(𝑀,𝑁)-Soft Intersection BL-Algebras and Their Congruences

Abstract

The purpose of this paper is to give a foundation for providing a new soft algebraic tool in considering many problems containing uncertainties. In order to provide these new soft algebraic structures, we discuss a new soft set-(M, N)-soft intersection set, which is a generalization of soft intersection sets. We introduce the concepts of (M, N)-SI filters of BL-algebras and establish some characterizations. Especially, (M, N)-soft congruences in BL-algebras are concerned.

1. Introduction

It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. In making inference levels, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. BL-algebra has been introduced by Hájek as the algebraic structures for his Basic Logic [1]. A well-known example of a BL-algebra is the interval endowed with the structure induced by a continuous -norm. In fact, the -algebras, Gödel algebras, and product algebras are the most known classes of BL-algebras. BL-algebras are further discussed by many researchers; see [2–12].

We note that the complexities of modeling uncertain data in economics, engineering, environmental science, sociology, information sciences, and many other fields cannot be successfully dealt with by classical methods. Based on this reason, Molodtsov [13] proposed a completely new approach for modeling vagueness and uncertainty, which is called soft set theory. We note that soft set theory emphasizes a balanced coverage of both theory and practice. Nowadays, it has promoted a breath of the discipline of information sciences, intelligent systems, expert and decision support systems, knowledge systems and decision making, and so on. For example, see [14–24]. In particular, Çaman et al., Sezgin et al., and Jun et al. applied soft intersection theory to groups [25], near-rings [26], and BL-algebras [27], respectively.

In this paper, we organize the recent paper as follows. In Section 2, we recall some concepts and results of BL-algebras and soft sets. In Section 3, we investigate some characterizations of -SI filters of BL-algebras. In particular, some important properties of -soft congruences of BL-algebras are discussed in Section 4.

2. Preliminaries

Recall that an algebra is a BL-algebra [1] if it is a bounded lattice such that the following conditions are satisfied:(i) is a commutative monoid;(ii) and form an adjoin pair; that is, if and only if for all ;(iii);(iv). In what follows, is a BL-algebra unless otherwise specified. In any BL-algebra , the following statements are true (see [1, 5, 6]):;;;;;;;;,where .

A nonempty subset of is called a filter of if it satisfies the following conditions:(I1),(I2), , . It is easy to check that a nonempty subset of is a filter of if and only if it satisfies(I3), ,(I4), , (see [6]).

From now on, we let be a BL-algebra, an initial universe, a set of parameters, and the power set of and .

Definition 1 (see [13, 16]). A soft set over is a set defined by such that if . Here is also called an approximate function. A soft set over can be represented by the set of ordered pairs . It is clear to see that a soft set is a parameterized family of subsets of . Note that the set of all soft sets over will be denoted by .

Definition 2 (see [16]). Let .(1) is said to be a soft subset of and denoted by if , for all . and are said to be soft equal, denoted by , if and .(2)The union of and , denoted by , is defined as , where , for all .(3)The intersection of and , denoted by , is defined as , where , for all .

Definition 3 (see [27]). A soft set over is called an - filter of over if it satisfies  for any ,  for all .

3. -SI Filters

In this section, we introduce the concept of - filters in BL-algebras and investigate some characterizations. From now on, we let .

Definition 4. A soft set over is called an -soft intersection filter (briefly, - filter) of over if it satisfies() for all ,() for all .

Remark 5. If is an - filter of over , then is an - filter of over . Hence every -filter of is an - filter of , but the converse need not be true in general. See the following example.

Example 6. Assume that , the symmetric 3-group is the universal set, and let , where . We define and and as follows: It is clear that is a BL-algebra. Let and . Define a soft set over by and . Then we can easily check that is an - filter of over , but it is not -filter of over since .

The following proposition is obvious.

Proposition 7. If a soft set over is an - filter of over , then

Define an ordered relation “” on as follows: for any , we define . And we define a relation “” as follows: and . Using this notion we state Definition 4 as follows.

Definition 8. A soft set over is called an -soft intersection filter (briefly, - filter) of over if it satisfies() for all ,() for all .

Proposition 9. If is an - filter of over , then is a filter of .

Proof. Assume that is an - filter of over . Then it is clear that . For any , . By Proposition 7, we have . Since is an - filter of over , we have Hence, , which implies . This shows that is a filter of .

Proposition 10. If a soft set over is an - filter of , then for any ,(1),(2),(3),(4),(5),(6),(7),(8).

Proof. (1) Let be such that . Then , and hence which implies .
(2) Let be such that . Then, that is, .
(3) By , we have for all . By (1), . Since , we obtain . It follows from that . Hence, .
Since and , we have and . Hence we have
, which implies . Thus .
(4) It is a consequence of (3), since .
(5) By .
(6) By .
(7) By .
(8) By .

By Definition 4 and Proposition 10, we can deduce the following result.

Proposition 11. A soft set over is an - filter of over if and only if it satisfies

Proposition 12. A soft set over is an - filter of over if and only if it satisfies , .

Proof. () By Proposition 10(1) and (3).
() Let . Since , by , we have . Hence holds. Since , by and , we have ; that is, holds. Therefore, is an - filter of over .

4. -Soft Congruences

In this section, we investigate -soft congruences, -soft congruences classes, and quotient soft BL-algebras.

Definition 13. A soft relation from to is called an -congruence in over if it satisfies , , , , .

Definition 14. Let be an -congruence in BL-algebra over and . Define in as . The set is called an -congruence class of by in . The set is called a quotient soft set by .

Lemma 15. If is an -congruence in over , then .

Proof. By and , we have .

Lemma 16. If is an -congruence in over , then is an - filter of over .

Proof. For any , we have This proves that holds.
For any , by and , we obtain It follows that that is, . This proves that holds. Thus, is an - filter of over .

Lemma 17. Let be an -SI filter of over . Then is an -soft congruence in .

Proof. For any , we have the following.  Consider This proves that holds.  It is clear that holds.  By Proposition 10(5), we have Thus holds.
Since and , we have Thus, we have which implies This implies that holds.
Finally, we prove condition : Thus, holds. Therefore is an -soft congruence in .

Let be an - filter of over and . In the following, let denote the -congruence class of by in and let be the quotient soft set by .

Lemma 18. If is an - filter of over , then if and only if for all .

Proof. If is an - filter of over , then ; that is, for all . If , then , and hence . Thus, .
Conversely, assume the given condition holds. By Proposition 10, we have and . If , then and . Thus . Similarly, we can prove that . This implies that for all . Hence, .

We denote by .

Corollary 19. If is an - filter of over , then if and only if , where if and only if and .
Let be an - filter of over . For any , we define

Theorem 20. If is an - filter of over , then is a BL-algebra.

Proof. We claim that the above operations on are well defined. In fact, if and , by Corollary 19, we have and , and so . Thus . Similarly, we prove , and . Then it is easy to see that is a BL-algebra. Especially, we prove the divisibility in as follows. Define a lattice ordered relation “” on as follows: By Corollary 19, we have . If , then

Theorem 21. If is an - filter of over , then .

Proof. Define by for all . For any , we have Hence, is an epic. Moreover, we have which shows that .

5. Conclusions

As a generalization of soft intersection filters of BL-algebras, we introduce the concept of - (implicative) filters of BL-algebras. We investigate their characterizations. In particular, we describe -soft congruences in BL-algebras.

To extend this work, one can further investigate - prime (semiprime) filters of BL-algebras. Maybe one can apply this idea to decision making, data analysis, and knowledge based systems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.