a subset of and is a mapping given by .
In general, for every , is an intuitionistic fuzzy subset of , and it is called an intuitionistic fuzzy value set. Clearly, can be written as an intuitionistic fuzzy set (IFS) such that , where and represent the degree of existence and nonexistence functions, respectively. If for every , , then will be generated as a standard fuzzy set, and then will be generated as a traditional fuzzy soft set.
3. -Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras
Definition 4. An IFSS in is called an intuitionistic fuzzy soft ideal (IFSID) of if satisfies the following:(1) and ,(2) and ,for each and .
Definition 5. Let be an IFSS of and .(1)The sets are called the a-upper cut and a-stronger upper cut of the IFSS , respectively.(2)The sets are called the a-lower cut and a-stronger lower cut of the IFSS , respectively.(3)The sets are called the a-upper -cut and a-stronger upper -cut of the IFSS , respectively.(4)The setsare called the a-lower -cut and a-stronger lower -cut of the IFSS , respectively.
Definition 6. (1)The degree of existence of is , and the degree of existence of is if it satisfies the following relations: and (2)The degree of existence of and is , and the degree of existence of or is if it satisfies the following relations: and (3)The degree of nonexistence of is , and the degree of nonexistence of is if it satisfies the following relations: and (4)The degree of nonexistence of and is , and the degree of nonexistence of or is if it satisfies the following relations: and Let denote the implication of Lukaswize triple-valued logic. Lukaswize truth table is presented in Table 1.
Let . Then, for , is a fuzzy soft point, and .
Definition 7. Let be an IFSS in . If for every and such that satisfies for every and ,(1),(2),then is called an -intuitionistic fuzzy soft ideal of .
Let be a set. Then, define a triple-valued fuzzy set mapping .
Definition 8. Let be an IFSS in . If for every and , the IFS satisfies for every ,(1),(2),then is an -intuitionistic fuzzy soft ideal of .
Example 1. Consider a -algebra with Table 2.
Let us consider the following IFS of :Then, it can be easily shown that is an IFID of . Hence, is an IFSID of .
Also, if we consider the IFS of ,then it can be easily proved that is an -IFID of . Hence, is an IFSID of .
Theorem 1. Let be an -intuitionistic fuzzy soft ideal of X. If , then is a fuzzy soft ideal of .
Proof. We shall prove that(1)(2) for every (1)We have to show . If , then . Let and for all and for all . Then, there exists such that . Thus, and . If or , then , and . Now, from Definition 8, . Therefore, either or either or If , then , and . Now, from Definition 8, . Therefore, either or either or Next, we have to show that . Suppose . Then, . Then, there exist such that . Then, , and so, . Thus, and . If or , then , and . Now, from Definition 8, . Therefore, either or either or If , then , and so, . Now, from Definition 8, . Therefore, either or either or Hence, for all and .(2)First, we show that . If , then and , and so, and . Let . Then, there exists such that . Thus, , , and .If or , then , and . Now, from Definition 8, . Therefore, either or either or If , then , and . Now, from Definition 8, . Therefore, either or either or Next, we show that . Suppose . Then, and and . Let such that . Then, and . Thus, and . Again, such that and . Thus, and .
If or , then , and . Now, from Definition 8, . Therefore, or or If , then , and . Now, from Definition 8, or or Hence, for every and . Therefore, is a fuzzy ideal of . Thus, is a fuzzy soft ideal of .
4. Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras with Thresholds
Definition 9. Let be an IFSS of X. Then, is an intuitionistic fuzzy soft ideal (IFSID) of with thresholds is an intuitionistic fuzzy ideals (IFIDs) of which satisfies the following for every and :(1) and (2) and
Example 2. Let us consider a -algebra with the Cayley table given in Table 3.
Let us suppose , , and , where , , and . Henceforth, is an IFID of with thresholds . Consequently, is an IFSID of with thresholds .
Theorem 2. An IFSS of is an -IFSID of is an IFSID of with thresholds .
Proof. Suppose is an -IFSID of . We have to show that is an IFSID of with thresholds . It is enough to show that(1) and (2) and for every and (1)Let , and let . Then, , and so, . Therefore, from Definition 8, or or Thus, for all . Now, let ; then, . Therefore, from Definition 8, or or Since , . Thus, .(2)Let , and let . Then, and , and so, and . Therefore, from Definition 8, or or or Thus, for all , and let ; then, and and and . Therefore, from Definition 8, or or Since , . Hence, .
Let be an IFSID of with thresholds . Let , and for and , we have to prove .(1)Let . Case 1: when , . If , then and . Thus, ; so, . This is a contradiction to . Hence, . Case 2: for , we have . If , then and and and . Thus, ; so, , which is a contradiction to . Therefore, . Hence, .(2)Let , and let . Case 1: at and and , if , then and . Thus,, and so, . This is a contradiction to . Thus, . Case 2: when , and and . Now, .If , then and . Now, which is a contradiction to . Thus, . Hence, . Therefore, is an IFID of . Hence, is an -IFSID of .
Theorem 3. An IFSS is an -IFSID of is an IFSID of with thresholds .
Proof. Suppose of is an IFSID of . We prove that is an IFSID of with thresholds . It is enough to show that(1) and (2) and for all and (1)Let and . If , then , which indicates that . Now, . This is a contradiction to . Hence, . Let ; then, . If and , then . This indicates . Thus, and . Now, . Therefore, . This contradicts the assumption . Hence, .(2)Let and . Now, if , we have, and and and . Therefore, . This contradicts the assumption . Hence, .Let ; then, and , but if , and , and then . Thus, and and and . Therefore, , which indicate that . This contradicts the assumption . Hence, .
Suppose is an IFSID of with thresholds . Let and . Case 1: let . Then, . Thus, . Case 2: let . Then, and . Thus, . Therefore, . Thus, . Hence, .Next, we prove that . Let . Case 1: let . Then, and and . Thus, ; we get . Case 2: let . Then, and and and . Thus, . Therefore, . Thus, . Hence, . Therefore, is an IFID of . Hence, is an IFSID of .
Theorem 4. An IFSS of is an -IFSID of is a FSID of for any .
Proof. Suppose is an -IFSID of . Let , and for , and and . Therefore, is a FID of . Hence, is a FSID of .
Assume for any , is a FSID of . Let , and for , we have to show that(1)(2)(1)Let . Case 1: for , and ; we get . Thus, . Case 2: for , . So, , and we get . Thus, . Therefore, . Next, to prove(2)Let . Case 1: at , and ; then, . Case 2: at , and . Therefore, , and we get . Thus, . Hence, . Therefore, is an -IFID of . Hence, is an -IFSID of .
Theorem 5. An IFSS of is an -IFSID of for any is a FSID of .
Proof. Suppose is an IFSID of . Then, for every and for , we have to show that(1)(2)As . Then, and . Therefore, and . Thus, and . Therefore, for any is a FID of . Hence, is a FSID of .
Let be a FSID of , for any . Let and .(1)If , then let . Case 1: for , . Hence, . Case 2: let . Then, . Thus, , from which . Hence, . If , then let be such that . Then, and . Therefore, . Hence, .(2). If , then let . Case 1: for , and , and we have . Thus, . Case 2: at , and ; so, . Therefore, . Hence, . If , then let be such that . Now, and . Thus, . Thus, . Therefore, is an -IFID of . Hence, is an -IFSID of .
Theorem 6. An IFSS of X is an -IFSID of for any is a FSID of .
Proof. Suppose is an -IFSID of . For every and for , . Then, and . Therefore, and . Thus, is a FID of . Hence, is a FSID of .
We claim that, for any is a FSID of . Let and for . Let . Case 1: let . Then, and . Let . For , we have and . Then, . Hence, . Case 2: let . Then, and . Thus, and . Let . Then, . Therefore, . Hence, .Next, we prove . Let . Case 1: let . Then, and . Therefore, . Let ; then, , and we get and and . Thus, , and so, . Hence, . Case 2: let . Then, and and . Then, and . Let . Then, . Therefore, as and . Thus, . Hence, . Thus, is an -IFID of . Hence, is an -IFSID of .
Theorem 7. An IFSS of is an -IFSID of for any and :(1) or (2) or (3) or (4) or
Proof. We prove only (2) and (4), and (1) and (3) can be similarly proved.(2)If , then . Then, and . Thus, or . Hence, .(4)If , then , and as and . Thus, or . Hence, . For every and , let . Case 1: let . Then, and . Suppose ; then, and . Then, and . This is a contradiction to (2). Hence, . Case 2: let . Then, . Thus, . If , then . Thus, , and so, and which is a contradiction to (4). Hence, . Therefore, . This shows that is an -IFID of . Hence, is an -IFSID of .
Theorem 8. An IFSS of is an IFSID of with thresholds for every , is a FSID of .
Proof. Suppose is an IFSID with thresholds of . Let and . Let . Case 1: let . Then, . Thus, . Case 2: let . Then, . Thus, . As , now, . Therefore, . Hence, .Next, we have to prove . Let . Case 1: let . Then, and and . Now, . Thus, . Case 2: let . Then, and or . Thus, , from which we get . Since and , therefore, , and so, . Thus, . Therefore, is a FID of . Hence, is a FSID of . We assume for every is a FSID of . We show .
If , then and . Thus, , and we have , and so, . This is a contradiction to . Hence, .
We have to show . If , then , and so, and . Thus, and . Therefore, , which is a contradiction to . Hence, .
Next, we have to show . If , then and . Thus, , and so, . This is a contradiction to . Hence, .
We have to show . If , then , and so, and and . Thus, , and so, . Therefore, , which is a contradiction to . Hence, . Therefore, is an IFID of with thresholds . Hence, is an IFSID of with thresholds .
5. Conclusion
The main goal of the present paper is to introduce the notion of -intuitionistic fuzzy soft ideal of -algebras, where are the membership values between an intuitionistic fuzzy soft point and intuitionistic fuzzy set. Moreover, intuitionistic fuzzy soft ideals with thresholds are introduced, and their related properties are investigated.
We hope that this work will give a deep impact on the upcoming research in this field and other soft algebraic studies to open up new horizons of interest and innovations. In future directions, these definitions and main results can be similarly extended to some other algebraic systems such as subtraction algebras, B-algebras, MV-algebras, d-algebras, and Q-algebras.
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the manuscript.