Abstract
This paper aims to make a combination between the quantum B-algebras (briefly, -s) and two interesting theories (e.g., soft set theory and fuzzy soft set theory). Firstly, we propose the novel notions of soft quantum B-algebras (briefly, -s), a soft deductive system of -s, and deducible soft quantum B-algebras (briefly, -s). Then, we discuss the relationship between -s and -s. Furthermore, we investigate the union and intersection operations of -s. Secondly, we introduce the notions of a fuzzy soft quantum B-algebras (briefly, -s), a fuzzy soft deductive system of -s, and present some characterizations of -s, along with several examples. Finally, we explain the basic properties of homomorphism image of -s.
1. Introduction
In 1999, Molodtsov [1] introduced the notion called soft sets (briefly, ) (i.e., which reduce the uncertainty and vagueness of knowledge). Maji et al. [2] presented the fuzzy soft sets (briefly, ). Since then, many researchers studied further on and as in the following published articles (e.g., [3–9]).
In 2014, Rump and Yang [10] proposed the notion of -s (i.e., a partial ordered implication algebras). Rump [11, 12] investigated many implication algebras (for example, pseudo-BCK-algebras, po-groups, BL-algebras, MV-algebras, GPE-algebras, and resituated lattices). Botur and Paseka [13] studied filters on integral -s, and Zhang et al. [14] established the quotient structures by using q-filters in -s and investigated the relation between basic implication algebras and -s. Han et al. [15] constructed the unitality of -s and explained the injective hulls of -s in [16]. By the framework of -s, there are many published papers on -s (e.g., [17–23]).
Regarding these developments, as the motivation of this paper, we will combine -s with and (i.e., enrich the previous work on hybrid soft set and fuzzy soft set theories algebras with quantum structures). We introduce the notions of -s and the soft deductive system of -s and consider the relation between -s and -s. Furthermore, some conditions are given to ensure the operations union and intersection holds of soft deductive of -s. Then, we investigate the homomorphism image of deductive -s. Lastly, we define -s and fuzzy soft deductive system of -s and give an example to illustrate its derive properties.
In the following, we have arranged the sections as follows. In Section 2, we briefly recall many notions related to -s, , and as indicated in Definitions 1–7, which are used in the sequel. In Section 3, we propose the notions of -s, soft deductive system of -s, and -s. In Section 4, we present the notions of -s and a fuzzy soft deductive system of -s and discuss the homomorphism image of -s. The conclusions are explained in Section 5.
2. Preliminaries
We give some basic notions of -s, , and before defining -s in Section 3.
Definition 1 (cf. [10]). (1)-s is a partially ordered set with two binary operations and which satisfy ():(2)- is a commutative (briefly, -) if .(3)A subset of a - is a subalgebra if .In what follows, denote by a - unless otherwise specified.
Definition 2 (cf. [10]). Let and be two -s. Then, is a morphism of -s if it satisfies ():We say morphism is exact if the inequalities become equations.
Definition 3 (cf. [1]). Assume that be a set and be a set of parameters. (called ) is a mapping given by (i.e., is the power set of ).
Definition 4 (cf. [3]). Assume that and are two over . is a subset of (denoted by ) if(1)(2)For every and are identical approximations
Definition 5 (cf. [3]). Assume that , and are three over . is the intersection of and (denoted by ) if(1)(2) or (as both are same sets)
Definition 6 (cf. [3]). Assume that , and are three over . is called the union of and (denoted by ) if(1).(2),
Definition 7 (cf. [2]). (called ) is a mapping given by (i.e., is the set of all fuzzy sets [24] of ).
3. -s
We define the -s and give several examples based on -s. Also, we will study the union and intersection operations between two -s as follows .
Definition 8. is a -s over if are subalgebras of (i.e., in case ).
Example 1. (1)Suppose (i.e., ) with the order . Now, we show, by Table 1, the binary operation . Clearly, is a -. We define (i.e., ) by From Table 1, we can get on , and then, are all subalgebras of . Consequently, is a -s over .(2)Suppose (i.e., ) with the order . Now, we show, by Table 2, the binary operation .Clearly, is a -. We define (i.e., ) byFrom Table 2, we can get on , and then, are all subalgebras of . Consequently, is a -s over .
We ensure the operations (i.e., union and intersection) are holding on -s by the following suggested theorem.
Theorem 1. Assume that and are -s over . Then,(1)If , then is called a - over (2)If , then is called a - over
Proof. (1)If and by Definition 5, we obtain or , for all . Since and are -s over , which implies that is a -s over , that is, or are both subalgebras of , therefore, is a - over .(2)If and by Definition 6, we obtainFor and since is a -, then we have is a subalgebra of . Similarly, for , then is a subalgebra of due to is a -. Again, for , so or , for all . Thus, is a - over .
Remark 1. If , then Theorem 1 (2) does not hold by the following example.
Example 2. Suppose (i.e., ). Now, we show, by Tables 3 and 4, the binary operations and , respectively.
Clearly, is a -. Then,(i)We define (i.e., ) by From Table 3, we can get , and then, are all subalgebras of . Consequently, is a -s over .(ii)We define (i.e., ) by From Table 4, we can get is the subalgebra of . Consequently, is a -s over .From (i) and (ii) and , then we have is not a subalgebra over . Thus, is not a -.
3.1. Soft Deductive Systems of -s
Based on Definition 8, we will propose the notion of soft deductive systems of -s as indicated below.
Definition 9. Assume that be a -. A nonempty subset is a deductive system of if it satisfies(1)(2)
Definition 10. Let be a - and a subalgebra of . A subset of is a deductive system of related to (i.e., -deductive system of ), denoted by ⋈, and satisfies the following two conditions:(1)(2)
Remark 2. According to Definitions 9 and 10, we obtain that any deductive system of is -deductive system if is a subalgebra of .
The converse of Remark 2 does not hold by Example 3 (i.e., is a subalgebra of and -deductive system is not a deductive system).
Example 3. Suppose (i.e., ) with partial order and . Now, we show, by Tables 5 and 6, the binary operations and , respectively.
Clearly, is a -. Consider a subalgebra and a subset ; we can see that ⋈. However, is not a deductive system of since and .
Definition 11. Assume that is a - over . (i.e., ) over is a soft deductive system of , denoted by , and satisfies the following two conditions:(1)(2)Now, we will give an example to illustrate Definition 11 as follows.
Example 4. Suppose (i.e., ) with partial order . Now, we show, by Tables 7 and 8, the binary operations and , respectively.
Clearly, is a -. We define (i.e., ) byFrom Tables 7 and 8, we can get on , and then, are all subalgebras of . Consequently, is a -s over .
Next, for a subset , we define byThen, we obtain and . Consequently, is a soft deductive system of .
Theorem 2. Assume that is a - over and and are two . Then,(1)If , then (2)If , then
Proof. (1)Follow from Definition 5.(2)If , then, by Definition 6, we have (i.e., ), , andSince , we obtain either or . Then, we have the following: Case 1: if , since , then Case 2: if and , then Consequently, for all , we have , which implies that .
Remark 3. If , then Theorem 2 (2) does not hold by the following example.
Example 5. Suppose (i.e., ). Now, we show, by Table 9, the binary operations .
Clearly, is a -. Then,(i)We define (i.e., ) by From Table 9, we can get , and then, are all subalgebras of . Consequently, is a -s over .(ii)We define (i.e., ) by Then, we can get . Therefore, is a soft deductive system over .(iii)We define (i.e., ) by Then, we can get . Therefore, is a soft deductive system over .From (i)–(iii), we have which is not a soft deductive system of , where is not a -deductive system because and .
3.2. -s
We will give the notion of -s and investigate homomorphism image of -s as indicated below.
Definition 12. Assume that is a - over . If is a deductive system of , then is called a - over .
Example 6 (continued from Example 1 (2)). Clearly, is - over .
Definition 13. (1)Suppose be a - with the greatest element 1 (i.e., just only a poset); for any , the order of element is defined as where is a natural number and .(2)If does not exist to satisfy the above condition (i), then is called infinite order.
Remark 4. Assume that and be two -s over such that . If is a - over , then is a -.
The converse of Remark 4 does not hold by the following Example 7.
Example 7 (continued from Example 2). We define (i.e., ) byThen, we get on . However, and imply that is not -. If we take and we define , then is -.
Definition 14. Assume that is - over with the greatest element 1. If , then is called whole -.
Example 8. Suppose (i.e., ) with partial order . Now, we show, by Tables 10 and 11, the binary operations and , respectively.
Clearly, is a -. We define (i.e., ) byFrom Tables 10 and 11, we can get on . Thus, is a whole - over .
Now, we will study homomorphism image of -s by the following two theorems.
Theorem 3. Assume that be a surjective exact morphism of - and is a -s. If is a - over , then is also- over .
Proof. Since is a deductive system of and is surjective, then is a deductive system of which implies that is a - over .
Theorem 4. Assume that be a surjective exact morphism of - and a - over . Then,(1)If , for all , then is the whole - over (2)If is whole - over , then is the whole - over
Proof. (1)Assume that , where . Since is surjective, then, from Theorem 3, we have . Thus, is the whole - over .(2)Clearly, since is whole - over . Thus, . By Theorem 3, we have is the whole - over
4. -s
We give the definition of -s; a concrete example is given to illustrate its derive properties. Furthermore, we study the homomorphism image and preimage of -s. Now, we first propose the definition of fuzzy quantum B-algebra (briefly, -) as indicated below.
Definition 15. We call - (or a fuzzy set in -) if it satisfies ( is -):
Definition 16. We call is a fuzzy deductive system of if it satisfies :
Definition 17. Assume that be a over . Then,(1)If there exists such that is a - (i.e., fuzzy deductive system) in a - over , then is called a - (i.e., fuzzy soft deductive system ) which depends on a parameter set over (2)If is a - (i.e., fuzzy deductive system) of based on all parameters, then we say that is a - (i.e., ) of In the following, a concrete example is given to illustrate Definition 17.
Example 9. Suppose that there are five-class cars:Let and be two soft machines to characterize two cars, defined by the following manner.
Then, is a -. Now, we consider a set of parameters: (Excellent, Good, Moderate). Then, we have the following:(1)We define over (i.e., [Excellent], [Good], and [Moderate] are fuzzy sets) by Table 12. Therefore, we can see that [Excellent], [Good], and [Moderate] are all -s based on parameters “Excellent,” “Good,” and “Moderate” over . Thus, is a - over .(2)We define over (i.e., [Excellent], [Good], and [Moderate] are fuzzy sets) by Table 13. However, is not a - based on a parameter “Excellent” over , where . Also, we obtain that is a - based on both the parameter “Good” and “Moderate” over .(3)We define over (i.e., and [Good] are fuzzy sets) by Table 14. Then, is a on parameters “Excellent.” However, is not a fuzzy deductive system of based on parameter “Good,” where [Good]() .(4)We define over (i.e., and are fuzzy sets) by Table 15. Then, is a of .Now, we will present several characterizations of -s.
By Definition 17, if is a - of - over based on all parameters, then we say that is a - of , that is,
Proposition 1. Assume be a -. If is - over , then, for all is the subalgebra of , in which
Proof. Let . Then, ; since is a -, then . So,Similarly, we have . Therefore, . This implies that is the subalgebra of .
Analogously, we can get Proposition 2 as follows.
Proposition 2. Assume that and are two - over . Then, and are -s over .
Definition 18. Let be a fuzzy soft map from - over to - over . Then,(1)If is an exact morphism from to , then is called a - exact morphism from to (2)If is an isomorphism from to and is a bijective from to , then is a called an isomorphism between -s
Proposition 3. Let and be two -s. is a - over and a - exact morphism from to ; then, is - over .
Proof. For ,Consequently, is a - over .
Similarly, we can get Proposition 4 as follows.
Proposition 4. Let and be two -s. is a -s over and a -s isomorphism from to ; then, is the -s over .
5. Conclusions
In this paper, we introduce the concept of -s, and some examples are given to illustrate this definition. Also, we investigate the union and intersection operations between two -s and give some conditions for the operation holds. With the help of the definition of -s, we define soft deductive systems of -s and then investigate the relation between them. As a further step, we define -s and investigate the homomorphism image of -s. Moreover, we define -s. Finally, a concrete example is given to illustrate its derive properties; besides, homomorphism image and preimage of -s are discussed.
As a future work, it makes sense to apply -s to medical diagnosis (for example, [25, 26]) in practice. Furthermore, it would be interesting if we study hybrid soft lattice-ordered quantum B-algebras.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Sultan Aljahdali acknowledges Taif University Researchers Supporting Project (no. TURSP2020/73), Taif University, Taif, Saudi Arabia. This work was supported by the science and technology project of Yulin City (CXY-2020-007).