Abstract
In this paper, the notion of hybrid structure is applied to the ideal theory in BCI-algebras. In fact, we introduce the notions of hybrid -ideal, hybrid h-ideal, and hybrid a-ideal in BCI-algebras and investigate their related properties. Furthermore, we show that every hybrid -ideal (or h-ideal or a-ideal) is a hybrid ideal in a BCI-algebra but converse need not be true in general and in support, and we exhibit counter examples for each case. Moreover, we consider characterizations of hybrid -ideal, hybrid h-ideal, and hybrid a-ideal in BCI-algebras.
1. Introduction
Imai and Iséki [1, 2] initiated the study of “BCK/BCI-algebras” in 1966 as a generalization of the notions of set-theoretical difference and propositional calculus. A great deal of literature has been developed on the theory of -algebras since then, in particular, more focus has been placed on the “ideal theory” of BCK/BCI-algebras. In -algebras, different kinds of ideals on different aspects have been studied (see, for example, [3–6]).
Fuzzy sets, introduced by Zadeh [7], deal with potential uncertainties, related to the imprecision of states, perceptions, and preferences. Molodtsov [8] proposed the concept of a “soft set” as a new mathematical framework for dealing with uncertainties, free of the difficulties that have disrupted normal theoretical approaches. Molodtsov pointed out a number of ways to develop soft sets. Molodtsov applied soft set theory in a variety of ways, such as smoothness of functions, game theory, operational research, Riemann integration, Perron integration, probability, and measurement theory (see [8–10]). Algebraic structures such as -algebra [11], -algebras [12], group [13], semigroup [14], ring [15], semiring [16], and decision-making [17, 18] are theoretically applied by soft set theory. Muhiuddin et al. (see, for example, [19–23]) investigated the fuzzy set theoretical approach to the -algebras on various aspects. Further concepts related to this analysis in different fields have also been studied in [24–34].
In a system of parameters, Jun et al. [35] proposed the concept of “hybrid structure” over the initial universe set by merging fuzzy sets and soft sets. The idea of a hybrid subalgebra, a hybrid field, and a hybrid linear space was introduced with this notion. The hybrid structure theory and its applications to -algebras and semigroups have recently been studied in (see [36–40] and references).
The objective of this paper is to introduce the notions of hybrid -ideal, hybrid h-ideal, and hybrid a-ideal in -algebras and investigate their related properties. Furthermore, we show that hybrid (-ideal, h-ideal, and a-ideal) are all hybrid ideals, but converse need not be true in general and in support, and we exhibit counter examples. Also, we provide conditions for a hybrid -ideal (or hybrid h-ideal or hybrid a-ideal) to be a hybrid ideal.
2. Preliminaries
An algebra of type is a -algebra if it satisfies for all :
If a -algebra satisfiesthen is a .
Any -algebra satisfies the following conditions:where if and only if . Note that is a partially ordered set.
Any -algebra satisfies the following conditions [41, 42]:
For more details on -algebras and -algebras, we refer the readers to [43–45].
A subset of a -algebra is called a subalgebra if and is called an of if and implies . Furthermore, a subset of -algebra is called -ideal (resp. -ideal and -ideal) if and implies (resp. implies , and implies ).
Definition 1 (see [35]). For a set of parameters , an initial universe set , a power set of the initial set , and the unit interval , a hybrid structure (briefly, HS) in over is defined to be a mapping , where and are mappings.
Definition 2 (see [35]). Let be a -algebra. For a HS in over , is said to be a hybrid subalgebra of if the following statements are valid:
Definition 3 (see [39]). A HS in over is said to be a hybrid ideal of over if
Proposition 1 (see [39]). Let be a hybrid ideal of . If the inequality holds in , then .
3. Hybrid Ideals in BCI-Algebras
Throughout the following sections denotes -algebra unless stated otherwise.
Definition 4. A HS in over is said to be a hybrid -ideal of over if it satisfies and .
Example 1. Let be the initial universe set. On a set of parameters , we define the operation in Table 1.
Then, is a -algebra. Let be a HS in over which is given by Table 2.
By routine verification is a hybrid -ideal of over .
Theorem 1. For any -algebra, every hybrid -ideal is a hybrid ideal.
Proof. Suppose that is a hybrid -ideal of over . Since for all , we havefor all . Hence, is a hybrid ideal of over .
As shown in the following example, the converse of Theorem 1 is not generally valid.
Example 2. Let be the initial universe set and be the set of parameters. We define the binary operation on by Table 3.
Then, is a -algebra. Now, define a HS in over which is given by Table 4.
It is easy to check that is a hybrid ideal of over but not hybrid -ideal as
Theorem 2. If is a hybrid -ideal of over , thenfor all .
Proof. Let be a hybrid -ideal of over . Then,for all . Substituting for and 0 for in (10), then
Theorem 3. If is a hybrid p-ideal of over , thenfor all .
Proof. By , . Therefore, by Proposition 1,Thus, by using (9), we have
Theorem 4. If is a hybrid -ideal of over , thenfor all .
Proof. Let be a hybrid -ideal of over . Note that holds in . It implies that . Since is a hybrid ideal of over , we havefor all .
The following theorem gives a condition for a hybrid ideal to be a hybrid -ideal.
Theorem 5. Let be a hybrid ideal of over satisfying the following condition:for all . Then, is a hybrid -ideal of over .
Proof. Suppose that is a hybrid ideal of satisfying (17). Then,for all , as required.
Definition 5. A HS in over is said to be a hybrid h-ideal of over if it satisfies and .
Example 3. Let be the initial universe set. As a set of parameters, we consider . The operation is defined on by Table 5.
Then, is a -algebra. Define a HS in over by the following Table 6.
It can be easily checked that is a hybrid h-ideal of over .
Theorem 6. For any -algebra, every hybrid h-ideal is a hybrid ideal.
Proof. Proof. Let be a hybrid h-ideal of over . Since for all , we havefor all . Hence, is a hybrid ideal of over .
In general, the converse of Theorem 6 is not valid, as the following example shows.
Example 4. Let be the initial universe set. For a set of parameters , we define a binary operation by the Cayley table in Table 7.
Then, is a -algebra. Define a HS in over by Table 8.
It is routine to verify that is a hybrid ideal but not hybrid h-ideal since
Theorem 7. Let be a hybrid h-ideal of over . Then,for all .
Proof. Let be a hybrid h-ideal of over . Then,Substituting by 0, by , and by , we have
Definition 6. A HS in over is said to be a hybrid a-ideal of over if it satisfies and .
Example 5. Consider represented in Example 1. It is routine to verify that is a hybrid a-ideal of over .
Theorem 8. Every hybrid a-ideal is both a hybrid subalgebra and a hybrid ideal of over .
Proof. Let be a hybrid a-ideal of over . By substituting in and using and , we obtainfor all .
Again setting in , using , and above equations, we obtainwhich implies which implies for all . Then, from and , it implies thatfor all . Hence, is a hybrid ideal of over .
Now, for any and using above equations, we haveTherefore, is a hybrid subalgebra of over . The converse of Theorem 8 is not generally valid as seen in the following example.
Example 6. Let be the initial universe set. For a set of parameters , we define a binary operation by the Cayley table in Table 9.
Then, is a -algebra. Let be a HS in over which is given by Table 10.
Then, it directs to show that is a hybrid subalgebra and hybrid ideal but not hybrid a-ideal as
Theorem 9. If is a hybrid a-ideal of over , then and for all .
Proof. Suppose that is a hybrid a-ideal of over . Since , .
By Theorem 8, is a hybrid ideal of over . Thus,for every .
Theorem 10. Let be a hybrid ideal of over satisfying the following inequality:for all , then is a hybrid a-ideal of over .
Proof. Since is a hybrid ideal of over ,for all . Hence, is a hybrid a-ideal of over .
4. Conclusion
In this paper, we apply the notion of hybrid structure to the ideal theory in -algebras. In fact, we have discussed the notions of hybrid -ideal, hybrid h-ideal, and a-ideal in -algebras and investigated several related properties. Furthermore, it has been shown that every hybrid -ideal (or h-ideal or a-ideal) is a hybrid ideal in a -algebra but converse need not be true in general, and in support, examples for each case are provided. Moreover, characterizations of hybrid -ideal, hybrid h-ideal, and hybrid a-ideal in -algebras are discussed.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.