On central soft sets: Definitions and basic operations 1

Abstract

The concept of a central soft set related with some common decision making problems in real life is introduced in this paper. Some basic operations on central soft sets and theirs properties such as distributive law and associative law are given. The study also shows that some classic operations between soft sets can be obtained by central soft sets with selecting different central sets. In the part of application, the concepts of an evaluation system for a parameter set and desired objects of a central soft set are initiated. An algorithm is presented to solve such decision making problems.

Keywords

Central soft set soft set intersection union evaluation system

1 Introduction

Molodtsov [1, 2] innovated a novel concept of soft sets as a new mathematical tool for describe some uncertainties which are appeared around everywhere. A soft set is a tuple which associates with a set of parameters and a mapping from the parameter set into the power set of an universe set. In fact, it is a parameterized family of subsets of the universe set. At present soft set theory has combined with several directions such as universal algebra [3 –5], relation analysis [6 –9] and other mathematical domains [10 , 14]. Soft set theory in application has its own significance. Especially the idea to solve decision-making problems [15 –18] is intuitive and directly. In this paper we try to solve some decision-making problems by a new type of soft sets.

This case is often encountered in our real life, for example, a jury composed of more than one person will make investigations on a project. The contents of this project needed to be examined are involved with multiple fields. Each member of the jury will make an appropriate judgment on the basis of certain sufficient knowledge background. However, everyone has a specialized area in which they excel generally. So naturally, their relevant scoring in non specialized field need for discretion. In the end we need to make a comprehensive evaluation based on these inspection results. Concerned about this phenomenon, the concept of central soft sets is proposed in this paper. It is different from the study which focuses on valuation objects of attribute sets [10 –12], that we pay attention to central attribute sets of soft sets, and examine what role they play in the operations between soft sets. In the part of theoretical application, we define an evaluation system of a parameter set consist of central soft sets. Then an algorithm for making corresponding decision of an evaluation system is given.

The rest of this paper is organized as follows. In the second section the concept of a central soft set is proposed firstly. Some properties of basic operations on central soft sets over a universe set are given in detail. We research the relationship between operations defined in this paper and some classic operations in soft set theory. It should be noted that there are some similar but different conclusions. Therefore central soft sets can not be simply considered as a simple repetition of soft sets. In the last section, we will study an evaluation system for a parameter set and give the method of obtaining its solution.

2 Preliminaries

First we present some basic definitions and notations used in what follows. In this paper, U is an initial universe set. The symbol $P (U)$ denotes the power set of U. Let E be a set of parameters which usually are initial attributes, characteristics, or properties of objects in U.

To try to solve problems by an intuitive, simple and practical way is an important and distinguishing feature in the study of soft sets. This is also a reason why we introduce the concept of a central soft set here.

Definition 2.1. Let A be a subset of the parameter set E. A pair (f, A) is called a soft set with a central set A, where f is a mapping given by $f : E \to P (U)$ . For simply, we call (f, A) a central soft set.

For two central soft sets (f, A) and (g, B), we say (f, A) = (g, B), if f = g and A = B.

In fact, a central soft set (f, A) over the universe set U gives a complete rather than partial parametrization of U by the mapping f. It is different from soft sets (see [1, 2]). Central sets are to illustrate some particularities, which can have a variety of meanings with different backgrounds of problems. For example, in the instance of choosing houses, let (f, A) be a central soft set presents information about the scoring given by Mr. X, where A is a parameter set related with his field of expertise. To take another instance, teachers in a school who are good at teaching some certain subjects will be chosen by students. Let E be a set of school subjects, and U consists of all teachers in this school. For each student X_i, we assume that A_i is the set of his excellent courses. A mapping $f_{i} : E \to P (U)$ denotes the evaluation result by X_i. Then (f_i, A_i) is the central soft set given by X_i. That is to say, in this paper, central sets look as high confidence regions of information given by mappings. Concretely, the function of central sets is mainly reflected in superiority of operations between soft sets, which will be shown in the next definitions.

Definition 2.2. The union of two central soft sets (f, A) and (g, B) over a common universe U is a central soft set (h, C), where C = A ∪ B and for all e ∈ E, $h (e) = {\begin{matrix} f (e), & if e \in A - B; \\ g (e), & if e \in B - A; \\ f (e) \cup g (e), & otherwise . \end{matrix}$

We write (f, A) ⊔ (g, B) = (h, C).

This operation union between central soft sets is given from the view of information synthesis. It is similar as the union $\tilde{\cup}$ of two soft sets shown in [2]. Of course, the differences also exist. Let e be in the complementary set of A ∪ B. h (e) is defined to be the union of f (e) and g (e). However, for the union of soft sets in [2], h (e) needs not to be defined.

Example 2.1. Suppose that U = {h₁, h₂, h₃, h₄, h₅} represents a set of houses. Let E = {I, II, III, IV, V, VI, VII, VIII}, where roman numerals I to VIII represent some attributes of houses respectively such as reasonable design space, green environment, excellent property management, convenient transportation. Consider two central soft sets (f, A) and (g, B) defined over U, where A = {IV, V} and B = {I, IV} are two central sets which are precisely the areas of expertise of Mr. X and Mr. Y in choosing houses respectively, the mappings f and g are defined as follows: $\begin{matrix} f (I) = {h_{2}, h_{3}, h_{4}, h_{5}}, f (II) = {h_{2}, h_{5}}, \\ f (III) = {h_{2}, h_{3}}, f (IV) = {h_{1}, h_{4}, h_{5}}, \\ f (V) = {h_{1}, h_{4}}, f (VI) = {h_{1}, h_{5}}, \\ f (VII) = {h_{2}, h_{5}}, f (VIII) = {h_{3}, h_{4}, h_{5}}; \end{matrix}$ and $\begin{matrix} g (I) = {h_{2}, h_{3}, h_{4}}, g (II) = {h_{1}, h_{5}}, \\ g (III) = {h_{3}, h_{4}}, g (IV) = {h_{2}, h_{4}, h_{5}}, \\ g (V) = {h_{2}, h_{4}}, g (VI) = {h_{4}, h_{5}}, \\ g (VII) = {h_{2}, h_{4}}, g (VIII) = {h_{1}, h_{4}, h_{5}} . \end{matrix}$

By definitions of union and intersection on central soft sets, we have (f, A) ⊔ (g, B) = (h, {I, IV, V}), where h is defined as: $\begin{matrix} h (I) = {h_{2}, h_{3}, h_{4}}, h (II) = {h_{1}, h_{2}, h_{5}}, \\ h (III) = {h_{2}, h_{3}, h_{4}}, h (IV) = {h_{1}, h_{2}, h_{4}, h_{5}}, \\ h (V) = {h_{1}, h_{4}}, h (VI) = {h_{1}, h_{4}, h_{5}}, \\ h (VII) = {h_{2}, h_{4}, h_{5}}, h (VIII) = {h_{1}, h_{3}, h_{4}, h_{5}} . \end{matrix}$

If we take a new central soft sets (f, C), where C = {III, IV}. Then (f, C) ⊔ (g, B) = (j, {I, III, IV}), where j is defined as: $\begin{matrix} j (I) = {h_{2}, h_{3}, h_{4}}, j (II) = {h_{1}, h_{2}, h_{5}}, \\ j (III) = {h_{2}, h_{3}}, j (IV) = {h_{1}, h_{2}, h_{4}, h_{5}}, \\ j (V) = {h_{1}, h_{2}, h_{4}}, j (VI) = {h_{1}, h_{4}, h_{5}}, \\ j (VII) = {h_{2}, h_{4}, h_{5}}, j (VIII) = {h_{1}, h_{3}, h_{4}, h_{5}} . \end{matrix}$

Since we choose two different central sets for the mapping f, two different central soft sets (h, {I, IV, V}) and (j, {I, III, IV}) are obtained. It demonstrates that central sets play an important role as mappings in the operation of union.

Definition 2.3. The intersection of two central soft sets (f, A) and (g, B) over a common universe U is a central soft set (h, C), where C = A ∩ B, and for all e ∈ E, $h (e) = {\begin{matrix} g (e), & if e \in A - B; \\ f (e), & if e \in B - A; \\ f (e) \cap g (e), & otherwise . \end{matrix}$

We write (f, A) ⊓ (g, B) = (h, C).

Clearly, (f, A) ⊔ (f, B) = (f, A ∪ B) and (f, A) ⊓ (f, B) = (f, A ∩ B).

The intersection of central soft sets is different from the extended intersection of soft sets defined in [2] completely. This definition may seem strange, however, actually it gives the maximum central soft set which is contained in two original central soft sets. It will be shown in the following conclusions.

Example 2.2. As it shown in Example 2.1, for the intersection operation, we have (f, A) ⊓ (g, B) = (k, {IV}), where k is defined as: $\begin{matrix} k (I) = {h_{2}, h_{3}, h_{4}, h_{5}}, k (II) = {h_{5}}, \\ k (III) = {h_{3}}, k (IV) = {h_{4}, h_{5}}, \\ k (V) = {h_{2}, h_{4}}, k (VI) = {h_{5}}, \\ k (VII) = {h_{2}}, k (VIII) = {h_{4}, h_{5}} . \end{matrix}$

Remark 2.1. In [13], the concept of a soft set is defined as follows: a soft set F_A on the universe U is defined by the set of ordered pairs $F_{A} = {(x, f_{A} (x)) : x \in E, f_{A} (x) \in P (U)}$ , where $f_{A} : E \to P (U)$ such that f_A (x) =∅ if and only if x ∉ A. The complete classification of universe sets is given by attribute sets of soft sets. Each mapping f_A in a soft set F_A is always corresponding to a parameter set A. In order to make the expression of mappings in soft sets consistent with central soft sets, we adopt this definition to describe soft sets.

First we assume that F_A and G_B are two soft sets defined as above. For mappings of soft sets, we will select central sets according to some correspondences respectively. We try to see more deeply the function of central sets in the operations of centralsoft sets.

(1) We choose two central sets A and B for the mappings f_A and g_B respectively. By Definition 2.2, the union of two central soft sets (f_A, A) and (g_B, B) is a new central soft set (h_C, C), i.e., (f_A, A) ⊔ (g_B, B) = (h_C, C), where C = A ∪ B and for all x ∈ E, $h_{C} (x) = {\begin{matrix} f_{A} (x), & if x \in A - B; \\ g_{B} (x), & if x \in B - A; \\ f_{A} (x) \cup g_{B} (x), & otherwise . \end{matrix}$

For all x ∉ C, we have h_C (x) =∅. Accordingly, for the mapping h_C a soft set H_C = {(x, h_C (x)) : x ∈ E} is obtained. In fact, for each a soft set F_A, there exists a correspondence with a central soft set (f_A, A). Then, by the operation defined on central soft sets, actually we define an operation $\tilde{\cup}$ between soft sets F_A and G_B, i.e., $F_{A} \tilde{\cup} G_{B} = H_{C}$ . This definition is consistent with the operation union given in [2]. We use a diagram to represent thisrelationship:

(2) Let E be the default central set. Then the union of two central soft sets (f_A, E) and (g_B, E) is a central soft set (l, E), where l is defined as: $l (x) = f_{A} (x) \cup g_{B} (x), {for all x \in E .$

Clearly f_A (x)∪ g_B (x) = ∅ for all x ∉ C, if C = A ∪ B. Thus a soft set L_C = {(x, l (x)) : x ∈ E} which corresponds to l is obtained. So, if we choose the set of all parameters as the default central set, the union of central soft sets defines a natural union operation ∪ between soft sets: $F_{A} \cup G_{B} = L_{C} .$

Similarly, we can consider the intersection of two central soft sets (f_A, E) and (g_B, E). Let (j, E) = (f_A, E) ⊓ (g_B, E), where j is defined as: $j (x) = f_{A} (x) \cap g_{B} (x), {for all x \in E .$

The mapping j corresponds to a soft set J_D = {(x, j (x)) : x ∈ E}, where D = {x ∈ E : j (x) ¬ = ∅}. Then the operation intersection of central soft sets just defines a natural intersection operation ∩ between soft sets: $F_{A} \cap G_{B} = J_{D} .$

(3) For each mapping f_A in a soft set F_A, we take the complement set of A as the central set. i.e., we choose two central sets E - A and E - B for the mappings f_A and g_B respectively.

Assume that (f_A, E - A) ⊓ (g_B, E - B) = (k, E - (A ∪ B)). For all x ∈ E, $k (x) = {\begin{matrix} f_{A} (x), & if x \in A - B; \\ g_{B} (x), & if x \in B - A; \\ f_{A} (x) \cap g_{B} (x), & otherwise . \end{matrix}$

Accordingly, for the mapping k a soft set K_D = {(x, k (x)) : x ∈ E} is obtained, where D = {x ∈ E : k (x) ¬ = ∅}. Then the operation union of central soft sets gives a new operation $F_{A} \hat{⊓} G_{B} = K_{D},$ which is similar to the extended intersection ⊓_ɛ between soft sets (see [6]).

The discussion above shows that, for the operations of union and intersection of central soft sets, different mappings in soft sets are obtained if central sets are not same. Central sets and mappings play same important role in operations between central soft sets. In one sense, central soft sets and operations defined above can be looked as a generalization of classic soft sets and some related operations.

Definition 2.4. For two central soft sets (f, A) and (g, B) over a common universe U, we write (f, A) ⊑ (g, B), and call it a central soft information order if (f, A) ⊔ (g, B) = (g, B).

The following equivalent forms of this ordering defined between central soft sets can be obtained immediately by definitions above.

Proposition 2.1. For two central soft sets (f, A) and (g, B) over a common universe U, (f, A) ⊑ (g, B), if and only if any of the following conditions are true

A ⊆ B and for all e ∉ B - A, f (e) ⊆ g (e);

(f, A) ⊓ (g, B) = (f, A).

By Proposition 2.1, we can directly show that the central soft information order is antisymmetric and transitive. Therefore, this ordering is a partialorder.

Example 2.3. As it shown in Example 2.1, let C = {I, IV, VI} be a central set, and the mapping $m : E \to P (U)$ is defined as follows: $\begin{matrix} m (I) = {h_{2}, h_{3}, h_{4}}, m (II) = {h_{1}, h_{5}}, \\ m (III) = {h_{3}, h_{4}}, m (IV) = {h_{2}, h_{4}, h_{5}}, \\ m (V) = {h_{2}, h_{4}}, m (VI) = {h_{1}}, \\ m (VII) = {h_{2}, h_{4}}, m (VIII) = {h_{1}, h_{4}, h_{5}} . \end{matrix}$

We have (g, B) ⊑ (m, C).

By the formula (g, B) ⊑ (m, C) showed in this example, we can intuitively understand the meaning of central soft information order from experience. Suppose that (m, C) means a result of evaluation of Mr. Z for choosing houses. The relation B ⊆ C says that Mr. Z has a wider professional knowledge than Mr. X. Meanwhile, for each parameter e not in the set C - B which is an exclusive advantage region of Mr. Z, the set m (e) is “bigger” than the set g (e).

Definition 2.5. The complement of a central soft set (f, A) is denoted by (f, A) ^c = (f^c, A^c), where A^c = E - A, f^c (e) = U - f (e) for all e ∈ E.

We use the same notation of complement of a soft set in [2], but please note that the substance of two definitions is different obviously.

Definition 2.6. For two central soft sets (f, A) and (g, B) over a common universe U, (f, A) - (g, B) is defined to be a central soft set (f, A) ⊓ (g, B) ^c.

In fact, let (f, A) ⊓ (g, B) ^c = (h, C), then C = A - B and $h (e) = {\begin{matrix} g^{c} (e), & if e \in A \cap B; \\ f (e), & if e \in (A \cup B)^{c}; \\ f (e) \cap G^{c} (e), & otherwise . \end{matrix}$

For e ∉ A ∩ B, h (e) ⊆ f (e). According to Definition 2.4, we have (f, A) - (g, B) ⊑ (f, A). It accords with the general characteristics of thisoperation.

Example 2.4. As it shown in Example 2.3, (m, C) - (g, B) = (n, {VI}), where the mapping $n : E \to P (U)$ is defined as follows: $\begin{matrix} n (I) = \emptyset, n (II) = {h_{1}, h_{5}}, \\ n (III) = {h_{3}, h_{4}}, n (IV) = \emptyset, \\ n (V) = {h_{2}, h_{4}}, n (VI) = {h_{2}, h_{3}, h_{4}, h_{5}}, \\ n (VII) = {h_{2}, h_{4}}, n (VIII) = {h_{1}, h_{4}, h_{5}} . \end{matrix}$

3 Properties of operations on central soft sets

In this part we study the properties of operations on central soft sets which are over a same universe set.

Theorem 3.1. Let (f, A) and (g, B) be two central soft sets over a same universe U. Then

[(f, A) ⊓ (g, B)] ^c = (f, A) ^c ⊔ (g, B) ^c;

[(f, A) ⊔ (g, B)] ^c = (f, A) ^c ⊓ (g, B) ^c.

Proof. Clearly the central sets in both sides of the first equation are (A ∩ B) ^c. We write (f, A) ⊓ (g, B) = (h, A ∩ B) and (f, A) ^c ⊔ (g, B) ^c = (k, A^c ∪ B^c).

Take any e ∈ E. If e ∈ A^c - B^c, i.e., e ∈ B - A, we have h^c (e) = U - h (e) = U - f (e) = f^c (e) = k (e). If e ∈ B^c - A^c, then h^c (e) = g^c (e) = k (e) similarly. Otherwise e ∉ (A - B) ∪ (B - A), then h^c (e) = U - h (e) = U - (f (e) ∩ g (e)) = f^c (e) ∪ g^c (e) = k (e). In summary, we have [(f, A) ⊓ (g, B)] ^c = (f, A) ^c ⊔ (g, B) ^c.

According to the definitions above, we can show that [(f, A) ⊔ (g, B)] ^c and (f, A) ^c ⊓ (g, B) ^c both represent the same central soft set (j, (A ∪ B) ^c), where the mapping j is defined as follows: $j (e) = {\begin{matrix} U - f (e), & if e \in A - B; \\ U - g (e), & if e \in B - A; \\ U - [f (e) \cup g (e)], & otherwise . \end{matrix}$

Therefore, the second equation is also established.

Remark 3.1. By the difference of Definition 2.5 in this paper and [2], we know that Theorem 3.1 is not just a repeat of Theorem 4.2 contained in [6].

The associative laws are also true for two operations union and intersection defined here.

Theorem 3.2. Let (f, A), (g, B) and (h, C) be central soft sets over a same universe set U. Then

(f, A) ⊓ [(g, B) ⊓ (h, C)] = [(f, A) ⊓ (g, B)] ⊓ (h, C);

(f, A) ⊔ [(g, B) ⊔ (h, C)] = [(f, A) ⊔ (g, B)] ⊔ (h, C).

Proof. Let write $\begin{matrix} (g, B) ⊓ (h, C) = (j, B \cap C); \\ (f, A) ⊓ (g, B) = (k, A \cap B); \\ (f, A) ⊓ (j, B \cap C) = (l, A \cap B \cap C); \\ (k, A \cap B) ⊓ (h, C) = (m, A \cap B \cap C) . \end{matrix}$

We will show that l (e) = m (e) for all e ∈ E in order to prove the associative law of the operation intersection.

If e ∈ (B ∩ C) - A, then l (e) = f (e). Since e ∈ C - (A ∩ B), m (e) = k (e) = f (e) . Thus m (e) = l (e).

If e ∈ A - (B ∩ C), it can be divided into three conditions:

If e ∈ A, e ∉ B and e ∉ C, then l (e) = j (e) = g (e) ∩ h (e) and m (e) = k (e) ∩ h (e) = g (e) ∩ h (e).

If e ∈ A, e ∉ B and e ∈ C, then l (e) = j (e) = g (e), m (e) = k (e) = g (e).

If e ∈ A, e ∉ C and e ∈ B, then l (e) = j (e) = h (e), m (e) = h (e).

If e ∉ A - (B ∩ C) and e ∉ (B ∩ C) - A, then l (e) = m (e) = f (e) ∩ g (e) ∩ h (e).

Thus we obtain that l (e) = m (e) for all e ∈ E.

The proof of the next equation is omitted here.

The distributive laws are false for soft sets shown in [6]. But they are true for central soft sets with operations defined here.

Theorem 3.3. Let (f, A), (g, B) and (h, C) be central soft sets over a same universe U. Then

(f, A) ⊓ [(g, B) ⊔ (h, C)] = [(f, A) ⊓ (g, B)] ⊔ [(f, A) ⊓ (h, C)];

(f, A) ⊔ [(g, B) ⊓ (h, C)] = [(f, A) ⊔ (g, B)] ⊓ [(f, A) ⊔ (h, C)].

Proof. Here we only prove the first equation. We write $\begin{matrix} (f, A) ⊓ (g, B) = (i, A \cap B); \\ (f, A) ⊓ (h, C) = (j, A \cap C); \\ (g, B) ⊔ (h, C) = (k, B \cup C); \\ (f, A) ⊓ (k, B \cup C) = (l, A \cap (B \cup C)); \\ (i, A \cap B) ⊔ (j, A \cap C) = (m, (A \cap B) \cup (A \cap C)) . \end{matrix}$

Then we need to show that l (e) = m (e) for all e ∈ E. It can be divided into the following severalkinds:

If e ∈ A - (B ∪ C), then l (e) = k (e) = g (e) ∪ h (e), i (e) = g (e) and j (e) = h (e). Since e ∉ A ∩ B and e ∉ A ∩ C, we have m (e) = i (e) ∪ j (e) = g (e) ∪ h (e) = l (e).

If e ∈ (B ∪ C) - A, then l (e) = f (e). Since e ∉ A ∩ B and e ∉ A ∩ C, we have m (e) = i (e) ∪ j (e). The set (B ∪ C) - A can be divided into three mutually disjoint partsS₁ = (B - A) - (B ∩ C), S₂ = (C - A) - (B ∩ C) and S₃ = (B ∩ C) - A.

If e ∈ S₁, i (e) = f (e) and j (e) = f (e) ∩ h (e). Then m (e) = i (e) ∪ j (e) = f (e).

If e ∈ S₂, j (e) = f (e) and i (e) = f (e) ∩ g (e). Then m (e) = i (e) ∪ j (e) = f (e).

If e ∈ S₃, i (e) = j (e) = f (e). Then m (e) = i (e) ∪ j (e) = f (e).

Thus we obtain that l (e) = f (e) = m (e) for all e ∈ (B ∪ C) - A.

Otherwise e is in the complementary set of [(B ∪ C) - A] ∪ [A - (B ∪ C)], then l (e) = f (e) ∩ k (e). The complementary set of [(B ∪ C) - A] ∪ [A - (B ∪ C)] can be divided into four mutually disjoint parts $\begin{matrix} S_{4} & = & (A \cap B) - (B \cap C), \\ S_{5} & = & (A \cap C) - (B \cap C), \\ S_{6} & = & A \cap B \cap C, and \\ S_{7} & = & E - (A \cup B \cup C) . \end{matrix}$

If e ∈ S₄, then k (e) = g (e) and m (e) = i (e) = f (e) ∩ g (e) = f (e) ∩ k (e) = l (e).

If e ∈ S₅, then k (e) = h (e) and m (e) = j (e) = f (e) ∩ h (e) = l (e).

If e ∈ S₆ or e ∈ S₇, then k (e) = g (e) ∪ h (e) and m (e) = i (e) ∪ j (e) = [f (e) ∩ g (e)] ∪ [f (e) ∩ h (e)] = f (e) ∩ [g (e) ∪ h (e)] = f (e) ∩ k (e) = l (e) .

Thus l (e) = m (e) for all e ∈ E - [(B ∪ C) - A] - [A - (B ∪ C)].

By the proof above we have l (e) = m (e) for all e ∈ E. So the first equation has been shown. By the same way the next distributive property of union with respect to intersection can be shown.

Next we get a natural property which similar as properties of operations of addition and subtraction of numbers.

Theorem 3.4. Let (f, A), (g, B) and (h, C) be central soft sets over a same universe U. Then $\begin{matrix} (f, A) - (g, B) - (h, C) \\ = (f, A) - [(g, B) ⊔ (h, C)] . \end{matrix}$

Proof. According to the definition of the operation subtraction - and Theorem 3.1, we have $\begin{matrix} (f, A) - (g, B) - (h, C) & = & (f, A) ⊓ (g, B)^{c} ⊓ (h, C)^{c} \\ = & (f, A) ⊓ [(g, B) ⊔ (h, C)]^{c} \\ = & (f, A) - [(g, B) ⊔ (h, C)] . \end{matrix}$

Theorem 3.5. Let {(f_i, A_i) : i ∈ I} be a set of central soft sets over a same universe U. Then $⨆_{i \in I} (f_{i}, A_{i}) = (h, ⋃_{i \in I} A_{i}),$ where $h : E \to P (U)$ is defined as follows:

∀e ∈ E, let I_e = {i ∈ I : e ∈ A_i}, we have $h (e) = {\begin{matrix} ⋃_{i \in I_{e}} f_{i} (e), & if I_{e} \neg = \emptyset; \\ ⋃_{i \in I} f_{i} (e), & otherwise . \end{matrix}$ Proof. First we show that $(h, ⋃_{i \in I} A_{i})$ is an upper bound of {(f_i, A_i) : i ∈ I} by Proposition 2.1. Let j ∈ I and $e \notin ⋃_{i \in I} A_{i} - A_{j}$ . If I_e¬ = ∅, then j ∈ I_e. We have $h (e) = ⋃_{i \in I_{e}} f_{i} (e) \supseteq f_{j} (e)$ . By the definition of h, we show that f_j (e) ⊆ h (e) for $e \notin ⋃_{i \in I} A_{i} - A_{j}$ . Then $(f_{j}, A_{j}) ⊑ (h, ⋃_{i \in I} A_{i})$ .

Suppose that (g, B) is another upper bound of the set {(f_i, A_i) : i ∈ I}. Clearly we have $⋃_{i \in I} A_{i} \subseteq B$ . In order to show that $(h, ⋃_{i \in I} A_{i}) ⊑ (g, B)$ , we only need to prove h (e) ⊆ g (e) for $e \notin B - ⋃_{i \in I} A_{i}$ . In fact, if e ∉ B, then e ∉ B - A_i for all i ∈ I. So f_i (e) ⊆ g (e). Thus $h (e) \subseteq ⋃_{i \in I} f_{i} (e) \subseteq g (e)$ . If $e \in ⋃_{i \in I} A_{i}$ , then e ∉ B - A_j for all j ∈ I_e. We obtain that f_j (e) ⊆ g (e) for all j ∈ I_e. Thus $h (e) = ⋃_{j \in I_{e}} f_{j} (e) \subseteq g (e)$ . So $(h, ⋃_{i \in I} A_{i}) ⊑ (g, B)$ . By the definition of supremum and what we have proved above, it follows that $⨆_{i \in I} (f_{i}, A_{i}) = (h, ⋃_{i \in I} A_{i}) .$

Theorem 3.6. Let {(f_i, A_i) : i ∈ I} be a set of central soft sets over a same universe U. Then $(f, A) ⊓ ⨆_{i \in I} (f_{i}, A_{i}) = ⨆_{i \in I} [(f, A) ⊓ (f_{i}, A_{i})],$

Proof. Let $\begin{matrix} ⨆_{i \in I} (f_{i}, A_{i}) = (g, ⋃_{i \in I} A_{i}), \\ (f, A) ⊓ (g, ⋃_{i \in I} A_{i}) = (k, A \cap (⋃_{i \in I} A_{i})), \\ (f, A) ⊓ (f_{i}, A_{i}) = (h_{i}, A \cap A_{i}), \\ ⨆_{i \in I} (h_{i}, A \cap A_{i}) = (h, A \cap (⋃_{i \in I} A_{i})) . \end{matrix}$

Following we need to show that k = h.

If e ∈ A - ⋃ _i∈IA_i, then k (e) = g (e). Since {i∈ I : e ∈ A ∩ A_i} = ∅ and {i∈ I : e ∈ A_i} = ∅, by Theorem 3.5 we have $h (e) = ⋃_{i \in I} h_{i} (e) = ⋃_{i \in I} f_{i} (e) = g (e)$ .

If e ∈ (⋃ _i∈IA_i) - A, then k (e) = f (e). Since {i∈ I : e ∈ A ∩ A_i} = ∅, by Theorem 3.5 we have $h (e) = ⋃_{i \in I} h_{i} (e)$ . For i ∈ I such that e ∈ A_i - A, h_i (e) = f (e). While, for i ∈ I such that e ∉ A_i - A, h_i (e) = f (e) ∩ f_i (e). So $h (e) = ⋃_{i \in I} h_{i} (e) = f (e)$ .

Otherwise, for e ∉ A - ⋃ _i∈IA_i and e ∉ (⋃ _i∈IA_i) - A, we have $\begin{matrix} k (e) & = & f (e) \cap g (e) = f (e) \cap ⋃_{i \in I} f_{i} (e) \\ = & ⋃_{i \in I} [f (e) \cap f_{i} (e)] = ⋃_{i \in I} h_{i} (e) = h (e) . \end{matrix}$

In any case we obtain that k (e) = h (e).

Definition 3.1. Let (f, A) be a central soft set over U and B ⊆ A. The projection of (f, A) over B, written (f, A) ^↓B, is defined to be a new central soft set (g, B) such that g (e) = f (e) for all e ∈ E.

The purpose of the projection operation is to constrain these central sets of central soft sets. While, mappings of central soft sets will not be changed.

Example 3.1. As it shown in Example 2.1, if S = {I, V}, then S ∩ A = {V} , S ∩ B = {I}. Thus (f, A) ^↓S∩A ⊔ (g, B) ^↓S∩B gives a new central soft set (k, S) defined as: $\begin{matrix} k (I) = {h_{2}, h_{3}, h_{4}}, k (II) = {h_{1}, h_{2}, h_{5}}, \\ k (III) = {h_{2}, h_{3}, h_{4}}, k (IV) = {h_{1}, h_{2}, h_{4}, h_{5}}, \\ k (V) = {h_{1}, h_{4}}, k (VI) = {h_{1}, h_{4}, h_{5}}, \\ k (VII) = {h_{2}, h_{4}, h_{5}}, k (VIII) = {h_{1}, h_{3}, h_{4}, h_{5}} . \end{matrix}$

According to the conclusion of Example 2.1, we have k = h. Then $[(f, A) ⊔ (g, B)]^{↓ S} = (f, A)^{↓ S \cap A} ⊔ (g, B)^{↓ S \cap B} .$

In fact, we can show the following general conclusion.

Proposition 3.1. For two central soft sets (f, A) and (g, B), $[(f, A) ⊔ (g, B)]^{↓ S} ⊑ (f, A)^{↓ S \cap A} ⊔ (g, B)^{↓ S \cap B},$ if S ⊆ A ∪ B. Especially, $[(f, A) ⊔ (g, B)]^{↓ S} = (f, A)^{↓ S \cap A} ⊔ (g, B)^{↓ S \cap B},$ if S = (A ∪ B) - D, where D ⊆ A ∩ B.

Proof. Let $(f, A) ⊔ (g, B) = (h, A \cup B)$ and ${(f, A)}^{↓ S \cap A} ⊔ {(g, B)}^{↓ S \cap B} = (h^{'}, S \cap (A \cup B)) .$

For all e ∈ E, we have $h^{'} (e) = {\begin{cases} f (e), if e \in (S \cap A) - (S \cap B); \\ g (e), if e \in (S \cap B) - (S \cap A); \\ f (e) \cup g (e), otherwise . \end{cases}$

Clearly (S ∩ A) - (S ∩ B) ⊆ A - B and (S ∩ B) - (S ∩ A) ⊆ B - A. Then we have $h (e) \subseteq h^{'} (e)$ for all e ∈ E by the definition of h. By Proposition 2.1, we obtain that $[(f, A) ⊔ (g, B)]^{↓ S} ⊑ (f, A)^{↓ S \cap A} ⊔ (g, B)^{↓ S \cap B} .$

If S = (A ∪ B) - D and D ⊆ A ∩ B, then we can show that (S ∩ A) - (S ∩ B) = A - B and (S ∩ B) - (S ∩ A) = B - A. By the definitions of h and h^′, we obtain that $[(f, A) ⊔ (g, B)]^{↓ S} = (f, A)^{↓ S \cap A} ⊔ (g, B)^{↓ S \cap B} .$

Similarly the conclusion on the intersection operation also can be obtained.

Proposition 3.2. For two central soft sets (f, A) and (g, B), if S ⊆ A ∩ B, then $(f, A)^{↓ S} ⊓ (g, B)^{↓ S} ⊑ [(f, A) ⊓ (g, B)]^{↓ S} .$

These properties of operations given in Propositions 3.1 and 3.2 show some basic regulation of information synthesis (union operation and intersection operation).

4 Evaluation systems for parameter sets

Let (f, A) be a central soft set and E be a set of initial parameters. If a parameter set B ⊆ E is such that

A ⊆ B;

|B| = max {|G| : A ⊆ G, ⋂ _e∈Gf (e) ¬ = ∅}, where |B| is the cardinality of B,

then x ∈ ⋂ _e∈Bf (e) is called a desired object of (f, A). It is clear that such parameter sets B are generally not unique. Especially, if ⋂_e∈Ef (e)¬ = ∅, we call x ∈ ⋂ _e∈Ef (e) a perfect object of (f, A).

Let A ⊆ E be a set of parameters, {(f_i, A_i) : i ∈ I} is a set of central soft sets. If A ⊆ ⋃ _i∈IA_i, then we call {(f_i, A_i) : i ∈ I} an evaluation system for the parameter set A. The desired objects of the central soft set [⨆ _i∈I (f_i, A_i)] ^↓A are called the desired objects of this evaluation system {(f_i, A_i) : i ∈ I}.

Following we give an algorithm for obtaining desired objects of a central soft set. Then the desired objects of an evaluation system are also obtained.

An algorithm for obtaining desired objects of a central soft set: Let U = {o₁, o₂, ⋯ , o_m} be a universal set and (f, A) be a central soft set with a central parameter set A ⊆ E.

Give an order for the elements of E, and we denote it by E = {e₁, e₂, ⋯ , e_n}.

Take a matrix L_n×m, where the elements l_ij (i = 1, 2, ⋯ , n ; j = 1, 2, ⋯ , m) are defined as follows: $l_{ij} = {\begin{matrix} 1, & if o_{j} \in f (e_{i}); \\ 0, & otherwise . \end{matrix}$

Compute ∑_{i:e_i∈A}l_ij and ∑_il_ij for each a fixed j, and denote it by b_j and a_j respectively. If the index set J = {j : b_j = |A|}¬ = ∅, let j^* ∈ J be a index such that a_j
^* = max {a_j : b_j = |A|}, then o_j
^* is a desired object of (f, A).

Example 4.1. As the example shown in Example 2.1, we can get two 8 × 5 matrices L, M shown as follows for the two central soft sets (f, A) and (g, B): $L = (\begin{matrix} 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 \end{matrix}), M = (\begin{matrix} 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \end{matrix}) .$

For the matrix L, we have b₁ = b₄ = 2, b₂ = b₃ = 0, b₅ = 1, and a₁ = 3, a₂ = 4, a₃ = 3, a₄ = 4, a₅ = 6.According to the definitions shown in the above, we know that h₄ is the desired object of (f, A).

For the matrix M, we have b₁ = 0, b₂ = b₄ = 2, b₃ = b₅ = 1, and a₁ = 2, a₂ = 4, a₃ = 2, a₄ = 7, a₅ = 4. Then h₄ is also the desired object of (g, B). Clearly, there is no any perfect objects for central soft sets (f, A) or (g, B).

Example 4.2. As the example shown in Example 2.1, the set {(f, A) , (g, B)} is an evaluation system for the parameter set D = {I, V}. We have that [(f, A) ⊔ (g, B)] ^↓D = (k, D), where k is defined as follows: $\begin{matrix} k (I) = {h_{2}, h_{3}, h_{4}}, k (II) = {h_{1}, h_{2}, h_{5}}, \\ k (III) = {h_{2}, h_{3}, h_{4}}, k (IV) = {h_{1}, h_{2}, h_{4}, h_{5}}, \\ k (V) = {h_{1}, h_{4}}, k (VI) = {h_{1}, h_{4}, h_{5}}, \\ k (VII) = {h_{2}, h_{4}, h_{5}}, k (VIII) = {h_{1}, h_{3}, h_{4}, h_{5}} . \end{matrix}$

The following matrix N is corresponded to the central soft set (k, D): $N = (\begin{matrix} 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \end{matrix}) .$

For the matrix N, we have b₁ = b₂ = b₃ = 1, b₄ = 2, b₅ = 0, and a₁ = a₂ = a₅ = 5, a₃ = 3, a₄ = 7. According to the algorithm given in the above, we know that h₄ is the desired object of this evaluation system.

5 Conclusions

The concept of a central soft set is introduced in this paper. It is shown some properties of operations such as union, intersection, complement and projection with central sets. The idea of an evaluation system for a parameter set and desired objects of a central soft set are proposed. An algorithm of giving desired objects of a soft set to solve such decision making problems is presented. More operators based on central soft sets can be given, for example the conservative union (for the complementary set of the union of central sets, the image set of each of its elements can be defined to be intersection of the corresponding sets). On the other hand, generalized soft sets with central sets can be considered for further studies.

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