Vague soft module

Abstract

In this paper, the study of vague soft structures of modules is initiated by introducing the concepts of vague soft module, vague soft module homomorphism and vague soft exactness. In the meantime, some of their properties and structural characteristics are investigated and discussed. Thereafter, several illustrative examples are given.

Keywords

Soft set vague set vague soft set soft module vague soft module vague soft module homomorphism vague soft exactness

1 Introduction

In the real world, there are some various uncertainties, but classical mathematical tools are not convenient for modeling these. Uncertain and imprecise data which are contained by economy, engineering, environmental science, social science, medical science, business administration and many other fields are common. Several theories like probability theory, fuzzy set theory, vague set theory, rough set theory and interval mathematics, can be considered as mathematical tools for modeling uncertainties (see [1 –3]). However, as pointed out by Molodtsov in [4], all of these theories have their own difficulties, and one of the major reasons for these difficulties is the inadequacy of the parametrization tools for these theories. Therefore, Molodtsov proposed the soft set theory, as a new mathematical tool for dealing with uncertainties, which is free from the difficulties existing in those theories mentioned above. Furthermore, he demonstrated that soft set theory has potential applications in many directions, including function smoothness, Riemann integration, Perron integration, probability theory, measurement theory, game theory and operations research.

In 1999, Molodstov [4] developed soft set theory which is considered a mathematical tool for working with uncertainties. In recent years, research on soft set theory, as well as its applications, especially the application in decision making, has received wide attention and achieved great progress. Maji et al. [5] defined some operations on soft sets and these operations are used soft sets of decision making problems. The application was improved by Chen et al. [6] with the help of a new definition of parameterization reduction. They made comparison between this definition and concept of restriction of property in the rough set theory. In theory, Maji et al. [7] worked various operator on soft set. Kong et al. [8] developed definition of parametrization reduction on soft set. Zou and Xiao [12] suggested some approach of data analysis in case of insufficient information on soft set. Jiang et al. [13] presented a unique approach of the semantic decision making by means of ontological thinking and ontology-based soft sets.

Besides, studies on soft module theory have continued and interesting results have been discovered recently. Sun et al. [9] presented the notion of soft set and soft module. Xiang [10] worked on soft module theory. T.Shah et al. [11] defined the notion of primary decomposition in a soft ring and soft module, and derived some related properties. Erami et al. [14] gave the concept of a soft MV-module and soft MV-submodule.

In 1993, Gau and Buehrer [15] proposed the theory of vague sets as an improvement of the theory of fuzzy sets in approximating the real life situation. Vague sets are higher order fuzzy sets. Bustince et al. [16] clarified that vague sets are intuitionistic fuzzy sets. Then, Chen et al. [17] presented the notion of measures of similarity between vague sets and between elements. Furthermore, they [18] described the concept of analyzing fuzzy system reliability using vague set theory. Besides Hong et al. [19] explained multicriteria fuzzy decision making problems based on vague set theory.

Soft set theory has been successfully applied in various areas of mathematics such as in classical algebra, fuzzy algebra, to develop various algebraic structures. However, an inherent weakness of soft set theory is that it is difficult to be used to represent the vagueness of problem parameters, particularly in problem-solving and decision making contexts. To overcome this disadvantage, various generalizations of soft sets such as fuzzy soft sets, vague soft sets, and intuitionistic fuzzy soft sets were introduced as better alternatives to the concept of soft sets. When it comes to vague soft set theory, Xu et al. [20] developed vague soft sets and their properties, which were a combination of the notion of vague sets and soft sets as an improvement to the notion of fuzzy soft sets as it can be used to better represent the vagueness of the parameters associated to the elements in a set. Hassan et al. [21 –23] presented the notion of possibility vague soft set and its application in decision making. After that, they introduced generalized vague soft set and showed its applications. They explained the concept of vague soft set relations and functions as well.

The main purpose of this paper is to deal with algebraic structure of module by applying vague soft set theory. The concept of vague soft module is introduced, their characterizations and algebraic properties of them are investigated by giving several examples. In addition, vague soft homomorphism, vague soft isomorphism and their properties are introduced. After all, vague soft exactness is investigated and illustrated with a related example.

2 Preliminaries

In this section, preliminary necessary information for introduction of vague soft modules are presented. First of all we give some basic concepts of soft set theory.

Definition 2.1. [24] Let X denotes an initial universe set and E is a set of parameters. The power set of X is denoted by P (X). A pair of (F, E) is called a soft set over X if and only if F is a mapping from E into the set of all subsets of X, i.e, F: E → P (X).

Definition 2.2. [24] Let (F, A) and (G, B) be two soft sets over a common universe X.

If $A \tilde{\subseteq}$ B and $F (a) \tilde{\subseteq}$ G (a) for all a∈ A, then we say that (F, A) is a soft subset of (G, B) and it is denoted by (F, A) $\tilde{\subseteq} (G, B)$ .

If (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A), then we say that (F, A) is a soft equal to (G, B) and it is denoted by $(F, A) \tilde{=}$ (G, B) .

The following definition is basic for fuzzy set theory and this is necessary for vague soft modules.

Definition 2.3. [25] Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function μ_A : X → [0, 1] and μ_A (x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ X. It is clear that A is completely determined by the set of tuples A = {(u, μ_A (u)) |u ∈ X}. Frequently we will write A (x) instead of μ_A (x). The family of all fuzzy sets in X is denoted by F (X) .

Secondly, in the following, we will give some basic definitions and properties of vague sets that are necessary for vague soft modules. Besides, we will define vague sets, unit vague value and zero vague value as well. Moreover, we will present unit vague set and zero vague set. Then, we will also demonstrate complement of vague set and then, notions of union and intersection of vague sets are presented.

Definition 2.4. [20] Let U be an initial universe set, U = {u₁, u₂,…, u_n} . A vague set over U is characterized by a truth-membership function t_v and a false-membership function f_v such that t_v : U → [0, 1] , f_v : U → [0, 1] , where t_v (u_i) is a lower bound on grade of membership of u_i derived from the evidence for u_i, f_v (u_i) is a lower bound on the negation of u_i derived from the evidence against u_i, and t_v (u_i) + f_v (u_i) ≤1 . The grade of membership of u_i in the vague set is bounded to a subinterval [t_v (u_i) , 1 - f_v (u_i)] of [0, 1] . The vague value [t_v (u_i) , 1 - f_v (u_i)] indicates that exact grade of membership μ_v (u_i) of u_i may be unknown, but it is bounded by t_v (u_i) ≤ μ_v (u_i) ≤1 - f_v (u_i) , where t_v (u_i) + f_v (u_i) ≤1 .

Definition 2.5. [20] When the universe U is continuous, a vague set A can be written as A = ∫_U [t_A (u_i) , 1 - f_A (u_i)] /u_i, u_i ∈ U . When the universe U is discrete, a vague set A can be written as $A = \sum_{i = 1}^{n} [t_{A} (u_{i}), 1 - f_{A} (u_{i})]$ /u_i, u_i ∈ U .

Below, we present some basic definitions and properties of vague sets such as unit vague value, zero vague value, unit vague set, zero vague set.

Definition 2.6. [20] Let x be a vague value, x = [t_x, 1 - f_x] , where t_x ∈ [0, 1] , f_x ∈ [0, 1] , and 0≤ t_x ≤ 1 - f_x ≤ 1 . If t_x = 1 and f_x = 0 (i.e. x = [1, 1]), then x is called a unit vague value. If t_x = 0 and f_x = 1 (i.e. x = [0, 0]), then x is called a zero vague value.

Definition 2.7. [20] Let x, y be two vague values, where x = [t_x, 1 - f_x] and y = [t_y, 1 - f_y] . If t_x = t_y and f_x = f_y, then vague values x and y are called equal (i.e. [t_x, 1 - f_x] = [t_y, 1 - f_y]).

Definition 2.8. [20] Let A be a vague set of the universe U . If ∀u_i ∈ U, t_A (u_i) =1 and f_A (u_i) =0, then A is called a unit vague set, where 1 ≤ i ≤ n . If ∀ u_i ∈ U, t_A (u_i) =0 and f_A (u_i) =1, then A is called a zero vague set, where 1 ≤ i ≤ n .

Now we describe the notions of complement of vague set, union and intersection of vague set.

Definition 2.9. [20] The complement of a vague set A is denoted by A^c and it is defined by t_{A
^c} = f_A, 1 - f_{A
^c} = 1 - t_A .

Definition 2.10. [20] Let A and B are two vague sets of the universe U. If ∀u_i ∈ U, [t_A (u_i) , 1 - f_A (u_i)] = [t_B (u_i) , 1 - f_B (u_i)] , then the vague set A and B are called equal, where 1 ≤ i ≤ n .

Definition 2.11. [20] Let A and B are two vague sets of the universe U. If ∀u_i ∈ U, t_A (u_i) ≤ t_B (u_i) and 1 - f_A (u_i) ≤1 - f_B (u_i) , then the vague set A is included in B, denoted by A ⊆ B, where 1 ≤ i ≤ n .

Definition 2.12. The union of two vague sets A and B is a vague set C, written as C = A ∪ B, whose truth-membership and false-membership functions are related to those of A and B by

$\begin{matrix} t_{C} & = & (t_{A} \lor t_{B}), \\ 1 - f_{C} & = & ((1 - f_{A}) \lor (1 - f_{B})) = 1 - (f_{A} \land f_{B}) . \end{matrix}$

Definition 2.13. [20] The intersection of two vague sets A and B is a vague set C, written as C = A ∩ B, whose truth-membership and false-membership functions are related to those of A and B by $\begin{matrix} t_{C} & = & (t_{A} \land t_{B}), \\ 1 - f_{C} & = & ((1 - f_{A}) \land (1 - f_{B})) = 1 - (f_{A} \lor f_{B}) . \end{matrix}$

Thirdly, we will introduce vague soft set and we will study some properties and theories of vague soft sets.

Definition 2.14. [20] Let U be a universe set, E be a set of parameters, P (U) is the power set of vague set on U and A ⊆ E . A pair (V, A) is called a vague soft set over U, where V is a mapping given by V : A → P (U) .

Definition 2.15. [20] For two vague soft sets (V, A) and (W, B) over a universe U, we say that (V, A) is vague soft subset of (W, B) if A ⊆ B and ∀ɛ ∈ A, V (ɛ) and W (ɛ) are identical approximations. This relationship is denoted by $(V, A) \tilde{\subseteq} (W, B)$ . Similarly, (V, A) is said to be vague soft superset of (W, B) , if (W, B) is a vague soft subset of (V, A) . We denote it by $(V, A) \tilde{\supseteq} (W, B)$ .

Definition 2.16. [20] Two vague soft sets (V, A) and (W, B) over a universe U are said to be vague soft equal if (V, A) is a vague soft subset of (W, B) and (W, B) is a vague soft subset of (V, A) .

Definition 2.17. [20] The complement of a vague soft set (V, A) is denoted by (V, A) ^c and it is defined by (V, A) ^c = (V^c, ¬ A) where V^c :¬ A → P (U) is a mapping given by ∀α∈ ¬ A and x ∈ U, $\begin{matrix} t_{V^{c} (α)} (x) & = & f_{V (\neg α)} (x), \\ 1 - f_{V^{c} (α)} (x) & = & 1 - t_{V (\neg α)} (x) . \end{matrix}$

Now we present two important definitions of vague soft sets. The one is null vague soft set and the other one is absolute vague soft set.

Definition 2.18. [20] A vague soft set (V, A) over U is said to be a null vague soft set denoted by φ, if ∀ɛ ∈ A, t_V(ɛ) (x) =0, 1 - f_V(ɛ) (x) =0 for x ∈ U .

Definition 2.19. [20] A vague soft set (V, A) over U is said to be an absolute vague soft set denoted by $\tilde{A}$ if ∀ɛ ∈ A, t_V(ɛ) (x) =1, 1 - f_V(ɛ) (x) =1 for x ∈ U .

In the following, four basic operations of vague soft sets such as union, intersection, and, or operations of vague soft sets are presented.

Definition 2.20. [20] The union of two vague soft sets of (V, A) and (W, B) over a universe U is a vague soft set (H, C) , where C = A ∪ B and ∀e ∈ C, $t_{H (e)} (x) = {\begin{matrix} t_{V (e)} (x), & if e \in A - B \\ t_{W (e)} (x), & if e \in B - A \\ (t_{V (e)} (x) \lor t_{W (e)} (x)), & if e \in A \cap B \end{matrix}}$ and 1 - f_H(e) (x) is defined by ${\begin{matrix} 1 - f_{V (e)} (x), & if e \in A - B \\ 1 - f_{W (e)} (x), & if e \in B - A \\ (1 - f_{V (e)} (x)) \lor (1 - f_{W (e)} (x)), & if e \in A \cap B \end{matrix}}$ We denoted it by $(V, A) \tilde{\cup} (W, B) = (H, C) .$

Definition 2.21. [20] The intersection of two vague soft sets of (V, A) and (W, B) over a universe U is a vague soft set (H, C) , where C = A ∪ B and ∀e ∈ C, $t_{H (e)} (x) = {\begin{matrix} t_{V (e)} (x), & if e \in A - B \\ t_{W (e)} (x), & if e \in B - A \\ (t_{V (e)} (x) \land t_{W (e)} (x)), & if e \in A \cap B \end{matrix}}$ and 1 - f_H(e) (x) is defined by ${\begin{matrix} 1 - f_{V (e)} (x), & if e \in A - B \\ 1 - f_{W (e)} (x), & if e \in B - A \\ (1 - f_{V (e)} (x)) \land (1 - f_{W (e)} (x)), & if e \in A \cap B \end{matrix}}$

We denote it by $(V, A) \tilde{\cap} (W, B) = (H, C) .$

Definition 2.22. [20] If (V, A) and (W, B) are two vague soft sets over U, “(V, A) and (W, B)” denoted by (V, A) ∧ (W, B), is defined by (V, A) ∧ (W, B) = (H, A × B) , where ∀ (α, β) ∈ A × B, x ∈ U, $\begin{matrix} t_{H (α, β)} (x) & = & {t_{V (α)} (x) \land t_{W (β)} (x)}, \\ 1 - f_{H (α, β)} (x) & = & {(1 - f_{V (α)} (x)) \land (1 - f_{W (β)} (x))} . \end{matrix}$

Definition 2.23. [20] If (V, A) and (W, B) are two vague soft sets over U, “(V, A) or (W, B)” denoted by (V, A) ∨ (W, B), is defined by (V, A) ∨ (W, B) = (O, A × B) , where ∀ (α, β) ∈ A × B, x ∈ U, $\begin{matrix} t_{O (α, β)} (x) & = & \lor {t_{V (α)} (x), t_{W (β)} (x)}, \\ 1 - f_{O (α, β)} (x) & = & {(1 - f_{V (α)} (x)) \lor (1 - f_{W (β)} (x))} . \end{matrix}$

Fourthly, the concepts of module, submodule, soft module, soft submodule and their basic properties are presented below.

Definition 2.24. [9] Let R be an arbitrary ring with identity and (M, +) an abelian group. M is said to be a left R -module if there is a left scalar multiplication λ : R × M → M via (a, x) ↦ ax satisfying the following axioms ∀r, r₁, r₂, 1 ∈ R ; m, m₁, m₂ ∈ M :

r (m₁ + m₂) = rm₁ + rm₂, (r₁ + r₂) m = r₁m + r₂m,

(r₁r₂) m = r₁ (r₂m) ,

1m = m .

Left R-module is denoted by _RM or M for short. Similarly we can define right R- module and denote it by M_R .

Definition 2.25. [9] Let R be a commutative ring with identity and M a module. Then N ⊂ M is called a submodule of M if m + n ∈ N for all m, n ∈ N and rn ∈ N for all r ∈ R, n ∈ N .

Definition 2.26. [9] Let M be a left R- module, A be a nonempty set and (F, A) is a soft set over M . (F, A) is said to be a soft module over M if and only if F (x) is submodule over M, for all x ∈ A .

Definition 2.27. [9] Let (F, A) and (G, B) be two soft modules over M .Then (G, B) is soft submodule of (F, A) if

B ⊂ A,

G (x)< F (x) , ∀ x ∈ B .

This is denoted by $(G, B) \tilde{<} (F, A) .$

Definition 2.28. [6] Let (F, A) be a soft module over M. Then

(F, A) is said to be a trivial soft module over M if F (x) =0 for all x ∈ A, where 0 is zero element of M .

(F, A) is said to be an whole soft module over M if F (x) = M for all x ∈ A .

Proposition 2.29. [9] Let (F, A) and (G, B) be two soft modules over M . Then the following statements hold:

$(F, A) \tilde{\cap} (G, B)$ is a soft module over M .

$(F, A) \tilde{\cup} (G, B)$ is a soft module over M if A ∩ B = ∅ .

Definition 2.30. [9] If (F, A) and (G, B) are two soft modules over M, then (F, A) $\tilde{+}$ (G, B) is defined as (H, A × B), where H (x, y) = F (x) $\tilde{+}$ G (y) for all (x, y) ∈ A × B .

Proposition 2.31. [9] Let (F, A) and (G, B) be two soft modules over M . Then (F, A) $\tilde{+}$ (G, B) is soft module over M .

Definition 2.32. [9] Let (F, A) and (G, B) be two soft modules over M and N respectively. Then (F, A) $\tilde{\times}$ (G, B) = (H, A × B) is defined as H (x, y) = F (x) $\tilde{\times}$ G (y) for all (x, y) ∈ A × B .

Proposition 2.33. [9] Let (F, A) and (G, B) be two soft modules over M and N respectively. Then (F, A) $\tilde{\times}$ (G, B) is soft module over M × N .

3 Vague soft module

In this section, firstly we introduce the concept of vague soft modules, then we give some operations on them. Throughout the section, M is a module and R is a ring.

Definition 3.1. Let (V, A) be a non-null vague soft set over M. Then, (V, A) is called vague soft module over M if and only if ∀a ∈ A, V (a) is a vague submodule over M i.e. V : A → P (M) by a → V (a) = V_a is vague submodule of M .

Theorem 3.2. Let (V, A) be a non-null vague soft set over M. Then, (V, A) is called vague soft module over M if and only if ∀a ∈ A, x, y ∈ M and r ∈ R the following conditions hold:

V_a (0) =1,

V_a (x + y) ≥ V_a (x) ∧ V_a (y) ,

V_a (rx) ≥ V_a (x) .

In the following, we give some examples of vague soft modules.

Example 3.3. Let M = {0, a, b, c} be a set with binary operation + defined as follows: $\begin{matrix} \begin{matrix} + & 0 & a & b & c \\ 0 & 0 & a & b & c \\ a & a & 0 & c & b \\ b & b & c & 0 & a \\ c & c & b & a & 0 \end{matrix} \end{matrix}$

Then (M, +) is an abelian group. If $R = ℤ_{2},$ then M is an R- module. Consider a set of parameters A = {x, y} . Let (V, A) be a vague soft set over M . Then, V_x, V_y are vague sets on M . We define that $\begin{matrix} V (x) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{1}, \bar{1}] / a, \\ [\bar{0.5}, \bar{0.5}] / \bar{b}, [\bar{0.5}, \bar{0.5}] / \bar{c}}, \\ V (y) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{1}, \bar{1}] / a, \\ [\bar{0.5}, \bar{0.5}] / \bar{b}, [\bar{0.5}, \bar{0.5}] / \bar{c}} . \end{matrix}$

Then, (V, A) is a vague soft module over M .

Example 3.4. Let $M = ℤ_{3} = {\bar{0},$ $\bar{1},$ $\bar{2}}$ is a $ℤ -$ module. Consider a set of A = {x, y, z} . Let (V, A) be a vague soft set over M . Suppose that $\begin{matrix} V (x) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.8}, \bar{0.9}] / \bar{1}, [\bar{0.8}, \bar{0.9}] / \bar{2}}, \\ V (y) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.5}, \bar{0.6}] / \bar{1}, [\bar{0.5}, \bar{0.6}] / \bar{2}}, \\ V (z) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.3}, \bar{0.4}] / \bar{1}, [\bar{0.3}, \bar{0.4}] / \bar{2}} . \end{matrix}$ It can be easily checked that (V, A) is a vague soft module over M .

Example 3.5. Let (V, A) be a vague soft set over M where A = {x, y, z} . Then, V_x, V_y, V_z are vague sets in M . We define that $\begin{matrix} V (x) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.8}, \bar{0.9}] / \bar{1}, [\bar{0.8}, \bar{0.9}] / \bar{2}}, \\ V (y) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.5}, \bar{0.6}] / \bar{1}, [\bar{0.5}, \bar{0.6}] / \bar{2}}, \\ V (z) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.2}, \bar{0.5}] / \bar{1}, [\bar{0.3}, \bar{0.5}] / \bar{2}} . \end{matrix}$

Then (V, A) is not a vague soft module over M because we know that $V_{z} (\bar{2} + \bar{2}) \geq V_{z} (\bar{2}) + V_{z} (\bar{2})$ , but

$t_{V_{Z}} (\bar{2} + \bar{2}) = t_{V_{Z}} (\bar{1}) = \bar{0.2} t_{V_{Z}} (\bar{2}) \land t_{V_{Z}} (\bar{2}) = \bar{0.3}$ .

Hence, take B = {x, y} and (V, B) is a vague soft module over M .

Definition 3.6. Let (V, A) and (W, B) be two vague soft modules over M. Then, (V, A) is a vague soft submodule of (W, B) , if

A ⊂ B,

∀x ∈ A, V (x) is a vague submodule of W (x) .

This is denoted by (V, A) $\tilde{\leq} (W, B) .$

Definition 3.7. Let (V, A) and (W, B) be two vague soft modules over M. Then, (V, A) and (W, B) are equal vague soft modules, if (V, A) is vague soft submodule of (W, B) and (W, B) is vague soft module of (V, A). This is denoted by (V, A) $\tilde{=} (W, B) .$

Example 3.8. Let A = {x, y, z} and B = {x, y} . We define $\begin{matrix} V (x) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.8}, \bar{0.9}] / \bar{1}, [\bar{0.8}, \bar{0.9}] / \bar{2}}, \\ V (y) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.5}, \bar{0.6}] / \bar{1}, [\bar{0.5}, \bar{0.6}] / \bar{2}}, \\ V (z) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.3}, \bar{0.4}] / \bar{1}, [\bar{0.3}, \bar{0.4}] / \bar{2}} . \\ W (x) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.8}, \bar{0.9}] / \bar{1}, [\bar{0.8}, \bar{0.9}] / \bar{2}}, \\ W (y) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.5}, \bar{0.6}] / \bar{1}, [\bar{0.5}, \bar{0.6}] / \bar{2}} . \end{matrix}$

(V, A) and (W, B) are vague soft modules over M . Then, A ⊆ B and ∀x ∈ B, W (x) ≤ V (x) . So, (V, A) $\tilde{\leq} (W, B) .$

The following results give valuable properties of vague soft modules such as notions of union, intersection, and operations of and, or over vague soft modules.

Theorem 3.9. Let (V, A) and (W, B) be two vague soft modules over M. Then, their intersection (V, A) $\tilde{\cap}$ (W, B) is a vague soft module over M .

Proof. We know that (V, A) $\tilde{\cap}$ (W, B) = (H, C) such that C = A ∪ B . For all x ∈ M, we can define the following statements: $t_{H (c)} (x) = {\begin{matrix} t_{V (c)} (x), & if c \in A - B \\ t_{W (c)} (x), & if c \in B - A \\ (t_{V (c)} (x) \land t_{W (c)} (x)), & if c \in A \cap B \end{matrix}}$ and 1 - f_H(c) (x) is defined by ${\begin{matrix} 1 - f_{V (c)} (x), & if c \in A - B \\ 1 - f_{W (c)} (x), & if c \in B - A \\ ((1 - f_{V (c)} (x)) \land (1 - f_{W (c)} (x))), & if c \in A \cap B \end{matrix}}$

Case1: If c ∈ A - B, for all x, y ∈ M and r ∈ R, then

t_H(c) (0) =1 because of t_V(c) (0) =1.

t_H(c) (x + y) = t_V(c) (x + y) ≥ t_V(c) (x) ∧ t_V(c) (y) = t_H(c) (x) ∧ t_H(c) (y) .

t_H(c) (rx) = t_V(c) (rx) ≥ t_V(c) (x) = t_H(c) (x) .

1 - f_H(c) (0) =1 because of 1 - f_V(c) (0) =1 .

1 - f_H(c) (x + y) =1 - f_V(c) (x + y)

≥(1 - f_V(c) (x)) ∧ (1 - f_V(c) (y))

= (1 - f_H(c) (x)) ∧ (1 - f_H(c) (y)) .

1 - f_H(c) (rx) =1 - f_V(c) (rx)

≥1 - f_V(c) (x) =1 - f_H(c) (x) .

Case2: If c ∈ B - A, for all x, y ∈ M and r ∈ R, then

t_H(c) (0) =1 because of t_W(c) (0) =1.

t_H(c) (x + y) = t_W(c) (x + y) ≥ t_W(c) (x) ∧ t_W(c) (y) = t_H(c) (x) ∧ t_H(c) (y) .

t_H(c) (rx) = t_W(c) (rx) ≥ t_W(c) (x) = t_H(c) (x) .

1 - f_H(c) (0) =1 because of 1 - f_W(c) (0) =1 .

1 - f_H(c) (x + y) =1 - f_W(c) (x + y)

≥(1 - f_W(c) (x)) ∧ (1 - f_W(c) (y))

= (1 - f_H(c) (x)) ∧ (1 - f_H(c) (y)) .

1 - f_H(c) (rx) =1 - f_W(c) (rx)

≥1 - f_W(c) (x) =1 - f_H(c) (x) .

Case3: If c ∈ A ∩ B, for all x, y ∈ M and r ∈ R, then

t_H(c) (0) =1 because of t_H(c) (0) = t_V(c) (0) ∧ t_W(c) (0) =1 ∧ 1 =1 .

t_H(c) (x + y) = (t_V(c) (x + y)) ∧ (t_W(c) (x + y))

≥ (t_V(c) (x) ∧ t_V(c) (y)) ∧ (t_W(c) (x) ∧ t_W(c) (y))

= (t_V(c) (x) ∧ t_W(c) (x)) ∧ (t_V(c) (y) ∧ t_W(c) (y))

= t_H(c) (x) ∧ t_H(c) (y) .

t_H(c) (rx) = (t_F(c) (rx)) ∧ (t_G(c) (rx))

≥ (t_F(c) (x)) ∧ (t_G(c) (x)) = t_H(c) (x) .

1 - f_H(c) (0) =1 because of (1 - f_V(c) (0)) ∧ (1 - f_W(c) (0)) =1 ∧ 1 =1 .

1 - f_H(c) (x + y) = (1 - f_V(c) (x + y)) ∧ (1 - f_W(c) (x + y))

≥ ((1 - f_V(c) (x)) ∧ (1 - f_V(c) (y))) ∧ ((1 - f_W(c) (x)) ∧ (1 - f_W(c) (y)))

= ((1 - f_V(c) (x)) ∧ (1 - f_W(c) (x))) ∧ ((1 - f_V(c) (y)) ∧ (1 - f_W(c) (y)))

= (1 - f_H(c) (x)) ∧ (1 - f_H(c) (y)) .

1 - f_H(c) (rx) = (1 - f_V(c) (rx)) ∧ (1 - f_W(c) (rx))

≥(1 - f_V(c) (x)) ∧ (1 - f_W(c) (x)) =1 - f_H(c) (x) . □

Theorem 3.10. Let (V, A) and (W, B) be two vague soft modules over M. Then, their union (V, A) $\tilde{\cup}$ (W, B) is a vague soft module over M if A ∩ B = ∅ .

Proof. We know that (V, A) $\tilde{\cup}$ (W, B) = (H, C) such that C = A ∪ B . For all x ∈ M, we can define the following statements: $t_{H (c)} (x) = {\begin{matrix} t_{V (c)} (x), & if c \in A - B, \\ t_{W (c)} (x), & if c \in B - A . \end{matrix}}$ and $1 - f_{H (c)} (x) = {\begin{matrix} 1 - f_{V (c)} (x), & if c \in A - B, \\ 1 - f_{W (c)} (x), & if c \in B - A . \end{matrix}}$ Case1: If c ∈ A - B, for all x, y ∈ M and r ∈ R, then

t_H(c) (0) =1 because of t_V(c) (0) =1.

t_H(c) (x + y) = t_V(c) (x + y) ≥ t_V(c) (x) ∨ t_V(c) (y) = t_H(c) (x) ∨ t_H(c) (y) .

t_H(c) (rx) = t_V(c) (rx) ≥ t_V(c) (x) = t_H(c) (x) .

1 - f_H(c) (0) =1 because of 1 - f_V(c) (0) =1 .

1 - f_H(c) (x + y) =1 - f_V(c) (x + y)

≥(1 - f_V(c) (x)) ∨ (1 - f_V(c) (y))

= (1 - f_H(c) (x)) ∨ (1 - f_H(c) (y)) .

1 - f_H(c) (rx) =1 - f_V(c) (rx)

≥1 - f_V(c) (x) =1 - f_H(c) (x) .

Case2: If c ∈ B - A, for all x, y ∈ M and r ∈ R, then

t_H(c) (0) =1 because of t_W(c) (0) =1.

t_H(c) (x + y) = t_W(c) (x + y) ≥ t_W(c) (x) ∨ t_W(c) (y) = t_H(c) (x) ∨ t_H(c) (y) .

t_H(c) (rx) = t_W(c) (rx) ≥ t_W(c) (x) = t_H(c) (x) .

1 - f_H(c) (0) =1 because of 1 - f_W(c) (0) =1 .

1 - f_H(c) (x + y) =1 - f_W(c) (x + y)

≥(1 - f_W(c) (x)) ∨ (1 - f_W(c) (y))

= (1 - f_H(c) (x)) ∨ (1 - f_H(c) (y)) .

1 - f_H(c) (rx) =1 - f_W(c) (rx) ≥1 - f_W(c) (x) =1 - f_H(c) (x) . □

If A ∩ B ≠ ∅ , then above theorem is not used. We illustrate this by an example.

Example 3.11. Let A = {x, y, z} and B = {x, y} . We define $\begin{matrix} V (x) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.8}, \bar{0.9}] / \bar{1}, [\bar{0.8}, \bar{0.9}] / \bar{2}}, \\ V (y) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.5}, \bar{0.6}] / \bar{1}, [\bar{0.5}, \bar{0.6}] / \bar{2}}, \\ V (z) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.3}, \bar{0.4}] / \bar{1}, [\bar{0.3}, \bar{0.4}] / \bar{2}} . \\ W (x) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.7}, \bar{0.8}] / \bar{1}, [\bar{0.7}, \bar{0.8}] / \bar{2}}, \\ W (y) & = & {[\bar{1}, \bar{1]} / \bar{0}, [\bar{0.4}, \bar{0.5}] / \bar{1}, [\bar{0.4}, \bar{0.5}] / \bar{2}} . \end{matrix}$

(V, A) and (W, B) are two vague soft modules over M. But union of $(V, A) \tilde{\cup} (W, B)$ is not a vague soft module over M because of A ∩ B ≠ ∅ .

Theorem 3.12. Let (V, A) and (W, B) be two vague soft modules over M. Then, (V, A) $\tilde{\land}$ (W, B) is a vague soft module over M .

Proof. We know that (V, A) $\tilde{\land}$ (W, B) = (H, C) such that C = A × B and $\begin{matrix} H & : & C \to P (M) : c \mapsto H (c) = H_{(c)} = H_{(a, b),} \end{matrix}$ where for all c = (a, b)∈ A × B ; x, y ∈ M and r ∈ R $\begin{matrix} H_{(a, b)} & = & {(t_{H (a, b)} (x) = t_{V (a)} (x) \land t_{W (b)} (x)), \\ (1 - f_{H (a, b)} (x) \\ = & (1 - f_{V (a)} (x)) \land (1 - f_{W (b)} (x)))} . \end{matrix}$

t_H(a,b) (0) =1 because of t_H(a,b) (0) = t_V(a) (0) ∧ t_W(b) (0) =1 ∧ 1 =1 .

t_H(a,b) (x + y) = t_V(a) (x + y) ∧ t_W(b) (x + y)

≥ (t_V(a) (x) ∧ t_V(a) (y)) ∧ (t_W(b) (x) ∧ t_W(b) (y))

= (t_V(a) (x) ∧ t_W(b) (x)) ∧ (t_V(a) (y) ∧ t_W(b) (y))

= t_H(a,b) (x) ∧ t_H(a,b) (y) .

t_H(a,b) (rx) = t_V(a) (rx) ∧ t_W(b) (rx)

≥t_V(a) (x) ∧ t_W(b) (x) = t_H(a,b) (x) .

1 - f_H(a,b) (0) = (1 - f_V(a) (0)) ∧ (1 - f_W(b) (0)) =1 ∧ 1 =1 .

1 - f_H(a,b) (x + y) = (1 - f_V(a) (x + y)) ∧ (1 - f_W(b) (x + y))

≥ ((1 - f_V(a) (x)) ∧ (1 - f_V(a) (y))) ∧ ((1 - f_W(b) (x) ∧ (1 - f_W(b) (y)))

= ((1 - f_V(a) (x)) ∧ (1 - f_W(b) (x))) ∧ ((1 - f_V(a) (y)) ∧ (1 - f_W(b) (y)))

= ((1 - f_H(a,b) (x)) ∧ (1 - f_H(a,b) (y))) .

1 - f_H(a,b) (rx) = (1 - f_V(a) (rx)) ∧ (1 - f_W(b) (rx)) ≥ (1 - f_V(a) (x)) ∧ (1 - f_W(b) (x))

= 1 - f_H(a,b) (x) . □

Theorem 3.13. Let (V, A) and (W, B) be two vague soft modules over M. Then, (V, A) $\tilde{\lor}$ (W, B) is a vague soft module over M .

Proof. We know that (V, A) $\tilde{\lor}$ (W, B) = (H, C) such that C = A × B and $\begin{matrix} H & : & C \to P (M) : c \mapsto H (c) = H_{(c)} = H_{(a, b),} \end{matrix}$ where for all c = (a, b)∈ A × B ; x, y ∈ M and r ∈ R $\begin{matrix} H_{(a, b)} & = & {(t_{H (a, b)} (x) = t_{V (a)} (x) \lor t_{W (b)} (x)), \\ (1 - f_{H (a, b)} (x) \\ = & (1 - f_{V (a)} (x)) \lor (1 - f_{W (b)} (x)))} . \end{matrix}$

t_H(a,b) (0) =1 because of t_H(a,b) (0) = t_V(a) (0) ∨ t_W(b) (0) =1 ∨ 1 =1 .

t_H(a,b) (x + y) = t_V(a) (x + y) ∨ t_W(b) (x + y)

≥ (t_V(a) (x) ∨ t_V(a) (y)) ∨ (t_W(b) (x) ∨ t_W(b) (y))

= (t_V(a) (x) ∨ t_W(b) (x)) ∨ (t_V(a) (y) ∨ t_W(b) (y))

= t_H(a,b) (x) ∨ t_H(a,b) (y) .

t_H(a,b) (rx) = t_V(a) (rx) ∨ t_W(b) (rx)

≥t_V(a) (x) ∨ t_W(b) (x) = t_H(a,b) (x) .

1 - f_H(a,b) (0) = (1 - f_V(a) (0))

∨(1 - f_W(b) (0)) =1 ∨ 1 =1 .

1 - f_H(a,b) (x + y) = (1 - f_V(a) (x + y)) ∨ (1 - f_W(b) (x + y))

≥ ((1 - f_V(a) (x)) ∨ (1 - f_V(a) (y)))

∨ ((1 - f_W(b) (x) ∨ (1 - f_W(b) (y)))

= ((1 - f_V(a) (x)) ∨ (1 - f_W(b) (x)))

∨ ((1 - f_V(a) (y)) ∨ (1 - f_W(b) (y)))

= ((1 - f_H(a,b) (x)) ∨ (1 - f_H(a,b) (y))) .

1 - f_H(a,b) (rx) = (1 - f_V(a) (rx)) ∨ (1 - f_W(b) (rx)) ≥ (1 - f_V(a) (x)) ∨ (1 - f_W(b) (x))

= 1 - f_H(a,b) (x) . □

Furthermore, the following theorems provide a new line of vision on vague soft modules.

Theorem 3.14. Let (V, A) be a vague soft set over M . Then (V, A) is a vague soft module over M if and only if ∀a ∈ A and arbitrary α ∈ (0, 1] with (V_a) ^α≠ ∅, the α- level soft set (V, A) ^α is a soft module over M .

Proof. (⇒) : Let (V, A) be a vague soft module over M . Then for each a ∈ A, V (a) = V_a is a vague submodule of M . Let a ∈ A and α ∈ (0, 1] be arbitrary such that (V_a) ^α ≠ ∅ . Then, x, y ∈ (V_a) ^α, so V_a (x) ≥ α and V_a (y) ≥ α . Hence V_a (x - y) ≥ V_a (x) ∧ V_a (y) ≥ α which implies that x - y ∈ (V_a) ^α . Now for any r ∈ R and x ∈ (V_a) ^α, V_a (rx) ≥ V_a (x) ≥ α which implies that rx ∈ (V_a) ^α . Hence (V_a) ^α is a submodule of M for each a ∈ A and (V, A) ^α is a soft module of M .

(⟸) : Let (V, A) ^α be a soft module over M for all α ∈ (0, 1] , (V_a) ^α ≠ ∅ . Then for each a ∈ A, (V_a) ^α is a submodule of M, ∀α ∈ (0, 1] . We have 0 ∈ (V_a) ¹, V_a (0) ≥1, hence V_a (0) =1, ∀a ∈ A . Let α = V_a (x) ∧ V_a (y) for x, y ∈ M, then x, y ∈ (V_a) ^α . (since V_a (x) ≥ α, V_a (y) ≥ α .) Then, x + y ∈ (V_a) ^α (since (V_a) ^α is submodule of M .) Hence V_a (x + y) ≥ α = V_a (x) ∧ V_a (y) , ∀a ∈ A . Now, for x ∈ M, let α = V_a (x) . Then, x ∈ (V_a) ^α implies that rx ∈ (V_a) ^α for any r ∈ R, which implies that V_a (rx) ≥ α = V_a (x) , ∀a ∈ A . Hence V_a is a vague submodule of M and (V, A) is a vague soft module over M. □

Theorem 3.15. Let (V, A) and (W, B) be two vague soft modules over M such that (V, A)⊑ (W, B). Then for any α ∈ (0, 1] , (V, A) ^α ⊆ (W, B) ^α .

Proof. Given that (V, A)⊑ (W, B) i.e. A ⊂ B and V_a ⊆ W_a, ∀a ∈ A . Now for any α ∈ (0, 1] , (V, A) ^α = (V^α, A) is soft set over M . Similarly, (W, B) ^α = (W^α, B) is also soft set over M . To show that (V, A) ^α ⊆ (W, B) ^α, we have to prove that A ⊂ B and (V_a) ^α ⊆ (W_a) ^α, ∀ a ∈ A . Firstly, A ⊂ B because of assumption. For any a ∈ A, let x ∈ (V_a) ^α which implies that V_a (x) ≥ α . Since W_a (x) ≥ V_a (x) , ∀a ∈ A, x ∈ M, we get W_a (x) ≥ α i.e. x ∈ (W_a) ^α . Hence (V_a) ^α ⊆ (W_a) ^α, ∀a ∈ A and (V, A) ^α ⊆ (W, B) ^α . □

Corollary 3.16. Let (V, A) and (W, B) be two vague soft modules over M . If (V, A) = (W, B), then (V, A) ^α = (W, B) ^α, ∀α ∈ (0, 1] .

Theorem 3.17. Let (V, A) be a vague soft module over M and let α, β ∈ (0, 1] with α ≤ β . Then, (V, A) ^β ⊆ (V, A) ^α, ∀α ∈ (0, 1] .

Proof. Let α, β ∈ (0, 1] with α ≤ β . We have (V, A) ^β = (V^β, A) and (V, A) ^α = (V^α, A). For any a ∈ A, let x ∈ (V_a) ^β i.e. V_a (x) ≥ β . Since β ≥ α, then we get V_a (x) ≥ α, i.e. x ∈ (V_a) ^α . Hence (V_a) ^β ⊆ (V_a) ^α, ∀a ∈ A and (V, A) ^β ⊆ (V, A) ^α . □

4 Vague soft homomorphism

In this section, we give some properties of vague soft function in module theory. Furthermore, we prove some theorems on vague soft homomorphism between vague soft modules.

Definition 4.1. Let (V, A) and (W, B) be two vague soft sets over X and Y respectively. Let f : X → Y and g : A → B be two functions. Then the pair (f, g) is called a vague soft function from X to Y, which means (f, g) is a vague soft function from the vague soft set (V, A) over X to the vague soft set (W, B) over Y .

Remark 4.2. In the case of taking (V, A) and (W, B) as soft sets over X and Y respectively, the pair (f, g) is called a soft function from X to Y .

Now we will take an example.

Example 4.3. Let $A = {\bar{0}, \bar{1}, \bar{2}},$ $B = {\bar{0}, \bar{2}, \bar{4}}$ and $X = ℤ_{2},$ $Y = ℤ_{3} .$ Suppose that f : X → Y be a function such that $k \to \bar{k},$ similarly g : A → B be a function such that k → 2k . We define $\begin{matrix} V_{\bar{0}} & = & {[\bar{1}, \bar{1}] / \bar{0}, [\bar{0.7}, \bar{0.9}] / \bar{1}, [\bar{0.7}, \bar{0.9}] / \bar{2}}, \\ V_{\bar{1}} & = & {[\bar{1}, \bar{1}] / \bar{0}, [\bar{0.6}, \bar{0.8}] / \bar{1}, [\bar{0.6}, \bar{0.8}] / \bar{2}}, \\ V_{\bar{2}} & = & {[\bar{1}, \bar{1}] / \bar{0}, [\bar{0.5}, \bar{0.7}] / \bar{1}, [\bar{0.5}, \bar{0.7}] / \bar{2}} . \\ W_{\bar{0}} & = & {[\bar{1}, \bar{1}] / \bar{0}, [\bar{0.7}, \bar{0.9}] / \bar{2}, [\bar{0.7}, \bar{0.9}] / \bar{4}}, \\ W_{\bar{2}} & = & {[\bar{1}, \bar{1}] / \bar{0}, [\bar{0.6}, \bar{0.8}] / \bar{2}, [\bar{0.6}, \bar{0.8}] / \bar{4}}, \\ W_{\bar{4}} & = & {[\bar{1}, \bar{1}] / \bar{0}, [\bar{0.5}, \bar{0.7}] / \bar{2}, [\bar{0.5}, \bar{0.7}] / \bar{4}}} . \end{matrix}$ It is clear that (f, g) is a vague soft function.

In the following, the image of (V, A) and pre-image of (W, B) under the vague soft function (f, g) are defined.

Definition 4.4. Let (V, A) and (W, B) be two vague soft sets over X and Y respectively. Let (f, g) is a vague soft function from X to Y .

the image of under the vague soft function (f, g) is denoted by (f, g) (V, A) = (f (V) , g (A)) is a vague soft set over Y where ∀b ∈ g (A) and ∀y ∈ Y $({(f (V))}_{b} (y) = {\begin{array}{l} \lor_{f (x) = y} \lor_{g (a) = b} V_{a} (x), & if f^{-1} (y) \neq \emptyset \\ 0, & otherwise . \end{array}}$

The pre-image of (W, B) under the vague soft function (f, g) is denoted by (f, g) ^-1 (W, B) is the vague soft set over X defined by (f, g) ^-1 (W, B) = (f^-1 (W) , g^-1 (B)) where $\begin{matrix} (f^{- 1} (W))_{a} (x) \\ = W_{g (a)} (f (x)), \forall a \in g^{- 1} (B), \forall x \in X . \end{matrix}$

Remark 4.5. i) Let f : X → Y be a function, I : A → A be the identity function and (V, A) be a vague soft set over X . Then, (f, I_A) (V, A) = (f (V) , A) is a vague soft set over Y, where for each a ∈ A, (f (V)) _a is a vague subset of Y defined by

\begin{matrix} (f (V))_{a} (y) \\ = {\begin{matrix} \lor {V_{a} (x) : y = f (x)}, & if f^{- 1} (y) \neq \emptyset \\ 0, & otherwise . \end{matrix}} . \end{matrix}

for all y ∈ Y.

ii) Let f : X → Y be a function and (W, A) be a vague soft set over Y . Then, (f, I_A) ^-1 (W, A) = (f^-1 (W) , A) is a vague soft set over X where for each a ∈ A, (f^-1 (W)) _a is a vague subset of X defined by $(f^{- 1} (W))_{a} (x) = W_{a} (f (x)), for all x \in X .$ The following theorem gives us basic properties of the image of (V, A) and pre-image of (W, B) under the vague soft function (f, g) .

Theorem 4.6. Let f : X → Y be a function and (V, A) and (W, B) be vague soft sets over X and Y respectively. Then, for any α ∈ (0, 1]

(f, I_A) (V, A) ^α ⊆ ((f, I_A) (V, A)) ^α,

(f, I_A) ^-1 (W, A) ^α = ((f, I_A) ^-1 (W, A)) ^α .

Proof. i) We know that (f, I_A) (V, A) ^α = (f, I_A) (V^α, A) = (f (V^α) , A) is a soft set over Y (Here (f, I_A) is a soft function from X to Y) defined by, for each a ∈ A, (f (V^α)) _a is a subset of Y where

\begin{matrix} (f (V^{α}))_{a} & = & f ((V_{a})^{α}) = {f (x) : x \in (V_{a})^{α}} \\ = & {f (x) : V_{a} (x) \geq α} . \end{matrix}

Now ((f, I_A) (V, A)) ^α = (f (V) , A) ^α = ((f (V)) ^α, A) is a soft set over Y defined by, for each a ∈ A,

(f (V))_{a}^{α} = ((f (V))_{a})^{α}

is a subset of Y, where

((f (V))_{a})^{α} = {y \in Y : (f (V))_{a} (y) \geq α} .

Let y ∈ (f (V^α)) _a = f ((V_a) ^α) → y = f (x) for some x ∈ (V_a) ^α → y = f (x) for some x with

\begin{matrix} V_{a} (x) & \geq & α \to \lor {V_{a} (x) : y = f (x)} \\ \geq & α \to (f (V))_{a} (y) \geq α . \end{matrix}

Hence y ∈ ((f (V)) _a) ^α and (f (V^α)) _a ⊆ ((f (V)) _a) ^α for all a ∈ A .In essence,

(f, I_{A}) (V, A)^{α} \subseteq ((f, I_{A}) (V, A))^{α} .

ii) We have (f, I_A) ^-1 (W, A) ^α = (f, I_A) ^-1 (W^α, A) = (f^-1 (W^α) , A) is a soft set over X defined by (f^-1 (W^α)) _a = f^-1 ((W_a) ^α) is a subset of X where

\begin{matrix} f^{- 1} ((W_{a})^{α}) & = & {x \in X : f (x) \in (W_{a})^{α}} \\ = & {x \in X : W_{a} (f (x)) \geq α} \\ = & {x \in X : f^{- 1} (W))_{a} (x) \geq α} \\ = & ((f^{- 1} (W))_{a})^{α} \end{matrix}

i.e. (f^-1 (W^α)) _a = ((f^-1 (W)) _a) ^α for all a ∈ A . Hence (f, I_A) ^-1 (W, A) ^α = ((f, I_A) ^-1 (W, A)) ^α. □

In the following, the vague soft homomorphism and vague soft isomorphism are defined.

Definition 4.7. Let (V, A) and (W, B) be vague soft sets over the R- modules M and N respectively and (f, g) be a vague soft function from M to N . If f is a module homomorphism from M to N, then (f, g) is said to be vague soft homomorphism from M to N . If f is an isomorphism from M to N and g is a one to one mapping from A onto B, then (f, g) is said to be a vague soft isomorphism.

Next theorem shows us that the image of a vague soft module is a vague soft module under a vague soft homomorphism. Similarly, the inverse image of a vague soft module is a vague soft module under a vague soft homomorphism.

Theorem 4.8. Let M and N be two R- modules. Let (V, A) be a vague soft module over M and (f, g) be a vague soft homomorphism from M to N . Then, (f, g) (V, A) is a vague soft module over N . Proof. We have (f, g) (V, A) = (f (V) , g (A)) is a vague soft set over N defined by ∀b ∈ g (A) , ∀ y ∈ N $({(f (V))}_{b} (y) = {\begin{array}{l} \lor_{f (x) = y} \lor_{g (a) = b} V_{a} (x), & if f^{-1} (y) \neq \emptyset \\ 0, & otherwise . \end{array}}$

Since f is homomorphism and V_a is a vague submodule of M for all a ∈ A, then we get $\begin{matrix} ((f (V))_{b} (0) & = & \lor_{f (x) = y} \lor_{g (a) = b} V_{a} (x) \\ = & \lor_{g (a) = b} V_{a} (0) = 1 . \end{matrix}$

Let y₁, y₂ ∈ N and b ∈ g (A) . If f^-1 (y₁) =∅ or f^-1 (y₂) =∅, then clearly, $(f (V))_{b} (y_{1} + y_{2}) \geq (f (V))_{b} (y_{1}) \land (f (V))_{b} (y_{2}) .$ If f^-1 (y₁)≠ ∅ and f^-1 (y₂)≠ ∅, then $\begin{matrix} (f (V))_{b} (y_{1} + y_{2}) = \lor_{f (x) = y_{1} + y_{2}} \lor_{g (a) = b} V_{a} (x) \\ \geq \lor_{g (a) = b} V_{a} (x_{1} + x_{2}) \\ (x = x_{1} + x_{2}, x_{1} \in f^{- 1} (y_{1}) and \\ x_{2} \in f^{- 1} (y_{2}) .) \\ \geq \lor_{g (a) = b} (V_{a} (x_{1}) \land V_{a} (x_{2})) \\ = (\lor_{g (a) = b} V_{a} (x_{1})) \land (\lor_{g (a) = b} V_{a} (x_{2})) . \end{matrix}$ for all x₁, x₂ ∈ M such that f (x₁) = y₁, f (x₂) = y₂ . Hence $\begin{matrix} (f (V))_{b} (y_{1} + y_{2}) \\ \geq (\lor_{f (x_{1}) = y_{1}} \lor_{g (a) = b} V_{a} (x_{1})) \\ \land (\lor_{f (x_{2}) = y_{2}} \lor_{g (a) = b} V_{a} (x_{2})) \\ = (f (V))_{b} (y_{1}) \land (f (V))_{b} (y_{2}) . \end{matrix}$

Now let y ∈ N, r ∈ R, b ∈ g (A) . If f^-1 (y) = ∅ , then clearly (f (V)) _b (ry) ≥ (f (V)) _b (y). If f^-1 (y) ≠ ∅ , then $\begin{matrix} (f (V))_{b} (ry) = \lor_{f (x) = ry} \lor_{g (a) = b} V_{a} (x) \\ \geq \lor_{f ({rx}_{1}) = ry} \lor_{g (a) = b} V_{a} ({rx}_{1}) \\ \geq \lor_{x_{1} \in f^{- 1} (y)} \lor_{g (a) = b} V_{a} ({rx}_{1}) \\ \geq \lor_{f (x_{1}) = y} \lor_{g (a) = b} V_{a} (x_{1}) \\ = (f (V))_{b} (y) . \end{matrix}$

Hence (f (V)) _b is a vague submodule of N for all b ∈ g (A) and (f, g) (V, A) = (f (V) , g (A)) is a vague soft module over N . □

Theorem 4.9. Let M and N be two R- modules. Let (W, B) be a vague soft module over N and (f, g) be a vague soft homomorphism from M to N . Then, (f, g) ^-1 (W, B) is a vague soft module over M .

Proof. We have (f, g) ^-1 (W, B) = ((f) ^-1 (W) , (g) ^-1 (B)) where (f^-1 (W)) _a (x) = W_g(a) (f (x)) for all a ∈ g^-1 (B) , x ∈ M . Using by Theorem 3.2,

For any a ∈ g^-1 (B) , $(f^{- 1} (W))_{a} (0) = W_{g (a)} (f (0)) = W_{g (a)} (0) = 1 .$ since f is a homomorphism from M to N and W_g(a) is a vague submodule of N .

Let x₁, x₂ ∈ M and for all a ∈ g^-1 (B) $\begin{matrix} (f^{- 1} (W))_{a} (x_{1} + x_{2}) = W_{g (a)} (f (x_{1} + x_{2})) \\ = W_{g (a)} (f (x_{1}) + f (x_{2})) \\ \geq W_{g (a)} (f (x_{1})) \land W_{g (a)} (f (x_{2})) \\ = (f^{- 1} (W))_{a} (x_{1}) \land (f^{- 1} (W))_{a} (x_{2}) . \end{matrix}$

Now let r ∈ R, x ∈ M and for all a ∈ g^-1 (B) $\begin{matrix} (f^{- 1} (W))_{a} (rx) \\ = W_{g (a)} (f (rx)) = W_{g (a)} (rf (x)) \\ \geq W_{g (a)} (f (x)) = (f^{- 1} (W))_{a} (x) . \end{matrix}$

Hence (f^-1 (W)) _a is a vague submodule of M for all a ∈ g^-1 (B) and

(f, g)^{- 1} (W, B) = ((f)^{- 1} (W), (g)^{- 1} (B))

is a vague soft module over M . □

Corollary 4.10. Let f : M → N be an R- module homomorphism and (V, A) , (W, A) be vague soft modules over M and N respectively. Then, (f (V) , A) is a vague soft module over N and (f^-1 (W) , A) is a vague soft module over M .

Proof. It is clear by taking (f, I_A) as vague soft homomorphism from M to N . □

5 Vague soft exactness

In this section, we introduce trivial and whole vague soft modules. Then, we investigate short exact and exact sequence of soft modules. Finally, we define vague soft exactness and some basic properties of them are obtained. Throughout this section M is a module.

Definition 5.1. Let (V, A) be a vague soft module over M, then

(V, A) is said to be trivial vague soft module over M if V (x) =0 for all x ∈ A, where 0 is zero element of M .

(V, A) is said to be whole vague soft module over M if V (x) = M for all x ∈ A .

Corollary 5.2. Let (V, A) be a vague soft module over M and f : M → N be a homomorphism. If V (x) = Kerf for all x ∈ A, then (f (V) , A) is the trivial vague soft module over N . Similarly, let (V, A) be a whole vague soft module over M and f : M → N be an epimorphism. Then (f (V) , A) is a whole vague soft module over N .

Definition 5.3. The homomorphism sequence of soft modules $. . . \to M_{n - 1} \to^{f_{n - 1}} M_{n} \to^{f_{n}} M_{n + 1} \to . . .$ is called exact sequence of soft modules if Imf_n-1 = Kerf_n for all $n \in ℕ$ and we call the exact sequence of soft modules form as $0 \to M_{1} \to^{f} M \to^{g} M_{2} \to 0$ the short exact sequence of soft modules.

Proposition 5.4. Let (V, A) be a trivial vague soft module over M₁ module and (W, B) be a whole vague soft module over M₂ module if $0 \to M_{1} \to^{f} \to M \to^{g} \to M_{2} \to 0$ is a short exact sequence, then $0 \to V (x) \to^{\tilde{f}} M \to^{\tilde{g}} \to W (y) \to 0$ is a short exact sequence for all x ∈ A, y ∈ B .

Proof. We have V (x) =0, ∀ x ∈ A, since (V, A) is a trivial vague soft module over M₁, so $\tilde{f}$ is a monomorphism. W, (y) = M₂, ∀ y ∈ B since (W, B) is a whole vague soft module over M₂ . g : M → M₂ is an epimorphism as 0 → M₁ → ^f → M → ^g → M₂ → 0 is a short exact sequence, so $\tilde{g}$ is an epimorphism. □

Proposition 5.5. Let (V, A) be a trivial vague soft module over M₁ module and (W, B) be a whole vague soft module over M module if

$0 \to M_{1} \to^{f} \to M \to^{g} \to M_{2} \to 0$

is a short exact sequence, then $0 \to f (V) (x) \to^{\tilde{f}} M \to^{\tilde{g}} g (W) (y) \to 0$

is a short exact sequence for all x ∈ A, y ∈ B .

Proof. We have V (x) =0, ∀ x ∈ A, since (V, A) is a trivial vague soft module over M₁. Kerf = 0, so Kerf = V (x) for all x ∈ A. Consequently (f (V) , A) is trivial vague soft module over M. (W, B) is a whole vague soft module over M and g : M → M₂ is an epimorphism, so (g (W) , B) is a whole vague soft module over M₂, thus

$0 \to f (V) (x) \to^{\tilde{f}} M \to^{\tilde{g}} g (W) (y) \to 0$ is a short exact sequence for all x ∈ A, y ∈ B . □

Definition 5.6. Let (V, A) , (W, B) and (U, C) be three vague soft modules over M, N and K modules respectively. Then we say vague soft exactness at (W, B) , if the following conditions are satisfied:

M → ^f
₁N → ^f
₂K is exact,

A → ^g
₁B → ^g
₂C is exact,

f₁ (V (x)) = W (g₁ (x)) for all x ∈ A,

f₂ (W (x)) = U (g₂ (x)) for all x ∈ B,

which is denoted by (V, A) → ^(f₁,g₁) (W, B) → ^(f₂,g₂) (U, C) .

In this definition, if every (V_i, A_i) , i ∈ I is vague soft exact, then we say that (V_i, A_i) _i∈I is vague soft exact.

Proposition 5.7. Let (V, A) and (W, B) be two vague soft modules over M and N modules respectively. If (V, A) → ^(f,g) (W, B) →0 is vague soft exact, then (f, g) is vague soft homomorphism. In particular, if

$0 \to (V, A) \to^{(f, g)} (W, B) \to 0$

is vague soft exact, then (f, g) is vague soft isomorphism.

Proof. Since (V, A) → ^(f,g) (W, B) →0 is vague soft exact, then we have M → ^fN → 0 and A → ^gB → 0 are exact. Thus f and g are epimorphisms, it is clear that (f, g) is homomorphism. If 0 → (V, A) → ^(f,g) (W, B) →0 is vague soft exact, then 0 → M → ^fN → 0 and 0 → A → ^gB → 0 are exact. Thus f and g are isomorphisms, it is clear that (f, g) is vague soft isomorphism. □

Definition 5.8. Let M = 0 and A = 0 . Then (V, A) =0 . We call (V, A) be a zero-vague soft module.

Proposition 5.9. Let (V, A) , (W, B) and (U, C) be three vague soft modules over M, N and K modules respectively. If

$(V, A) \to^{(f_{1}, g_{1})} (W, B) \to^{(f_{2}, g_{2})} (U, C)$

is vague soft exact with f₁, g₁ epimorphism and f₂, g₂ monomorphism, then (W, B) is a zero-vague soft module.

Proof. Since (V, A) → ^(f₁,g₁) (W, B) → ^(f₂,g₂) → (U, C) is vague soft exact with f₁, g₁ epimorphism and f₂, g₂ monomorphism, then we have M → ^{f
₁}N → ^{f
₂}K and A → ^{g
₁}B → ^{g
₂}C, hence N = 0 and B = 0, it is clear that (W, B) is zero-vague soft module. □

Theorem 5.10. Let (V, A) and (W, B) be two vague soft modules over M and N modules respectively. For any M ⊂ N, A ⊂ B and M ⊂ W (x) where x ∈ B . If (V, A) → ^(f,g) (W, B) is vague soft homomorphism, then

$0 \to (V, A) \to^{(f, g)} (W, B) \to^{(f_{1}, g_{1})} (I, B / A) \to 0$

is vague soft exact, where I (x + A) = W (x)/M for all x ∈ B .

Proof. We know that 0 → M → ^fN → ^{f
₁}N/M → 0 and 0 → A → ^gB → ^{g
₁}B/A → 0 are exact. It is clear that M is a soft submodule of N, so that N/M is a soft module and M is a soft submodule of W (x) and W (x)/M is always a soft submodule of N/M . This shows that (I, B/A) is a vague soft module over N/M .

For all x ∈ B/A . Define f₁ : N → N/M by f₁ (n) = n + M, for all n ∈ N . Meanwhile, we define g₁ : B → B/A by g₁ (b) = b + A, for all b ∈ B . Therefore, it gives that f₁ (W (x)) = W (x) + M, I (g₁ (x)) = I (x + A) = W (x) + M for all x ∈ B, and hence f₁ (W (x)) = I (g₁ (x)) .

This implies 0 → (V, A) → ^(f,g) (W, B) → ^(f₁,g₁) (I, B/A) →0 is vague soft exact. □

Theorem 5.11. Let (V, A₂) , (W, A₁) and (U, A) be three vague soft modules over M₂, M₁ and M modules respectively. If M₁ and M₂ are soft submodules of M with M₂ ⊂ M₁, A₁ and A₂ be soft submodules of A with A₂ ⊂ A₁, where M₁ ⊂ U (x) , for all x ∈ A and M₂ ⊂ W (x) for all x ∈ A₁, then

$0 \to (I, A_{1} / A_{2}) \to^{(f_{1}, g_{1})} (J, A / A_{1}) \to^{(f_{2}, g_{2})} (P, A / A_{1}) \to 0$

is vague soft exact, where I (x + A₂) = W (x)/M₂, for all x ∈ A₁, J (x + A₂) = U (x)/M₂, for all x ∈ A, P (x + A₁) = U (x)/M₁, for all x ∈ A .

Proof. Since M₁ and M₂ are soft submodules of M with M₂ ⊂ M₁, then we have a short exact sequence 0 → M₁/M₂ → ^{f
₁}M/M₂ → ^{f
₂}M/M₁ → 0 . Since A₁ and A₂ are soft submodules of A with A₂ ⊂ A₁, there is a short exact sequence 0 → A₁/A₂ → ^{g
₁}A/A₂ → ^{g
₂}A/A₁ → 0 . It is clear that M₂ is a soft submodule of M₁, so that M₁/M₂ is a soft module. It gives that W (x)/M₂ is a soft module for all x ∈ A₁ from M₂ is a soft submodule of W (x) . However W (x)/M₂ is always a soft submodule of M₁/M₂ . This shows that (I, A₁/A₂) is a vague soft module over M₁/M₂ for all x ∈ A₁/A₂ . It is clear that (J, A/A₂) and (P, A/A₁) be a vague soft module over M/M₂ and M/M₁ respectively.

Define f₁ : M₁/M₂ → M/M₂ by f₁ (m₁ + M₁) = m + M₂, for all m₁ ∈ M₁ . Meanwhile, we define g₁ : A₁/A₂ → A/A₂ by g₁ (a₁ + A₂) = a + A₂, for all a₁ ∈ A₁ . Therefore, we have f₁ (I (x)) = f₁ (W ((x)/M₂) = U (x) + M₂, J (g₁ (x)) = J (x + A₂) = U (x) + M₂ for all x ∈ A₁/A₂, so f₁ (I (x)) = J (g₁ (x)) for all x ∈ A₁/A₂ .

Define f₂ : M/M₂ → M/M₁ by f₂ (m + M₂) = m + M₁, for all m ∈ M . Let g₂ : A/A₂ → A/A₁ be defined by g₂ (a + A₂) = a + A₁, for all a ∈ A . Also, we have f₂ (J (x)) = f₂ (U ((x)/M₂) = U (x) + M₁ for all x ∈ A/A₂, so f₂ (J (x)) = P (g₂ (x)) for all x ∈ A/A₂ . Hence 0 → (I, A₁/A₂) → ^(f₁,g₁) (J, A/A₁) → ^(f₂,g₂) (P, A/A₁) →0 is vague soft exact. □

Theorem 5.12. Let (V_i, A_i) , i = 1, 2, 3, 4, 5 be vague soft modules over M_i modules, i = 1, 2, 3, 4, 5 respectively. If

$\begin{matrix} 0 \to (V_{1}, A_{1}) \to^{(f_{1}, g_{1})} (V_{2}, A_{2}) \to^{(f_{2}, g_{2})} (V_{3}, A_{3}) \to 0 \end{matrix}$

and $\begin{matrix} 0 \to (V_{3}, A_{3}) \to^{(f_{3}, g_{3})} (V_{4}, A_{4}) \to^{(f_{4}, g_{4})} (V_{5}, A_{5}) \to 0 \end{matrix}$

are vague soft exact, then $\begin{matrix} 0 \to (V_{1}, A_{1}) \to^{(f_{1}, g_{1})} (V_{2}, A_{2}) \to^{(f_{3} f_{2}, g_{3} g_{2})} (V_{4}, A_{4}) \to^{(f_{4}, g_{4})} (V_{5}, A_{5}) \to 0 \end{matrix}$ is vague soft exact.

Proof. Since 0 → (V₁, A₁) → ^(f₁,g₁) (V₂, A₂) → ^(f₂,g₂) (V₃, A₃) →0 and 0 → (V₃, A₃) → ^(f₃,g₃) (V₄, A₄) → ^(f₄,g₄) (V₅, A₅) →0 are vague soft exact, we have 0 → M₁ → ^{f
₁}M₂ → ^{f
₂}M₃ → 0 and 0 → M₃ → ^{f
₃}M₄ → ^{f
₄}M₅ → 0 are exact. It is clear that 0 → M₁ → ^{f
₁}M₂ → ^{f
₃
f
₂}M₄ → ^{f
₄}M₅ → 0 is exact. Since 0 → A₁ → ^{g
₁}A₂ → ^{g
₂}A₃ → 0 and 0 → A₃ → ^{g
₃}A₄ → ^{g
₄}A₅ → 0 are exact. It is clear that 0 → A₁ → ^{g
₁}A₂ → ^{g
₃
g
₂}A₄ → ^{g
₄}A₅ → 0 is exact. Since f₂ (V₂ (x)) = V₃ (g₂ (x)) for all x ∈ A₂ and f₃ (V₃ (x)) = V₄ (g₃ (x)) for all x ∈ A₃ . We have f₃f₂ (V₂ (x)) = f₃ (V₃ (g₂ (x))) = V₄ (g₃g₂ (x)) for all x ∈ A₂ . This implies 0 → (V₁, A₁) → ^(f₁,g₁) (V₂, A₂) → ^{(f₃ f₂,g₃g₂)} (V₄, A₄) → ^(f₄,g₄) (V₅, A₅) →0 is vague soft exactness. □

6 Conclusion

In this work, the theoretical point of view of vague soft module was discussed. The work was focused on vague soft module, vague soft module homomorphism and vague soft exactness. By using these concepts, we studied the algebraic properties of vague soft sets in module structure. One could extends this work by studying other algebraic structures.

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