Subalgebras of BCK/BCI-Algebras Based on Cubic Soft Sets

Operations of cubic soft sets including “AND” operation and “OR” operation based on P-orders and R-orders are introduced and some related properties are investigated. An example is presented to show that the R-union of two internal cubic soft sets might not be internal. A sufficient condition is provided, which ensure that the R-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.


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][12][13][14][15]. Soft set based decision making was first considered by Maji et al. [9]. Ç aǧman and Enginoglu [16] developed the -decision making method in virtue of soft sets. Feng et al. [17] improve and further extend Ç aǧman and Enginoglu's approach using choice value soft sets and ksatisfaction 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 Ç aǧman [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], andalgebras [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 2 The Scientific World Journal (internal, external) cubic soft sets, -cubic (resp., -cubic) soft subsets, -union (resp., -intersection, -union, andintersection) 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.

Preliminary
In this section we include some elementary aspects that are necessary for this paper.
An algebra ( ; * , 0) of type (2, 0) is called a BCI-algebra if it satisfies the following axioms: (i) (∀ , , ∈ ) ((( * ) * ( * )) * ( * ) = 0), If a BCI-algebra satisfies the following identity: is called a BCK-algebra. Any BCK/BCI-algebra satisfies the following conditions: A fuzzy set in a set is defined to be a function : → , where = [0, 1]. 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 { | ∈ Λ} of fuzzy sets in , we define the join (∨) and meet (∧) operations as follows: respectively, for all ∈ .
By an interval number we mean a closed subinterval̃= and similarly we may havẽ1 ⪯̃2 and̃1 =̃2. To saỹ1 ≻̃2 For anỹ∈ [ ], its complement, denoted bỹ, is defined to be the interval number: Let be a nonempty set.
The complement of ∈ [ ] is defined as follows: ( ) = ( ) for all ∈ ; that is, The Scientific World Journal 3 For a family { | ∈ Λ} of IVF sets in where Λ is an index set, the union = ⋃ ∈Λ and the intersection = ⋂ ∈Λ 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 P( ) 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].

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 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 (F, ) where F is a mapping from to C and ⊂ . Note that the pair (F, ) can be represented as the following set: Definition 5. Let (F, ) and (G, ) be cubic soft sets over . Then "(F, ) AND (G, ) based on the -order" is denoted by (F, )∧ (G, ) and is defined by where H( , ) = F( ) ⋂ G( ) for all ( , ) ∈ × .
Definition 6. Let (F, ) and (G, ) be cubic soft sets over . Then "(F, ) AND (G, ) based on the -order" is denoted by (F, )∧ (G, ) and is defined by where Definition 7. Let (F, ) and (G, ) be cubic soft sets over . Then "(F, ) OR (G, ) based on the -order" is denoted by (F, )∨ (G, ) and is defined by where Definition 8. Let (F, ) and (G, ) be cubic soft sets over . Then "(F, ) OR (G, ) based on the -order" is denoted by (F, )∨ (G, ) and is defined by where The Scientific World Journal  Table 2: Tabular representation of the cubic soft set (G, ).
Example 9. Suppose that there are six houses in the universe given by 1 stands for the parameter "expensive, " 2 stands for the parameter "beautiful, " 3 stands for the parameter "wooden, " 4 stands for the parameter "cheap, " 5 stands for the parameter "in the green surroundings. "  The cubic soft set (F, ) can be represented in the tabular form of Table 1 (see [29]).
For a subset = { 2 , 5 } ⊆ , consider the cubic soft set (G, ) with the tabular representation in Table 2.
with the tabular representation in Table 3.
(2) "(F, ) OR (G, ) based on the -order" is a soft set with the tabular representation in Table 4.
with the tabular representation in Table 5.
with the tabular representation in Table 6.
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.
For a subset = { 3 , 5 } ⊆ , consider the cubic soft set (G, ) with the tabular representation in Table 8.
Then the -union (F, ) ⋓ (G, ) of (F, ) and (G, ) is the soft set over with the tabular representation in Table 9.
Note that ( Next, we provide a condition for the -union of two internal cubic soft sets to be an internal cubic soft set. The Scientific World Journal 5 Table 3: Tabular representation of the cubic soft set (F, )∨ (G, ) = (H, × ).  Table 4: Tabular representation of the cubic soft set (F, )∨ (G, ) = (H, × ).

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 (F, ) 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 (F, ) is an -cubic soft subalgebra over for all ∈ , one says that (F, ) is a cubic soft subalgebra over .
be a universe, and consider a binary operation ✠ which produces the following products: Then, ( , ✠, white) is a BCI-algebra (see [30]). Consider sets of parameters: The Scientific World Journal 7  where 1 stands for the parameter "beautiful, " 2 stands for the parameter "fine, " 3 stands for the parameter "moderate, " 4 stands for the parameter "smart, " 5 stands for the parameter "chaste. " Then and are not disjoint, and a soft set (F, ) over with the tabular representation in Table 10 is a cubic soft subalgebra over .
Also a soft set (G, ) over with the tabular representation in Table 11 is a cubic soft subalgebra over .

Proposition 17.
Given a parameter ∈ , if a cubic soft set (F, ) over is an -cubic soft subalgebra over , then Proof. For every ∈ , we have Proof. The proof is straightforward.

Conclusion
In this paper, we first have considered operations of cubic soft sets, that is, "AND" operation and "OR" operation based on the -order and the -order. In [29], Muhiuddin and Alroqi have posed a question: is the -union of two internal The Scientific World Journal 9 cubic soft sets an internal cubic soft set? We have given an example to show that the answer to this question is negative, and then we have provided a condition for the -union of two internal cubic soft sets to be an internal cubic soft set. We also have investigated several properties of cubic soft subalgebras of BCK/BCI-algebras based on any given parameter. Some important issues to be explored in the future include (1) developing strategies for obtaining more valuable results, (2) applying these notions and results for studying related notions in other (soft) algebraic structures.