A novel concept of ( m,n )-ary subhypermodules in the framework of fuzzy sets

Abstract

This paper provides a continuation of ideas presented by Davvaz and Corsini in [20]. We introduce the concepts of (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodules and $(\bar{\in}, \bar{\in} \lor {\bar{q}}_{k})$ -fuzzy (m, n)-ary subhypermodules in an (m, n)-ary hypermodule, which are generalization of the concepts as given in [20]. Different classes of (m, n)-ary hypermodules are characterized by the properties of these fuzzy (m, n)-ary subhypermodules. Using the notion of fuzzy (m, n)-ary subhypermodule with thresholds, characterization of fuzzy (m, n)-ary subhypermodules, (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodules and $(\bar{\in}, \bar{\in} \lor {\bar{q}}_{k})$ -fuzzy (m, n)-ary subhypermodules are discussed.

1 Introduction

Algebraic hyperstructures which is based on the notion of hyperoperation was introduced by Marty [25] and studied extensively by many mathematicians. Several books have been written on hyperstructure theory, see [9 , 35]. A book on hyperstructures [9] points out on their applications in fuzzy and rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. Another book [16] is devoted especially to the study of hyperring theory. Several kinds of hyperrings are introduced and analyzed. The first paper concerning the theory of n-ary groups was written (under the inspiration of Emmy Noether) by DÃürnte in 1928 (see [12]). Then, the concept of an n-ary hypergroup is defined by Davvaz and Vougiouklis in [15], which is a generalization of the concept of hypergroup in the sense of Marty and a generalization of an n-ary group, too. Then this concept was studied by Ghadiri and Waphare [23], Leoreanu-Fotea [28], Leoreanu-Fotea and Davvaz [29, 30], Davvaz et al. [21, 22] and others [3, 19].

Recently, the notion of Krasner (m, n)-hyperrings which are generalization of (m, n)-rings and generalization of Krasner hyperrings was defined by Mirvakili and Davvaz [26] and they obtained (m, n)-rings from (m, n)-ary hyperrings using fundamental relations [27]. After than the concept of n-ary prime and n-ary primary hyperideals in Krasner (m, n)-hyperring is given in [1]. The reseearch of (m, n)-ary hypermodules over (m, n)-ary hyperrings has been initiated by Anvariyeh, Mirvakili and Davvaz who introduced these hyperstructures in [4]. In addition, in [5], Anvariyeh et al. defined a strongly compatible relation on an (m, n)-ary hypermodule and determined a sufficient condition such that the strongly compatible relation is transitive. In [6], Belali et al. defined free and canonical (m, n)-hypermodules and studied some results. Also, the concept of the quotient (m, n)-ary hypermodules and some properties of subhypermodules of an (m, n)-ary hypermodule were given in [2].

The theory of fuzzy sets which was introduced by Zadeh [36] is applied to many mathematical branches. On the other hand, few years after the inception of the notion of fuzzy set, Rosenfeld started the pioneer work in the domain of fuzzification of the algebraic objects, with his work on fuzzy groups [34]. This work is a contribution to the theory founded on the ideas of those authors and their followers. Das [11] characterized fuzzy subgroups by their level subgroups. A new type of fuzzy subgroup (viz, (∈, ∈ ∨ q)-fuzzy subgroup) was introduced in an earlier paper of Bhakat and Das [7, 8] by using the combined notions of “belongingness” and “quasicoincidence” of fuzzy points and fuzzy sets. In fact, (∈, ∈ ∨ q)-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. Also, a generalization of Rosenfeld’s fuzzy subgroup, and Bhakat and Das’s fuzzy subgroup is given in [33].

The study of fuzzy hyperstructures is an interesting research topic of fuzzy sets. Fuzzy sets and hyperstructures introduced by Zadeh [36] and Marty [25], respectively, are now used extensively from both the theoretical point of view and their many applications. The relationships between the fuzzy sets and algebraic hyperstructures have been considered by Corsini, Davvaz, Leoreanu, Zahedi and others. The reader is refereed to [10 , 32]. In [19], Davvaz introduced the concept of a fuzzy hyperideal of a Krasner ((m, n)-ary hyperring and the fuzzy (m, n)-ary subhypermodule in an (m, n)-ary hypermodule with respect to fuzzy sets within fuzzy points were studied by Davvaz et al. [20].

Now, by using these papers, we introduce the notion of an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule and an $(\bar{\in}, \bar{\in} \lor {\bar{q}}_{k})$ -fuzzy (m, n)-ary subhypermodule which are generalizations of an (∈, ∈ ∨ q)-fuzzy (m, n)-ary subhypermodule in an (m, n)-ary hypermodule. A generalization of ideas introduced in [20] and [7 , 24] is presented. We discuss the implication-based fuzzy (m, n)-ary subhypermodule of a hypermodule. The important achievement of the study with an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule is that the notion of an (∈, ∈ ∨ q)-fuzzy (m, n)-ary subhypermodule is a special case of an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule, and thus many results in the paper [20] are corollaries of our results obtained in this paper.

2 Preliminaries

We start by giving some known and useful definitions and notations. Let H be a non-empty set and f be a mapping f : H × H → P^* (H), where P^* (H) is the set of all non-empty subsets of H. Then f is called a binary hyperoperation on H. We denote by Hⁿ the cartesian product H × . . . × H where H appears n times. An element of Hⁿ will be denoted by (x₁, . . . , x_n) where x_i ∈ H for any i with 1 ≤ i ≤ n. In general, a mapping f : Hⁿ → P^* (H) is called an n-ary hyperoperation and n is called the arity of the hyperoperation f. Let f be an n-ary hyperoperation on H and A₁, . . . , A_n subsets of H. We define f (A₁, . . . , A_n) = ∪ {f (x₁, . . . , x_n) ∣ x_i ∈ A_i, i = 1, . . . , n} . We shall use the following abbreviated notation: The sequence x_i, x_i+1, . . . , x_j will be denoted by $x_{i}^{j}$ . For j < i, $x_{i}^{j}$ is the empty set. Thus f (x₁, . . . , x_i, y_i+1, . . . , y_j, z_j+1, . . . , z_n) will be written as $f (x_{1}^{i}, y_{i + 1}^{j}, z_{j + 1}^{n})$ . A non-empty set H with an n-ary hyperoperation f : H × H → P^* (H) will be called an n-ary hypergroupoid and will be denoted by (H, f). An n-ary hypergroupoid (H, f) will be called an n-ary semihypergroup if and only if the following associative axiom holds: $f (x_{1}^{j - 1}, f (x_{i}^{n + i - 1}), x_{n + i}^{2 n - 1}) = f (x_{1}^{j - 1}, f (x_{j}^{n + j - 1}), x_{n + j}^{2 n - 1})$ for every i, j ∈ {1, 2, . . . , n} and x₁, x₂, . . . , x_2n-1 ∈ H. If for all (a₁, a₂, . . . , a_n) ∈ Hⁿ, the set f (a₁, a₂, . . . , a_n) is singleton, then f is called an n-ary operation and (H, f) is called an n-ary groupoid (rep. n-ary semigroup). An n-ary semihypergroup (H, f) in which the equation $b \in f (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{n})$ has a solution x_i ∈ H for every a₁, . . . , a_i-1, a_i+1, . . . , a_n, b ∈ H and 1 ≤ i ≤ n, is called an n-ary hypergroup. If f is n-ary operation then the equation becomes: $b = f (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{n})$ . In this case (H, f) is an n-ary group. Let (H, f) be an n-ary hypergroup and B be a non-empty subset of H. Then B is an n-ary subhypergroup of H if the following conditions hold:

B is closed under the n-ary hyperoperation f, i.e., for every $x_{1}^{n} \in B$ we have $f (x_{1}^{n}) \subseteq B$ .

Equation $b \in f (b_{1}^{i - 1}, x_{i}, b_{i + 1}^{n})$ has a solution x_i ∈ B for every b₁, …, b_i-1, b_i+1, …, b_n, b ∈ B and 1 ≤ i ≤ n.

Definition 2.1. [27]. An (m, n)-ary hyperring is an algebraic hyperstructure (R, f, g), which satisfies the following axioms:

(R, f) is an m-ary hypergroup,

(R, g) is an n-ary hypersemigroup,

the n-ary hyperoperation g is distributive with respect to the m-ary hyperoperation f, i.e.,

\begin{matrix} g (a_{1}^{i - 1}, f (x_{1}^{m}), a_{1 + i}^{n}) \\ = f (g (a_{1}^{i - 1}, x_{1}, a_{i + 1}^{n}), . . ., g (a_{1}^{i - 1}, x_{m}, a_{i + 1}^{n})) \end{matrix}

for every

a_{1}^{i - 1}, a_{i + 1}^{n}, x_{1}^{m} \in R, 1 \leq i \leq n

. (R, f, g) is called an m-ary hyperring if m = n. An m-ary hyperring R is a hyperring if m = 2 [25].

Example 2.2. [27]. Let (R, + , ·) be a Krasner hyperring. Let f be an m-ary hyperoperation and g an n-ary operation on R as follows: $\begin{matrix} f (x_{1}^{m}) & = & \sum_{i = 1}^{m} x_{i} for all x_{1}^{m} \in R, \\ g (x_{1}^{n}) & = & \prod_{i = 1}^{n} x_{i} for all x_{1}^{n} \in R . \end{matrix}$ Then (R, f, g) is a Krasner (m, n)-hyperring and denoted by (R, f, g) = der_(m,n) (R, + , ·) .

Definition 2.3. [5]. A non-empty set M = (M, h, k) is an (m, n)-ary hypermodule over an (m, n)-ary hyperring (R, f, g), if (M, h) is an m-ary hypergroup and there exists the n-ary hyperoperation k : R^n-1 × M → P^* (M) such that

$k (r_{1}^{n - 1}, h (x_{1}^{m})) = h (k (r_{1}^{n - 1}, x_{1}), . . ., k (r_{1}^{n - 1}, x_{m}))$

$k (r_{1}^{i - 1}, f (s_{1}^{m}), r_{i + 1}^{n - 1}, x) =$ $h (k (r_{1}^{i - 1}, s_{1}, r_{i + 1}^{n - 1}, x), . . ., k (r_{1}^{i - 1}, s_{m}, r_{i + 1}^{n - 1}, x))$

$k (r_{1}^{i - 1}, g (r_{i}^{i + n - m}), r_{i + n}^{2 n - 2}, x) = k (r_{1}^{n - 1}, k (r_{n}^{2 n - 2}, x))$

If k is a scalar n-ary hyperoperation on M, S₁, . . . , S_n-1 be non-empty subsets of R and M₁ ⊆ M, we set k (S₁, . . . , S_n-1, M₁) = ⋃ {k (r₁, . . . , r_n-1, x) ∣ r_i ∈ S_i, i = 1, . . . , n - 1, x ∈ M₁} .

Definition 2.4. [4]. Let (M, h, k) be an (m, n)-ary hypermodule over an (m, n)-ary hyperring (R, f, g). A non-empty subset N ⊆ M is called an (m, n)-ary subhypermodule of M, if (N, h, k) is an (m, n)-ary hypermodule over the (m, n)-ary hyperring (R, f, g).

A mapping μ : X → [0, 1], where X is an arbitrary non-empty set, is called a fuzzy subset of X. The complement of μ, denoted by μ^c is the fuzzy subset given by μ^c (x) =1 - μ (x) for all x ∈ X. In 1971, Rosenfeld [34] applied the concept of fuzzy sets to the theory of groups and studied fuzzy subgroups of a group. Davvaz [13] applied fuzzy sets to the theory of algebraic hyperstructures and defined the concept of fuzzy subhypergroups. Davvaz and Corsini [14] introduced the notion of fuzzy n-ary subhypergroups of n-ary hypergroup. We shall use the following abbreviated notation: the sequence μ (a_i) , μ (a_i+1) , . . . , μ (a_j) will be denoted by $μ_{a_{i}}^{a_{j}}$ .

Definition 2.5. [20]. Let (M, h, k) be an (m, n)-ary hypermodule over (m, n)-ary hyperring (R, f, g) and μ a fuzzy subset of M. Then μ is said to be a fuzzy (m, n)-ary subhypermodule of M if the following axioms hold:

$min {μ_{x_{1}}^{x_{m}}} \leq inf_{z \in h (x_{1}^{m})} {μ (z)}$ for all $x_{1}^{m} \in M$ ,

for all $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in M$ and 1 ≤ i ≤ m, there exist x_i ∈ M such that $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ and $min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i} + 1}^{a_{m}}, μ (b)} \leq μ (x_{i}) .$

$μ (x) \leq inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)}$ for all x ∈ M, $r_{1}^{n - 1} \in R$ .

Example 2.6. Let M = {0, 1, 2} and $(R, f, g) = {der}_{(3, 2)} (ℤ, +, \cdot)$ (see Example 2.2). We define the commutative hyperoperations h and k as follows: $\begin{matrix} h (0, 0, 0) & = & h (0, 0, 2) = h (2, 2, 2) \\ = & h (0, 2, 2) = h (2, 1, 1) = {0, 2}, \\ h (0, 0, 1) & = & h (0, 2, 1) = h (2, 2, 1) \\ = & h (1, 1, 1) = 1, h (0, 1, 1) = {0, 2} \end{matrix}$ and $\begin{matrix} k : R \times M \to P^{*} (M), \\ k (r, x) = {\begin{matrix} {0, 2} & if x \in {0, 2} or r \in 2 ℤ, \\ 1 & if otherwise . \end{matrix} \end{matrix}$ Then (M, h, k) is an (3, 2)-ary hypermodule over an (3, 2)-ary hyperring (R, f, g). If μ : M → [0, 1] is defined by μ (0) =0.3, μ (1) =0.5, μ (2) =0.3.

Then it is easy to see that μ is a fuzzy (m, n)-ary sub-hypermodule of M.

Let (M, h, k) be an (m, n)-ary hypermodule over (m, n)-ary hyperring (R, f, g) and B ⊆ M. Then the characteristic function χ_B is a fuzzy (m, n)-ary sub-hypermodule of M if and only if B is an (m, n)-ary sub-hypermodule of M. For any fuzzy subset μ of a non-empty set X and any t ∈ (0, 1], we define the set μ_t = {x ∈ X ∣ μ (x) ≥ t}.

Theorem 2.7. [20]. Let (M, h, k) be an (m, n)-ary hypermodule over (m, n)-ary hyperring (R, f, g) and and μ a fuzzy subset of M. Then μ is a fuzzy (m, n)-ary subhypermodule of M if and only if for every t ∈ (0, 1], μ_t (≠ ∅) is an (m, n)-ary subhypermodule of M.

3 (∈, ∈ ∨ q_k)( $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ )-fuzzy (m, n)-ary sub-hypermodule

For any fuzzy subset μ of M, the set {x ∈ M|0 < μ (x)} is called the support of μ, and is denoted by supp μ. A fuzzy set μ on M which takes the value t ∈ (0, 1] at some x ∈ M and takes the value 0 for all y ∈ M expect x is called a fuzzy point and is denoted by x_t, where the point x is called its support point and t is called its value.

In what follows let k denote an arbitrary element of [0, 1) unless otherwise specified. For a fuzzy point x_t is said to be

belong to a fuzzy set μ, written as x_t ∈ μ if μ (x) ≥ t.

k-quasi-coincident with a fuzzy set μ, written as x_tq_kμ if μ (x) + t + k > 1.

x_t ∈ μ or x_tq_kμ, then we write x_t ∈ ∨ q_kμ.

The formula $x_{t} \bar{α} μ$ means that x_tαμ does not hold for α ∈ {∈, q_k, ∈ ∨ q_k}.

Definition 3.1. A fuzzy subset μ of M is called an (∈, ∈ ∨ q_k)-fuzzy (m,n)-ary subhypermodule of M if for for all $t, t_{1}^{m} \in (0, 1]$ and $x, x_{1}^{m} \in M$ , $r_{1}^{n} \in R$ ,

(x₁) _t
₁, (x₂) _t
₂, …, (x_m) _{t
_m} ∈ μ implies $z_{min {t_{1}^{m}}} \in \lor q_{k} μ$ for all $z \in h (x_{1}^{m})$ ,

(a₁) _t
₁, (a₂) _t
₂, . . . , (a_i-1) _{t
_i-1}, (a_i+1) _{t
_i+1}, . . . , (a_m) _{t
_m}, b_s ∈ μ and 1 ≤ i ≤ m implies $(x_{i})_{min {t_{1}^{i - 1}, t_{i + 1}^{m}, s}} \in \lor q_{k} μ for some x_{i} \in M with b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ ,

x_t ∈ μ implies z_t ∈ ∨ q_kμ for all $z \in k (r_{1}^{n - 1}, x)$ .

An (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule of M with k = 0 is called an (∈, ∈ ∨ q)-fuzzy (m, n)-ary subhypermodule of M (see [20], Definition 2.1).

Theorem 3.2. Conditions (i)-(iii) in Definition 3.1 are equivalent to the following conditions, respectively.

$min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}} \leq inf_{z \in h (x_{1}^{m})} {μ (z)}$ for all $x_{1}^{m} \in M$ ,

for all $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in M$ and 1 ≤ i ≤ m, there exist x_i ∈ M such that $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ and $min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}} \leq μ (x_{i})$ ,

$min {μ (x), \frac{1 - k}{2}} \leq inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)}$ for all x ∈ M, $r_{1}^{n - 1} \in R$ .

Proof. (i ⇒1): Suppose $x_{1}^{m} \in M$ . We consider the following cases:

$min {μ_{x_{1}}^{x_{m}}} < \frac{1 - k}{2}$ ,

$min {μ_{x_{1}}^{x_{m}}} \geq \frac{1 - k}{2}$ .

Case a: Assume that there exist $z \in h (x_{1}^{m})$ such that $μ (z) < min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}}$ , which implies that $μ (z) < min {μ_{x_{1}}^{x_{m}}}$ . Choose t such that $μ (z) < t < min {μ_{x_{1}}^{x_{m}}}$ . Then (x₁) _{t
₁}, (x₂) _{t
₂}, . . . , (x_m) _{t
_m} ∈ μ, but $z_{t} \bar{\in \lor q μ}$ , which contradicts (i).

Case b: Assume that $μ (z) < \frac{1 - k}{2}$ for some $z \in h (x_{1}^{m})$ . Then $(x_{1})_{\frac{1 - k}{2}}, (x_{2})_{\frac{1 - k}{2}}, . . ., (x_{m})_{\frac{1 - k}{2}} \in μ$ , but $z_{\frac{1 - k}{2}} \bar{\in \lor q μ}$ , which is a contradiction. Therefore (1) holds.

(ii ⇒2): Suppose that $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in M$ and 1 ≤ i ≤ m. We consider the following cases:

$min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)} < \frac{1 - k}{2}$ ,

$min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)} \geq \frac{1 - k}{2}$ .

Case a: Assume that for all x_i with $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ , we have $μ (x_{i}) < min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)} .$ Choose t such that $μ (x_{i}) < t < \min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}$ and t + μ (x_i) <1. Then (a₁) _t, . . . , (a_i-1) _t, (a_i+1) _t, . . . , (a_m) _t, b_t ∈ μ, but $(x_{i})_{t} \bar{\in \lor q μ}$ , which contradicts (ii).

Case b: Assume that for all x_i with $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ , we obtain $μ (x_{i}) < min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}},$ then $(a_{1})_{\frac{1 - k}{2}}, . . ., (a_{i - 1})_{\frac{1 - k}{2}}, (a_{i + 1})_{\frac{1 - k}{2}}, . . ., (a_{m})_{\frac{1 - k}{2}}, b_{\frac{1 - k}{2}} \in μ$ , but $(x_{i})_{\frac{1 - k}{2}} \bar{\in \lor q μ}$ , which contradicts (ii). Hence the condition (2) holds.

(iii ⇒3): Suppose x ∈ M. We consider the following cases:

$μ (x) < \frac{1 - k}{2}$ ,

$μ (x) \geq \frac{1 - k}{2}$ .

Case a: Assume that there exist $z \in h (x_{1}^{m})$ such that $μ (z) < min {μ (x), \frac{1 - k}{2}}$ , which implies that μ (z) < μ (x). Choose t such that μ (z) < t < μ (x). Then (x) _t ∈ μ, but $z_{t} \bar{\in \lor q μ}$ , which contradicts (i).

Case b: Assume that $μ (z) < \frac{1 - k}{2}$ for some $z \in h (x_{1}^{m})$ . Then $(x)_{\frac{1 - k}{2}} \in μ$ , but $z_{\frac{1 - k}{2}} \bar{\in \lor q μ}$ , which is a contradiction. Therefore (1) holds.

(1⇒ i): Let (x₁) _{t
₁}, (x₂) _{t
₂}, . . . , (x_m) _{t
_m} ∈ μ. Then μ (x₁) ≥ t₁, μ (x₂) ≥ t₂, . . . , μ (x_m) ≥ t_m. For every $z \in h (x_{1}^{m})$ , we have $μ (z) \geq min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}} \geq min {t_{1}^{m}, \frac{1 - k}{2}}$ .

If $min {t_{1}^{m}} > \frac{1 - k}{2}$ , then $μ (z) \geq \frac{1 - k}{2}$ which implies that $μ (z) + min {t_{1}^{m}} > 1 .$ If $min {t_{1}^{m}} \leq \frac{1 - k}{2}$ , then $μ (z) \geq min {t_{1}^{m}}$ . Therefore $z_{min {t_{1}^{m}}} \in \lor q_{k} μ$ for all $z \in h (x_{1}^{m}) .$

(2⇒ ii): Let (a₁) _{t
₁}, (a₂) _{t
₂}, . . . , (a_i-1) _{t
_i-1}, (a_i+1) _{t
_i+1}, . . . , (a_m) _{t
_m}, b_s ∈ μ. Then μ (a₁) ≥ t₁, μ (a₂) ≥ t₂, . . . , μ (a_i-1) ≥ t_i-1, μ (a_i+1) ≥ t_i+1, . . . , μ (a_m) ≥ t_m, μ (b) ≥ s. Now, for some x_i with $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ , we have $\begin{matrix} μ (x_{i}) & \geq & min {{min {μ_{1}^{a_{i - 1}}, μ_{i + 1}^{a_{m}}, μ (b)}, \frac{1 - k}{2}} \\ \geq & min {t_{1}^{i - 1}, t_{i + 1}^{m}, s, \frac{1 - k}{2}} \end{matrix}$ If $min {t_{1}^{i - 1}, t_{i + 1}^{m}, s} > \frac{1 - k}{2}$ , then $μ (x_{i}) \geq \frac{1 - k}{2}$ which implies that $μ (x_{i}) + (min {t_{1}^{i - 1}, t_{i + 1}^{m}, s}) > 1 .$ If $min {t_{1}^{i - 1}, t_{i + 1}^{m}, s} \leq \frac{1 - k}{2}$ , then $μ (x_{i}) \geq min {t_{1}^{i - 1}, t_{i + 1}^{m}, s} .$ Therefore $(x_{i})_{min {t_{1}^{i - 1}, t_{i + 1}^{m}, s}} \in \lor q_{k} μ$ . Hence (ii) holds.

(3 ⇒ iii): Let x_t ∈ μ. Then μ (x) ≥ t. For every $z \in h (x_{1}^{m})$ , we have $μ (z) \geq min {μ (x), t, \frac{1 - k}{2}}$ .

If $t > \frac{1 - k}{2}$ , then $μ (z) \geq \frac{1 - k}{2}$ which implies μ (z) + (t) >1.

If $t \leq \frac{1 - k}{2}$ , then μ (z) ≥ t. Therefore z_t ∈ ∨ q_kμ for all $z \in h (x_{1}^{m})$ .

The above theorem is a generalization of Proposition 4.2 in [20].

The following corollary is exactly obtained from Definition 3.1 and Theorem 3.2.

Corollary 3.3. Let μ be a fuzzy subset of (m, n)-ary hypermodule M. Then μ is an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule M if and only if the conditions (1), (2) and (3) in Theorem 3.2 hold.

Remark 3.4. Every fuzzy (m, n)-ary subhypermodule and (∈, ∈ ∨ q)-fuzzy (m, n)-ary subhypermodule μ of M is an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule of M, but, as the following example shows, the converse is not necessarily true.

Example 3.5. Consider the (3, 2)-ary hypermodule (M, h, k) over (3, 2)-hyperring (R, f, g) defined Example 2.6. Let μ : M → [0, 1] be defined by: $μ (0) = 0.41, μ (1) = 0.42, μ (2) = 0.43 .$ Then

μ is an (∈, ∈ ∨ q_0.2)-(3, 2)-ary fuzzy subhypermodule,

μ is not an (∈, ∈ ∨ q)-(3, 2)-ary fuzzy subhypermodule,

μ is not an (3, 2)-ary fuzzy subhypermodule of M since $0.42 = min {μ (2), μ (1), μ (1)} ≰ inf_{z \in h (2, 1, 1)} {μ (z)} = \inf {μ (0), μ (2)} = 0.41$ .

Theorem 3.6. A non-empty subset I of M is an (m, n)-ary subhypermodule of M if and only if χ_I is an is an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule of M.

Proof. Straightforward.

The following theorem is a more updated result than [20, Theorem 4.4].

Theorem 3.7. A fuzzy subset μ of M is an (∈, ∈ ∨ q_k)-fuzzy (m,n)-ary subhypermodule of M if and only if the set μ (t) (≠ ∅) is an (m, n)-ary subhypermodule of M for all $0 < t \leq \frac{1 - k}{2}$ .

Proof. Let μ be an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule of M and $0 < t \leq \frac{1 - k}{2}$ . Let $x_{1}^{m} \in μ_{t}$ . Then μ (x₁) ≥ t, μ (x₂) ≥ t, . . . , μ (x_m) ≥ t. Now $inf_{z \in h (x_{1}^{m})} {μ (z)} \geq min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}} \geq min {t, \frac{1 - k}{2}} = t .$ Therefore for every $z \in h (x_{1}^{m})$ we have μ (z) ≥ t, that is z ∈ μ_t, so $h (x_{1}^{m}) \subseteq μ_{t}$ . Now, let $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in μ_{t}$ and 1 ≤ i ≤ m, then there exists x_i ∈ M such that $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ and $min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}} \leq μ (x_{i})$ . From, $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in μ_{t}$ , we have $t = min {t, \frac{1 - k}{2}} \leq min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}} \leq μ (x_{i}) .$ Hence x_i ∈ μ_t. Now, let $r_{1}^{n - 1} \in R$ and x ∈ μ_t. Then μ (x) ≥ t. Since μ is an (∈, ∈ ∨ q_k)-fuzzy (m,n)-ary subhypermodule of M, we have $t \leq min {μ (x), \frac{1 - k}{2}} \geq inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)}$ , which implies for all $z \in k (r_{1}^{n - 1}, x)$ , t ≤ μ (x). Hence $k (r_{1}^{n - 1}, x) \subseteq μ_{t}$ . This proves that μ_t is an (m, n)-ary subhypermodule of M for all $0 < t \leq \frac{1 - k}{2}$ .

Conversely, let μ be a fuzzy subset of M such that μ_t (≠ ∅) is an (m, n)-ary subhypermodule of M for all $0 < t \leq \frac{1 - k}{2}$ . For every $x_{1}^{m} \in M$ we can write $μ (x_{i}) \geq min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}} = t_{0},$ then $x_{1}^{m} \in μ_{t_{0}}$ , so $h (x_{1}^{m}) \subseteq μ_{t_{0}}$ . Therefore for every $z \in h (x_{1}^{m})$ we have μ (z) ≥ t₀ which implies $inf_{z \in h (x_{1}^{n})} {μ (z)} \geq t_{0} .$ and in this way the condition (1) of Theorem 3.2 is verified.

To verify the second condition, if for every $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in M$ , we put $t_{1} = min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}},$ then $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in μ_{t_{1}}$ . So there exists x_i ∈ μ_{t
₁} such that $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ . Since x_i ∈ μ_{t
₁}, we have μ (x_i) ≥ t₁ or $μ (x_{i}) \geq min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}}$ and in this way the condition (2) of Theorem 3.2 is verified.

To verify the third condition, now, let $r_{1}^{n - 1} \in R$ and x ∈ M. We put $t_{2} = min {μ (x), \frac{1 - k}{2}}$ . Then x ∈ μ_{t
₂}. Since μ_{t
₂} is an (m, n)-ary sub-hypermodule of M, we obtain $k (r_{1}^{n - 1}, x) \subseteq μ_{t_{2}}$ , which implies that for every $z \in k (r_{1}^{n - 1}, x), μ (z) \geq t_{2}$ . Therefore, we obtain $inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)} \geq t_{2} = min {μ (x), \frac{1 - k}{2}}$ and in this way the condition (3) of Theorem 3.2 is verified.

Definition 3.8. A fuzzy subset μ of M is called an $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m,n)-ary subhypermodule of M if for for all $t, t_{1}^{m} \in (0, 1]$ and $x, x_{1}^{m} \in M$ , $r_{1}^{n} \in R$ ,

$z_{min {t_{1}^{m}}} \bar{\in} μ$ implies there exists 1 ≤ i ≤ m such that $(x_{i})_{t_{i}} \bar{\in} \lor \bar{q_{k}} μ$ for all $z \in h (x_{1}^{n})$ ,

$(x_{i})_{min {t_{1}^{i - 1}, t_{i + 1}^{m}, s}} \bar{\in} μ$ implies there exists 1 ≤ i ≤ m such that $(a_{1})_{t_{1}}, (a_{2})_{t_{2}}, . . ., (a_{i - 1})_{t_{i - 1}}, (a_{i + 1})_{t_{i + 1}}, . . ., (a_{m})_{t_{m}}, b_{s} \bar{\in} \lor \bar{q_{k}} μ for some x_{i} \in M with b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m}) .$

$z_{t} \bar{\in} μ$ implies $x_{t} \bar{\in} \lor \bar{q_{k}} μ$ for all $z \in k (r_{1}^{n - 1}, x) .$

An $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule of M with k = 0 is called an $(\bar{\in}, \bar{\in} \lor \bar{q})$ -fuzzy (m, n)-ary subhypermodule of M.

Every $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule is an $(\bar{\in}, \bar{\in} \lor \bar{q})$ -fuzzy (m, n)-ary subhypermodule of M, but the converse may not be true as seen in the following example.

Example 3.9. Consider the (3, 2)-ary hypermodule M which is given in Example 2.6. Let μ be a fuzzy subset of M defined by: $μ (0) = \frac{1}{2}, μ (1) = \frac{1}{3}, μ (2) = \frac{1}{4} .$ Then it is easy to check that μ is an $(\bar{\in}, \bar{\in} \lor \bar{q})$ -(3, 2)-ary fuzzy subhypermodule of M, but it is not an $(\bar{\in}, \bar{\in} \lor \bar{q_{\frac{1}{2}}})$ -fuzzy (3,2)-ary subhypermodule since $\begin{matrix} \frac{1}{3} & = & min {μ (0), μ (1), μ (1)} \\ ≰ & inf_{z \in h (0, 1, 1)} sup {μ (z), \frac{1}{4}} \\ = & inf {sup {μ (0), \frac{1}{4}}, sup {μ (2), \frac{1}{4}}} = \frac{1}{4} . \end{matrix}$

Theorem 3.10. A fuzzy subset μ of (m, n)-ary hypermodule M is an $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule of M if and only if the set μ_t (≠ ∅) is an (m, n)-ary subhypermodule of M for all $\frac{1 - k}{2} < t \leq 1$ .

Proof. It is similar to the proof of Theorem 3.7.

Corollary 3.11. For a fuzzy subset μ of (m, n)-ary hypermodule M, the following are equivalent.

μ is an $(\bar{\in}, \bar{\in} \lor \bar{q})$ -fuzzy (m, n)-ary subhypermodule of M,

μ_t (≠ ∅) is a (m, n)-ary subhypermodule of M for all t ∈ (0.5, 1].

Proof. The proof follows from Theorem 3.10, if we take k = 0.

For a fuzzy subset μ of M, we consider the following set:

K_t = {t ∈ (0, 1] ∣ μ_t (¬ = ∅) ⇒ μ_t is an (m, n)-ary subhypermodule of M}. Then

If K_t = (0, 1], then μ is a fuzzy (m, n)-ary subhypermodule of M,

If $K_{t} = (0, \frac{1 - k}{2}]$ , then μ is an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary sub-hypermodule of M,

If $K_{t} = (\frac{1 - k}{2}, 1]$ , then μ is an $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule of M.

An obvious question is whether μ is a kind of fuzzy (m, n)-ary subhypermodule of M or not when K_t≠ ∅ (e.g., K_t = (α, β] , α, β ∈ (0, 1) , α < β).

Definition 3.12. [20, Definition 4.6]. Let α, β ∈ [0, 1] and α < β. Let μ be a fuzzy subset of M. Then μ is called a fuzzy (m, n)-ary subhypermodule with thresholds (α, β) of M, iff

$min {μ_{x_{1}}^{x_{m}}} \land β \leq inf_{z \in h (x_{1}^{m})} (μ (z) \lor α)$ for all $x_{1}^{m} \in M$ ,

For all $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in M$ and 1 ≤ i ≤ m, there exist x_i ∈ M such that $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ and $min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)} \land β \leq μ (x_{i}) \lor α,$

$μ (x) \land β \leq inf_{z \in k (r_{1}^{n - 1}, x)} (μ (z) \lor α)$ for all x ∈ M, $r_{1}^{n - 1} \in R$ .

The following example shows that there exist α, β ∈ [0, 1] with α < β such that μ is a fuzzy (m, n)-ary subhypermodule with thresholds (α, β) which is not an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule with thresholds.

Example 3.13. Consider the (3, 2)-ary hypermodule M which is given in Example 2.6. Let μ be a fuzzy subset of M defined by: $μ (0) = 0.2, μ (1) = 0.4, μ (2) = 0.3 .$ Then it is easy to check that μ is a fuzzy (m, n)-ary subhypermodule with thresholds α = 0.4, β = 0.6. But it is not an (∈, ∈ ∨ q_0.2)-fuzzy (m, n)-ary subhypermodule of M since $0.3 = {min {μ (2), μ (1), μ (1)}, 0.4} ≰ inf_{z \in h (2, 1, 1)} {μ (z)} = \inf {μ (0), μ (2)} = 0.2 .$

Now, we characterize fuzzy (m, n)-ary subhypermodules with thresholds by their level (m, n)-ary subhypermodules.

Theorem 3.14. [20, Theorem 4.7.] A fuzzy subset μ of a (m, n)-ary subhypermodule M is a fuzzy (m, n)-ary subhypermodule with thresholds (α, β) of M if and only if μ_t (¬ =∅) is a (m, n)-ary subhypermodule of M for all t ∈ (α, β].

Proof. The proof is similar to the proof of Theorem 3.7.

The following example shows that there exist α, β ∈ (0, 1] with α < β such that μ is a fuzzy (m, n)-ary subhypermodule with thresholds (α, β) which is not an $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule of M.

Example 3.15. Consider the (3, 2)-ary hypermodule M which is given in Example 2.6. Let μ be a fuzzy subset of M defined by: $μ (0) = 0.5, μ (1) = 0.6, μ (2) = 0.4 .$ Then it is easy to check that μ is a fuzzy (m, n)-ary subhypermodule with thresholds α = 0.6, β = 0.7. If we take k = 0.2, then $0.5 = \min {μ (0), μ (1), μ (1)} ≰ inf_{z \in h (0, 1, 1)} (μ (z) \lor 0.4) = \inf {μ (0) \lor 0.4, μ (1) \lor 0.4} = 0.4 .$ Hence μ is not an $(\bar{\in}, \bar{\in} \lor \bar{q_{0.2}})$ -fuzzy (3, 2)-ary subhypermodule of M.

Theorem 3.16. Let μ be a fuzzy subset of M and α, β ∈ (0, 1] with α < β. Then

μ is a fuzzy (m, n)-ary subhypermodule of M if and only if μ is a fuzzy (m, n)-ary subhypermodule with thresholds α = 0 and β = 1.

μ is an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule of M if and only if μ is a fuzzy (m, n)-ary subhypermodule with thresholds α = 0 and $β = \frac{1 - k}{2}$ .

μ is an $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule of M if and only if μ is a fuzzy (m, n)-ary subhypermodule with thresholds $α = \frac{1 - k}{2}$ and β = 1.

Proof. Straightforward.

4 Implication-based fuzzy (m, n)-ary subhypermodule

Fuzzy logic is an extension of set theoretic multivalued logic in which the truth values are linguistic variables or terms of the linguistic variable truth. Some operators, for example, ∧, ∨, ¬, → in fuzzy logic are also defined by using truth tables and the extension principle can be applied to derive definitions of the operators. In fuzzy logic, truth value of fuzzy proposition P is denoted by [P]. For a universe U of discourse, we display, the fuzzy logical and corresponding set-theoretical notations used in this paper [33].

[x ∈ F] = F (x);

[x∉ F] =1 - F (x) ;

[P∧ Q] = min {[P] , [Q]} ;

[P∨ Q] = max {[P] , [Q]} ;

[P → Q] = min {1, 1 - [P] + [Q]};

[∀ xP (x)] = inf [P (x)] ;

⊨P if and only if [P] =1 for all valuations.

The truth valuation rules given in the above are those in the £ukasiewicz system of continuous-valued logic. Of course, various implication operators have been defined. We only show a selection of them in the next table. α denotes the degree of truth (or degree of membership) of the premise, β the respective values for the consequence, and I the resulting degree of truth for the implication.

The “quality” of these implication operators could be evaluated either empirically or axiomatically.

In the following, we consider the definition of implication operator in the £ukasiewicz system of continuous-valued logic.

Definition 4.1. A fuzzy subset μ of a (m, n)-ary hypermodule M is called a fuzzifying (m, n)-ary subhypermodule of M if and only if it satisfies:

For any $x_{1}^{m} \in M$ $⊨ [[x_{1} \in μ] \land [x_{2} \in μ] \land . . . \land [x_{m} \in μ] ⟶ [\forall z \in h (x_{1}^{m}), z \in μ]]$ ,

For any $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in M$ and 1 ≤ i ≤ m, there exist x_i ∈ M with $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ ⊨ [[a₁ ∈ μ] ∧ [a₂ ∈ μ] ∧ . . . ∧ [a_i-1 ∈ μ] ∧ [a_i+1 ∈ μ] ∧ . . . ∧ [a_m ∈ μ] ∧ [b ∈ μ] ⟶ [x_i ∈ μ]].

For all x ∈ M, $r_{1}^{n - 1} \in R$ , $⊨ [x \in μ] ⟶ [\forall z \in k (r_{1}^{n - 1}, x), z \in μ]]$ ,

Now, we introduce the concept of t-tautology, i.e., ⊨_t P if and only if [P] ≥ t for all valuations. So, we can extend the concept of implication-based fuzzy (m, n)-ary subhypermodule in the following way:

Definition 4.2.

For any $x_{1}^{m} \in M$ $⊨_{t} [[x_{1} \in μ] \land [x_{2} \in μ] \land . . . \land [x_{m} \in μ] ⟶ [\forall z \in h (x_{1}^{n}), z \in μ]]$

For any $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in H$ and 1 ≤ i ≤ m, there exist x_i ∈ M with $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ ⊨_t [[a₁ ∈ μ] ∧ [a₂ ∈ μ] ∧ . . . ∧ [a_i-1 ∈ μ] ∧ [a_i+1 ∈ μ] ∧ . . . ∧ [a_m ∈ μ] ∧ [b ∈ μ] ⟶ [x_i ∈ μ]].

For all x ∈ M, $r_{1}^{n - 1} \in R$ , $⊨_{t} [x \in μ] ⟶ [\forall z \in k (r_{1}^{n - 1}, x), z \in μ]]$

Corollary 4.3. A fuzzy subset μ of an (m, n)-ary hypermodule M is a t-implication-based fuzzy (m, n)-ary subhypermodule with respect to implication I if and only if

$I (min {μ_{x_{1}}^{x_{m}}}, inf_{z \in h (x_{1}^{m})} μ (z)) \geq t$ for all $x_{1}^{m} \in M$ ,

For any $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in M$ and 1 ≤ i ≤ m, there exist x_i ∈ M with $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ and $I (min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, μ (x_{i})) \geq t$ .

$I (μ (x), inf_{z \in k (r_{1}^{n - 1}, x)} μ (z)) \geq t$ . for all x ∈ M, $r_{1}^{n - 1} \in R$ .

Theorem 4.4. Let μ be fuzzy subset of an (m, n)-ary hypermodule M. Then

Let I = I_gr. Then μ is an 0.5-implication-based fuzzy (m, n)-ary subhypermodule of M if and only if μ is a fuzzy (m, n)-ary subhypermodule with thresholds α = 0 and β = 1 of M.

Let I = I_g. Then μ is an $\frac{1 - k}{2}$ -implication-based fuzzy (m, n)-ary subhypermodule of M if and only if μ is a fuzzy (m, n)-ary subhypermodule with thresholds α = 0 and $β = \frac{1 - k}{2}$ of M.

Let I = I_cg. Then μ is an $\frac{1 - k}{2}$ -implication-based fuzzy (m, n)-ary subhypermodule with thresholds if and only if μ is a fuzzy (m, n)-ary subhypermodule with thresholds $α = \frac{1 - k}{2}$ and β = 1 of M.

Proof. We only prove (2) and the proofs (1) and (3) are similar. Assume that μ is a $\frac{1 - k}{2}$ -implication-based fuzzy sublattice of M. Then by Corollary 4.3(i), we have

$I_{g} (min {μ_{x_{1}}^{x_{m}}}, inf_{z \in h (x_{1}^{m})} {μ (z)}) \geq \frac{1 - k}{2}$ for all $x_{1}^{m} \in M$ ,

For any $a_{1}^{i - 1}, a_{i + 1}^{m}, b \in M$ and 1 ≤ i ≤ m, there exist x_i ∈ M with $b \in h (a_{1}^{i - 1}, x_{i}, a_{i + 1}^{m})$ and $I_{g} (min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, μ (x_{i})) \geq \frac{1 - k}{2}$ .

$I_{g} (μ (x), inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)}) \geq \frac{1 - k}{2}$ for all x ∈ M, $r_{1}^{n - 1} \in R$ .

From (i), we have

min {μ_{x_{1}}^{x_{m}}} \leq inf_{z \in h (x_{1}^{m})} {μ (z))} or \frac{1 - k}{2} \leq inf_{z \in h (x_{1}^{m})} {μ (z)} < min {μ_{x_{1}}^{x_{m}}}

Then $min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}} \leq inf_{z \in h (x_{1}^{m})} {μ (z)}$ which implies that $min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}} \leq inf_{z \in h (x_{1}^{m})} (μ (z) \lor 0) .$

From (ii), we have $min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)} \leq μ (x_{i})$ or $\frac{1 - k}{2} \leq μ (x_{i}) < min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}$ . Thus $min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}} \leq μ (x_{i})$ which implies that $min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}} \leq μ (x_{i}) \lor 0 .$

From (ii), we have $μ (x) \leq inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)} or \frac{1 - k}{2} \leq inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)} < μ (x)$ .

Then $min {μ (x), \frac{1 - k}{2}} \leq inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)}$ which implies that $min {μ (x), \frac{1 - k}{2}} \leq inf_{z \in k (r_{1}^{n - 1}, x)} (μ (z) \lor 0) .$ Therefore μ is a fuzzy (m, n)-ary subhypermodule with thresholds α = 0 and $β = \frac{1 - k}{2}$ hence μ is an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule of M by Theorem 3.16.

Conversely, Assume that μ is a fuzzy (m, n)-ary subhypermodule with thresholds α = 0 and $β = \frac{1 - k}{2}$ of M. Then $inf_{z \in h (x_{1}^{m})} {μ (z)} = inf_{z \in h (x_{1}^{m})} (μ (z) \lor 0) \geq min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}},$ $\begin{matrix} μ (x_{i}) & = & μ (x_{i}) \lor 0 \\ \geq min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}}, \end{matrix}$ $\begin{matrix} inf_{z \in k (r_{1}^{n - 1}, x)} {μ (z)} \\ = inf_{z \in k (r_{1}^{n - 1}, x)} (μ (z) \lor 0) \geq min {μ (x), \frac{1 - k}{2}} . \end{matrix}$

For the first case, if $min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}} = min {μ_{x_{1}}^{x_{m}}}$ then $inf_{z \in h (x_{1}^{m})} {μ (z)} \geq min {μ_{x_{1}}^{x_{m}}}$ and thus $I_{g} (min {μ_{x_{1}}^{x_{m}}}, inf_{z \in h (x_{1}^{m})} {μ (z)}) = 1 \geq \frac{1 - k}{2}$ .

Suppose that $min {min {μ_{x_{1}}^{x_{m}}}, \frac{1 - k}{2}} = \frac{1 - k}{2}$ . Then $inf_{z \in h (x_{1}^{m})} {μ (z)} \geq \frac{1 - k}{2}$ and hence $I_{g} (min {μ_{x_{1}}^{x_{m}}}, inf_{z \in h (x_{1}^{m})} {μ (z)}) \geq inf_{z \in h (x_{1}^{m})} {μ (z)} \geq \frac{1 - k}{2}$ .

For the second case, if $min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}} = min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)},$ then $μ (x_{i}) \geq min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}$ and thus $I_{g} (min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, μ (x_{i})) = 1 \geq \frac{1 - k}{2}$ .

Suppose that $min {min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, \frac{1 - k}{2}} = \frac{1 - k}{2}$ . Then $μ (x_{i}) \geq \frac{1 - k}{2}$ and hence $I_{g} (min {μ_{a_{1}}^{a_{i - 1}}, μ_{a_{i + 1}}^{a_{m}}, μ (b)}, μ (x_{i})) \geq μ (x_{i}) \geq \frac{1 - k}{2}$ .

For the third case, if $min {μ (x), \frac{1 - k}{2}} = μ (x),$ then $inf_{z \in k (x_{1}^{n - 1}, x)} {μ (z)} \geq μ (x)$ and thus $I_{g} (μ (x), inf_{z \in k (x_{1}^{n - 1}, x)} {μ (z)}) = 1 \geq \frac{1 - k}{2}$ .

Suppose that $min {μ (x), \frac{1 - k}{2}} = \frac{1 - k}{2}$ . Then $inf_{z \in k (x_{1}^{n - 1}, x)} {μ (z)} \geq \frac{1 - k}{2}$ and hence $I_{g} (μ (x), inf_{z \in k (x_{1}^{n - 1}, x)} {μ (z)}) \geq inf_{z \in k (x_{1}^{n - 1}, x)} {μ (z)} \geq \frac{1 - k}{2}$ .

Therefore μ is an $\frac{1 - k}{2}$ -implication-based fuzzy (m, n)-ary subhypermodule of M.

5 Conclusion

To obtain a general type of an (∈, ∈ ∨ q)-fuzzy (m, n)-ary subhypermodule of a hypermodule, we have introduced the notion of an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule. We have provided examples which are (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule but not (∈, ∈ ∨ q) -fuzzy (m, n)-ary subhypermodule. We have dealt with characterizations of an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule and an $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule. We have investigated conditions for an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule (resp. $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule) to be a fuzzy (m, n)-ary subhypermodule. Different classes of (m, n)-ary subhypermodules have been characterized by the properties of these fuzzy (m, n)-ary subhypermodule. Using the notion of a fuzzy (m, n)-ary subhypermodule with thresholds, we have discussed characterizations of a fuzzy (m, n)-ary sub- hypermodule, an (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule and an $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule. In this manner, we have given some characterizations of three particular cases of (m, n)-ary hypermodules by these generalized fuzzy (m, n)-ary subhypermodules. We finally have considered characterizations of a fuzzy (m, n)-ary subhypermodule, (∈, ∈ ∨ q_k)-fuzzy (m, n)-ary subhypermodule and an $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -fuzzy (m, n)-ary subhypermodule by using implication operators and the notion of implication-based fuzzy (m, n)-ary subhypermodule.

References

Ameri

and Norouzi

, Prime and primary hyperideals in Krasner (m, n)-hyperrings, European J Combin 34 (2013), 379–390.

Ameri

, Norouzi

and Leoreanu-Fotea

, On prime and primary subhypermodules of (m, n)-hypermodules, European J Combin 44 (2015), 175–190.

Ameri

, Amiri-Bideshki

, Saeid

A.B.

and Hoskova-Mayerova

, Prime filters of hyperlattices, An. St. Univ. Ovidius Constanta Seria mathematica XXIV 2 (2016), 3–12.

Anvariyeh

S.M.

and Mirvakili

, Canonical (m, n)-hypermodules over Krasner (m, n)-hyperrings, Iranian J Math Sci Inf 7(2) (2012), 17–34.

Anvariyeh

S.M.

, Mirvakili

and Davvaz

, Fundamental relation on (m, n)-ary hypermodules on (m, n)-ary hyperring, Ars Combinatoria 94 (2010), 273–288.

Belali

, Anvariyeh

S.M.

and Mirvakili

, Free and cyclic (m, n)-hypermodules, Tamkang J Math 42 (2011), 105–118.

Bhakat

S.K.

and Das

, On the definition of a fuzzy subgroup, Fuzzy Sets Syst 51 (1992), 235–241.

Bhakat

S.K.

and Das

, (∈ ∈ ∨ q)-fuzzy subgroups, Fuzzy Sets Syst 80 (1996), 359–368.

Corsini

and Leoreanu

, Applications of hyperstructures theory, Advanced in Mathematics, Kluwer Academic Publishers, 2003.

10.

Cristea

, Hyperstructures and fuzzy sets endowed with two membership functions, Fuzzy Sets Syst 160 (2009), 1114–1124.

11.

Das

, Fuzzy groups and level subgroups, J Math Anal Appl 85 (1981), 264–269.

12.

Dörnte

, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Mathematische Zeitschrift 29 (1928), 1–19.

13.

Davvaz

, Fuzzy Hv-submodules, Fuzzy Sets Syst 117 (2001), 477–484.

14.

Davvaz

and Corsini

, Fuzzy n-ary hypergroups, J Intell Fuzzy Syst 18(4) (2007), 377–382.

15.

Davvaz

and Vougiouklis

, n-ary hypergroups, Iran J Sci Technol Trans A Sci 30(A2) (2006), 165–174.

16.

Davvaz

and Leoreanu-Fotea

, Hyperring Theory and Applications, Hadronic Press, Inc., 115, Palm Harber, USA, 2007.

17.

Davvaz

and Corsini

, Redefined fuzzy Hv-submodules and many valued implications, Inform Sci 177 (2007), 865–875.

18.

Davvaz

, Fathi

and Salleh

A.R.

, Fuzzy hyperrings (H_v – rings) based on fuzzy universal sets, Inform Sci 180 (2010), 3021–3032.

19.

Davvaz

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

20.

Davvaz

and Corsini

, Fuzzy (m, n)-ary subhypermodules (with thresholds), J Intell Fuzzy Syst 23 (2012), 1–8.

21.

Davvaz

, Dudek

W.A.

and Mirvakili

, Neutral elements, fundamental relations and n-ary hypersemigroups, Int J Algebr Comput 19(4) (2009), 567–583.

22.

Davvaz

, Dudek

W.A.

and Vougiouklis

, A generalization of n-ary algebraic systems, Comm Algebra 37 (2009), 1248–1263.

23.

Ghadiri

and Waphare

B.N.

, n-ary polygroups, Iran J Sci Technol Trans A Sci 33(2) (2009), 145–158.

24.

Kazanci

, Davvaz

and Yamak

, A new characterization of fuzzy n-ary polygroups, Neural Comput and Applic 21 (2012), 59–68.

25.

Marty

, Sur une generalization de la notion de group, 8th Congress Math Scandenaves, Stockholm, 1934, pp. 45–49.

26.

Mirvakili

and Davvaz

, Constructions of (m, n)-hyperrings, Matematiqki Vesnik 67(1) (2015), 1–16.

27.

Mirvakili

and Davvaz

, Relations on Krasner (m, n)-hyperrings, European J Combin 31 (2010), 790–802.

28.

Leoreanu-Fotea

, Canonical n-ary hypergroup, Italian J Pure Appl Math 24 (2008), 247–254.

29.

Leoreanu-Fotea

and Davvaz

, Roughness in n-ary hypergroups, Inform Sci 178 (2008), 4114–4124.

30.

Leoreanu-Fotea

and Davvaz

, n-hypergroups and binary relations, European J Combin 29 (2008), 1027–1218.

31.

Leoreanu-Fotea

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

32.

Leoreanu-Fotea

and Davvaz

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

33.

Yuan

X.H.

, Zhang

and Ren

Y.H.

, Generalized fuzzy groups and many valued implications, Fuzzy Sets Syst 138 (2003), 205–211.

34.

Rosenfeld

, Fuzzy groups, J Math Anal Appl 35 (1971), 512–517.

35.

Vougiouklis

, Hyperstructures and their representations, Hadronic Press, Inc, 115, Palm Harber, USA, 1994.

36.

Zadeh

L.A.

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

37.

Zahedi

M.M.

, Bolurian

and Hasankhani

, On polygroups and fuzzy subpolygroups, J Fuzzy Math 3 (1995), 1–15.

A novel concept of ( m,n )-ary subhypermodules in the framework of fuzzy sets

Abstract

1 Introduction

2 Preliminaries

3 (∈, ∈ ∨ q k )( ( ∈ ¯ , ∈ ¯ ∨ q k ¯ ) )-fuzzy (m, n)-ary sub-hypermodule

4 Implication-based fuzzy (m, n)-ary subhypermodule

5 Conclusion

References

3 (∈, ∈ ∨ q_k)( $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ )-fuzzy (m, n)-ary sub-hypermodule