Introduction to linear representations of soft groups

1 Introduction

Soft set theory initiated by Molodtsov [17] is an entirely new mathematical tool, which could be capable of dealing with various uncertainties and vagueness existing economics, engineering, and environment etc. A soft set in essence is a parameterized family of subsets of the universal set. It can be illustrated as a neighborhood system or a generalization of fuzzy set. Strikingly different with fuzzy sets, the problem of setting the membership function together with other related problems did not arise in soft set theory. Compared with traditional methods such as theory of probability, theory of fuzzy sets [24], interval mathematics and theory of rough sets [19], soft set theory shows obvious advantage, which embodied in the absence of any inherent limitation. According to Molodtsov’s view, the key reason may be the inadequacy of the parametrization tool of traditional methods, but in contrast, soft sets use adequate parameters, so that the difficulties existing in these classical methods are avoided. As soon as the novel concept was put forward, many scholars were attracted and immediately began a variety of related studies from both theoretical and practical levels. Maji et al. [16] further investigated the operations on soft sets and presented a detailed theoretical study of soft set theory. In order to better extract useful information from synthesis of two soft sets, Ali and Feng et al. [3] proposed some new operations on soft sets, such as the restricted intersection, the restricted union, the restricted difference and the extended intersection, and discussed their De Morgan’s laws. Sezgin and Atagün [20] extended the theoretical aspect of operations on soft sets through clarifying their interconnections between these operations. Chen et al. [8] gave a comparative study of parametrization reduction of soft sets and attributes reduction of rough sets. Zou and Xiao [26] proposed a data analysis approach based on soft set theory to refer to information system under incomplete. Çağman et al. [6] defined approximate function, which is a set-valued mapping from the set of parameters to the power set of the universe set, to describe a soft set, and constructed a new approach named uni-int decision making method to apply the problem of company recruitment. In algebraic structure of soft set theory, after Aktaş and Çağman [2] proposed the new concept of soft group which is a combination of soft sets and groups, there emerged a series of related research outcomes. These including soft semiring by Feng et al. [9], soft BCK/BCI-algebras by Jun [11], soft rings by Acar et al. [1], fuzzy soft groups by Aygünoğlu and Aygün [5], and combinations of soft sets and some other algebraic structures. (e.g.,[4, 7, 12–15, 21, 23, 25, 4, 7, 12–15, 21, 23, 25]).

Representation theory [22] is focused on the way of writing a group as a corresponding group of matrices. It provided one of the keys to not only readily but also precisely understanding of groups. However, as to the structure of soft group proposed by by Aktaş and Çağman (see [2]), which is not a group in traditional sense but a collection of subgroups of the initial group. Hence, the classic representation theory of groups seems unappropriate to describe the structure of soft group. To adapt the fact, this paper is devoted to constructing the basic framework of linear representations of soft groups. We give some primal definitions of the linear representations of soft groups together with some illustrative examples. Based on these definitions, some theorems are established to reveal the relationships among these concepts. This work is provided an effective mathematical tool to investigate the structure of soft group, especially when the soft group has a rather complex structure, and may be beneficial to enrich classic representation theory.

The remainder of this paper is organized as follows. We first review some previous knowledge on linear representations of groups and some basic results on soft groups. We then introduce some definitions such as linear and matrix representations of soft groups, equivalent representations, faithful representations, trivial representations of soft groups, and provide some appropriate examples. We continue to propose the notions of soft invariant space, irreducible representation, completely reducible representation. We investigate the relationships among these concepts through establishing some related theorems.

2 Basic results on linear representations of groups

The theory of linear representations of groups plays an important role to investigate the structure of groups. It provides an effective technique to a proper understanding of groups, especially when the group has a rather complicated structure. Throughout the whole paper, G is a group and K is a field. V is a vector space over K. GL (V) and GL _n  (K) denote the group of all invertible linear transformations of the vector space V and the group of nonsingular matrices of order n over the common field K, respectively.

Definition 2.1. ([22]) A linear representation of the group G over the field K is a homomorphism of G into the group GL (V) of V over K. V is called the representation space, and its dimension is called the dimension or the degree of the representation.

Definition 2.2. [22] A matrix representation of the group G over the field K is a homomorphism

of G into the group GL _n  (K) over K. The number n is called the dimension of the representation φ.

Definition 2.3. ([22]) Two matrix representations φ ₁ and φ ₂ of G are equivalent, written by φ ₁ ≃ φ ₂, if they have the same dimension and there exists a nonsingular matrix C such that

Every matrix S of order n with entries in the field K defines a linear transformation g ↦ Sg of the space K ⁿ of column vectors. On the contrary, suppose that the space V has a finite dimension n. Then each linear representation of the group G in V can associate with a class of equivalent n-dimensional matrix representations. Meanwhile, these matrix representations are equivalent.

Definition 2.4. ([22]) The set of all elements of G which are taken by φ into the identity matrix is a normal subgroup of G, called the kernel of the representation φ and denoted by ker φ. Then we have

The representation φ is said to be faithful if ker φ reduces to the identity element of G. In this case G is isomorphic to the subgroup φ (G) of GL _n  (K).

Theorem 2.1. ([22]) Let φ : G → GL (V) be a linear representation of the group G, let H be a group and φ : H → G be a homomorphism. Then the composed mapping φ ∘ φ is a linear representation of the group H.

Definition 2.5. ([22]) Let φ : G → GL (V) be a linear representation of the group G in a vector space V. A subspace U ⊂ V is said to be invariant under representation φ if

Theorem 2.2. ([22]) Suppose that the space V is finite-dimensional and that (e) = (e₁, …, e _n ) is some basis in V such that U = 〈e₁, …, e _k 〉. Then the invariance of U under the given representation φ of G means that in the basis (e) each φ (g) is given by a matrix of the form

Here M (g) is a nonsingular matrix with order k, L (g) and N (g) mean k × (n - k) and (n - k) × (n - k) matrices, respectively.

Definition 2.6. ([22]) Every invariant subspace U can associate two linear representations of G, acting on the spaces U and V/U respectively.

φ _U is said to be a subrepresentation of φ, is defined as φ _U  (g) = φ (g) _arrowvertU for all g ∈ G.

Definition 2.7. ([22]) Let φ : G → GL (V) be a linear representation of the group G. Then

φ is said to be irreducible if there are no nontrivial (different from 0 and V) subspaces U ⊂ V invariant under φ.

Theorem 2.3. ([22]) Every subrepresentation of a completely reducible representation is completely reducible.

Theorem 2.4. ([22]) The representation space of any completely reducible finite-dimensional representation admits a decomposition into a direct sum of minimal invariant subspaces. Where an invariant subspace is called minimal if it is minimal among the nonzero invariant subspaces.

Theorem 2.5. ([22]) Let φ : G → GL (V) be a linear representation. And let

be a decomposition of the space V into a (not necessarily direct) sum of minimal invariant subspaces. Then φ is completely reducible. Moreover, for every invariant subspace U there exist i₁, …, i _p such that

more details regarding linear representations of groups, one could refer to the books [22] and [10].

3 Basic results on soft groups

Molodtsov [17] originates the concept of soft sets in the following manner. Let U be an initial universe and E be a set of parameters. Let 𝒫 (U) refer to the power set of U and A ⊆ E.

Definition 3.1. ([17]) A pair (F, A) is called a soft set over U iff F is a mapping given by A → 𝒫(U) In other words, the soft set over U is a parameterized family of subsets of the set U. For each ε ∈ A, F (ε) may be considered as the set of ε-elements or the set of ε-approximate elements of the soft set.

Molodtsov [18] pointed that the natural generalization of a soft set is a soft mapping. Let X be a certain set.

Definition 3.2. ([18]) A pair (F, A) is called a soft mapping over U if F is a mapping from the set X × A to 𝒫 (U). As we can seen, if (F, A) is a soft mapping from X to 𝒫(U), then (F, X × A) is a soft set over U, where the Cartesian product X × A can be seen the set of parameters. For convenience, we can write by (F _X , A) instead of (F, X × A), F _X denotes the mapping F of restrictions on the set X.

From the angle of soft mapping, here we reconsider Molodtsov’s example as quoted from [17].

Example 3.1. Let a soft mapping (F, E) describe the attractiveness of the houses.

Suppose that U = {u ₁, u ₂, u ₃, u ₄, u ₅, u ₆} is the set of houses under consideration. And E = {ε ₁, ε ₂, ε ₃, ε ₄} is the set of parameters. Here ε _i   {i = 1, 2, 3, 4} stand for the parameters “expensive”, “beautiful”, “wooden” and “cheap”, respectively.

The soft mapping is defined in the following way

It is easy to obtain a soft set (F _U , E) that means to point out expensive houses, beautiful houses, wooden houses and cheap houses in U. Suppose that F _U  (ε ₁) = {h ₁, h ₄}, F _U  (ε ₂)   =   {h ₂, h ₃, h ₆}, F _U  (ε ₃)  =   ∅,F _U  (ε ₄) = {h ₆}. Then we can rewrite the soft set (F _U , E) by means of a collection of parameterized approximations:

Definition 3.3. ([2]) Let (F, A) be a soft set over G. Then (F, A) is called a soft group over G if for all ε ∈ A,

F (ε) is a nonempty subset of G;

Definition 3.4. ([2]) Let (F, A) and (H, K) be two soft groups over G. Then (H, K) is a soft subgroup of (F, A), denoted by ( H , K ) < ˜ ( F , A ), if

K ⊂ A;

4 Linear representations of soft groups

In what follows, we now introduce the concept of the linear representation of a soft group and present some relevant properties. Let (F, A) be a soft group over G.

Definition 4.1. A pair (φ, A) is called a linear representation of the soft group (F, A) if for each ε ∈ A, φ is a homomorphism from subgroup F (ε) of G to GL (V). The dimension of V is called the dimension of the representation (φ, A). In essence, the linear representation of a soft group is a parameterized collection of homomorphisms over G. Clearly, it is a soft mapping over GL (V), where φ is a mapping from the set

From Definition 3.2, (φ, A) can be viewed a soft homomorphism over GL (V) in Molodtsov’s sense in which φ is a homomorphism from F (ε) to GL (V) for all ε ∈ A.

Definition 4.2. A soft mapping (φ, A) is called a matrix representation of the soft group (F, A) if for each ε ∈ A, φ is a homomorphism from the group F (ε) to GL _n  (K). The number n is called the dimension of therepresentation.

Example 4.1. Let G be the dihedral group D ₈ = 〈s, t : s ⁴ = t ² = 1,  t ^-1 st = s ^-1〉, and let E = {1, s, t}. We define the set-valued function F ( ε ) = { g ∈ G : g = ε n , n ∈ ℕ }. Then the soft set (F, E) is a soft group. In this case, it is easy to obtain that F (1) = {1}, F (s) = {1, s, s ², s ³}, F (t) = {1, t}. We define the matrices S and T by

and check that

Define φ : F ( ε ) → GL 2 ( ℝ ) for ∀ε ∈ E (ℝ is the set of real numbers) given by

It is easy to verify that φ is a homomorphism from F (ε) to GL 2 ( ℝ ) for all ε ∈ E. Then (φ, E) is a matrix representation of (F, E), the dimension of (φ, E) is 2.

The matrices φ (g _ε ) are given in the following two tables:

Proposition 4.1. Assume that (F, A) and (H, K) are two soft groups over G, with ( H , K ) < ˜ ( F , A ), let (φ, A) be a matrix representation of (F, A). Then (φ, K) is a matrix representation of (H, K).

Proof. By Definition 4.2, φ is a matrix representation of F (ε) for ∀ε ∈ A. In particular, it also hold for ∀ε ∈ K ⊂ A. Using Definition 3.4 and Definition 4.2, hence (φ, K) is a matrix representation of (H, K). □

Definition 4.3. Suppose that (φ ₁, A) and (φ ₂, A) are two matrix representations of (F, A), then they are equivalent, written by (φ, A) ≈ (φ, A), if they have the same dimension and there exists a set of nonsingular matrices denoted by {C _ε |ε ∈ A} such that

Example 4.2. Let G  =  〈g : g ²  =  1〉 with the field K = ℝ.Let (F, A) be a soft group over G defined as follows

Give a matrix

And check that S ² = I ₂. Hence φ : 1 → I ₂, g → S is a matrix representation of F (ε) for ∀ε ∈ {1, g}. Moreover, (φ, A) is a matrix representation of soft group (F, A). Fix a nonsingular matrix

Then

Hence we obtain a matrix representation (φ, A) of (F, A) for which

Clearly, from Definition 4.3, (φ, A) is equivalent to (φ, A).

Remark 4.1. With regard to a soft group (F, A), a set of matrices (S, A), in which each matrix S (ε) with entries in the field K is of order n, define a collection of linear transformations g _ε  ↦ S (ε) g _ε of the space K ⁿ of column vectors for all ε ∈ A, g _ε  ∈ F (ε) and S (ε) ∈ (S, A). Moreover, these correspondences turn the multiplication of matrices into the multiplication of linear transformations. Conversely, assume that the space V has a finite dimension n, each linear representation (φ, A) of the soft group (F, A) in V can associate with a class of n-dimensional matrix representations. From Definition 4.3, these matrix representations of (F, A) are equivalent.

Definition 4.4. Let (φ, A) be a matrix representation of (F, A), it is called faithful if and only if

Where “1 _ε ” is the identity element of group F (ε) for all ε ∈ A, obviously, according to the uniqueness of the identity element in G, it also the identity element of G,written by “1”.

Example 4.3. Consider a group G = 〈g : g ³ = 1〉. It is a cyclic group of order 3. Let A = {1, g}, and let

Then (F, A) is a soft group over G. Suppose that (φ, A) is a matrix representation of (F, A), and define

Following from Definition 4.2, it is obvious that (φ, A) is a matrix representation of (F, A) over ℂ (ℂ is the set of complex numbers). And one can obtain easily that φ (g _ε ) = I ₃ if and only if g _ε  = 1 for all ε ∈ A. Therefore (φ, A) is faithful.

Definition 4.5. Let (φ, A) be a matrix representation of (F, A), it is called trivial if and only if

Example 4.4. Reconsider the cyclic group G = 〈g : g ³ = 1〉. Similarly, define a soft group (F, A) as

Suppose that (φ, A) is a matrix representation of (F, A), and define

Then (φ, A) is the trivial representation of (F, A).

Aktaş and Çağman [2] first defined the concept of soft homomorphism, which can be addressed as follows.

Definition 4.6. [2] Let (F, A) and (H, B) be two soft groups over groups G and K respectively, and let f : G → K and h : A → B be two functions. Then (f, h) is called a soft homomorphism, and that (F, A) is soft homomorphic to (H, B), written by (F, A) ∼ (G, B), if the following conditions are satisfied:

f is a homomorphism from G to K,

Theorem 4.1. Let (φ, B) be a linear representation of the soft group (H, B) over a group K. Suppose that (F, A) is a soft group over G and (f, A) is a soft homomorphism from (F, A) to (H, B). Then the soft mapping (φ ∘ f, A), defined by

is a linear representation of the soft group (F, A).

Proof. Certainly φ ∘ f is a homomorphism from F (ε) to GL (V) for all ε ∈ A. Hence (φ ∘ f, A) is a linear representation of the soft group (F, A). □

Definition 4.7. Let (φ, A) be a linear representation of the soft group (F, A) in a vector space V. A parameterized set of subspaces written by (U, A), defined by U _ε  ⊂ V for all ε ∈ A, is called to be invariant under representation (φ, A) if

We call (U, A) as a soft invariant space under representation (φ, A).

Example 4.5. Let G = ℝ be a additive group and Let V denote the linear space of all polynomials with real coefficients. Suppose that (F, A) is a soft group over G, defined as F (ε) =< ε > for all ε ∈ A. Given A = {1, - 1}, Hence the soft group (F, A) can be written by

Let (φ, A) be the representation of soft group (F, A) in the space of all polynomials, given by the rule

for t 1 ∈ F ( 1 ) = ℤ and f ( x 1 ) ∈ U 1.

And

for t - 1 ∈ F ( - 1 ) = ℤ and f ( x - 1 ) ∈ U - 1.

Where U ₁ and U _-1 are two subspaces of all polynomials with integral coefficients, and all polynomials in U ₁ and U _-1 are degree ≤n.

Then it is readily checked that the parameterized set of subspaces {U ₁, U _-1} denoted by (U, A) is invariant under (φ, A) for every n.

Proposition 4.2. Let (φ, A) be a linear representation of the soft group (F, A) in a vector space V. Assume that both two parameterized sets of subspaces (U ₁, A) and (U ₂, A) are invariant under representation of (φ, A). Then

The sum of (U ₁, A) and (U ₂, A), written by (U ₁, A) + (U ₂, A), which defined as U _{1 ε}  + U _{2 ε} for all ε ∈ A, are invariant under representation of (φ, A);

Proof. Straightforward. □

Theorem 4.2. Suppose that V is a finite-dimensional space. Take a basis in V written by (e) = (e₁, …, e_n) such that U_ε = 〈e₁, …, e_k(ε)〉 for all ε ∈ A, where k (ε) means that k is a function of ε. Then the invariance of (U, A) under the given representation (φ, A) of (F, A) means that in the basis (e) each φ (g_ε) for all g_ε ∈ F (ε) and ε ∈ A, is given by a matrix of the form

for all g _ε  ∈ F (ε)  and ε ∈ A.

Where M (g _ε ) is a nonsingular matrix with order k _ε , L (g _ε ) and N (g _ε ) are k _ε   ×   (n  -  k _ε ) and (n  -  k _ε )   ×   (n  -  k _ε )matrices, respectively.

Proof. Straightforward. □

Definition 4.8. Every soft invariant subspace (U, A) can associate two linear representation of (F, A), acting on the two parameterized sets of subspaces (U, A) and (V/U, A), respectively.

(φ _(U,A), A) is said to be a subrepresentation of (φ, A), is defined as φ _{U ε}  (g _ε ) = φ (g _ε ) _{arrowvert Uε} for all g _ε  ∈ F (ε), U _ε  ∈ (U, A) and ε ∈ A.

Remark 4.2. Assume that the space V is finite-dimensional, the linear representations (φ _(U,A), A) and (φ _(V/U,A), A) can be easily translated into matrices representations correspondingly. For each ε ∈ A, pick a basis (e ₁, …, e _n ) of V such that U _ε  = 〈e ₁, …, e _k(ε)〉. Then the linear transformation φ (g _ε ), g _ε  ∈ F (ε), are described by the matrices of the form

for all g _ε  ∈ F (ε)  and ε ∈ A.

Here M (g _ε ) and N (g _ε ) are the matrices of the transformations φ _{U ε}  (g _ε ) and φ _{V/U ε}  (g _ε ) in the basis (e ₁, …, e _k(ε)) of U _ε and (e _k(ε)+1 + U _ε , …, e _n  + U _ε ) of V/U _ε respectively.

Definition 4.9. Suppose that (φ, A) is a linear representation of (F, A). Then

(φ, A) is said to be irreducible if and only if for all ε ∈ A, there are no nontrivial (different from 0 and V) subspaces U _ε invariant under (φ, A).

Example 4.6. Let G = GL (V) and (F, A) be a soft group defined by

Suppose that (φ, A) is an identity matrix representation of (F, A), defined by

Where I _n and g _ε are identity matrix and nonsingular matrix of order n respectively. Based on Definition 4.7. We obtain that

Where v is an arbitrary nonnull vector in V, due to v can be take into any other nonnull vector by an invertible linear transformation g _ε for every ε ∈ A. Therefore (φ, A) is irreducible.

Example 4.7. Let G = ℝ, and let (F, A) be a soft group over ℝ defined by

Consider a two-dimensional linear representation (φ, A) of (F, A), given in the basis (e) = (e ₁, e ₂) by the matrices

Pick a new basis (e ₁, e ₁ - e ₂), One can check that

So one can obtain easily that

It is readily verified that 〈e ₁〉 and 〈e ₁ - e ₂〉 are the only nontrivial subspaces invariant under φ for all ε ∈ A. Therefore (φ, A) is completely reducible.

Theorem 4.3. Assume that (φ, A) is a completely reducible representation of (F, A). Then every subrepresentation of (φ, A) is completely reducible.

Proof. Let (U, A) be a soft invariant space under representation (φ, A). And let (U ₁, A) be an arbitrary soft invariant space contained in (U, A), i.e., for all ε ∈ A, U _{1 ε} is an invariant subspace of U _ε . Since (φ, A) is completely reducible, from Definition 4.9 (ii), (U ₁, A) has an soft invariant complement (W, A), where W _ε contained in V, and satisfied V = U _{1 ε}  ⊕ W _ε for all ε ∈ A. Hence, it is readily verified that (W ∩ U, A) is soft invariant such that

That is, for all ε ∈ A, (W ∩ U)  _ε is a complement of U _{1 ε} in U _ε . This completes the proof of the theorem. □

Theorem 4.4. Assume that (φ, A) is a completely reducible representation of (F, A) and V is the finite-dimensional representation space. Then V can admit a decomposition into a direct sum of minimal invariant subspaces for all ε ∈ A.

Proof. For all ε ∈ A, firstly, if (φ, A) is irreducible. Then theorem is obvious (because the sum reduces to one term). Suppose that (U, A) be an arbitrary soft minimal invariant subspace, let (W, A) be a soft invariant complement of (U, A). On the basis of Theorem 4.3, the subrepresentation (φ _(U,A), A) is completely reducible. Continue this way, finally we can obtain a decomposition of V into a direct sum of minimal invariant subspaces. □

Theorem 4.5. Assume that (φ, A) is a linear representation of (F, A). For all ε ∈ A, let

be a decomposition of V into a sum of minimal invariant subspaces. Then (φ, A) is completely reducible. Furthermore, take an arbitrary soft invariant subspace (U, A) there exist indices i_1ε, …, i_pε such that

Proof. Because of the second assertion of the theorem is a stronger version of the first. We just devote to prove the second assertion. Suppose that (U, A) is a soft invariant subspace, let {i _{1 ε} , …, i _{p ε} } relating to ε be a maximal set of indices such that the subspaces U _ε , V _{i 1 ε} , …, V _{i p ε} are linearly independent. To prove the second assertion in this case, it suffices to show that

for each ε ∈ A and i _ε  ∉ {i _{1 ε} , …, i _{p ε} }. So for all ε ∈ A,

is an invariant subspace contained in V _{i ε} , due to V _{i ε} is a minimal invariant subspace, so either (i) holds or

Yet (iii) is false, because it would imply the linear independence of the subspaces U _ε , V _{i 1 ε} , …, V _{i p ε} , V _{i ε} and contradict the choice of the set {i _{1 ε} , …, i _{p ε} } for all ε ∈ A. This completes the proof of the theorem. □

5 Conclusion

Soft group is a fundamental concept of algebraic theory of soft sets. In this paper, we have made an initial study of the linear representations of soft groups, which would be a powerful tool to investigate the structures of soft groups. We introduce a series of basic definitions such as linear and matrix representation of a soft group, equivalent representations, faithful representations, trivial representations, etc. To illustrate these notions vividly, we give some corresponding examples. We then present the notion of soft invariant space, and the notions of irreducible and completely reducible representations respectively. Based on these results, we establish several related theorems which elaborate the relationships among these concepts. To extend the work, one could further study the properties of finite-dimensional representations of soft groups.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (No. 71171209) and Major Consulting Research Project of Chinese Academy of Engineering (No. 2012-ZD-15). The authors are highly grateful to the editor and anonymous referees for their valuable comments and constructive suggestions which help to improve the present form of this paper.