# The Combination of Soft Sets and𝒩-Structures with Applications

Authors: Young Bae Jun; Seok Zun Song; Kyoung Ja Lee
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Using the notions of soft sets and𝒩-structures, 𝒩-soft set theory is introduced. We apply it to both a decision making problem and a BCK/BCI algebra.


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## Body

## 1. Introduction

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 [1]. Maji et al. [2] and Molodtsov [1] 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 [1] 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. [2] described the application of soft set theory to a decision making problem. Maji et al. [3] also studied several operations on the theory of soft sets. Chen et al. [4] 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 [5]. Roy and Maji [6] presented some results on an application of fuzzy soft sets in decision making problem. Aygünoglu and Aygün [7] introduced the notion of fuzzy soft group and studied its properties. Ali et al. [8] discussed new operations in soft set theory. Jun [9] applied the notion of soft set to BCK/BCI-algebras, and Jun et al. [10] considered applications of soft set theory in the ideals of d-algebras.A (crisp) setA in a universe X can be defined in the form of its characteristic function μA:X→{0,1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A. So far most of the generalization of the crisp set has been conducted on the unit interval [0,1], 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 {1} into the interval [0,1]. 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. [11] introduced a new function which is called negative-valued function and constructed 𝒩-structures. They applied 𝒩-structures to BCK/BCI-algebras, and discussed 𝒩-subalgebras and 𝒩-ideals in BCK/BCI-algebras. Jun et al. [12] considered closed ideals in BCH-algebras based on 𝒩-structures.In this paper we introduce the notion of𝒩-soft sets which are a soft set based on 𝒩-structures by using the notions of soft sets and 𝒩-structures, and then we apply it to both a decision making problem and a BCK/BCI-algebra.


## 2. Preliminaries

ABCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki and was extensively investigated by several researchers.An algebra(X;*,0) of type (2,0) is called a BCI-algebra if it satisfies the following conditions:(I)
(∀x,y,z∈X)(((x*y)*(x*z))*(z*y)=0),(II)
(∀x,y∈X)((x*(x*y))*y=0),(III)
(∀x∈X)(x*x=0),(IV)
(∀x,y∈X)(x*y=0,y*x=0⇒x=y).If a BCI-algebraX satisfies the following identity:(V)
(∀x∈X)(0*x=0),thenX is called a BCK-algebra. Any BCK-algebra X satisfies the following axioms:(a1)
(∀x∈X)(x*0=x),(a2)
(∀x,y,z∈X)(x≤y⇒x*z≤y*z,z*y≤z*x),(a3)
(∀x,y,z∈X)((x*y)*z=(x*z)*y),(a4)
(∀x,y,z∈X)((x*z)*(y*z)≤x*y),wherex≤y if and only if x*y=0.Any BCI-algebraX satisfies the following axioms:(a5)
(∀x,y,z∈X)(0*(0*((x*z)*(y*z)))=(0*y)*(0*x)),(a6)
(∀x,y∈X)(0*(0*(x*y))=(0*y)*(0*x)).A nonempty subsetS of a BCK/BCI-algebra X is called a BCK/BCI-subalgebra of X if x*y∈S for all x,y∈S.For any family{ai∣i∈Λ} of real numbers, we define
(1)⋁{ai∣i∈Λ}:={max{ai∣i∈Λ}ifΛisfinite,sup{ai∣i∈Λ}otherwise,⋀{ai∣i∈Λ}:={min{ai∣i∈Λ}ifΛisfinite,inf{ai∣i∈Λ}otherwise.Denote byF(S,[-1,0]) the collection of functions from a set S to [-1,0]. We say that an element of F(S,[-1,0]) is a negative-valued function from S to [-1,0] (briefly, 𝒩-function on S). By an 𝒩-structure we mean an ordered pair (S,f) of S and an 𝒩-function f on S.


## 3.𝒩-Soft Sets

Definition 1.
LetX be an initial universe set and E a set of attributes. By an 𝒩-soft set over X we mean a pair (f,A) where A⊂E and f is a mapping from A to ℱ(X,[-1,0]); that is; for each a∈A,f(a):=fa is an 𝒩-function on X.Denote by𝒩(X,E) the collection of all 𝒩-soft sets over X with attributes from E and we call it an 𝒩-soft class.We provide an example of an𝒩-soft set.Example 2.
As an initial universe set and a set of attributes, we considerX={x1,x2,x3,x4,x5,x6} consists of six houses, and E={e1,e2,e3,e4,e5}, respectively, where e1
stands for the attribute “cheap,”e2
stands for the attribute “messy,”e3
stands for the attribute “brick,”e4
stands for the attribute “expensive,”e5
stands for the attribute “in the flooded area.”
Letfe1,fe2,fe3,fe4, and fe5 be 𝒩-functions on X defined by
(2)fe1=(x1x2x3x4x5x6-0.50-0.60-0.7-1),fe2=(x1x2x3x4x5x60-0.60-0.6-0.4-0.2),fe3=(x1x2x3x4x5x6-0.8-0.7000-1),fe4=(x1x2x3x4x5x60-10-0.80-0.8),fe5=(x1x2x3x4x5x60-0.9-0.5-0.7-0.8-0.7).
The𝒩-soft set (f,E) is an attributed family {fei∣i=1,2,…,5} of all 𝒩-functions on the set X and gives us a collection of approximate description of an object. The 𝒩-function f* here is “* houses,” where * is to be filled up by an attribute e∈E. Therefore fe1 means “cheap houses” whose functional value is represented by {x1/-0.5,x2/0,x3/-0.6,x4/0,x5/-0.7,x6/-1}, that is,
(3)cheaphouses={x1-0.5,x20,x3-0.6,x40,x5-0.7,x6-1}.
Therefore, we can represent the𝒩-soft set (f,E) as follows:
(4)(f,E)={cheaphouses={x1-0.5,x20,x3-0.6,x40,x5-0.7,x6-1},messyhouses={x10,x2-0.6,x30,x4-0.6,x5-0.4,x6-0.2},brickhouses={x1-0.8,x2-0.7,x30,x40,x50,x6-1},expensivehouses={x10,x2-1,x30,x4-0.8,x50,x6-0.8},housesinthefloodedarea={x10,x2-0.9,x3-0.5,x4-0.7,x5-0.8,x6-0.7}},
where each approximation has two parts: (i)
a predicatep,(ii)
an approximate value𝒩-set v.
For example, for the approximation “cheaphouses={x1/-0.5,x2/0,x3/-0.6,x4/0,x5/-0.7,x6/-1},” we have (i)
the predicate name is “cheap houses,”(ii)
the approximate value𝒩-set is {x1/-0.5, x2/0, x3/-0.6, x4/0, x5/-0.7, x6/-1}.
Therefore, an𝒩-soft set (f,E) can be viewed as a collection of 𝒩-approximations as follows:
(5)(f,E)={p1=v1,p2=v2,…,pn=vn}.
For the purpose of storing an 𝒩-soft set in a computer, we could represent an 𝒩-soft set, which is described in the above, in the form of Table 1.
For convenience of explanation, we can represent the𝒩-soft set, which is described in the above, in matrix form as follows:
(6)x1x2x3x4x5x6(f,E)=e1e2e3e4e5(-0.500-0.60-0.7-10-0.60-0.6-0.4-0.2-0.8-0.7000-10-10-0.80-0.80-0.9-0.5-0.7-0.8-0.7).Table 1
X
Cheap
Messy
Brick
Expensive
In the flooded area
x
1
−0.5
0
−0.8
0
0
x
2
0
−0.6
−0.7
−1
−0.9
x
3
−0.6
0
0
0
−0.5
x
4
0
−0.6
0
−0.8
−0.7
x
5
−0.7
−0.4
0
0
−0.8
x
6
−1
−0.2
−1
−0.8
−0.7Definition 3.
Let(f,A) and (g,B) be 𝒩-soft sets in 𝒩(X,E). Then (f,A) is called an 𝒩-soft subset of (g,B), denoted by (f,A)⊆~(g,B), if it satisfies as following: (i)
A⊆B,(ii)
(∀e∈A)(fe⊆ge,thatis,fe(x)≤ge(x)forallx∈X).Example 4.
LetX={x1,x2,x3,x4,x5,x6} and
(7)E={cheap(e1),beautiful(e2),messy(e3),brick(e4),beautiful(e5),expensive(e6),inthefloodedarea(e7),inthegreensurrounding(e8)}.
Let (f,A) and (g,B) be 𝒩-soft sets in 𝒩(X,E) given by
(8)A={cheap(e1),beautiful(e2),messy(e3)},B={cheap(e1),beautiful(e2),messy(e3),expensive(e6),inthefloodedarea(e7)},x1x2x3x4x5x6(f,A)=e1e2e3(-0.6-0.4-0.5-0.2-0.300-0.5-0.50-0.4-0.3-0.5-0.4-0.7-0.2-0.6-0.3),x1x2x3x4x5x6(g,B)=e1e2e3e6e7(-0.4-0.3-0.4-0.1-0.200-0.4-0.50-0.30-0.4-0.4-0.50-0.30-0.6-0.4-0.5-0.2-0.70-10-10-10).
Then (f,A) is an 𝒩-soft subset of (g,B).Let(f,A) be an 𝒩-soft set in (X,E). The complement of (f,A), denoted by (f,A)c, is defined to be an 𝒩-soft set (fc,⌉A), where ⌉A is not the set of A, that is, ⌉A={¬a∣a∈A}, and fc:⌉A→ℱ(X,[-1,0]) is an 𝒩-function given by fc(¬a) is 𝒩-complement of f(a) for all ¬a∈⌉A, that is,
(9)(∀¬a∈⌉A)(∀x∈X)(fc(¬a)(x)+f(a)(x)=-1).Example 5.
Consider the𝒩-soft set (f,E) in Example 2. Then the complement of (f,E) is represented as follows:
(10)(f,E)c={{x1-1,x2-0.1,x3-0.5,x4-0.3,x5-0.2,x6-0.3}notcheaphouses={x1-0.5,x2-1,x3-0.4,x4-1,x5-0.3,x60},notmessyhouses={x1-1,x2-0.4,x3-1,x4-0.4,x5-0.6,x6-0.8},notbrickhouses={x1-0.2,x2-0.3,x3-1,x4-1,x5-1,x60},notexpensivehouses={x1-1,x20,x3-1,x4-0.2,x5-1,x6-0.2},nothousesinthefloodedarea={x1-1,x2-0.1,x3-0.5,x4-0.3,x5-0.2,x6-0.3}}.For any𝒩-soft sets (f,A) and (g,B) in (X,E), we define(i)
“(f,A) AND (g,B),” denoted by (f,A)∧~(g,B), to be an 𝒩-soft set (f,A)∧~(g,B)=(h,A×B), where h(α,β)=⋀{fα,gβ} for all (α,β)∈A×B; that is, h(α,β)(x)=⋀{fα(x),gβ(x)} for all (α,β)∈A×B and x∈X.(ii)
“(f,A) OR (g,B),” denoted by (f,A)∨~(g,B), to be an 𝒩-soft set (f,A)∨~(g,B)=(λ,A×B), where λ(α,β)=⋁{fα,gβ} for all (α,β)∈A×B; that is, λ(α,β)(x)=⋁{fα(x),gβ(x)} for all (α,β)∈A×B and x∈X.Example 6.
Consider two𝒩-soft sets (f,A) and (g,B) in (X,E) which describes the “cost of houses” and the “attractiveness of houses.” Suppose that X={x1,x2,…,x10} and E={e1,e2,e3,e4,e5}, where e1
stands for the attribute “cheap,”e2
stands for the attribute “costly,”e3
stands for the attribute “very costly,”e4
stands for the attribute “beautiful,”e5
stands for the attribute “in the green surroundings.”
TakeA={e1,e2,e3} and B={e1,e4,e5}, and define(11)x1x2x3x4x5x6x7x8x9x10(f,A)=e1e2e3(-0.6-0.8-0.5-0.6-0.70-0.9-0.4-0.3-0.2-0.30-0.20-0.1-0.700-0.6-0.500-0.20-0.1-0.700-0.7-0.3),x1x2x3x4x5x6x7x8x9x10(g,B)=e1e4e5(-0.6-0.7-0.5-0.6-0.90-0.9-0.4-0.5-0.4-0.5-0.3-0.2-0.400-0.10-0.7-0.1-0.400-0.2-0.4-0.20-0.2-0.5-0.3).Then (f,A)∧~(g,B) and (f,A)∨~(g,B) are represented as follows:(12)x1x2x3x4x5x6x7x8x9x10(f,A)∧~(g,B)=(h,A×B)=(e1,e1)(e1,e4)(e1,e5)(e2,e1)(e2,e4)(e2,e5)(e3,e1)(e3,e4)(e3,e5)(-0.6-0.8-0.5-0.6-0.90-0.9-0.4-0.5-0.4-0.6-0.8-0.5-0.6-0.70-0.9-0.4-0.7-0.2-0.6-0.8-0.5-0.6-0.7-0.2-0.9-0.4-0.5-0.3-0.6-0.7-0.5-0.6-0.9-0.7-0.9-0.4-0.6-0.5-0.5-0.3-0.2-0.4-0.1-0.7-0.10-0.7-0.5-0.40-0.2-0.2-0.4-0.70-0.2-0.6-0.5-0.6-0.7-0.5-0.6-0.9-0.7-0.9-0.4-0.7-0.4-0.5-0.3-0.2-0.4-0.1-0.7-0.10-0.7-0.3-0.40-0.2-0.2-0.4-0.70-0.2-0.7-0.3),(13)x1x2x3x4x5x6x7x8x9x10(f,A)∨~(g,B)=(λ,A×B)=(e1,e1)(e1,e4)(e1,e5)(e2,e1)(e2,e4)(e2,e5)(e3,e1)(e3,e4)(e3,e5)(-0.6-0.7-0.5-0.6-0.70-0.9-0.4-0.3-0.2-0.5-0.3-0.2-0.400-0.10-0.3-0.1-0.400-0.2-0.400-0.2-0.3-0.2-0.30-0.20-0.1000-0.5-0.4-0.30-0.200000-0.6-0.1-0.3000-0.1-0.200-0.5-0.300-0.20-0.1000-0.5-0.300-0.200000-0.7-0.10000-0.1-0.200-0.5-0.3).


## 4. Application in a Decision Making Problem

The problem in an𝒩-soft class is to choose an object from the initial universe set of given objects with respect to a set of choice attribute P. 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 the𝒩-soft sets (f,A),(g,B), and (h,C).(2)
Input the attribute setP as observed by the observe.(3)
Compute the corresponding resultant𝒩-soft set (ω,P) from the 𝒩-soft sets (f,A),(g,B), and  (h,C) and place it in matrix form.(4)
Construct the comparison table of the𝒩-soft set (ω,P) 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 namesx1,x2,…,xn of the universe,(iii)
the entries arecij, where cij is determined by the number of attributes for which the membership value of object xi is less than or equal to the membership value of object xj.(5)
Compute the row sumri for each xi and column sum tj for each xj which are calculated by using the formula: ri=∑j=1n‍cij and tj=∑i=1n‍cij.(6)
Compute the scoreSi of each xi, which is given as Si=ri-ti.(7)
The decision isSk if Sk=miniSi.(8)
Ifk has more than one value, then any one of xk may be chosen.LetX={x1,x2,x3,x4,x5,x6} and E={e1,e2,…,e13} be a set of six houses and a set of attributes, respectively, wheree1 stands for the attribute “black roof,”e2 stands for the attribute “brown roof,”e3 stands for the attribute “yellow roof,”e4 stands for the attribute “red roof,”e5 stands for the attribute “large size,”e6 stands for the attribute “small size,”e7 stands for the attribute “very small size,”e8 stands for the attribute “average size,”e9 stands for the attribute “very large size,”e10 stands for the attribute “cheap,”e11 stands for the attribute “expensive,”e12 stands for the attribute “very cheap,”e13 stands for the attribute “very expensive.”Consider three subsetsA,B, and C of E as follows:A={e1,e2,e3,e4} which represents the color of roof of the house,B={e5,e6,e7,e8,e9} which represents the size of the house,C={e10,e11,e12,e13} which represents the cost of the house.Let(f,A),(g,B), and (h,C) be 𝒩-soft sets in (X,E) defined by
(14)x1x2x3x4x5x6(f,A)=e1e2e3e4(-0.7-0.7-0.6-0.2-0.3-0.1-0.6-0.1-0.5-0.8-0.7-0.8-0.4-0.7-0.2-0.6-0.4-0.6-0.1-0.5-0.3-0.2-0.5-0.7),x1x2x3x4x5x6(g,B)=e5e6e7e8e9(-0.6-0.2-0.4-0.1-0.8-0.7-0.8-0.4-0.6-0.2-0.9-0.8-0.2-0.7-0.6-0.8-0.1-0.2-0.4-0.9-0.9-0.9-0.2-0.4-0.5-0.3-0.3-0.6-0.3-0.5),x1x2x3x4x5x6(h,C)=e10e11e12e13(-0.7-0.4-0.5-0.3-0.4-0.2-0.6-0.5-0.4-0.4-0.4-0.3-0.9-0.6-0.7-0.4-0.5-0.3-0.1-0.5-0.4-0.7-0.6-0.1),
respectively. We perform “(f,A) AND (g,B)” and it is represented as follows:
(15)(f,A)∧~(g,B)x1x2x3x4x5x6=(e1,e5)(e1,e6)(e1,e7)(e1,e8)(e1,e9)(e2,e5)(e2,e6)(e2,e7)(e2,e8)(e2,e9)(e3,e5)(e3,e6)(e3,e7)(e3,e8)(e3,e9)(e4,e5)(e4,e6)(e4,e7)(e4,e8)(e4,e9)(-0.7-0.7-0.6-0.2-0.8-0.7-0.8-0.7-0.6-0.2-0.9-0.8-0.7-0.7-0.6-0.8-0.3-0.2-0.7-0.9-0.9-0.9-0.3-0.4-0.7-0.7-0.6-0.6-0.3-0.5-0.6-0.2-0.5-0.8-0.8-0.8-0.8-0.4-0.6-0.8-0.9-0.8-0.6-0.7-0.6-0.8-0.7-0.8-0.6-0.9-0.9-0.9-0.7-0.8-0.6-0.3-0.5-0.8-0.7-0.8-0.6-0.7-0.4-0.6-0.8-0.7-0.8-0.7-0.6-0.6-0.9-0.8-0.4-0.7-0.6-0.8-0.4-0.6-0.4-0.9-0.9-0.9-0.4-0.6-0.5-0.7-0.3-0.6-0.4-0.6-0.6-0.5-0.4-0.2-0.8-0.7-0.8-0.5-0.6-0.2-0.9-0.8-0.2-0.7-0.6-0.8-0.5-0.7-0.4-0.9-0.9-0.9-0.5-0.7-0.5-0.5-0.3-0.6-0.5-0.7).
If we require the 𝒩-soft set for the attributes D={d1,d2,d3,d4,d5,d6,d7}, where
(16)d1=(e1,e5),d2=(e1,e9),d3=(e2,e5),d4=(e2,e8),d5=(e3,e7),d6=(e4,e8),d7=(e4,e9),
then the resultant 𝒩-soft set for the 𝒩-soft sets (f,A) and (g,B) will be (k,D), say, which is represented as follows:
(17)x1x2x3x4x5x6(k,D)=d1d2d3d4d5d6d7(-0.7-0.7-0.6-0.2-0.8-0.7-0.7-0.7-0.6-0.6-0.3-0.5-0.6-0.2-0.5-0.8-0.8-0.8-0.6-0.9-0.9-0.9-0.7-0.8-0.4-0.7-0.6-0.8-0.4-0.6-0.4-0.9-0.9-0.9-0.5-0.7-0.5-0.5-0.3-0.6-0.5-0.7).
If we perform “(k,D) AND (h,C),” then we will have 7×4=28 attributes. Let
(18)P={(d1,e10),(d2,e12),(d3,e11),(d4,e13),(d5,e12),(d6,e12),(d7,e12)}
be the set of choice attributes of an observer, where (di,ejk)=max{di,ejk}. Then the resultant 𝒩-soft set (ω,P) is represented as follows:
(19)(ω,P)x1x2x3x4x5x6=(d1,e10)(d2,e12)(d3,e11)(d4,e13)(d5,e12)(d6,e12)(d7,e12)(-0.7-0.4-0.5-0.2-0.4-0.2-0.7-0.6-0.6-0.4-0.3-0.3-0.6-0.2-0.4-0.4-0.4-0.3-0.1-0.5-0.4-0.7-0.6-0.1-0.4-0.6-0.6-0.4-0.4-0.3-0.4-0.6-0.7-0.4-0.5-0.3-0.5-0.5-0.3-0.4-0.5-0.3).
The comparison table for the 𝒩-soft set (ω,P) is given by Table 2.Table 2
Comparison table.
x
1
x
2
x
3
x
4
x
5
x
6
x
1
7
4
4
6
5
7
x
2
4
7
4
5
5
6
x
3
3
5
7
5
5
7
x
4
3
2
3
7
4
7
x
5
4
4
3
5
7
7
x
6
1
1
1
1
1
7We now compute the raw sum, column sum, and the score for eachxi, and it is given by Table 3.Table 3
Raw-sum, column-sum, and the score.
Row-sum (ri)
Column-sum(ti)
Score(Si)
x
1
33
22
11
x
2
31
23
8
x
3
32
22
10
x
4
26
29
−3
x
5
30
27
3
x
6
12
41
−29From the score table (Table3), it is clear that the minimum score is -29, scored by x6, and the decision is in favor of selecting x6.


## 5. Application inBCK/BCI-Algebras

In what follows letE 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 [11]).
By asubalgebra of a BCK/BCI-algebra X based on 𝒩-function f (briefly, 𝒩-subalgebra of X), we mean an 𝒩-structure (X,f) in which f satisfies the following assertion:
(20)(∀x,y∈X)(f(x*y)≤⋁{f(x),f(y)}).Definition 9.
Let(f,A) be an 𝒩-soft set over a BCK/BCI-algebra X, where A is a subset of E. If there exists an attribute u∈A for which the 𝒩-structure (X,fu) is an 𝒩-subalgebra of X, then we say that (f,A) is an 𝒩-softBCK/BCI-algebra related to the attribute u (briefly, 𝒩u-softBCK/BCI-algebra). If (f,A) is an 𝒩u-soft BCK/BCI-algebra for all u∈A, we say that (f,A) is an 𝒩-softBCK/BCI-algebra.Example 10.
Suppose there are five colors in the universeU, that is,
(21)U:={white,blackish,reddish,green,yellow}
and E:={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:
(22)white♡x=white∀x∈U,blackish♡y={whiteify∈{blackish,green,yellow},blackishify∈{white,reddish},reddish♡z={whiteifz∈{reddish,yellow},reddishifz∈{white,blackish,green},green♡u={whiteifu∈{green,yellow},greenifu∈{white,blackish,reddish},yellow♡v={whiteifv=yellow,reddishifv=green,greenifv=reddish,yellowifv∈{white,blackish}.
Then (U,♡,white) is a BCK-algebra. Consider a set of attributes
(23)A:={beautiful,fine,moderate}⊆E
and define an 𝒩-soft set (f,A) over the BCK-algebra (U,♡,white) as follows:
(24)(f,A)={fbeautiful(white)=-0.7,fbeautiful(blackish)=-0.7,fbeautiful(reddish)=-0.7,fbeautiful(green)=-0.4,fbeautiful(yellow)=-0.4,ffine(white)=-0.9,ffine(blackish)=-0.7,ffine(reddish)=-0.3,ffine(green)=-0.6,ffine(yellow)=-0.3,fmoderate(white)=-0.8,fmoderate(blackish)=-0.2,fmoderate(reddish)=-0.6,fmoderate(green)=-0.2,fmoderate(yellow)=-0.2}.
The tabular representation of (f,A) is given by Table 4.Table 4
Tabular representation of(f,A).
(
f
,
A
)
White
Blackish
Reddish
Green
Yellow
Beautiful
−0.7
−0.7
−0.7
−0.4
−0.4
Fine
−0.9
−0.7
−0.3
−0.6
−0.3
Moderate
−0.8
−0.2
−0.6
−0.2
−0.2Then(f,A) is an 𝒩-soft BCK-algebra over the BCK-algebra (U,♡,white).Now let(g,A) be an 𝒩-soft set over the BCK-algebra (U,♡,white) with the tabular representation which is given by Table 5.Table 5
Tabular representation of(g,A).
(
g
,
A
)
White
Blackish
Reddish
Green
Yellow
Beautiful
−0.6
−0.6
−0.6
−0.06
−0.06
Fine
−0.2
−0.5
−0.4
−0.4
−0.3
Moderate
−0.7
−0.2
−0.6
−0.2
−0.2Then(g,A) is not an 𝒩fine-soft BCK-algebra over (U,♡,white) since
(25)gfine(blackish♡green)=gfine(white)=-0.2>-0.4=⋁{gfine(blackish),gfine(green)}.
Hence (g,A) is not an 𝒩-soft BCK-algebra over (U,♡,white). But we can verify that (g,A) is both an 𝒩beautiful-soft BCK-algebra and 𝒩moderate-soft BCK-algebra over (U,♡,white).Proposition 11.
Every𝒩-soft BCK/BCI-algebra (f,A) over a BCK/BCI-algebra X satisfies the following inequality:
(26)(∀x∈X)(∀u∈A)(fu(0)≤fu(x)).Proof.
For anyx∈X and u∈A, we have
(27)fu(0)=fu(x*x)≤⋁{fu(x),fu(x)}=fu(x).
This completes the proof.The problem we now discuss is as follows.If(g,B) is an 𝒩-soft BCK/BCI-algebra over a BCK/BCI-algebra X, then is every 𝒩-soft subset of (g,B) an 𝒩-soft BCK/BCI-algebra over X?Unfortunately this is not true as seen in the following example.Example 12.
Consider the universe:(28)U:={white,blackish,reddish,green,yellow}
which is considered in Example 10. Consider a soft machine $ which produces the following products:
(29)white$x={whiteifx∈{white,blackish,reddish},greenifx∈{green,yellow},(30)blackish$y={whiteify=blackish,blackishify∈{white,reddish},greenify=yellow,yellowify=green,(31)reddish$z={whiteifz=reddish,reddishifz∈{white,blackish},greenifz∈{green,yellow},(32)green$u={whiteifu∈{green,yellow},greenifu∈{white,blackish,reddish},(33)yellow$v={whiteifv=yellow,blackishifv=green,greenifv=blackish,yellowifv∈{white,reddish}.
Then (U,$,white) is a BCI-algebra. Take
(34)B={beautiful,fine,moderate,elegant,smart}
and let (g,B) be an 𝒩-soft set over the BCI-algebra (U,$,white) with the tabular representation which is given by Table 6.Table 6
Tabular representation of(g,B).
(
g
,
B
)
White
Blackish
Reddish
Green
Yellow
Beautiful
−0.8
−0.5
−0.3
−0.2
−0.2
Fine
−0.5
−0.3
−0.5
−0.4
−0.3
Moderate
−0.6
−0.2
−0.4
−0.3
−0.2
Elegant
−0.7
−0.7
−0.6
−0.3
−0.3
Smart
−0.8
−0.2
−0.7
−0.5
−0.2Then(g,B) is an 𝒩-soft BCI-algebra over (U,$,white). Now let (f,A) be an 𝒩-soft subset of (g,B), where
(35)A={fine,elegant,smart}⊂B,
and the tabular representation of (f,A) is given by Table 7.Table 7
Tabular representation of(f,A).
(
f
,
A
)
White
Blackish
Reddish
Green
Yellow
Fine
−0.6
−0.3
−0.6
−0.5
−0.3
Elegant
−0.9
−0.8
−0.7
−0.7
−0.4
Smart
−0.8
−0.1
−0.5
−0.3
−0.1Then(36)felegent(blackish$green)=felegent(yellow)=-0.4>-0.7=⋁{felegent(blackish),felegent(green)},
and so (f,A) is not an 𝒩elegent-soft BCI-algebra over (U,$,white). Hence (f,A) is not an 𝒩-soft BCI-algebra over (U,$,white).But, we have the following theorem.Theorem 13.
For any subsetA of E, let (f,A) be an 𝒩-soft BCK/BCI-algebra over a BCK/BCI-algebra X. If B is a subset of A, then (f∣B,B) is an 𝒩-soft BCK/BCI-algebra over X.Proof.
Straightforward.The following example shows that there exists an𝒩-soft set (f,A) over a BCK/BCI-algebra X such that(i)
(f,A) is not an 𝒩-soft BCK/BCI-algebra over a BCK/BCI-algebra X,(ii)
there exists a subsetB of A such that (f∣B,B) is an 𝒩-soft BCK/BCI-algebra over a BCK/BCI-algebra X.Example 14.
Let(U,♡,white) be a BCK-algebra as in Example 10. Consider a set of attributes A:={beautiful,fine,moderate,smart,chaste}⊆E. Let (f,A) be an 𝒩-soft set over (U,♡,white) with the tabular representation which is given by Table 8.Table 8
Tabular representation of(f,A).
(
f
,
A
)
White
Blackish
Reddish
Green
Yellow
Beautiful
−0.6
−0.6
−0.6
−0.3
−0.3
Fine
−0.8
−0.7
−0.2
−0.5
−0.2
Moderate
−0.7
−0.1
−0.5
−0.1
−0.1
Smart
−0.1
−0.2
−0.3
−0.4
−0.5
Chaste
−0.3
−0.2
−0.6
−0.7
−0.2Then(f,A) is neither an 𝒩smart-soft BCK-algebra nor an 𝒩chaste-soft BCK-algebra over (U,♡,white). Hence (f,A) is not an 𝒩-soft BCK-algebra over (U,♡,white). But if we take
(37)B:={beautiful,fine,moderate}⊆A,
then (f∣B,B) is an 𝒩-soft BCK-algebra over (U,♡,white).Definition 15.
Let(f,A) and (g,B) be two 𝒩-soft sets in (X,E). The union of (f,A) and (g,B) is defined to be the 𝒩-soft set (h,C) in (X,E) satisfying the following conditions: (i)
C=A∪B,(ii)
for allx∈C,
(38)hx={fxifx∈A∖B,gxifx∈B∖A,fx∩gxifx∈A∩B.In this case, we write(f,A)∪~(g,B)=(h,C).Lemma 16 (see [11]).
If(X,f) and (X,g) are 𝒩-subalgebras of a BCK/BCI-algebra X, then the union (X,f∪g) of (X,f) and (X,g) is an 𝒩-subalgebra of X.Theorem 17.
If(f,A) and (g,B) are 𝒩-soft BCK/BCI-algebras over a BCK/BCI-algebra X, then the union of (f,A) and (g,B) is an 𝒩-soft BCK/BCI-algebra over X.Proof.
Let(f,A)∪~(g,B)=(h,C) be the union of (f,A) and (g,B). Then C=A∪B. For any x∈C, if x∈A∖B (resp. x∈B∖A), then (X,hx)=(X,fx) (resp. (X,hx)=(X,gx)) is an 𝒩-subalgebra of X. If A∩B≠∅, then (X,hx)=(X,fx∪gx) is an 𝒩-subalgebra of X for all x∈A∩B by Lemma 16. Therefore (h,C) is an 𝒩-soft BCK/BCI-algebra over a BCK/BCI-algebra X.Definition 18.
Let(f,A) and (g,B) be two 𝒩-soft sets in (X,E). The intersection of (f,A) and (g,B) is the 𝒩-soft set (h,C) in (X,E) where C=A∪B and, for every x∈C,
(39)hx={fxifx∈A∖B,gxifx∈B∖A,fx∩gxifx∈A∩B.In this case, we write(f,A)∩~(g,B)=(h,C).Theorem 19.
Let(f,A) and (g,B) be 𝒩-soft BCK/BCI-algebras over a BCK/BCI-algebra X. If A and B are disjoint, then the intersection of (f,A) and (g,B) is an 𝒩-soft BCK/BCI-algebra over X.Proof.
Let(f,A)∩~(g,B)=(h,C) be the intersection of (f,A) and (g,B). Then C=A∪B. Since A∩B=∅, if x∈C, then either x∈A∖B or x∈B∖A. If x∈A∖B, then (X,hx)=(X,fx) is an 𝒩-subalgebra of X. If x∈B∖A, then (X,hx)=(X,gx) is an 𝒩-subalgebra of X. Hence (h,C) is an 𝒩-soft BCK/BCI-algebra over a BCK/BCI-algebra X.The following example shows that Theorem19 is not valid if A and B are not disjoint.Example 20.
Suppose there are five colors in the universeU, that is,
(40)U:={white,blackish,reddish,green,yellow}.
Let Ξ be a soft machine to mix two colors according to order in such a way that we have the following results:
(41)xΞwhite=x∀x∈U,xΞblackish={whiteifx∈{white,blackish},xifx∈{reddish,green,yellow},xΞreddish={reddishifx∈{white,blackish},whiteifx=reddish,yellowifx=green,greenifx=yellow,xΞgreen={greenifx∈{white,blackish},yellowifx=reddish,whiteifx=green,reddishifx=yellow,xΞyellow={yellowifx∈{white,blackish},greenifx=reddish,reddishifx=green,whiteifx=yellow.
Then (U,Ξ,white) is a BCI-algebra. Consider sets of attributes:
(42)A={beautiful,fine,elegant,smart},B={elegant,smart,chaste}.
Then A and B are not disjoint. Let (f,A) and (g,B) be 𝒩-soft sets over (U,Ξ,white) having the tabular representations which are given in Tables 9 and 10, respectively.Table 9
Tabular representation of(f,A).
(
f
,
A
)
White
Blackish
Reddish
Green
Yellow
Beautiful
−0.7
−0.6
−0.3
−0.3
−0.3
Fine
−0.6
−0.5
−0.4
−0.2
−0.2
Elegant
−0.8
−0.5
−0.1
−0.3
−0.1
Smart
−0.5
−0.5
−0.2
−0.2
−0.4Table 10
Tabular representation of(g,B).
(
g
,
B
)
White
Blackish
Reddish
Green
Yellow
Elegant
−0.8
−0.6
−0.3
−0.1
−0.1
Smart
−0.7
−0.6
−0.3
−0.3
−0.5
Chaste
−0.9
−0.5
−0.2
−0.4
−0.2Then(f,A) and (g,B) are 𝒩-soft BCI-algebras of (U,Ξ,white). But the intersection (f,A)∩~(g,B)=(h,C) of (f,A) and (g,B) is not an 𝒩-soft BCI-algebra of (U,Ξ,white) since
(43)(felegant∩gelegant)(greenΞreddish)=(felegant∩gelegant)(yellow)=-0.1>-0.3=⋁{(felegant∩gelegant)(green),(felegant∩gelegant)(reddish)},
that is, (U,felegant∩gelegant) is not an 𝒩-subalgebra of (U,Ξ,white).Theorem 21.
If(f,A) and (g,B) are two 𝒩-soft BCK/BCI-algebras over a BCK/BCI-algebra X, then (f,A)∨~(g,B) is a fuzzy soft BCK/BCI-algebra over X.Proof.
We note that(44)(f,A)∨~(g,B)=(h,A×B),
where h(u,v)=⋁{fu,gv} for all (u,v)∈A×B. For any x,y∈X, we have
(45)h(u,v)(x*y)=⋁{fu(x*y),gv(x*y)}≤⋁{⋁{fu(x),fu(y)},⋁{gv(x),gv(y)}}=⋁{⋁{fu(x),gv(x)},⋁{fu(y),gv(y)}}=⋁{h(u,v)(x),h(u,v)(y)}.
Hence (h,A×B)=(f,A)∨~(g,B) is an 𝒩-soft BCK/BCI-algebra based on (u,v). Since (u,v) is arbitrary, we know that (h,A×B)=(f,A)∨~(g,B) is an 𝒩-soft BCK/BCI-algebra over X.


## 6. Conclusions

Using the notions of soft sets and𝒩-structures, we have introduced the concept of 𝒩-soft sets and considered its application in both a decision making problem and a BCK/BCI-algebra. In an imprecise environment, the importance of the problem of decision making has been emphasized in recent years. We have presented an 𝒩-soft set theoretic approach towards solution of the decision making problem. We have taken the algorithm that involves Construction of Comparison Table from the resultant 𝒩-soft set and the final decision based on the minimum score computed from the Comparison Table (see Tables 2 and 3). Through the application in a BCK/BCI-algebra, we have introduced the notion of 𝒩-soft BCK/BCI-algebras and have investigated related properties.



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