The Combination of Soft Sets and -Structures with Applications
Using the notions of soft sets and -structures, -soft set theory is introduced. We apply it to both a decision making problem and a BCK/BCI algebra.
To solve complicated problems in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as the probability theory, the theory of (intuitionistic) fuzzy sets, the theory of vague sets, the theory of interval mathematics, and the theory of rough sets. However, all of these theories have their own difficulties which are pointed out in . Maji et al.  and Molodtsov  suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov  introduced the concept of soft set as a new mathematical tool for dealing with uncertainties which is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al.  described the application of soft set theory to a decision making problem. Maji et al.  also studied several operations on the theory of soft sets. Chen et al.  presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh . Roy and Maji  presented some results on an application of fuzzy soft sets in decision making problem. Aygünoglu and Aygün  introduced the notion of fuzzy soft group and studied its properties. Ali et al.  discussed new operations in soft set theory. Jun  applied the notion of soft set to -algebras, and Jun et al.  considered applications of soft set theory in the ideals of -algebras.
A (crisp) set in a universe can be defined in the form of its characteristic function yielding the value for elements belonging to the set and the value for elements excluded from the set . So far most of the generalization of the crisp set has been conducted on the unit interval , and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point into the interval . Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al.  introduced a new function which is called negative-valued function and constructed -structures. They applied -structures to BCK/BCI-algebras, and discussed -subalgebras and -ideals in BCK/BCI-algebras. Jun et al.  considered closed ideals in BCH-algebras based on -structures.
In this paper we introduce the notion of -soft sets which are a soft set based on -structures by using the notions of soft sets and -structures, and then we apply it to both a decision making problem and a -algebra.
A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki and was extensively investigated by several researchers.
An algebra of type is called a -algebra if it satisfies the following conditions: (I), (II), (III), (IV).
If a BCI-algebra satisfies the following identity: (V),
then is called a -algebra. Any BCK-algebra satisfies the following axioms:(a1), (a2), (a3), (a4),
where if and only if .
Any BCI-algebra satisfies the following axioms:(a5), (a6).
A nonempty subset of a BCK/BCI-algebra is called a BCK/BCI-subalgebra of if for all .
For any family of real numbers, we define
Denote by the collection of functions from a set to . We say that an element of is a negative-valued function from to (briefly, -function on ). By an -structure we mean an ordered pair of and an -function on .
3. -Soft Sets
Definition 1. Let be an initial universe set and a set of attributes. By an -soft set over we mean a pair where and is a mapping from to ; that is; for each is an -function on .
Denote by the collection of all -soft sets over with attributes from and we call it an -soft class.
We provide an example of an -soft set.
Example 2. As an initial universe set and a set of attributes, we consider consists of six houses, and , respectively, where stands for the attribute “cheap,”stands for the attribute “messy,”stands for the attribute “brick,”stands for the attribute “expensive,”stands for the attribute “in the flooded area.”
Let , and be -functions on defined by
The -soft set is an attributed family of all -functions on the set and gives us a collection of approximate description of an object. The -function here is “ houses,” where is to be filled up by an attribute . Therefore means “cheap houses” whose functional value is represented by , that is,
Therefore, we can represent the -soft set as follows: where each approximation has two parts: (i)a predicate ,(ii)an approximate value -set .
For example, for the approximation “,” we have (i)the predicate name is “cheap houses,”(ii)the approximate value -set is , , , , , .
Therefore, an -soft set can be viewed as a collection of -approximations as follows: For the purpose of storing an -soft set in a computer, we could represent an -soft set, which is described in the above, in the form of Table 1.
For convenience of explanation, we can represent the -soft set, which is described in the above, in matrix form as follows:
Definition 3. Let and be -soft sets in . Then is called an -soft subset of , denoted by , if it satisfies as following: (i), (ii).
Example 4. Let and Let and be -soft sets in given by Then is an -soft subset of .
Let be an -soft set in . The complement of , denoted by , is defined to be an -soft set , where is not the set of , that is, , and is an -function given by is -complement of for all , that is,
Example 5. Consider the -soft set in Example 2. Then the complement of is represented as follows:
For any -soft sets and in , we define(i)“ AND ,” denoted by , to be an -soft set , where for all ; that is, for all and .(ii)“ OR ,” denoted by , to be an -soft set , where for all ; that is, for all and .
Example 6. Consider two -soft sets and in which describes the “cost of houses” and the “attractiveness of houses.” Suppose that and , where stands for the attribute “cheap,”stands for the attribute “costly,”stands for the attribute “very costly,”stands for the attribute “beautiful,”stands for the attribute “in the green surroundings.”
Take and , and defineThen and are represented as follows:
4. Application in a Decision Making Problem
The problem in an -soft class is to choose an object from the initial universe set of given objects with respect to a set of choice attribute . We present an algorithm for identification of an object based on multiobserves input data characterized by the color of roofs, size, and cost.
Algorithm 7. Consider the following.(1)Input the -soft sets , and .(2)Input the attribute set as observed by the observe.(3)Compute the corresponding resultant -soft set from the -soft sets , and and place it in matrix form.(4)Construct the comparison table of the -soft set where the comparison table is a square table in which(i)the number of rows and the number of columns are equal,(ii)rows and columns both are labelled by the object names of the universe,(iii)the entries are , where is determined by the number of attributes for which the membership value of object is less than or equal to the membership value of object .(5)Compute the row sum for each and column sum for each which are calculated by using the formula: and .(6)Compute the score of each , which is given as .(7)The decision is if .(8)If has more than one value, then any one of may be chosen.
Let and be a set of six houses and a set of attributes, respectively, where stands for the attribute “black roof,” stands for the attribute “brown roof,” stands for the attribute “yellow roof,” stands for the attribute “red roof,” stands for the attribute “large size,” stands for the attribute “small size,” stands for the attribute “very small size,” stands for the attribute “average size,” stands for the attribute “very large size,” stands for the attribute “cheap,” stands for the attribute “expensive,” stands for the attribute “very cheap,” stands for the attribute “very expensive.”
Consider three subsets , and of as follows: which represents the color of roof of the house, which represents the size of the house, which represents the cost of the house.
Let , and be -soft sets in defined by respectively. We perform “ AND ” and it is represented as follows: If we require the -soft set for the attributes , where then the resultant -soft set for the -soft sets and will be , say, which is represented as follows: If we perform “ AND ,” then we will have attributes. Let be the set of choice attributes of an observer, where . Then the resultant -soft set is represented as follows: The comparison table for the -soft set is given by Table 2.
We now compute the raw sum, column sum, and the score for each , and it is given by Table 3.
From the score table (Table 3), it is clear that the minimum score is , scored by , and the decision is in favor of selecting .
5. Application in BCK/BCI-Algebras
In what follows let denote a set of attributes unless otherwise specified. We will use the terminology “soft machine” which means that it produces a BCK/BCI-algebra.
Definition 8 (see ). By a subalgebra of a -algebra based on -function (briefly, -subalgebra of ), we mean an -structure in which satisfies the following assertion:
Definition 9. Let be an -soft set over a -algebra , where is a subset of . If there exists an attribute for which the -structure is an -subalgebra of , then we say that is an -soft -algebra related to the attribute (briefly, -soft -algebra). If is an -soft -algebra for all , we say that is an -soft -algebra.
Example 10. Suppose there are five colors in the universe , that is, and beautiful, fine, moderate, delicate, elegant, smart, chaste} be a set of attributes. Let be a soft machine to mix two colors according to order in such a way that we have the following results: Then is a BCK-algebra. Consider a set of attributes and define an -soft set over the -algebra as follows: The tabular representation of is given by Table 4.
Then is an -soft -algebra over the -algebra .
Now let be an -soft set over the -algebra with the tabular representation which is given by Table 5.
Then is not an -soft -algebra over since Hence is not an -soft -algebra over . But we can verify that is both an -soft -algebra and -soft -algebra over .
Proposition 11. Every -soft -algebra over a -algebra satisfies the following inequality:
Proof. For any and , we have This completes the proof.
The problem we now discuss is as follows.
If is an -soft -algebra over a -algebra , then is every -soft subset of an -soft -algebra over ?
Unfortunately this is not true as seen in the following example.
Example 12. Consider the universe: which is considered in Example 10. Consider a soft machine which produces the following products: Then is a BCI-algebra. Take and let be an -soft set over the -algebra with the tabular representation which is given by Table 6.
Then is an -soft -algebra over . Now let be an -soft subset of , where and the tabular representation of is given by Table 7.
Then and so is not an -soft -algebra over . Hence is not an -soft -algebra over .
But, we have the following theorem.
Theorem 13. For any subset of , let be an -soft -algebra over a -algebra . If is a subset of , then is an -soft -algebra over .
The following example shows that there exists an -soft set over a -algebra such that(i) is not an -soft -algebra over a -algebra ,(ii)there exists a subset of such that is an -soft -algebra over a -algebra .
Then is neither an -soft -algebra nor an -soft -algebra over . Hence is not an -soft -algebra over . But if we take then is an -soft -algebra over .
Definition 15. Let and be two -soft sets in . The union of and is defined to be the -soft set in satisfying the following conditions: (i), (ii)for all ,
In this case, we write .
Lemma 16 (see ). If and are -subalgebras of a -algebra , then the union of and is an -subalgebra of .
Theorem 17. If and are -soft -algebras over a -algebra , then the union of and is an -soft -algebra over .
Proof. Let be the union of and . Then . For any , if (resp. ), then (resp. ) is an -subalgebra of . If , then is an -subalgebra of for all by Lemma 16. Therefore is an -soft -algebra over a -algebra .
Definition 18. Let and be two -soft sets in . The intersection of and is the -soft set in where and, for every ,