Abstract
We introduce the notion of soft filters in residuated lattices and investigate their basic properties. We investigate relations between soft residuated lattices and soft filter residuated lattices. The restricted and extended intersection (union), and -intersection, cartesian product, and restricted and extended difference of the family of soft filters residuated lattices are established. Also, we consider the set of all soft sets over a universe set and the set of parameters with respect to , (), and we study its structure.
1. Introduction
In economics, engineering, environmental science, medical science, and social science, there are complicated problems which to solve them methods in classical mathematics may not be successfully used because of various uncertainties arising in these problems. Alternatively, mathematical theories such as probability theory, fuzzy set theory [1], rough set theory [2, 3], vague set theory [4], and the interval mathematics [5] were established by researchers to modelling uncertainties appearing in the above fields. In 1992, Molodtsov [6] introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields. At present, works on soft set theory are progressing rapidly. Some authors, for example, Maji et al. [7], discussed the application of soft set theory to a decision making problem. Chen et al. [8] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. In theoretical aspects, Maji et al. [9] and Ali et al. [10] defined and studied several operations on soft sets. The algebraic structure of the soft sets has been studied by some authors. Aktaş and Çağman [11] studied the basic concepts of soft set theory and compared soft sets to the related concepts of fuzzy sets and rough sets. Soft set relations are defined and studied in [12] and some new operations are introduced in [13]. Jun et al. [14] introduced and investigated the notion of soft -algebras. Zhan and Jun [15] studied soft BL-algebras on fuzzy sets. Also, Feng et al. [16] combined soft sets theory, fuzzy sets, and rough sets. Feng et al. [17] studied deeply the relation between soft set theory and rough set theory. Recently, Yamak et al. in [18] introduce and study the notion of soft hyperstructure.
Residuated lattices were introduced in by Krull in [19] who discussed decomposition into isolated component ideals. After him, they were investigated by Ward and Dilworth in , as the main tool in the abstract study of ideal theory in rings. Residuated lattices are the algebraic counterpart of logics without contraction rule.
An important class of residuated lattices is BL-algebras. BL-algebras constitute the algebraic structures for Hájeks basic logic [20]. MV-algebras, Gödel algebras, and product algebras are particular cases of BL-algebras. Also, there is an interesting connection between MV-algebras and BCK-algebras.
In this paper, we study the concept of soft residuated lattices. The paper is organized in four sections. In Section 2, we gather the definitions and basic properties of residuated lattices and some basic notions relevant to soft set theory will be used in the next sections, and we prove that is a bounded commutative BCK-algebra with respect to suitable operations. In Section 3, we introduce the notion of soft filters in residuated lattices and study their properties. Section 4 is a conclusion.
2. A Brief Excursion into Residuated Lattices and Soft Sets
2.1. Residuated Lattices
In the following, we recall some basic definitions and properties of residuated lattices and give some examples in this concept.
Definition 1. A residuated lattice is an algebra of type satisfying the following: () is a bounded lattice, () is a commutative monoid, () and form an adjoint pair; that is, if and only if , for all . is called a divisible residuated lattice if it satisfies the following: (div) . is called an MTL-algebra if it satisfies the following: (prel) . is called a BL-algebra if it satisfies the and conditions.
A residuated lattice is nontrivial if and only if . We denote the set of natural numbers by and define and an for . In a bounded residuated lattice the order of , , in symbols is the smallest such that an ; if no such exists, then . A BL-algebra is called locally finite if all nonunit elements in it has finite order. Also, in a bounded residuated lattice we define a negation, , by , for all . For any bounded residuated lattice we denote by . A bounded residuated lattice verifying (double negation), that is, , condition is also called a “Girard monoid”. An algebra is an MV-algebra if is a commutative monoid, . and , for all . It is well known that a BL-algebra is an MV-algebra if and only if satisfies the . Also, according to [21], a residuated lattice is an MV-algebra if and only if satisfies the additional condition . Let be an MV-algebra. We define and . One can see that is an MV-algebra, too.
Also the structure of type is called a BCK-algebra if the following axioms are satisfied for all . (BCK1) . (BCK2) . (BCK3) . (BCK4) and imply .
By a bounded BCK-algebra we mean an algebra , where is a BCK-algebra and , for each . A commutative BCK-algebra is a BCK-algebra that satisfies the identity . By [22], MV-algebras are known to be term-wise equivalent to bounded commutative BCK-algebras.(a)Let be an MV-algebra. We define and . Then, is a bounded commutative BCK-algebra in which and .(b)Let be a bounded commutative BCK-algebra. We define and . Then, is an MV-algebra in which .
In the following, we give some examples of residuated lattice.
Example 2. (i) Assume that is a commutative ring with unit and let be the collection of all ideals of . This set, ordered by inclusion, is a lattice. The meet of two ideals is their intersection and their join is the ideal generated by the union. We define multiplication of two ideals in the usual way:
Then, forms a residuated lattice with unit of the ring itself and divisions given by . It was in this setting that residuated lattices were first defined by Ward and Dilworth [23].
(ii) Define on the real unit interval the binary operations “” and “” by
Then, is a bounded residuated lattice.
(iii) Let denote the set of real numbers and denote the set of rationals. Then, the unit interval of endowed with the following operations
for all becomes an MV-algebra which is called the standard MV-algebra. Also, for each , if we set , , then and are MV-algebras where and .
Let be a residuated lattice and be a nonempty subset of . is called a filter of if it satisfies the following conditions for all :(fi1) implies ,(fi2) and imply .
Trivial examples of filters are and . We will denote by the set of filters.
Leustean in [24] introduced the notion of coannihilator of BL-algebras. Let be a filter of and . The coannihilator of relative to is the set . For any , we will denote by the coannihilator in which is the generated principle ideal of .
Proposition 3 (see [24]). Let and be filters of BL-algebra and . Then, (1) is a filter of ; (2); (3) implies ; (4) implies ; (5) if and only if ; (6); (7) and ; (8). (9); (10); (11).
For any nonempty subset of , the coannihilator of is the set . It is easy to see that and . For any subset of , is denoted by .
Proposition 4 (see [24]). Let and be two nonempty subsets of BL-algebra , and let be a nonempty family subset of and . Then, (1) is a filter of ; (2)If , then and ; (3), , ; (4), ; (5) is a prime filter if and only if is a chain and ; (6).
2.2. Soft Sets
In this subsection, we recall some basic notions relevant to soft set. Let be an initial universe set and let (simply denoted by ) be the set of parameters with respect to . Usually, parameters are attributes, characteristics, or properties of the objects in . The family of all subsets of is denoted by .
Definition 5 (see [6]). A pair is called a soft set over , when , and is a set-valued mapping.
In [25], for a soft set , the set is called the support of the soft set . The soft set is called nonnull if , and it is called a relative null soft set (with respect to the parameter set ), denoted by, , if . is called the empty soft set over . The soft set is called relative whole soft set (with respect to the parameter set ), denoted by , if , for all, . is called the whole soft set. In the following, for a soft set by we mean the parameterized set .
For illustration, Molodtsov considered several examples in [6]. These examples were also discussed in [9, 11]. Now, we give an example of a soft set.
Example 6. Let . Define on the following operations:
One can see that is a divisible residuated lattice. Furthermore, , , but , are incomparable; thus is not a chain. Also, so is not an MTL-algebra. Moreover, . Now, let and which is defined by , where is a permutation on . Then, is a soft set over .
Maji et al. [9], Feng et al. [25], and Ali et al. [10] introduced and investigated several binary operations.
Definition 7 (see [6]). Let and be two soft sets over a common universe . (i) is said to be a soft subset of and is denoted by if and for all .(ii) and are said to be soft equal and is denoted by if and .
Example 8. Consider divisible residuated lattice in Example 6. Let and . Now, we define by and by for each , where , , and . By Proposition 3, we obtain that .
In the following, let be a nonempty family of soft sets over a common universe .
Definition 9 (see [10]). Let be a family of soft sets over a common universe . (1)The restricted intersection (union) of is defined as the soft set (), where and (), for all . If , we define and .(2)The extended intersection (union) of is defined as the soft set (), where and (), where .
In order to make the above definition more clear, we present the following example.
Example 10. Consider divisible residuated lattice in Example 6. Let , , and . Now, we define with for each , where . Now, we assume that and . Also, we suppose that in which and in which . By Proposition 3, we can obtain that , , and , where . Also, if we let , we get , , , , , and .
Definition 11 (see [25]). Let be a family of soft sets over a common universe .(1)The -intersection of is defined as the soft set , where and , for all .(2)The -intersection of is defined as the soft set , where and , for all .(3)Let be a soft set over universe , where . The cartesian product of is defined as the soft set , where and , for all .
Example 12. Consider divisible residuated lattice in Example 6. Let and . Now, we define with . Also, we define with and . Now, we assume that , in which and , in which and , and in which and . Therefore, we have , , and .
Definition 13 (see [10]). Let and be two soft sets over a common universe such that . (i)The restricted difference of and is defined as the soft set , where and , for all . If , we define (ii)The extended difference of and is defined as the soft set , where , and we have
Example 14. Consider Example 10. Let and . We suppose that in which and in which and . Then, we can obtain that and .
Definition 15 (see [10]). The complement of a soft set over is denoted by and is defined by where is a mapping given by , for all . Clearly, and .
In the following, the set of all soft sets over , in which and is a map, is denoted by .
Let be a universal set and let be the set of parameters with respect to . One can see that , where and , is a distributive bounded complete lattice if is closed under and . Furthermore, the partial relation defined by lattice operations coincides with . Since is a distributive bounded complete lattice, we can define a new operation as follows: Also, we let . Clearly, we have . Hence, we obtain the following corollary.
Corollary 16. Let be an universal set and let be the set of parameters with respect to . Then, is a bounded residuated lattice.
Example 17. Consider divisible residuated lattice in Example 6. Let , , , , and . Now, we define with , for and . Also, let with , for . If we let and we consider the soft set with and , then we have . Also, it is clear that .
In the next proposition, we show that is a divisible residuated lattice.
Proposition 18. Let be an universal set and let be the set of parameters with respect to . Then, for each soft sets , we have (i), (ii).
Proof. (i) It is obvious.
(ii) First we show that . Obviously, We have . On the other hand, , so and it shows the equality. Similarly, we can show that, for each , we get . Hence, the result holds.
Example 19. Consider Example 17. Clearly, we have and . Hence, is not an MTL-algebra. Also, and it implies that is not a Girard monoid. Therefore, is not an MV-algebra.
Recently, Ali et al. in [13] show that is an MV-algebra. Also, they show directly that is an MV-algebra and is a bounded BCK-algebra whose every element is an involution.
By [10], if and are two soft sets over the same universe , then So is an MV-algebra.
Lemma 20. Let and be two soft sets over the same universe . Then, (1).
Proof. Let . Then, , and for each we have . The result holds.
Corollary 21. Let be an universal set and let be the set of parameters with respect to . Then, is a bounded commutative BCK-algebra.
Proof. By Lemma 20, it is straightforward.
3. Soft Residuated Lattices and Soft Filters
In what follows, let and be a residuated lattice and a nonempty set, respectively, and will refer to a arbitrary relation between an element of and an element of ; that is is a subset of unless otherwise specified. A set valued function can be defined as , for all . Obviously, the pair is a soft set over which is derived from the relation .
Definition 22. Let be a nonnull soft set over residuated lattice . Then, is called a soft residuated lattice over , if is a subalgebra of for each .
Example 23. Let . Define on the following operations:
Then, is a distributive divisible residuated lattice which it is not an MTL-algebra since . Now, let be a soft set over residuated lattice , where and is a set valued function defined by
for all . Then, , , , and . So is a subalgebra of , for each . Therefore, is a soft residuated lattice over .
Example 24. Consider the standard MV-algebra . Let . We define , in which . Obviously, the pair is a soft MV-algebra over .
Definition 25. Let be a soft residuated lattice over residuated lattice . A nonnull soft set is called a soft filter of if it satisfies the following conditions: (SF1) ; (SF2) is a filter of , for each .
Example 26. Let be a residuated lattice. Suppose that . We set given by . By Proposition 4, is a soft filter over the soft residuated lattice .
Example 27. Consider soft residuated lattice over in Example 23. We define in which we have . Therefore, one can see that soft set is a soft filter of .
Let be a residuated lattice and let be a filter of and . Then, and imply that is a subalgebra of . Thus, each soft filter of soft residuated lattice is a soft residuated lattice.
Theorem 28. Let be a soft residuated lattice over and let be a nonempty family of soft filter of . Then, the restricted intersection and the extended intersection are soft filters of if they are nonnull.
Proof. Suppose that is a nonempty family of soft filters of . By Definition 9, let where and , for all . Clearly, we have . Now, let , so for each . By hypothesis is a filter of , for each and . It follows that is a filter of . Hence, is a soft filter of . Similarly, we can show that the extended intersection is a soft filter of .
Theorem 29. Let be a soft residuated lattice over and let be a nonempty family of soft filters of . Then, the restricted union and the extended union are soft filters of if or , for all , and they are nonnull.
Proof. Suppose that is a nonempty family of soft filters of . By Definition 9, let where and , for all . Clearly, we have . First, note that is nonnull since . Let . Then, and so we have for some . Now, let . Therefore, and , for some . We can suppose that . So by hypothesis we obtain that . On the other hand, let , , and . Then, , for some , and it implies that . Thus, is a filter of , for each . Hence, is a soft filter of .
Let be a soft residuated lattice over and let be a nonempty family of soft filters of . Assume that for all , . Therefore, has only one element. Let . So for each , , and it implies that, for each and , either or . Hence, by Theorem 29 we can conclude that the union is a soft filter of .
In the following, we give an example to see that Theorem 29 is not established in general.
Example 30. Consider divisible residuated lattice in Example 6. Let and . Now, we define by and by for each , where . Obviously, the restricted union and the extended union are not soft filters of , where .
Theorem 31. Let be a soft residuated lattice over and let be a nonempty family of soft filters of . Then, the -intersection is a soft filter of if is nonnull.
Proof. By Definition 11, we can let , where and , for all . Also, we let , where and , for all . Clearly, we have . Now, let be nonnull and . We have , and since each is a filter of , we obtain that is a filter of .
Theorem 32. Let be a soft residuated lattice over and let be a nonempty family of soft filters of . If or , for all and , then the -intersection is a soft filter of if is nonnull, for some .
Proof. Suppose that , in which and , for all . Also, we let , where and , for all . Clearly, we have . Now, let be nonnull, for some . Assume that . Obviously, is a nonnull soft set because, for each , where , we have . Now, suppose that . We have . Similar to the proof of Theorem 29, we can obtain that is a filter of . Hence, the -intersection is a soft filter of .
Theorem 33. Let be a soft residuated lattice over and let be a nonempty family nonnull soft filters of . Then, the cartesian product is a soft filter of .
Proof. Suppose that , in which and , for all . Also, let , where and , for all . Clearly, we have . By [20], for each , is a residuated lattice so is a soft residuated lattice over . Now, let . Therefore, by [20] is a filter of . Hence, is a soft filter of on .
In the following, we study the connection between soft sets and residuated lattice homomorphisms. Also, we consider a function between two soft residuated lattices and investigate its properties.
Proposition 34. Let be a residuated lattice homomorphism. If is a soft filter over , then is a soft filter over . Also, if is onto and is a soft filter over , then is a soft filter over .
Proof. Since is a nonnull soft set by Definition 25 and is a soft filter over , we observe that is a nonnull soft set over . We see that, for all , . Since the nonempty set is a filter of and is a homomorphism, so is a filter of . Therefore, is a filter of for all . Consequently, is a soft filter over .
Similarly, we can show that is a soft filter over where is a soft filter over and is an epimorphism.
A soft filter over a residuated lattice is said to be trivial if for every . A soft filter over is said to be whole if for each .
Theorem 35. Let be a soft filter over and let be a residuated lattice epimorphism. (i)If for all , then is the trivial soft filter over .(ii)If is whole, then is the whole soft filter over .
Proof. (i) Suppose that for all . Then, for all . So, is the trivial soft filter over .
(ii) Assume that is whole. Then, for each . Hence, for all . Hence, the result holds.
4. Conclusion
In this study, we have proposed the new concept of soft residuated lattice and have introduced their initial basic properties such as soft filters by using soft set theory. Also, the study of algebraic structures of soft sets with respect to new operations gives us a deep insight into their application. In fact, we establish a connection between the set of all soft set on a common universe and its lattice structure. It also provides new examples of these structures on the other hand. Residuated lattices, MV-algebras, and BCK-algebras of soft sets are indicated towards possible applications of soft sets in classical and nonclassical logic. To extend this work, one could study the properties of soft sets in other algebraic structures.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of the paper.