On fuzzy subsupermodule

Abstract

In this paper we introduce the concept of (weak) fuzzy subsupermodules based on (thin) supermodules different from fuzzy subhypermodules. In this study, the concept of α-cuts play a main role for constructing of extended (weak) fuzzy subsupermodules. In final, we introduce a notation of residual quotients of (weak) fuzzy subsupermodules and obtain some conditions to be a (weak) fuzzy subsupermodule. Also obtained some applied results in residual quotients of (weak) fuzzy subsupermodules of superrings as specially subsupermodules.

Keywords

Superrings (thin) supermodule (weak) fuzzy subsupermodules

1 Introduction

The hypercompositional structure theory as an extension of classic structures, was firstly introduced, by F. Marty in 1934 [21]. In algebraic hypercompositional system, output from the hyperoperation on elements is a set and so any algebraic system is an algebraic hypercompositional system. Marty extended the concept of groups to hypergroups and other researchers presented the algebraic hypercompositional structures concepts such as hyperring, hypermodule, hyperfield, hypergraph, polygroup, multiring, etc in this similar way [18]. Algebraic hypercompositional structures are applied in several branches of sciences such as artificial intelligence and (hyper)complex network [9]. The foundations of the theory of hypermodules were provided by Massouros in 1988, when he introduced the concept of free hypermodules and cyclic hypermodules [17]. Recently, based on various types of morphisms, the authors introduced some categories consisting of these hypermodules such as the categories _Rhmod, _{_Rs}hmod, $_{R} H mod$ , and $_{R_{s}} H mod$ (some types of category of all hypermodules over a hyperring R, in the sense of Krasner) [1, 2], fundamental relation and hypermodule [5, 6], support of a hypermodules [7], graded hypermodules [12], complete part of fuzzy hypermodules [15], (M, N)-hypermodule [22], free hypermodules [25], normal subfuzzy (m, n)-hypermodules [26] and fuzzy hypermodules [30].

In classical (hyper)set theory, could not easily characterize complex phenomena, so as a generalization of the classical (hyper) set theory, fuzzy (hyper) set theory (a theory that plays an important role in modeling and controlling unsure complex (hyper) system) was introduced by Zadeh to deal with uncertainties [32]. After the work of Zadeh, among other fields, an important extension is extracted in the field of fuzzy algebraic hypercompositional structures and fuzzy subalgebraic hypercompositional structures such as fuzzy hypergroup [28], fuzzy subhypergroup [8], fuzzy hyperring [13] and etc. The concept of fuzzy hypermodules, is introduced by V. L. Fotea based on left hypermodule over hyperring [14] and other researchers investigated related notations on fuzzy hypermodules such as fundamental relation on fuzzy hypermodules, isomorphism theorems of fuzzy hypermodule and L-fuzzy hypermodules [11 , 31]. In 1973 J. Mittas introduced the concept of the superring (superanneau) as a result of his study on polynomials with coefficients from a hyperring [23, 24] and Massouros in [19, 20] introduced the concept of the supermodule. Recently, Ameri et al. in [4] introduced the multiplicative hyperring of fractions and coprime hyperideals and Hamidi et al. presented a new concept of the thin supermodules, which are a special case of the supermodules. They showed that the category RGKHmod (thin supermodules and all homomorphisms) is an abelian category and thin supermodules have a normal injective resolution [16].

Regarding these points we introduce the concept of (weak) fuzzy subsupermodules based on (thin) supermodules. The main motivation of this work is a generalisation of fuzzy submodules to fuzzy subsupermodules and so this investigation is based on extension of the structures of modules to supermodules. We investigated some properties of (weak) fuzzy subsupermodules via hyperproduct of elements and an external multiplication of its elements. We established the concept of (weak) fuzzy subsupermodules via values-cuts and showed that there exists a corresponding relation between of (weak) fuzzy subsupermodules and supermodules. In this regard, we work on extension of (weak) fuzzy subsupermodules to a larger class of (weak) fuzzy subsupermodules via union, intersection and direct sum of (weak) fuzzy subsupermodules. A notation of residual quotients of (weak) fuzzy subsupermodules as a result of binary operation on (weak) fuzzy subsupermodules is introduced and discussed in this paper and under some conditions is proved that the residual quotients of (weak) fuzzy subsupermodules is a (weak) fuzzy subsupermodules via the concept of α-cuts.

2 Preliminaries

In what follows, we recall some results from [3 , 24], that need in our work.

Let R be a nonempty set and $P^{*} (R) = {S | \emptyset \neq S \subseteq R}$ . Every map $+_{R} : R \times R ⟶ P^{*} (R)$ , is said to be a hyperoperation (hypercomposition), an algebraic hypercompositional structure (R, + _R) is called a hypergroupoid and for all nonempty subsets A, B of R, A + _RB = ⋃ _a∈A,b∈B (a + _Rb)(generally, the singleton {x} is identified with its member x, so x + _RB = {x} + _RB, where x ∈ R). Recall that a hypergroupoid (R, + _R) is called a semihypergroup, if for all x, y, z ∈ R, (x + _Ry) + _Rz = x + _R (y + _Rz) and a semihypergroup (R, + _R) is called a hypergroup, if for all x ∈ R, x + _RR = R + _Rx = R (reproduction axiom). A commutative hypergroup (R, + _R) (for all x, y ∈ R, x + _Ry = y + _Rx) is called a canonical hypergroup provided that

there exists a unique element 0_R ∈ R such that for all x ∈ R, 0_R + _Rx = x + _R0_R = {x},

for all x ∈ R, there exists a unique element -x ∈ R such that 0_R ∈ (x + _R (- x)) ∩ ((- x) + _Rx),

for all x, y, z ∈ R, x ∈ y + _Rz implies y ∈ x + _R (- z) and z ∈ x + _R (- y),

and we will denote it by (R, + _R, 0_R). A system (R, + _R, 0_R, · _R) is called a weakly distributive superring or simply superring whenever

(R, + _R, 0_R) is a canonical hypergroup,

(R, · _R) is a semihypergroup such that for all x ∈ R, x · _R0_R = 0_R · _Rx = {0_R},

for all x, y, z ∈ R, we have and .

A superring (R, + _R, 0_R, · _R) is called commutative (with unit element), if for all x, y ∈ R, x · _Ry = y · _Rx (if there exists an element 1 ∈ R such that for all x ∈ R, 1 · _Rx = x · _R1 = {x}). For a given superring (R, + _R, 0_R, · _R), a canonical hypergroup (A, + _A, 0_A) together with a left external multiplication

* : R \times A ⟶ P^{*} (A)

, is called a R-supermodule or simplify supermodule over superring R ( we denote it by (A, + _A, 0_A, *)), if for all r, s ∈ R and for all a, b ∈ A,

r * (a + _Ab) ⊆ (r * a) + _A (r * b) ,

(r + _Rs) * a ⊆ (r * a) + _A (s * a) ,

(r · _Rs) * a ⊆ r * (s * a),

0_R * a = {0_A}.

A supermodule A is called unitary if there exists 1_R ∈ R such that for all a ∈ A we have 1_R * a = {a} and is called thin, if for all r ∈ R and for all a ∈ A, we have |r * a|=1, otherwise it is called a non- thin supermodule. If in (iii), (r · _Rs) * a = r * (s * a) will call it is an associative supermodule. A map f : A → A′ is called a (an inclusion) strong or good homomorphism of supermodules if, for all x, y ∈ A and for all r ∈ R, (f (x + _Ay) ⊆ f (x) + _A′f (y)) f (x + _Ay) = f (x) + _A′f (y) and (f (r * x) ⊆ r * ′f (x)) f (r * x) = r * ′f (x). A map f : A → A′ is called a weak homomorphism of supermodules, if for all x, y ∈ A and for all r ∈ R, f (x+ _Ay) ∩ (f (x) + _A′f (y)) ≠ ∅ and f (r * x) ∩ (r * ′f (x)) ≠ ∅ . A nonempty subset B of A is said to be a subsupermodule of A (is denoted by B ≤ A), if for all x, y ∈ B and for all r ∈ R, x + _A (- y) ⊆ B and r * x ⊆ B, also B is called a normal general subsupermodule (is denote by B ⊴ A), if for all x ∈ A, we have x + _AB + _A (- x) ⊆ B.

Theorem 2.1. [16] Assume (R, + _R, 0_R, · _R, 1_R) is a superring and (A, + _A, *) is a supermodule. Then for all r ∈ R and a ∈ A:

0_A ∈ r * 0_A;

- (r* a) ∩ (- r * a) ≠ ∅;

if (A, + _A, *) is a thin supermodule, then - (r* a) = (- r) * a ;

- (r* a) ∩ (r * (- a)) ≠ ∅;

if (A, + _A, *) is a thin supermodule, then - (r* a) = r * (- a) ;

if (A, + _A, *) is a thin supermodule, then 0_A = r * 0_A.

3 Fuzzy subsupermodule

In this section, we introduce a concept of (weak) fuzzy subsupermodule and investigate their basic properties. Also based on thin and unitary supermodule, it presented some weak fuzzy subsupermodules and is compared the effect of fuzzy subsupermodule on zero or unit element of any given supermodule. Finally, we try to find a relation between of fuzzy subsupermodule and subsupermodule via level subsets and to extend fuzzy subsupermodules with respect to generated fuzzy subsupermodule and residual quotients.

Theorem 3.1. Let (A, + _A, 0_A, *) be a thin supermodule, r ∈ R, C ⊆ R, B, B₁, B₂ be subsupermodules of A.

if A is unitary and 1 ∈ C, then C*B is a subsupermodule of A,

B₁ + _AB₂ is a subsupermodule of A.

Proof. (i) Let x ∈ C * B. Then there exist r ∈ C and b ∈ B such that x ∈ r * b. Since (A, + _A, 0_A, *) is a thin supermodule and B is a subsupermodule of A, by Theorem 2.1, -x ∈ - (r * b) = r * (- b) ⊆ C * B. If x, y ∈ C * B, then there exist r, r′ ∈ C and b, b′ ∈ B such that x + _Ay ⊆ r * b + _Ar′ * b′ ⊆ B ⊆ C * B. In addition, for s ∈ R and x ∈ C * B, there exist r ∈ C and b ∈ B such that s * x ⊆ s * (r * b) ⊆ s * B ⊆ C * B. Therefore, C * B is a subsupermodule of A.

(ii) It is similar to part (i).

Definition 3.2. Let (A, + _A, 0_A, *) be a supermodule. A fuzzy subset μ of A is called a fuzzy subsupermodule of A if the following conditions hold:

for any x, y∈ A, ⋀ _{z∈x+_A(-y)}μ (z) ≥ T_min (μ (x) , μ (y)) (simplify ⋀ μ (x + _A (- y)) ≥ T_min (μ (x) , μ (y))) ;

for any x ∈ A, and r ∈ R, ⋀ _z∈r*xμ (z) ≥ μ (x) (simplify ⋀ μ (r * x) ≥ μ (x)).

From now on for simplify, consider $F (A), F (R), F_{gs} (A)$ as the set of all fuzzy subsets of A, R, and fuzzy subsupermodule of A, respectively.

Example 3.3. Set R = {0, 1, a}. Then (R, + _R, · _R) is a commutative superring and (R, + _R, *) is an associative non-thin supermodule. $\begin{matrix} \begin{matrix} +_{R} & 0 & 1 & a 0 & 0 & 1 & a \\ 1 & 1 & {0, a} & 1 \\ a & a & 1 & {0, a} \end{matrix}, \begin{matrix} \cdot_{R} & 0 & 1 & a 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & a \\ a & 0 & a & {0, a} \end{matrix} and \\ \begin{matrix} * & 0 & 1 & a 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & a \\ a & 0 & a & {0, a} \end{matrix} . \end{matrix}$ Now, μ is a fuzzy subsupermodule as follows:

μ	0	1	a
	0.5	0.2	0.3

Theorem 3.4. Let (A, + _A, 0_A, *) be a supermodule, x, y ∈ A, B, C ⊆ A and μ a fuzzy subsupermodule of A. Then

⋀μ (x + _A (- B)) ≥ T_min (μ (x) , ⋀ μ (B)) .

⋀μ (C + _A (- B))) ≥ T_min (⋀ μ (C) , ⋀ μ (B)) .

⋀μ (r * B) ≥ ⋀ μ (B) .

If (A, + _A, 0_A, *) is a unitary supermodule, then ⋀μ (1 * x) = μ (x) .

If (A, + _A, 0_A, *) is a unitary supermodule, then ⋀μ (1 * B) = μ (B) .

Proof. (i) Let z ∈ x + _A (- B) be an arbitrary element. Then there exists y ∈ B such that z ∈ x + _A (- y) and so μ (z) ≥ ⋀ (μ (x + _A (- y))) ≥ T_min (μ (x) , μ (y)) ≥ T_min (μ (x) , ⋀ μ (B)) . It follows that ⋀_{z∈x+_A(-B)}μ (z) ≥ T_min (μ (x) , ⋀ μ (B)).

(ii) It is similar to (i).

(iii) Let z ∈ r * B be an arbitrary element. Then there exists y ∈ B such that z ∈ r * y and so μ (z) ≥ ⋀ μ (r * y) ≥ μ (y) ≥ ⋀ μ (B).

(iv) , (v) Are clear by definition.

Theorem 3.5. Let (A, + _A, 0_A, *) be a supermodule, x ∈ A and μ be a fuzzy subsupermodule of A. Then

μ (0_A) ≥ μ (x);

μ (- x) ≥ μ (x);

if there exists r ∈ R suchthat r * x = A, then μ is a constant;

if there exists r, s ∈ R suchthat r + _A (- s) = A, then μ is a constant;

for any r ∈ R, ⋀μ (r * 0_A) = μ (0_A).

Proof. (i) Let x ∈ A. Then by definition, μ (0_A) = ⋀ μ (0_R * x) ≥ μ (x).

(ii) Let x ∈ A. Then by definition, μ (- x) = μ (0_A + _A (- x)) ≥ T_min (μ (0_A) , μ (x)) ≥ μ (x).

(iii) Let x ∈ A. Since there exists r ∈ R such that r * x = A, by definition of μ we get that μ (x) ≥ ⋀ μ (A) ≥ μ (x). It follows that ⋀μ (A) = μ (x) and so μ is a constant map.

(iv) It is similar to (iii).

(v) Let r ∈ R . Then by definition and item (i), μ (0_A) ≤ ⋀ μ (r * 0_A) ≤ μ (0_A). Hence μ (r * 0_A) = μ (0_A).

Corollary 3.6. Let (A, + _A, 0_A, *) be a supermodule, B ⊆ A and μ be a fuzzy subsupermodule of A. Then

μ (0_A) ≥ ⋀ μ (B + _A (- B));

⋀μ (- B) ≥ ⋀ μ (B).

Let $μ \in F (A)$ . Then μ is called weak, if for all x ∈ A, μ (- x) = μ (x).

Lemma 3.7. Let (A, + _A, 0_A, *) be a supermodule, B ≤ A, and 0 ≤ α ≤ 1. Then

α_B is a weak fuzzy subsupermodule of A,

if A is unitary, then 1_A is a weak fuzzy subsupermodule of A.

Proof. (i) Let x, y ∈ A. Then ⋀α_B (x + _A (- y)) = ⋀ {α_B (t) | t ∈ x + _A (- y)}. If x + _A (- y) ⊆ B, then ⋀ {α_B (t) | t ∈ x + _A (- y)} = α ≥ T_min (α_B (x) , α_B (y)) . Suppose there exists t ∈ x + _A (- y) such that t ∉ B. In this case, if T_min (α_B (x) , α_B (y)) = α ≠ 0, we get that α_B (x) = α_B (y) = α, which implies that x + _A (- y) ⊆ B and it is a contradiction. In addition, -x ∈ B, concludes that x = - (- x) ∈ B and so α_B (- x) = α_B (x) = α. In similar way, -x ∉ B, concludes that x = - (- x) ∉ B and so α_B (- x) = α_B (x) =0.

(ii) By item (i), is clear.

Theorem 3.8. Let (A, + _A, 0_A, *) be a supermodule, x, y, z ∈ A and μ a fuzzy subsupermodule of A. Then

⋀_{z∈x+_Ay}μ (z) ≥ T_min (μ (x) , μ (- y)) ,

⋀_{z∈x+_Ay}μ (z) ≥ T_min (μ (x) , μ (y)) ,

if z ∈ x + _A (- y), then μ (z) ≥ T_min (μ (x) , μ (y)) ,

for any x ∈ A, ⋀ μ (x + _A (- x)) ≥ μ (x) ,

if ⋀μ (x + _Ay) = μ (0_A), then T_min (μ (x) , μ (y)) ≥ T_min (μ (- x) , μ (- y)) ,

if ⋀μ (x + _Ay) = μ (0_A), then μ (x) = μ (y).

Proof. (i) Let x, y ∈ A. Since y = - (- y), we get that ⋀_{z∈x+_Ay}μ (z) = ⋀ _{z∈x+_A(-(-y))}μ (z) ≥ T_min (μ (x) , μ (- y)).

(ii) It is clear by (i).

(iii) Let x, y, z ∈ A. Since z ∈ x + _A (- y), get μ (z) ≥ ⋀ _{t∈x+_A(-y)}μ (t) ≥ T_min (μ (x) , μ (y)), and so μ (z) ≥ T_min (μ (x) , μ (y)) .

(iv) Let x ∈ A . Then by Theorem 3.5, ⋀μ (x + _A (- x)) ≥ T_min (μ (x) , μ (- x)) ≥ T_min (μ (x) , μ (x)) = μ (x).

(v) Let x, y ∈ A. Since y ∈ ((x + _Ay) + _A (- x)), by Theorem 3.4, we get that μ (y) ≥ T_min (⋀ μ (x + _Ay) , μ (- x)) = T_min (μ (0_A) , μ (- x)) = μ (- x). In a similar a way can see that μ (x) ≥ μ (- y) and so T_min (μ (x) , μ (y)) ≥ T_min (μ (- x) , μ (- y)) .

(vi) It is routine by (v).

Corollary 3.9. Let (A, + _A, 0_A, *) be a supermodule, B, C ⊆ A and μ a fuzzy subsupermodule of A. Then

⋀μ (C + _AB) ≥ T_min (⋀ μ (C) , ⋀ μ (B)) ,

for all $m \in ℕ$ , $⋀ μ (\underset{m - time}{\underset{︸}{x +_{A} x +_{A} \dots +_{A} x)}} \geq μ (x),$

for all $m \in ℕ$ , $⋀ μ (\underset{m - time}{\underset{︸}{B +_{A} B +_{A} \dots +_{A} B)}} \geq ⋀ μ (B) .$

Theorem 3.10. Let (A, + _A, 0_A, *) be a supermodule, x₁, x₂, …, x_n ∈ A and μ a fuzzy subsupermodule of A. Then

$⋀ μ (x_{1} +_{A} x_{2} +_{A} \dots +_{A} x_{n}) \geq ⋀_{i = 1}^{n} μ (x_{i})$ .

If $⋀_{i = 1}^{n - 1} μ (x_{i}) \geq μ (x_{n})$ , then ⋀μ (x₁ + _Ax₂ + _A … + _Ax_n) = μ (x_n).

Proof. (i) It is obvious by Corollary 3.9.

(ii) By item (i), we get that $⋀ μ (x_{1} +_{A} x_{2} +_{A} \dots +_{A} x_{n - 1}) \geq ⋀_{i = 1}^{n - 1} μ (x_{i}) \geq μ (x_{n})$ . Since x_n ∈ ((x₁ + _Ax₂ + _A … + _Ax_n-1) + _A (- (x₁ + _Ax₂ + _A … + _Ax_n-1)) + _Ax_n) , we get $\begin{matrix} μ (x_{n}) \geq ⋀ μ ((x_{1} +_{A} x_{2} +_{A} \dots +_{A} x_{n - 1}) +_{A} x_{n}) +_{A} \\ (- (x_{1} +_{A} x_{2} +_{A} \dots +_{A} x_{n - 1}))) \geq T_{\min} (⋀ μ (x_{1} +_{A} x_{2} +_{A} \\ \dots +_{A} x_{n - 1} +_{A} x_{n}), ⋀ μ (x_{1} +_{A} x_{2} +_{A} \dots +_{A} x_{n - 1})) \\ \geq T_{\min} (⋀ μ (x_{1} +_{A} x_{2} +_{A} \dots +_{A} x_{n - 1}), μ (x_{n})) \\ \geq T_{\min} (μ (x_{n}), μ (x_{n})) = μ (x_{n}) . \end{matrix}$ Thus T_min (⋀ μ ((x₁ + _Ax₂ + _A … + _Ax_n-1 + _Ax_n) , ⋀ μ (x₁ + _Ax₂ + _A … + _Ax_n-1)) = μ (x_n). In addition, ⋀μ (x₁ + _Ax₂ + _A … + _Ax_n-1) ≥ μ (x_n), so ⋀μ (x₁ + _Ax₂ + _A … + _Ax_n-1 + _Ax_n) = μ (x_n).

Example 3.11. Consider the ring $(ℤ, +, .)$ and A = {0, e, a, b, c}. Then (A, + _A, *) is a non-thin supermodule as follows; $\begin{matrix} \begin{matrix} +_{A} & 0 & e & a & b & c \\ 0 & {0, e} & 0 & a & b & c \\ e & 0 & e & a & b & c \\ a & a & a & {0, e} & c & b \\ b & b & b & c & {0, e} & a \\ c & c & c & b & a & {0, e} \end{matrix}, \begin{matrix} * & 0 & e & a & b & c \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ - 2 & {e, 0} & e & {e, 0} & {e, 0} & {e, 0} \\ - 1 & {e, 0} & e & {a, e} & {b, e} & {e, c} \\ 0 & e & e & e & e & e \\ 1 & {e, 0} & e & {a, e} & {b, e} & {c, e} \\ 2 & {e, 0} & e & {e, 0} & {e, 0} & {e, 0} \\ ⋮ & ⋮ & ⋮ & ⋮ \end{matrix} . \end{matrix}$ Thus μ is a fuzzy subsupermodule as follows.

μ	e	a	b	c	0
	0.2	0.2	0.2	0.2	0.2

Remark. By Example 3.11, we found that the converse of Theorem 3.5, is not necessarily true.

Theorem 3.12. Let μ be a fuzzy subsupermodule of A. If A is thin and unitary, then μ is a weak fuzzy subsupermodule of A.

Proof. Let x ∈ X. Since A is a unitary and thin supermodule, we get that -x = - (1 * x) = (-1) * x. It implies that μ (- x) = μ (x). Thus μ is a weak fuzzy subsupermodule of A.

Theorem 3.13. Let A be a thin and unitary supermodule and μ a fuzzy subset of A. Then μ is a fuzzy subsupermodule of A if and only if ∀ x, y ∈ A, ∀ r, s ∈ R, ⋀ μ (r * x + _As * y) ≥ T_min (μ (x) , μ (y)) .

Proof. Let x, y ∈ A and r, s ∈ R. Since ⋀μ (r * x + _As * y) ≥ T_min (μ (x) , μ (y)), by Theorem 3.12, we get that ⋀μ (x + _A (- y)) = ⋀ μ (1 * x + _A (-1) * y) ≥ T_min (μ (x) , μ (y)). In addition, ⋀μ (r * x) = ⋀ μ (r * x + _A0_R * y) ≥ T_min (μ (x) , μ (0_A)) = μ (x), implies that ⋀μ (r * x) ≥ μ (x). Conversely, for x, y ∈ A and r, s ∈ R, since μ is a fuzzy subsupermodule of A, by Corollary 3.9, get that ⋀μ (r * x + _As * y) ≥ T_min (⋀ μ (r * x) , ⋀ μ (r * y)) ≥ T_min (μ (x) , μ (y)).

3.1 α-cuts and fuzzy subsupermodule

In this subsection, considered the notation of α-cuts on fuzzy subsupermodules and tried to obtain an equal condition to introduce and extend of fuzzy subsupermodules.

Theorem 3.14. Let (A, + _A, 0_A, *) be a supermodule and μ be a fuzzy subset of A and 0 ≤ α ≤ 1. Then the following statements are equivalent:

μ is a fuzzy subsupermodule of A;

μ^α is a subsupermodule of A, where μ^α = {x ∈ A | μ (x) ≥ α}.

Proof. Let x, y ∈ μ^α and r ∈ R. Since μ is a fuzzy subsupermodule of A, we get that ⋀μ (x + _A (- y)) ≥ T_min (μ (x) , μ (y)) ≥ α and so x + _A (- y) ⊆ μ^α. In addition, ⋀μ (r * x) ≥ μ (x) ≥ α concludes that r * x ⊆ μ^α and so μ^α is a subsupermodule of A, where μ^α = {x ∈ A | μ (x) ≥ α}.

Conversely, for x, y ∈ A, let α₁ = μ (x) , α₂ = μ (y) and α = T_min (α₁, α₂). Hence T_min (μ (x) , μ (y)) ≥ α and so x, y ∈ μ^α. But μ^α is a subsupermodule of A, it follows that x + _A (- y) ⊆ μ^α and so ⋀μ (x + _A (- y)) ≥ α ≥ T_min (μ (x) , μ (y)). In addition, if α = μ (x), then x ∈ μ^α. Since μ^α is a subsupermodule of A, for all r ∈ R, we get that r * x ⊆ μ^α. Thus ⋀μ (r * x) ≥ α = μ (x). Therefore, μ is a fuzzy subsupermodule of A.

Corollary 3.15. Let μ be a fuzzy subsupermodule of A. Then

0 ∈ μ^α,

if 0 ≤ α ≤ α′ ≤ 1, then μ^α′ ⊆ μ^α.

Let A be a supermodule, S ≤ A and α, α′ ∈ [0, 1]. Define $μ^{[α, α^{'}]} (x) = {\begin{matrix} α^{'}, & if x \in S, \\ α, & if x \notin S, \end{matrix} .$ Thus have the following theorem.

Theorem 3.16. Let A be a supermodule and S ≤ A. Then μ^[α,α′] is a fuzzy subsupermodule of A.

Proof. Let x, y ∈ A, then ⋀_{z∈x+_A(-y)}μ^[α,α′] (z) = α or α′. If x, y ∈ S, then x + _A (- y) ⊆ S and so ⋀_{z∈x+_A(-y)}μ^[α,α′] (z) = α′ ≥ T_min (μ^[α,α′] (x) , μ^[α,α′] (y)). If (x ∈ S and y ∉ S) , (x ∉ S and y ∈ S) and (x ∉ S and y ∉ S), then ⋀_{z∈x+_A(-y)}μ^[α,α′] (z) ≥ T_min (α, α′) ≥ T_min (μ^[α,α′] (x) , μ^[α,α′] (y)). In similar way, ∀ x ∈ A, ∀ r ∈ R, ⋀ _z∈r*xμ^[α,α′] (z) ≥ μ^[α,α′] (x) . It follows that μ^[α,α′] is a fuzzy subsupermodule of A.

Theorem 3.17. Let A be a supermodule and μ a fuzzy subsupermodule of A. Then A⁽¹⁾ = {x | x ∈ A, μ (x) =1} is a subsupermodule of A.

Proof. Let x, y ∈ A⁽¹⁾. Then ⋀μ (x + _Ay) ≥ T_min (μ (x) , μ (- y)) ≥ T_min (1, 1) =1. It follows that ⋀μ (x + _A (- y)) =1 and so x + _A (- y) ⊆ A⁽¹⁾. In addition, for r ∈ R, ⋀μ (r * x) ≥ μ (x) ≥1, implies that r * x ⊆ A⁽¹⁾ and so A⁽¹⁾ = {x | x ∈ A, μ (x) =1} is a subsupermodule of A.

Theorem 3.18. Let A be a supermodule and {μ_i} _i∈I a family of fuzzy subsupermodule of A. Then

⋂_i∈Iμ_i is a fuzzy subsupermodule of A.

If {μ_i} _i∈I is a chain, then ⋃_i∈Iμ_i is a fuzzy subsupermodule of A.

Proof. (i) Let x, y ∈ A . Then $\begin{matrix} ⋀_{x, y \in A} ⋂_{i \in I} μ_{i} (x +_{A} (- y)) = ⋀_{x, y \in A} ⋀_{i \in I} μ_{i} (x +_{A} (- y)) \\ = ⋀_{i \in I} ⋀_{x, y \in A} μ_{i} (x +_{A} (- y)) \\ \geq ⋀_{i \in I} (T_{\min} (μ_{i} (x), μ_{i} (y))) = T_{\min} (⋀_{i \in I} μ_{i} (x), ⋀_{i \in I} μ_{i} (y)) \\ = T_{\min} (⋂_{i \in I} μ_{i} (x), ⋂_{i \in I} μ_{i} (y)) . \end{matrix}$ Let x ∈ A, r ∈ R. In similar a way, ⋀ ⋂ _i∈Iμ_i (r * x) ≥ ⋂ _i∈Iμ_i (x), so ⋂_i∈Iμ_i is a fuzzy subsupermodule of A.

(ii) Let x, y ∈ A. Since {μ_i} _i∈I is a chain, there exists t ∈ I such that $\begin{matrix} ⋀_{x, y \in A} ⋃_{i \in I} μ_{i} (x +_{A} (- y)) = ⋀_{x, y \in A} μ_{t} (x +_{A} (- y)) \\ \geq T_{\min} (μ_{t} (x), μ_{t} (y)) = T_{\min} (⋁_{i \in I} μ_{i} (x), ⋁_{i \in I} μ_{i} (y)) \\ = T_{\min} ((⋃_{i \in I} μ_{i}) (x), (⋃_{i \in I} μ_{i}) (y)) . \end{matrix}$ Let x ∈ A, r ∈ R. In similar a way, Since {μ_i} _i∈I is a chain, there exists t ∈ I that $\begin{matrix} ⋀ ⋃_{i \in I} μ_{i} (r * x) = ⋀ μ_{t} (r * x) \\ \geq μ_{t} (x) = ⋁_{i \in I} μ_{i} (x) = ⋃_{i \in I} μ_{i} (x) . \end{matrix}$

Let μ be a fuzzy subset of A. Define $〈 μ 〉 = ⋂ {ν \in F_{gs} (A) | μ \subseteq ν}$ , so have the following clear results, by Theorem 3.18 and definition.

Corollary 3.19. Let A be a supermodule and $μ, ν \in F (A)$ and 0 ≤ α ≤ 1. Then

$〈 μ 〉 \in F_{gs} (A)$ (call it by fuzzy subsupermodules generated by μ),

$μ \in F_{gs} (A)$ if and only if 〈μ〉 = μ,

if μ ⊆ ν, then 〈μ〉 ⊆ 〈μ〉,

〈μ〉^α = 〈μ^α〉.

4 Extension of fuzzy subsupermodules via α-cut

In this section, genersted some fuzzy subsupermodules via some operations on fuzzy subsupermodules and by the concept of α-cut showed that operations on fuzzy subsupermodules is a fuzzy subsupermodule.

Definition 4.1. Let (A, + _A, 0_A, *) be a supermodule and μ, ν fuzzy subsupermodules of A. For any x, y, z ∈ A, define (μ ⊕ _Aν) (z) = ⋁ _{z∈(x+_Ay)} ((μ (x) ∧ ν (y)).

Theorem 4.2. Let A be a supermodule and μ, ν fuzzy subsupermodules of A.

for 0 ≤ α ≤ 1, have (μ ⊕ _Aν) ^α = μ^α + _Aν^α,

if μ, ν are weak, then μ ⊕ _Aν is weak,

μ ⊕ _Aν is a fuzzy subsupermodule of A.

Proof. (i) Let z ∈ (μ ⊕ _Aν) ^α. Then (μ ⊕ _Aν) (z) ≥ α and so ⋁_{z∈(x+_Ay)} ((μ (x) ∧ ν (y)) ≥ α. It concludes that there exists a, b ∈ A such that z ∈ (a + _Ab) and (μ (a) ∧ ν (b)) ≥ α. Thus a ∈ μ^α and b ∈ ν^α and so (μ ⊕ _Aν) ^α ⊆ μ^α + _Aμ^α. In similar way, one can see that μ^α + _Aν^α ⊆ (μ ⊕ _Aν) ^α and so (μ ⊕ _Aν) ^α = μ^α + _Aμ^α. (ii) Let x ∈ A. Then $\begin{matrix} (μ \oplus_{A} ν) (- x) = ⋁_{- x \in (a +_{A} b)} ((μ (a) \land ν (b)) \\ = ⋁_{x \in (- a +_{A} (- b))} ((μ (- a) \land ν (- b)) = (μ \oplus_{A} ν) (x), \end{matrix}$ so μ ⊕ _Aν is weak.

(iii) By Corollary 3.15, (μ⊕ _Aν) ^α ≠ ∅. Thus by Proposition 3.1 and Theorem 3.14, μ ⊕ _Aν is a fuzzy subsupermodule of A.

Let (A, + _A, 0_A, *) be a supermodule, $ξ \in F (R)$ and $μ \in F (A)$ . Then for any x, y ∈ A and r ∈ R, define (ξ ⊛ μ) (y) = ⋁ _y∈(r*x) ((ξ (r) ∧ μ (x)) and (r * μ) (y) = ⋁ _y∈r*xμ (x). Then we have the following results.

Theorem 4.3. Let (A, + _A, 0_A, *) be a supermodule, $ξ \in F (R)$ , $μ, ν \in F (A)$ and 0 ≤ α ≤ 1. Then

(ξ ∪ ξ′) ⊛ ν = (ξ ⊛ ν) ∪ (ξ′ ⊛ ν) ,

(ξ ∩ ξ′) ⊛ ν = (ξ ⊛ ν) ∩ (ξ′ ⊛ ν) ,

ξ ⊛ (ν ∪ ν′) = (ξ ⊛ ν) ∪ (ξ ⊛ ν′) ,

ξ ⊛ (ν ∩ ν′) = (ξ ⊛ ν) ∩ (ξ ⊛ ν′) ,

if ν ⊆ ν′, then ξ ⊛ ν ⊆ ξ ⊛ ν′,

if ξ ⊆ ξ′, then ξ ⊛ ν ⊆ ξ′ ⊛ ν.

Proof. (i) Let y ∈ A. Then $\begin{matrix} ((ξ \cup ξ^{'}) ⊛ ν) (y) = ⋁_{y \in r * x} ((ξ \cup ξ^{'}) (r) \land ν (x)) \\ = ⋁_{y \in r * x} ((ξ (r) \lor ξ^{'} (r)) \land ν (x)) \\ = ⋁_{y \in r * x} ((ξ (r) \land ν (x)) \lor (ξ^{'} (r) \land ν (x))) \\ = (⋁_{y \in r * x} (ξ (r) \land ν (x)) \lor (⋁_{y \in r * x} (ξ^{'} (r) \land ν (x))) \\ = ((ξ ⊛ ν) \cup (ξ^{'} ⊛ ν)) (y) . \end{matrix}$

(ii) It is similar to (i).

(iii) Let y ∈ A. Then $\begin{matrix} (ξ ⊛ (ν \cup ν^{'})) (y) = ⋁_{y \in r * x} (ξ (r) \land (ν \cup ν^{'}) (x)) \\ = ⋁_{y \in r * x} (ξ (r) \land (ν (x) \lor ν^{'} (x))) \\ = ⋁_{y \in r * x} ((ξ (r) \land ν (x)) \lor (ξ (r) \land ν^{'} (x))) \\ = (⋁_{y \in r * x} ((ξ (r) \land ν (x))) \lor ⋁_{y \in r * x} ((ξ (r) \land ν^{'} (x)) \\ = ((ξ ⊛ ν) \cup (ξ ⊛ ν^{'})) (y) \end{matrix}$

(iv) It is similar to (iii).

(v, vi) Are clear by definition.

Theorem 4.4. Let (A, + _A, 0_A, *) be a supermodule, $ξ \in F (R)$ and $μ \in F (A)$ . Then

If (A, + _A, 0_A, *) is thin and μ is weak, then ξ ⊛ μ is weak,

If 0 ≤ α ≤ 1, then (ξ ⊛ μ) ^α = ξ^α * μ^α.

Proof. (i) Let y ∈ A. Then by Theorem 2.1, $\begin{matrix} (ξ ⊛ μ) (- y) = ⋁_{- y \in (r * x)} ((ξ (r) \land μ (x)) \\ = ⋁_{y \in r * (- x)} ((ξ (r) \land μ (- x)) = (ξ ⊛ μ) (y) . \end{matrix}$ It follows that ξ ⊛ μ is (very) weak.

(ii) Let 0 ≤ α ≤ 1, and y ∈ (ξ ⊛ μ) ^α . Then ⋁_y∈(r*x) ((ξ (r) ∧ μ (x)) = (ξ ⊛ μ) (y) ≥ α and so there exist r ∈ R and x ∈ A such that y ∈ r * x and ξ (r) ∧ μ (x) ≥ α. Hence there exist r ∈ R and x ∈ A such that ξ (r) ≥ α and μ (x) ≥ α. Consequently, y ∈ r * x ⊆ ξ^α * μ^α.

Conversely, let y ∈ ξ^α * μ^α. Then there exist r ∈ ξ^α and x ∈ μ^α such that y ∈ r * x and ξ (r) ≥ α and μ (x) ≥ α. It follows that (ξ ⊛ μ) (y) ≥ (ξ (r) ∧ μ (x)) ≥ α and so (ξ ⊛ μ) ^α = ξ^α * μ^α.

Theorem 4.5. Let (A, + _A, 0_A, *) be a unitary thin supermodule, $ξ \in F (R)$ and $μ \in F_{gs} (A)$ . If 1 ∈ ξ^α, then $ξ ⊛ μ \in F_{gs} (A)$ .

Proof. Since 1 ∈ ξ^α, μ^α≠ ∅ (by Corollary 3.15) and 0_A ∈ 1 *0_A, by Theorem 4.4, we get (ξ⊛ μ) ^α ≠ ∅. Let y ∈ ξ^α * μ^α. Then there exist r ∈ ξ^α and x ∈ μ^α such that y ∈ r * x. Since μ is a fuzzy subsupermodule, we get that μ (- x) ≥ μ (x) ≥ α and so -x ∈ μ^α. Because (A, + _A, 0_A, *) is a thin superrmodule, by Theorem 2.1, -y ∈ - (r * x) = r * (- x) ∈ ξ^α * μ^α. Suppose x, y ∈ ξ^α * μ^α, because 1 ∈ ξ^α, by Proposition 3.1, x + _Ay ⊆ ξ^α * μ^α and so x + _A (- y) ⊆ ξ^α * μ^α. If r ∈ R and x ∈ ξ^α * μ^α, then there exists s ∈ ξ^α and y ∈ μ^α such that x ∈ s * y. Hence r * x ⊆ r * (s * y) ⊆ r * μ^α ⊆ μ^α. Applying 1 ∈ ξ^α, conclude that r * (ξ^α * μ^α) ⊆ ξ^α * μ^α and so ξ^α * μ^α is a supermodule of A. Therefore, by Theorems 3.14 and 4.4, ξ ⊛ μ is a fuzzy subsupermodule.

Theorem 4.6. Let A be a supermodule, $μ \in F (A), r \in R$ and x ∈ A. Then

for any 0 ≤ α ≤ 1, r_α ⊛ μ ⊆ α;

for any 0 ≤ α, β ≤ 1, r_α ⊛ x_β = T_min (α, β);

for any 0 ≤ α ≤ 1, ξ ⊛ x_α ⊆ α;

for all r ∈ R, 1_{r} ⊛ μ = r * μ.

Proof. Let x, y ∈ A and r ∈ R. Then for any 0 ≤ α, β ≤ 1,

(i) (r_α ⊛ μ) (y) = ⋁ _y∈s*x (r_α (s) ∧ μ (x)) = ⋁ _y∈r*x (α ∧ μ (x)) ≤ α ≤ α (y). It follows that r_α ⊛ μ ⊆ α.

(ii) (r_α ⊛ x_β) (y) = ⋁ _y∈s*z (r_α (s) ∧ x_β (z)) = ⋁ _y∈r*x (α ∧ β) = α ∧ β. It follows that r_α ⊛ x_β = T_min (α, β).

(iii) (ξ ⊛ x_α) (y) = ⋁ _y∈r*z (x_α (z) ∧ ξ (r)) = ⋁ _y∈r*x (α ∧ ξ (r)) ≤ α = α (y). It follows that ξ ⊛ x_α ⊆ α.

$\begin{matrix} (iv) (1_{{r}} ⊛ μ) (y) = ⋁_{y \in (s * x)} ((1_{{r}} (s) \land μ (x)) \\ = (⋁_{y \in (s * x) r \neq s} ((1_{{r}} (s) \land μ (x))) \lor (⋁_{y \in (s * x) r = s} ((1_{{r}} (s) \land μ (x))) \\ = ⋁_{y \in (r * x)} ((1_{{r}} (r) \land μ (x)) = ⋁_{y \in (r * x)} μ (x) = (r * μ) (y) . \end{matrix}$

Since every superring R is a supermodule (in some following items we can consider A = R), we have the following results.

Corollary 4.7. Let A be a supermodule and μ a fuzzy subset of A. Then

1_R ⊛ μ = r * μ;

1_R ⊛ 1_A = 1_A;

1_R ⊛ α_A = α;

1_R ⊛ 1_R = 1_R.

Definition 4.8. Let (A, + _A, 0_A, *) be a supermodule, $ξ, η \in F (R)$ and $μ, ν, τ \in F (A)$ . Define μ : ν = ⋃ _η⊛ν⊆μη and μ : ξ = ⋃ _ξ⊛τ⊆μτ as residual quotients. So have the following results.

Theorem 4.9. Let (A, + _A, 0_A, *) be a supermodule, $ξ \in F (R)$ , $μ, ν \in F (A)$ and 0 ≤ α ≤ 1. Then

μ : ν = ∪ {r_α | r ∈ R, r_α ⊛ ν ⊆ μ};

μ : ξ = ∪ {x_α | x ∈ A, ξ ⊛ x_α ⊆ μ}.

Proof. (i) Let x, y ∈ A, r ∈ R and 0 ≤ α ≤ 1. Since r_α is a fuzzy subset of R, get that ∪ {r_α | r ∈ R, r_α ⊛ ν ⊆ μ} ⊆ μ : ν. Conversely, let η be a fuzzy subset of R, η ⊛ ν ⊆ μ and for r ∈ R, η (r) = α. Then ∀ y ∈ A, $\begin{matrix} (r_{α} ⊛ ν) (y) = ⋁_{y \in (s * x)} (r_{α} (s) \land ν (x)) = ⋁_{y \in (r * x)} (α \land ν (x)) \\ = ⋁_{y \in (r * x)} (η (r) \land ν (x)) \leq ⋁_{y \in (s * x)} ⋁_{y \in (r * x)} (η (r) \land ν (x)) \\ = ⋁_{y \in (s * x)} (η ⊛ ν) (y) \leq μ (y) . \end{matrix}$ (ii) The proof is similar to (i).

Theorem 4.10. Let (A, + _A, 0_A, *) be a supermodule, $ξ \in F (R)$ , and $μ, ν \in F (A)$ . Then

(μ : ν) ⊛ ν ⊆ μ;

ξ ⊛ (μ : ξ) ⊆ μ.

Proof. (i) Let x, y ∈ A, r ∈ R and η be a fuzzy subset of R. Then $\begin{matrix} ((μ : ν) ⊛ ν) (y) = ⋁_{y \in (r * x)} ((μ : ν) (r) \land ν (x)) \\ = ⋁_{y \in (r * x)} (⋁_{η ⊛ ν \subseteq μ} η (r) \land ν (x)) = ⋁_{y \in (r * x)} {(η ⊛ ν) (y) \\ | η ⊛ ν \subseteq μ} \leq ⋁_{y \in (r * x)} μ (y) = μ (y) . \end{matrix}$

(ii) The proof is similar to (i).

Theorem 4.11. Let (A, + _A, 0_A, *) be a supermodule, $ξ, ξ^{'} \in F (R)$ , $μ, μ^{'} \in F (A)$ and 0 ≤ α ≤ 1. Then

(μ : ξ ∪ ξ′) = (μ : ξ) ∩ (μ : ξ′) ,

(μ : ξ ∩ ξ′) = (μ : ξ) ∪ (μ : ξ′) ,

(μ ∩ μ′) : ξ = (μ : ξ) ∩ (μ′ : ξ) ,

(μ ∪ μ′) : ξ = (μ : ξ) ∪ (μ′ : ξ) .

Proof. (i) Let y ∈ A. Then $(μ : ξ \cup ξ^{'}) = \cup {ν \in F (A) | (ξ \cup ξ^{'}) ⊛ ν \subseteq μ}$ . Applying Theorem 4.3, (ξ ∪ ξ′) ⊛ ν ⊆ μ, implies that (ξ ⊛ ν) ∪ (ξ′ ⊛ ν) ⊆ μ and so (ξ ⊛ ν) ⊆ μ and (ξ′ ⊛ ν) ⊆ μ. It follows that $\begin{matrix} (μ : ξ \cup ξ^{'}) \subseteq (\cup {ν \in F (A) | (ξ ⊛ ν) \subseteq μ}) \\ \cap (\cup ({ν \in F (A) | (ξ^{'} ⊛ ν) \subseteq μ} = (μ : ξ) \cap (μ : ξ^{'})) . \end{matrix}$ Because (ξ ⊛ (μ : ξ)) ⊆ μ and (ξ′ ⊛ (μ : ξ′)) ⊆ μ, by Theorem 4.3, get that $\begin{matrix} (ξ \cup ξ^{'}) ⊛ ((μ : ξ) \cap (μ : ξ^{'})) = (ξ ⊛ ((μ : ξ) \\ \cap (μ : ξ^{'})) \cup (ξ^{'} ⊛ ((μ : ξ) \cap (μ : ξ^{'})) \\ \subseteq (ξ ⊛ (μ : ξ)) \cup (ξ^{'} ⊛ (μ : ξ^{'})) \subseteq μ \cup μ = μ \end{matrix}$ and so (μ : ξ ∪ ξ′) = (μ : ξ) ∩ (μ : ξ′) .

(ii) Let y ∈ A. Then $(μ : ξ \cap ξ^{'}) = \cup {ν \in F (A) | (ξ \cap ξ^{'}) ⊛ ν \subseteq μ}$ . Since (ξ ∩ ξ′) ⊛ ν ⊆ ξ ⊛ ν and (ξ ∩ ξ′) ⊛ ν ⊆ ξ′ ⊛ ν, we get that $\begin{matrix} (μ : ξ \cap ξ^{'}) \subseteq (\cup {ν \in F (A) | (ξ ⊛ ν) \subseteq μ}) \\ \cup (\cup ({ν \in F (A) | (ξ^{'} ⊛ ν) \subseteq μ} = (μ : ξ) \cup (μ : ξ^{'})) . \end{matrix}$ Because (ξ ⊛ (μ : ξ)) ⊆ μ and (ξ′ ⊛ (μ : ξ′)) ⊆ μ, by Theorem 4.3, get that $\begin{matrix} (ξ \cap ξ^{'}) ⊛ ((μ : ξ) \cup (μ : ξ^{'})) = ((ξ \cap ξ^{'}) ⊛ ((μ : ξ)) \\ \cup ((ξ \cap ξ^{'}) ⊛ (μ : ξ^{'})) \subseteq (ξ ⊛ (μ : ξ)) \cup (ξ^{'} ⊛ (μ : ξ^{'})) \\ \subseteq μ \cup μ = μ \end{matrix}$ and so (μ : ξ ∩ ξ′) = (μ : ξ) ∪ (μ : ξ′) .

(iii) Let y ∈ A. Since ξ ⊛ τ ⊆ μ ∩ μ′ implies that ξ ⊛ τ ⊆ μ and ξ ⊛ τ ⊆ μ′, get that $\begin{matrix} ((μ \cap μ^{'}) : ξ) (y) = ⋁ {τ (y) | ξ ⊛ τ \subseteq μ \cap μ^{'}} \\ = (⋁ {τ (y) | ξ ⊛ τ \subseteq μ}) \land (⋁ {τ (y) | ξ ⊛ τ \subseteq μ^{'}}) \\ = (μ : ξ) (y) \land (μ^{'} : ξ) (y) = ((μ : ξ) \cap (μ^{'} : ξ)) (y) . \end{matrix}$

(iv) Let y ∈ A. By Theorem 4.3, (μ : ξ) ⊆ (μ ∪ μ′) : ξ and (μ′ : ξ) ⊆ (μ ∪ μ′) : ξ, thus (μ : ξ) ∪ (μ′ : ξ) ⊆ (μ ∪ μ′) : ξ .

Because (ξ ⊛ (μ : ξ)) ⊆ μ and (ξ′ ⊛ (μ : ξ′)) ⊆ μ, by Theorem 4.3, get that $\begin{matrix} ξ ⊛ ((μ : ξ) \cup (μ^{'} : ξ)) = (ξ ⊛ (μ : ξ)) \\ \cup (ξ ⊛ (μ^{'} : ξ)) \subseteq μ \cup μ = μ \end{matrix}$ and so (μ ∪ μ′) : ξ = (μ : ξ) ∪ (μ′ : ξ) .

In [16], Davvaz introduced the concept of fuzzy hyperideal in superring. We apply this conception in the following theorem.

Theorem 4.12. Let (A, + _A, 0_A, *) be a associative supermodule, ξ be a fuzzy hyperideal of R and $μ, τ \in F (A)$ .

⋃_r∈R (ξ ⊛ (r * x) _α) ⊆ ξ ⊛ (x) _α,

⋃_r∈R (ξ ⊛ (r * x) _α) = ξ ⊛ ⋃ _r∈R ((r * x) _α),

$μ : ξ = \cup {τ \in F_{gs} (A) | ξ ⊛ τ \subseteq μ},$

If μ is a fuzzy subsupermodule of A, then μ : ξ is a fuzzy subsupermodule.

Proof. (i) Let y ∈ A. Since (A, + _A, 0_A, *) is associative and ξ is a fuzzy hyperideal of R, $\begin{matrix} (ξ ⊛ (r * x)_{α}) (y) = ⋁_{y \in s * z} (ξ (s) \land (r * x)_{α} (z)) \\ = ⋁_{y \in s * z z \in r * x} (ξ (s) \land α) \leq ⋁_{y \in s * (r * x)} (ξ (s) \land α) \\ = ⋁_{y \in (s \cdot_{R} r) * x} (ξ (s) \land α) \leq ⋁_{y \in (s \cdot_{R} r) * x} (ξ (s \cdot_{R} r) \land α) \\ = (ξ ⊛ (x)_{α}) (y) . \end{matrix}$ (ii) It is clear by Theorem 4.3.

(iii) Let 0 ≤ α ≤ 1, x ∈ A and for $μ \in F (A),$ have ξ ⊛ (x) _α ⊆ μ. Then for all r ∈ R, by item (i) , ⋃_r∈R (ξ ⊛ (r * x) _α) ⊆ ξ ⊛ (x) _α ⊆ μ. Using Theorem 4.9, we have $\cup {τ \in F (A) | ξ ⊛ τ \subseteq μ} \subseteq \cup {τ \in F_{gs} (A) | ξ ⊛ τ \subseteq μ}$ and so $μ : ξ = \cup {τ \in F_{gs} (A) | ξ ⊛ τ \subseteq μ} .$

(iv) Let y ∈ (μ : ξ) ^α. Then (μ : ξ) (y) ≥ α and so $⋁ {ν (y) | ν \in F_{gs} (A), ξ ⊛ ν \subseteq μ} \geq α$ . Thus there exists $τ \in F_{gs} (A)$ such that ξ ⊛ τ ⊆ μ and τ (y) ≥ α. Since τ is a fuzzy subsupermodule of A, we get that -y ∈ τ (y) ^α and so $⋁ {ν (- y) | ν \in F (A), ξ ⊛ ν \subseteq μ} \geq τ (- y) \geq α$ . It follows that (μ : ξ) (- y) ≥ α and so -y ∈ (μ : ξ) ^α. Let x, y ∈ A. Then $\begin{matrix} ⋀ (μ : ξ) (x +_{A} y) = ⋀_{z \in x +_{A} y} (μ : ξ) (z) \\ = ⋀_{z \in x +_{A} y} ⋁ {τ (z) | ξ ⊛ τ \subseteq μ} = ⋁ {⋀_{z \in x +_{A} y} τ (z) | \\ ξ ⊛ τ \subseteq μ} \geq ⋁ {τ (x) \land τ (y) | ξ ⊛ τ \subseteq μ} \\ \geq ⋁ {τ (x) | ξ ⊛ τ \subseteq μ} \land ⋁ {τ (y) | ξ ⊛ τ \subseteq μ} \\ = (μ : ξ) (x) \land (μ : ξ) (y) . \end{matrix}$ Let r ∈ R and x ∈ A. Then $\begin{matrix} ⋀ (μ : ξ) (r * x) = ⋀_{z \in r * x} (μ : ξ) (z) \\ = ⋀_{z \in r * x} ⋁ {τ (z) | ξ ⊛ τ \subseteq μ} = ⋁ {⋀_{z \in r * x} τ (z) | \\ ξ ⊛ τ \subseteq μ} \geq ⋁ {τ (x) | ξ ⊛ τ \subseteq μ} = (μ : ξ) (x) . \end{matrix}$ Therefore, μ : ξ is a fuzzy subsupermodule.

Since every superring R is a supermodule (in some following items we can consider A = R), we have the following results.

Theorem 4.13. Let (A, + _A, 0_A, *) be a supermodule, $ξ \in F (R)$ , $ν \in F (A)$ and 0 ≤ α, β, γ ≤ 1. Then

α_A : ν = ∪ {r_β | r ∈ R, (β_A ∩ ν) ⊆ α_A};

α_A : β_A = ∪ {r_γ | r ∈ R, γ ∧ β ≤ α};

α_A : α_A = ∪ {r_γ | r ∈ R};

ν : r_α = ∪ {x_β | x ∈ A, (α_A ∩ β) ⊆ ν};

s_β : r_α = ∪ {t_γ | t ∈ R, α ∧ γ ≤ β}.

r_α : r_α = ∪ {r_β | r ∈ R}.

Proof. (i) Let x, y ∈ A, r ∈ R and 0 ≤ α, β, γ ≤ 1. Then by Theorem 4.9, $\begin{matrix} (α_{A} : ν) (s) = ⋁ {r_{β} (s) | r \in R, (r_{β} ⊛ ν) (y) \leq α_{A} (y)} \\ = ⋁ {r_{β} (s) | r \in R, ⋁_{y \in (t * x)} ((r_{β} (t) \land ν (x)) \leq α} \\ = ⋁ {r_{β} (s) | r \in R, ⋁_{y \in (r * x)} (β \land ν (x)) \leq α} \\ = ⋁ {r_{β} (s) | r \in R, (β \land ν (x)) \leq α, y \in (r * x)} . \end{matrix}$ (ii) By item (i), $\begin{matrix} (α_{A} : β_{A}) (s) = ⋁ {r_{γ} (s) | r \in R, \\ ⋁_{y \in (r * x)} (γ \land β_{A} (x)) \leq α} = ⋁ {r_{γ} | γ \land β \leq α} . \end{matrix}$ (iii) It is clear by item (ii).

(iv) Let x, y, z ∈ A and 0 ≤ α, β, γ ≤ 1. Then by Theorem 4.9, $\begin{matrix} (ν : r_{α}) (y) = ⋁ {x_{β} (y) | x \in A, (r_{α} ⊛ x_{β}) (y) \leq ν (y)} \\ = ⋁ {x_{β} (y) | x \in A, ⋁_{y \in (s * w)} ((r_{α} (s) \land x_{β} (w)) \leq ν (y)} \\ = ⋁ {x_{β} (y) | x \in A, ⋁_{y \in (r * x)} (α \land β) \leq ν (y)} \\ = ⋁ {x_{β} (y) | x \in A, (α \land β) \leq ν (y), y \in (r * x))} . \end{matrix}$ (v) Let x, y, z ∈ A and 0 ≤ α, β, γ ≤ 1. Then by item (iv), $\begin{matrix} (s_{β} : r_{α}) (y) = ⋁ {t_{γ} (y) | t \in R, (α \land γ) \leq β, y \in (r * x))} . \end{matrix}$ (vi) By item (v), it is clear.

Since every superring R is a supermodule (in some following items we can consider A = R), we have the following results.

Corollary 4.14. Let (A, + _A, 0_A, *) be a supermodule, $ξ \in F (R)$ , $ν \in F (A)$ and 0 ≤ α, β ≤ 1. Then

1_A : ν = ∪ {r_α | r ∈ A};

ν : 1_R = ∪ {x_α | x ∈ A, ν ⊇ α_R};

ν : 1_A = ∪ {r_α | r ∈ R, ν ⊇ α_A};

1_R : r_α = ∪ {s_β | s ∈ R};

1_A : r_α = ∪ {x_β | x ∈ A}.

Let (A, + _A, 0_A, *) be a supermodule, B a subsupermodule of A and ∅ ≠ C ⊆ A. Define B : C = {r ∈ R | r * C ⊆ B} and so have the following results.

Theorem 4.15. Let (A, + _A, 0_A, *) be a supermodule, B a subsupermodule of A, $ζ \in F (R)$ , $μ, ν \in F (A)$ and 0 ≤ α, β ≤ 1. Then

ζ ⊛ ν ⊆ μ, if and only if ζ ⊆ μ : ν;

υ ⊆ μ : ζ, if and only if ζ ⊆ μ : ν;

(1_B:A ∪ α_R) ⊛1_A ⊆ 1_B ∪ α_A;

(1_B ∪ α_A) :1_A = 1_B:A ∪ α_R.

Proof. (i, ii) They are clear by definition.

(iii) Let y ∈ A. Then $\begin{matrix} ((1_{B : A} \cup α_{R}) ⊛ 1_{A}) (y) \\ = & ⋁_{y \in r * x} ((1_{B : A} \cup α_{R}) (r) \land 1_{A} (x)) \\ = & ⋁_{y \in r * x} ((1_{B : A} (r) \lor α_{R} (r)) \land 1_{A} (x)) \\ = & ⋁_{y \in r * x} ((1_{B : A} (r) \lor α) \land 1) \\ = & ⋁_{y \in r * x} ((1_{B : A} (r) \lor α) \in {1, α} . \end{matrix}$ In other hand, (1_B ∪ α_A) (y) =1_B (y) ∨ α_A (y) =1_B (y) ∨ α ∈ {1, α}. Now, if y ∈ B, then (1_B ∪ α_A) (y) =1 ≥ ((1_B:A ∪ α_R) ⊛1_A) (y). But if y ∉ B, since y ∈ r * x, we get that r ∉ B : A and so (1_B ∪ α_A) (y) = α = ((1_B:A ∪ α_R) ⊛1_A) (y). Thus in any cases, we have (1_B:A ∪ α_R) ⊛1_A ⊆ 1_B ∪ α_A.

(iv) By (iii), we have (1_B:A ∪ α_R) ⊛1_A ⊆ 1_B ∪ α_A. Now by (i), (1_B:A ∪ α_R) ⊆ (1_B ∪ α_A) :1_A. Let s ∈ R, x, y ∈ A. Then by Theorem 4.9, $\begin{matrix} ((1_{B} \cup α_{A}) : 1_{A}) (s) = ⋁ {r_{β} (s) | r \in R, (r_{β} ⊛ 1_{A}) \\ \subseteq (1_{B} \cup α_{A})} = ⋁ {r_{β} (s) | r \in R, ⋁_{y \in t * x} (r_{β} (t) \\ \land 1_{A} (x)) \leq (1_{B} (y) \lor α_{A} (y)} = ⋁ {r_{β} (s) | r \in R, \\ ⋁_{y \in r * x} (β \land 1) \leq (1_{B} (y) \lor α_{A} (y)} = ⋁ {r_{β} (s) | r \in R, \\ β \leq (1_{B} (y) \lor α), y \in r * x} . \end{matrix}$

If s ∉ B : A, then (1_B:A ∪ α_R) (s) = α. It follows that ((1_B ∪ α_A) :1_A) (s) = ⋁ {r_β (s) | r ∈ R, β ≤ α} = ⋁ {r_α (s) | r ∈ R} ≥ (1_B:A ∪ α_R) (s) and so (1_B ∪ α_A) :1_A ⊆ 1_B:A ∪ α_R.

If s ∈ B : A, then (1_B:A ∪ α_R) (s) =1. It follows that ((1_B ∪ α_A) :1_A) (s) = ⋁ {r_β (s) | r ∈ R, β ≤ α} = ⋁ {r_α (s) | r ∈ R} ≤1 = (1_B:A ∪ α_R) (s) and in this case, (1_B ∪ α_A) :1_A ⊆ 1_B:A ∪ α_R. Thus in any cases (1_B ∪ α_A) :1_A ⊆ 1_B:A ∪ α_R and by item (ii), we get (1_B ∪ α_A) :1_A = 1_B:A ∪ α_R.

Corollary 4.16. Let (A, + _A, 0_A, *) be a supermodule, B a subsupermodule of A, $ζ \in F (R)$ , $μ, ν \in F (A)$ and 0 ≤ α, β ≤ 1. Then

(1_B ∪ 1_A) :1_A = 1_R,

(1_A ∪ α_A) :1_A = 1_R,

(1_B ∪ α_B) :1_B = 1_R.

5 Conclusion

The current paper has defined and considered the notion of (thin) supermodule, and has introduced the concept of fuzzy subsupermodules and investigated of their properties. We applied the notation of valued-cuts to obtain of the equal conditions between of fuzzy subsupermodules and supermodules. Moreover for especial extension of fuzzy subsupermodules, it is introduced a concept of residual quotients and is showed that some conditions to be fuzzy subsupermodule. We considered superrings as special subsupermodules and investigated some important results in residual quotients of its (weak) fuzzy subsupermodules. These results can be helpful in the further study on fuzzy module theory. In fuzzy homology supermodules, primary fuzzy subsupermodules, π-primary fuzzy subsupermodules, there remain many open problems to be answered and applications to be developed.

Declaration of interest statement

The authors declare that they have no conflict of interest.

References

Ameri

, On Categories of Hypergroups and Hypermodules, JDiscrete Math Sci Cryptography 6(2–3) (2003), 121–132.

Ameri

, Shojaei

, Projective and Injective KrasnerHypermodules, J Algebra Appl 20(10) (2021), 2150186–.

Ameri

, Eyvazi

, Hoskova-Mayerova

, Superring ofPolynomials Over a Hyperring, Mathematics 7 (2019), 902–.

Ameri

, Kordi

, Hoskova-Mayerova

, Multiplicative Hyperringof Fractions and Coprime Hyperideals, An Şt. Univ. OvidiusConstanţa 25(1) (2017), 5–23.

Anvariyeh

S.M.

, Mirvakili

, Davvaz

, θ^*-Relation onHypermodules and Fundamental Modules Over Commutative FundamentalRings, Comm Algebra 36 (2008), 622–631.

Anvariyeh

S.M.

, Mirvakili

, Davvaz

, Transitivity ofθ^*-Relation on Hypermodules, Iran J Sci Technol Trans ASci 32(A3) (2008), 188–205.

Bordbar

, Novak

, Cristea

, A Note On The Support of aHypermodule, J Algebra Appl 19(1) (2020), 2050019–.

Davvaz

, Fuzzy H_v-Groups, Fuzzy Sets Syst 101(1) (1999), 191–195.

Davvaz

and Leoreanu-Fotea

, Hyperring Theory and Applications, Int. Acad. Press, USA, 2007.

10.

Davvaz

, Fuzzy Krasner (m, n)-Hyperrings, Comput Math withAppl 59 (2010), 3879–3891.

11.

Davvaz

, Firouzkouhi

, Fundamental Relation on FuzzyHypermodules, Soft Comput 23 (2019), 13025–13033.

12.

Farzalipour

, Ghiasvand

, On Graded Hyperrings and GradedHypermodules, Algebr Struct their Appl 7(2) (2020), 15–28.

13.

Fotea

V.L.

, Davvaz

, Fuzzy Hyperrings, Fuzzy Sets Syst 160(16) (2009), 2366–2378.

14.

Fotea

V.L.

, Fuzzy Hypermodules, Comput Math with Appl 57 (2009), 466–475.

15.

Firouzkouhi

and Davvaz

, New Fundamental Relation and Complete Part of Fuzzy Hypermodules, J Discrete Math Sci Cryptography (Published online), (2020).

16.

Hamidi

, Faraji

, Ameri

and Ahmadi-Amoli

Kh.

, Normal Injective Resolution of General Krasner Hypermodule, Journal of Algebraic Systems 10(1) (2022), 121–145.

17.

Massouros

C.G.

, Free and Cyclic Hypermodules, Ann Mat PuraAppl 150(1) (1988), 153–166.

18.

Massouros

C.H.G.

, Massouros

G.G.

, An Overview of the Foundationsof the Hypergroup Theory, Mathematics 9(9) (2021), 1014–.

19.

Massouros

CH.G.

and Massouros

G.G.

, Operators and Hyperoperators Acting on Hypergroups, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics ICNAAM 2008 Kos, American Institute of Physics (AIP) Conference Proceedings, 380–383.

20.

Massouros

G.G.

, Automata and Hypermoduloids, Proceedings of the 5th International Congress on AHA, Iasi 1993, Hadronic Press 1994, 251–265.

21.

Marty

, Sur Une Generalization De La notion De Groupe, In 8th Congress Math. Scandenaves., Stockholm, (1934), 45–49.

22.

Mirvakili

, Anvariyeh

S.M.

, Davvaz

, Construction of (M, N)-Hypermodule Over (R, S)-Hyperring, Acta UniversitatisSapientiae, Mathematica 11(1) (2019), 131–143.

23.

Mittas

J.D.

, Sur Certaines Classes De StructuresHypercompositionnelles, Proceedings of the Academy of Athens 48 (1973), 298–318.

24.

Mittas

J.D.

, Sur Les Structures Hypercompositionnelles, In Proceedings of the 4th International Congress on AHA, Xanthi, Greece, 27–30 June 1990; World Scientific: Singapore, 1991, 137–147.

25.

Sh.

, Mousavi, Free hypermodules, a categorical approach, Commun Algebra 48(8) (2020), 3184–3203.

26.

Norouzi

, Normal Subfuzzy (m, n)-hypermodules, J DiscreteMath Sci Cryptography 22(3) (2019), 433–451.

27.

Norouzi

, Fotea

V.L.

, Isomorphism Theorems of FuzzyHypermodules, An. Şt Univ Ovidius Constanţa 25(1) (2017), 145–161.

28.

Sen

M.K.

, Ameri

, Chowdhury

, Fuzzy Hypersemigroups, Soft Comput 12 (2008), 891–900.

29.

Shojaei

, Ameri

, Hoskova-Mayerova

, On Properties ofVarious Morphisms in the Categories of General Krasner Hypermodules, Ital J Pure Appl Math 39 (2018), 475–484.

30.

Zhan

J.M.

, Davvaz

, Shum

K.P.

, A New View on FuzzyHypermodules, Acta Math Sin Engl Ser 23 (2007), 1345–1356.

31.

Yin

, Zhan

, Xu

, Wang

, The L-Fuzzy Hypermodules, Comput Math with Appl 59 (2010), 953–963.

32.

Zadeh

L.A.

, Fuzzy sets, Inf Control 8(3) (1965), 338–353.