Abstract

Operations of cubic soft sets including “AND” operation and “OR” operation based on -orders and -orders are introduced and some related properties are investigated. An example is presented to show that the -union of two internal cubic soft sets might not be internal. A sufficient condition is provided, which ensure that the -union of two internal cubic soft sets is also internal. Moreover, some properties of cubic soft subalgebras of BCK/BCI-algebras based on a given parameter are discussed.

1. Introduction

Zadeh [1] made an extension of the concept of a fuzzy set by an interval-valued fuzzy set, that is, a fuzzy set with an interval-valued membership function. Using a fuzzy set and an interval-valued fuzzy set, Jun et al. [2] introduced a new notion, called a (internal, external) cubic set, and investigated several properties. They dealt with -union, -intersection, -union, and -intersection of cubic sets and investigated several related properties. Later on, Jun et al. [3] applied the notion of cubic set theory to BCI-algebras.

To solve complicated problems in economics, engineering, and environment, we cannot successfully make use of classical methods because of various uncertainties typical for those real-word problems. On the contrary, uncertainties could be dealt with the help of a wide range of contemporary mathematical theories such as probability theory, theory of fuzzy sets [4], interval mathematics [5], and rough set theory [6, 7]. However, all of these theories have their own difficulties which were pointed out in [8]. Further, Maji et al. [9] and Molodtsov [8] suggested that one reason for these difficulties might be due to the inadequacy of the parameterization tool of the theory. To overcome these difficulties, Molodtsov [8] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties based on the viewpoint of parameterization. It has been demonstrated that soft sets have potential applications in various fields such as the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory, and measurement theory [8, 10]. Since then, many researchers around the world have contributed to soft set theory from various aspects [9, 11–15]. Soft set based decision making was first considered by Maji et al. [9]. Çaǧman and Enginoglu [16] developed the - decision making method in virtue of soft sets. Feng et al. [17] improve and further extend Çaǧman and Enginoglu’s approach using choice value soft sets and k-satisfaction relations. It is interesting to see that soft sets are closely related to many other soft computing models such as rough sets and fuzzy sets [18, 19]. Aktaş and Çaǧman [20] defined the notion of soft groups and derived some related properties. This initiated an important research direction concerning algebraic properties of soft sets in miscellaneous kinds of algebras such as BCK/BCI-algberas [21], -algebras [22], semirings [23], rings [24], Lie algebras [25], and -algebras [26, 27]. In addition, Feng and Li [28] ascertained the relationships among five different types of soft subsets and considered the free soft algebras associated with soft product operations. It has been shown that soft sets have some nonclassical algebraic properties which are distinct from those of crisp sets or fuzzy sets.

Recently, combining cubic sets and soft sets, the first author together with Al-roqi [29] introduced the notions of (internal, external) cubic soft sets, -cubic (resp., -cubic) soft subsets, -union (resp., -intersection, -union, and -intersection) of cubic soft sets, and the complement of a cubic soft set. They investigated several related properties and applied the notion of cubic soft sets to BCK/BCI-algebras.

In this paper, we consider several basic operations of cubic soft sets, namely, “AND” operation and “OR” operation based on the -order and the -order. We provide an example to illustrate that the -union of two internal cubic soft sets might not be internal. Then we discuss the condition for the -union of two internal cubic soft sets to be an internal cubic soft set. We also investigate several properties of cubic soft subalgebras of BCK/BCI-algebras based on a given parameter.

2. Preliminary

In this section we include some elementary aspects that are necessary for this paper.

An algebra of type is called a BCI-algebra if it satisfies the following axioms:(i), (ii), (iii), (iv).

If a BCI-algebra satisfies the following identity:(v), then is called a BCK-algebra. Any BCK/BCI-algebra satisfies the following conditions: (a1), (a2), (a3), (a4).

A fuzzy set in a set is defined to be a function , where . Denote by the collection of all fuzzy sets in a set . Define a relation on as follows: The join () and meet () of and are defined by respectively, for all . The complement of , denoted by , is defined by For a family of fuzzy sets in , we define the join () and meet () operations as follows: respectively, for all .

By an interval number we mean a closed subinterval of , where . Denote by the set of all interval numbers. Let us define what is known as refined minimum and refined maximum (briefly, and ) of two elements in . We also define the symbols “,” “,” and “” in case of two elements in . Consider two interval numbers and . Then, and similarly we may have and . To say (resp., ), we mean and (resp., and ). Let , where . We define For any , its complement, denoted by , is defined to be the interval number:

Let be a nonempty set. A function is called an interval-valued fuzzy set (briefly, an IVF set) in . Let stand for the set of all IVF sets in . For every and is called the degree of membership of an element to , where and are fuzzy sets in which are called a lower fuzzy set and an upper fuzzy set in , respectively. For simplicity, we denote . For every , we define The complement of is defined as follows: for all ; that is, For a family of IVF sets in where is an index set, the union and the intersection are defined as follows: for all , respectively.

Molodtsov [8] defined the soft set in the following way: let be an initial universe set and let be a set of parameters. Let denote the power set of and .

A pair is called a soft set over , where is a mapping given by

In other words, a soft set over is a parameterized family of subsets of the universe . For may be considered as the set of -approximate elements of the soft set . Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [8].

3. Cubic Soft Sets

Definition 1 (see [2]). Let be a universe. By a cubic set in one means a structure in which is an IVF set in and is a fuzzy set in .

Definition 2 (see [2]). Let and be cubic sets in a universe . Then one defines the following. (a)(Equality) and .(b)(-order) and .(c)(-order) and .

Definition 3 (see [2]). For any where , one defines (a) (-union);(b) (-intersection);(c) (-union);(d) (-intersection).
The complement of is defined to be the cubic soft set: Obviously, , , , and . For any we have and . Also we have

In what follows, a cubic set is simply denoted by , and denote by the collection of all cubic sets in .

Definition 4 (see [29]). Let be an initial universe set and let be a set of parameters. A cubic soft set over is defined to be a pair where is a mapping from to and . Note that the pair can be represented as the following set:

Definition 5. Let and be cubic soft sets over . Then “ AND based on the -order” is denoted by and is defined by where for all .

Definition 6. Let and be cubic soft sets over . Then “ AND based on the -order” is denoted by and is defined by where for all .

Definition 7. Let and be cubic soft sets over . Then “ OR based on the -order” is denoted by and is defined by where for all .

Definition 8. Let and be cubic soft sets over . Then “ OR based on the -order” is denoted by and is defined by where for all .

Example 9. Suppose that there are six houses in the universe given by and , where   stands for the parameter “expensive,”  stands for the parameter “beautiful,”  stands for the parameter “wooden,”  stands for the parameter “cheap,”  stands for the parameter “in the green surroundings.”For , the set is a cubic soft set over where The cubic soft set can be represented in the tabular form of Table 1 (see [29]).
For a subset , consider the cubic soft set with the tabular representation in Table 2.(1)“ OR based on the -order” is a soft set with the tabular representation in Table 3.(2)“ OR based on the -order” is a soft set with the tabular representation in Table 4.(3)“ AND based on the -order” is a soft set with the tabular representation in Table 5.(4)“ AND based on the -order” is a soft set with the tabular representation in Table 6.
In [29], Muhiuddin and Al-roqi posed the following question.

Question 1. Is the -union of two internal cubic soft sets an internal cubic soft set?

The following example shows that the answer to this question is negative.

Example 10. Suppose that there are three houses in the universe given by and is the set of parameters, where   stands for the parameter “expensive,”  stands for the parameter “beautiful,”  stands for the parameter “wooden,”  stands for the parameter “cheap,”  stands for the parameter “in the green surroundings.”For a subset , consider the cubic soft set with the tabular representation in Table 7.
For a subset , consider the cubic soft set with the tabular representation in Table 8.
Then the -union of and is the soft set over with the tabular representation in Table 9.
Note that and are internal cubic soft sets over . But the -union is not an internal cubic soft set over since .

Next, we provide a condition for the -union of two internal cubic soft sets to be an internal cubic soft set.

Theorem 11. Let and be internal cubic soft sets over such that for all and . Then the -union of and is an internal cubic soft set over .

Proof. Since and are internal cubic soft sets over , if or , then clearly the -union of and is an internal cubic soft set over . Note that for all , and . It follows from (26) that for all and . Therefore the -union of and is an internal cubic soft set over .

4. Cubic Soft Subalgebras of BCK/BCI-Algebras

In what follows, let be an initial universe set which is a BCK/BCI-algebra.

Definition 12 (see [29]). A cubic soft set over is said to be a cubic soft BCK/BCI-algebra over based on a parameter (briefly, -cubic soft subalgebra over ) if there exists a parameter such that for all . If is an -cubic soft subalgebra over for all , one says that is a cubic soft subalgebra over .

Definition 13 (see [29]). The -union of cubic soft sets and over is a cubic soft set , where and for all . This is denoted by .