A new type of fuzzy subsemihypermodules

Abstract

In this paper, combining the notions of “belongingness” and “quasi coincidence” of fuzzy points and fuzzy subsets, a new type of fuzzy subsemihypermodules, namely (∝, ⋐_k)-fuzzy subsemihypermodules of general hypermodules, is introduced and characterized, using the “max” operator. Their properties and connections with other (α, β)-fuzzy subsemihypermodules are investigated.

Keywords

General hypermodule (∝,∝)-fuzzy subsemihypermodule

1 Introduction

The first fuzzification approach of an algebraic concept dates back away to 1971, when Rosenfeld [17] introduced the fuzzy subgroups. After the definition of the relations of “belongingness ∈” and “quasi-coincidence q” of a fuzzy point to a fuzzy subset, a new generalization of the fuzzy subgroups was introduced. In particular, Bhakat and Das [7] defined and studied the (∈, ∈ ∨ q)-fuzzy subgroups, extended later on to other fuzzy algebraic structures/hyperstructures, like: fuzzy subalgebras of BCK/BCI-algebras [15], fuzzy subhypermodules [2, 22], fuzzy n-ary subhypergroups [11], fuzzy H_v-ideals of H_v-rings [14], fuzzy ideals of hemirings [14], fuzzy subhypernearrings [21], etc.

On the other hand, Ameri et al. [2] generalized the notion of quasi-coincidence of a fuzzy point with a fuzzy subset in the hypermodules framework. In particular, in the mentioned paper, considering the fuzzy subsets of hypermodules defined over a Krasner hyperring, the (∈, q_k)-fuzzy subhypermodules and(∈, ∈ ∨ q_k)-fuzzy subhypermodules were defined and studied. Their definitions are based on the triangular norm “min”.

Having this in mind, in the current note, we propose another generalization of the fuzzy subhypermodules, making two significant changes in the previous definitions. First, instead of Krasner hyperrings, we consider the general hyperrings, having addition and multiplication defined as hyperoperations. Secondly, we consider the “max” operator. Combining these two ideas, we introduce, in Section 3, the concept of (∝, ⋐_k)-fuzzy subsemihypermodules of hypermodules over general hyperrings. Several characterizations are stated, motivated by concrete examples, grouped in Section 4. The paper ends with a conclusive part, together with a suggestion for new lines of research to deepen this argument.

2 Preliminaries

Throughtout this paper, unless otherwise stated, R denotes a general hyperring, where both addition and multiplication are hyperoperations. For convenience, we call it, by short, a hyperring. Different aspects concerning hypermodule theory can be found in [1 , 23], while fuzzy hypermodules have been studied, for example, in [9 , 20], or in the recently published book on fuzzy algebraic hyperstructures [8]. In the following, we recall the main and basic notions that will be used in the paper.

Definition 2.1. A hyperring is an algebraic hyperstructure (R, + , ·) which satisfies the following axioms:

(R, +) is a hypergroup.

(R, ·) is a semihypergroup.

The multiplication “·” is distributive with respect to the addition “+”.

If (R, + , ·) is a hyperring and A a nonempty subset of R, then A is called a subhyperring of R, if (A, + , ·) itself is a hyperring, too.

Definition 2.2. The algebraic hyperstructure (M, ⊕ , ⊙) is called a left hypermodule over the hyperring (R, + , ·), by short, a left R-hypermodule, if (M, ⊕) is a hypergroup and there exists the map $⊙ : R \times M ⟶ P^{*} (M)$ , defined by (r, m) ↦ r ⊙ m, such that, for all r₁, r₂ ∈ R and m₁, m₂ ∈ M, we have

r₁ ⊙ (m₁ ⊕ m₂) = (r₁ ⊙ m₁) ⊕ (r₁ ⊙ m₂);

(r₁ + r₂) ⊙ m₁ = (r₁ ⊙ m₁) ⊕ (r₂ ⊙ m₁);

(r₁ · r₂) ⊙ m₁ = r₁ ⊙ (r₂ ⊙ m₁).

In what follows, all hypermodules are left hypermodules. Recall that a nonempty subset A of a hypermodule (M, ⊕ , ⊙) is a subhypermodule if (A, ⊕ , ⊙) is a hypermodule, too. Also, we say that A is a subsemihypermodule of M if x ⊕ y ⊆ A and r ⊙ x ⊆ A, for all x, y ∈ A and r ∈ R.

Based on these two general definitions, we can introduce now the notion of fuzzy subsemihypermodule of a hypermodule.

Definition 2.3. Let (M, ⊕ , ⊙) be a hypermodule over the hyperring (R, + , ·). A fuzzy subset μ is called a fuzzy subsemihypermodule of M, if

$inf_{z \in x \oplus y} μ (z) \geq min {μ (x), μ (y)}$ , for allx, y ∈ M.

μ (a ⊙ x) = ⋁ _t∈a⊙x μ (t) ≥ μ (x), for all a ∈ R and x ∈ M.

For a fuzzy subset μ of an R-hypermodule M, the level subset μ_t is defined by $μ_{t} = {x \in M ∣ μ (x) \geq t}, t \in [0, 1] .$

The next result gives us a characterization of a fuzzy subsemihypermodule using level sets.

Theorem 2.4. Let μ be a fuzzy subset of an R-hypermodule M. Then the following statements are equivalent:

μ is a fuzzy subsemihypermodule of M.

Each nonempty level subset of μ is a subsemihypermodule of M.

A fuzzy subset μ of an R-hypermodule M having the form $μ (y) = {\begin{array}{l} t (\neq 0), & if y = x \\ 0, & if y \neq x \end{array}$

is said to be a fuzzy point with the support x and the value t and is denoted by (x) _t. A fuzzy point (x) _t is said “to belong to” (resp. “quasi-coincident with”) a fuzzy subset μ, written as (x) _t ∈ μ (resp. (x) _t qμ), if μ (x) ≥ t (resp. μ (x) + t > 1). If (x) _t ∈ μ or (x) _t qμ, then we write (x) _t ∈ ∨ qμ. If (x) _t ∈ μ and (x) _tqμ, then we write (x) _t ∈ ∧ qμ.

For an arbitrary real number k of [0, 1), we say that (x) _tq_kμ, if μ (x) + t + k > 1. Also, for α ∈ {∈, ∈ ∨ q, ∈ ∧ q, …}, the notation $(x)_{t} \bar{α} μ$ means that (x) _tαμ does not hold.

3 (∝, ⋐_k)-fuzzy subsemihypermodules

In this section, a new type of fuzzy subsemihypermodule of R-hypermodules, stated by the opposite concepts of the symbols “belong to” (∈) and “quasi coincidence” (q), is introduced. Moreover, some characterizations of these fuzzy subsemihypermodules are proposed, being supported by some related examples.

We say that (x) _t ∝ μ, if μ (x) < t (that is, $(x)_{t} \bar{\in} μ$ ). Also, we say that (x) _t⋐_k μ, if μ (x) + t + k ≤ 1 or (x) _t ∝ μ (that is, $(x)_{t} \bar{q_{k}} μ$ or $(x)_{t} \bar{\in} μ$ ).

In the following, let (M, ⊕ , ⊙) be a hypermodule over the hyperring (R, + , ·). Based on these statements, we define the following new type of fuzzy subsemihypermodules.

Definition 3.1. A fuzzy subset μ is called an (∝, ⋐_k)-fuzzy subsemihypermodule of M if:

(x) _t
₁ ∝ μ and (y) _t
₂ ∝ μ imply that (z) _{max{t₁,t₂}}⋐_k μ, for all z ∈ x ⊕ y.

(x) _t
₁ ∝ μ implies that (a) _t
₁⋐_k μ, for all a ∈ r ⊙ x,

for all t₁, t₂ ∈ [0, 1), r ∈ R and x, y ∈ M.

In the next two theorems, we present characterizations of this type of fuzzy subsemihypermodule.

Theorem 3.2. A fuzzy subset μ of M is an (∝, ⋐_k)-fuzzy subsemihypermodule if and only if, for all x, y ∈ M and r ∈ R, the following conditions hold:

$μ (z) \leq max {μ (x), μ (y), \frac{1 - k}{2}}$ , for all zinx ⊕ y.

$μ (r ⊙ x) \leq max {μ (x), \frac{1 - k}{2}}$ .

Proof. Let μ be an (∝, ⋐_k)-fuzzy subsemihypermodule of M and suppose that assertion (i) is not valid. Then, there exist a, b ∈ M and z ∈ a ⊕ b such that: $μ (z) > max {μ (a), μ (b), \frac{1 - k}{2}} .$

Hence, we can take t ∈ (0, 1), with $μ (z) \geq t > max {μ (a), μ (b), \frac{1 - k}{2}}$ . It follows that (a) _t ∝ μ and (b) _t ∝ μ. Then, by hypothesis, we have (z) _t⋐_k μ. Since μ (z) ≥ t, it follows that (z) _t ∈ μ and therefore $(z)_{t} \bar{q_{k}} μ$ . Hence, μ (z) + t + k ≤ 1, which implies that 2t ≤ μ (z) + t ≤ 1 - k. We get that $t \leq \frac{1 - k}{2}$ , which is a contradiction. Similarly, we can show that assertion (ii) is valid, too.

Conversely, suppose that both assertions (i) and (ii) are valid. Let consider x, y ∈ M and t₁, t₂ ∈ (0, 1], such that we have (x) _{t
₁} ∝ μ and (y) _{t
₂} ∝ μ. Then, according with assertion (i), for all z ∈ x ⊕ y, we have $μ (z) \leq max {μ (x), μ (y), \frac{1 - k}{2}} < max {t_{1}, t_{2}, \frac{1 - k}{2}} .$

If $t_{1} \geq \frac{1 - k}{2}$ or $t_{2} \geq \frac{1 - k}{2}$ , then clearly μ (z) < max {t₁, t₂}, which implies that (z) _{max{t₁,t₂}} ∝ μ. Similarly, if $t_{1}, t_{2} \leq \frac{1 - k}{2}$ , then $μ (z) < \frac{1 - k}{2}$ . Hence, we have $μ (z) + max {t_{1}, t_{2}} < \frac{1 - k}{2} + \frac{1 - k}{2} = 1 - k,$ meaning that $(z)_{max {t_{1}, t_{2}}} \bar{q_{k}} μ$ . Therefore, (z) _{max{t₁,t₂}} ∝ μ or $(z)_{max {t_{1}, t_{2}}} \bar{q_{k}} μ$ , that is, (z) _{max{t₁,t₂}}⋐_k μ. Similarly, we can show that, for x ∈ M, t ∈ (0, 1] and r ∈ R, (x) _t ∝ μ implies that (r ⊙ x) _t⋐_k μ. Hence, μ is an (∝, ⋐_k)-fuzzy subsemihypermodule of M. □

Now we give another characterization of an(∝, ⋐_k)-fuzzy subsemihypermodule, using the complementary of a level subset.

Theorem 3.3. Let μ be a fuzzy subset of M. Then μ is an (∝, ⋐_k)-fuzzy subsemihypermodule of M if and only if the subset $\bar{μ_{t}} = {x \in M ∣ μ (x) < t}$ is a subsemihypermodule of M, for all $t \in (\frac{1 - k}{2}, 1]$ .

Proof. Let μ be an (∝, ⋐_k)-fuzzy subsemihypermodule of M. Assume that $t \in (\frac{1 - k}{2}, 1]$ and $x, y \in \bar{μ_{t}}$ . Then μ (x) < t and μ (y) < t, and so, by hypothesis and Theorem 3.2, for all z ∈ x ⊕ y $\begin{matrix} μ (z) \leq max {μ (x), μ (y), \frac{1 - k}{2}} \\ < max {t, \frac{1 - k}{2}} = t . \end{matrix}$

Hence, $z \in \bar{μ_{t}}$ . This implies that $x \oplus y \subseteq \bar{μ_{t}}$ .

Now, let $x \in \bar{μ_{t}}$ and r ∈ R. Then, by Theorem 3.2, for all b ∈ r ⊙ x, we have $\begin{matrix} μ (b) & \leq & ⋁_{t \in r ⊙ x} μ (t) = μ (r ⊙ x) \\ \leq & max {μ (x), \frac{1 - k}{2}} \\ < & max {t, \frac{1 - k}{2}} = t . \end{matrix}$

Hence, $b \in \bar{μ_{t}}$ and so $r ⊙ x \subseteq \bar{μ_{t}}$ . Therefore $\bar{μ_{t}}$ is a subsemihypermodule of M.

Conversely, suppose that $\bar{μ_{t}}$ is a subsemihypermodule of M, for all $t \in (\frac{1 - k}{2}, 1]$ . Let suppose that assertion (i) in Theorem 3.2 is not valid; then there exist a, b ∈ M and z ∈ a ⊕ b such that $μ (z) > max {μ (a), μ (b), \frac{1 - k}{2}} .$

Hence, for t ∈ (0, 1), we have $μ (z) \geq t > max {μ (a), μ (b), \frac{1 - k}{2}} .$

So, $t \in (\frac{1 - k}{2}, 1]$ and $a, b \in \bar{μ_{t}}$ . Since $\bar{μ_{t}}$ is a subsemihypermodule of M, it follows that $a \oplus b \subseteq \bar{μ_{t}}$ and so $z \in \bar{μ_{t}}$ , that is μ (z) < t, which is a contradiction. Therefore assertion (i) in Theorem 3.2 is valid. Similarly, we can show that assertion (ii) in Theorem 3.2 is valid, too. Consequently, μ is an (∝, ⋐_k)-fuzzy subsemihypermodule of M, by Theorem 3.2. □

Example 3.4. Consider M = {0, a, b, c} and R = {0, 1, 2} with the following hyperoperations:

⊕	0	a	b	c
0	{0}	{0, a}	{0, b}	{0, c}
a	{0, a}	{a}	{a, b}	{a, c}
b	{0, b}	{a, b}	{b}	{b, c}
c	{0, c}	{a, c}	{b, c}	{c}

⊙	0	a	b	c
0	{0}	{0}	{0}	{0}
1	{0}	{a}	{b}	{c}
2	{0}	{a}	{b}	{c}

⊎	0	1	2
0	{0}	{0, 1}	{0, 2}
1	{0, 1}	{1}	{1, 2}
2	{0, 2}	{1, 2}	{2}

○	0	1	2
0	{0}	{0}	{0}
1	{0}	{1}	{2}
2	{0}	{1}	{2}

Then (M, ⊕ , ⊙) is a hypermodule over the hyperring (R, ⊎ , ∘). We define the fuzzy subset μ on M as μ (0) =0.42, μ (a) =0.44 and μ (b) = μ (c) =0.35.

(1) If k = 0.1, then $\bar{μ_{t}} = {x \in M ∣ μ (x) < t} = M$ , for all $t \in (\frac{1 - k}{2}, 1] = (0.45, 1]$ , which is a trivial subsemihypermodule of M. Hence, μ is an (∝, ⋐_0.1)-fuzzy subsemihypermodule of M, by Theorem 3.3.

(2) If k = 0.14, then $\bar{μ_{t}} = {\begin{matrix} {0, b, c}, & t \in (0.43, 0.44] \\ M, & t \in (0.44, 1] \end{matrix}$

Since {0, b, c} and M are subsemihypermodules of M, then μ is an (∝, ⋐_0.14)-fuzzy subsemihypermodule of M, by Theorem 3.3.

(3) If k = 0.2, then $\bar{μ_{t}} = {\begin{matrix} {b, c}, & t \in (0.4, 0.42] \\ {0, b, c}, & t \in (0.42, 0.44] \\ M, & t \in (0.44, 1] \end{matrix}$

Note that $\bar{μ_{t}}$ is not a subsemihypermodule of M, for t ∈ (0.4, 0.42]. Hence, μ is not an (∝, ⋐_0.2)-fuzzy subsemihypermodule of M, by Theorem 3.3.

4 Properties of (∝, ⋐_k)-fuzzy subsemihypermodules

In this section, some connections of (α, β)-fuzzy subsemihypermodules are investigated, for $α, β \in {\in, \bar{q_{k}} \propto, ⋐_{k}}$ , and also examples for invalidity of some relations are stated.

Definition 4.1. A fuzzy subset μ is an (α, β)-fuzzy subsemihypermodule of M, if, for all (x) _{t
₁}αμ, (y) _{t
₂}αμ, z ∈ x ⊕ y, a ∈ r ⊙ x, we have (z) _{max{t₁,t₂}}βμ and (a) _{t
₁}βμ, for any x, y ∈ M, r ∈ R and t₁, t₂ ∈ (0, 1].

By this general definition, we state the following results for (∝, ⋐_k)-fuzzy subsemihypermodules.

Theorem 4.2. Every (∝, ∝)-fuzzy subsemihypermodule of M is an (∝, ⋐_k)-fuzzy subsemihypermodule of M.

Proof. Let μ be an (∝, ∝)-fuzzy subsemihypermodule of M, and suppose that assertion (i) in Theorem 3.2 is not valid. Then, similar to the proof of Theorem 3.2, there exists t ∈ (0, 1) such that, for a, b ∈ M and z ∈ a ⊕ b, we have $μ (z) \geq t > max {μ (a), μ (b), \frac{1 - k}{2}} .$

Hence, (a) _t ∝ μ and (b) _t ∝ μ which, by hypothesis, implies that (z) _t ∝ μ and so μ (z) < t. This is a contradiction. Therefore, assertion (i), and similarly (ii), in Theorem 3.2 is valid. Consequently μ is an (∝, ⋐_k)-fuzzy subsemihypermodule of M, by Theorem 3.2. □

The converse of Theorem 4.2 is not true, as shown in the following example.

Example 4.3. Consider the (∝, ⋐_0.1)-fuzzy subsemihypermodule of M given in Example 3.4. Then μ is not an (∝, ∝)-fuzzy subsemihypermodule of M, since, for b ∈ M and 2 ∈ R, we have {0} =2 ⊙ b and μ (b) =0.35 < 0.4, but μ (0) =0.42 > 0.4.

Adding a supplementary condition on the fuzzy subset μ, we get also the converse of Theorem 4.2.

Theorem 4.4. Let μ be an (∝, ⋐_k)-fuzzy subsemihypermodule of M such that $μ (x) \geq \frac{1 - k}{2}$ , for all x ∈ M. Then, μ is an (∝, ∝)-fuzzy subsemihypermodule of M.

Proof. Let (x) _{t
₁}αμ and (y) _{t
₂}αμ, for x, y ∈ M. By hypothesis, for z ∈ x ⊕ y, we have $\begin{matrix} μ (z) & \leq max {μ (x), μ (y), \frac{1 - k}{2}} \\ = max {μ (x), μ (y)} < max {t_{1}, t_{2}} . \end{matrix}$

Hence, (z) _{max(t₁,t₂)}αμ. Similarly, it can be shown that, if (x) _{t
₁}αμ, then (z) _{t
₁}αμ, for all z ∈ r ⊙ x, with r ∈ R. Therefore, μ is an (∝, ∝)-fuzzy subsemihypermodule of M and the proof is completed. □

Theorem 4.5. Every $(\propto, \bar{q_{k}})$ -fuzzy subsemihypermodule of M is an (∝, ⋐_k)-fuzzy subsemihypermodule of M.

Proof. Let μ be an $(\propto, \bar{q_{k}})$ -fuzzy subsemihypermodule of M, and suppose that assertion (i) in Theorem 3.2 is not valid. Then, similar to the proof of Theorem 3.2, we have $μ (z) \geq t > max {μ (a), μ (b), \frac{1 - k}{2}}$ , for a, b ∈ M, t ∈ (0, 1) and z ∈ a ⊕ b. Hence, we have (a) _t ∝ μ and (b) _t ∝ μ which, by hypothesis, implies that $(z)_{t} \bar{q_{k}} μ$ and so μ (z) + t + k ≤ 1.

On the other hand, $μ (z) + t \geq t + t > \frac{1 - k}{2} + \frac{1 - k}{2} = 1 - k,$ that is μ (z) + t + k > 1, which is a contradiction. Similarly, it can be shown that also assertion (ii) in Theorem 3.2 is valid. Therefore, μ is an (∝, ⋐_k)-fuzzy subsemihypermodule of M. □

The converse of Theorem 4.5 is not valid in general, as we will show in the next example.

Example 4.6. Consider the (∝, ⋐_0.14)-fuzzy subsemihypermodule of M given in Example 3.4. Note that (a) _0.5 ∝ μ and (b) _0.4 ∝ μ, but for a ∈ a ⊕ b = {a, b} we have $\begin{matrix} μ (a) + max {0.5, 0.4} + 0.14 \\ = 0.44 + 0.5 + 0.14 > 1 . \end{matrix}$

Hence, μ is not an -fuzzy subsemihypermodule of M.

Theorem 4.7. If 0 ≤ r < k < 1, then every (∝, ⋐_k)-fuzzy subsemihypermodule of M is an (∝, ⋐_r)-fuzzy subsemihypermodule.

Proof. Suppose that μ is an (∝, ⋐_k)-fuzzy subsemihypermodule of M and, for t₁, t₂ ∈ (0, 1] and x, y ∈ M, we have (x) _{t
₁} ∝ μ and (y) _{t
₂} ∝ μ. By hypothesis, for all z ∈ x ⊕ y, we have $\begin{matrix} μ (z) + max {t_{1}, t_{2}} + r \\ < μ (z) + max {t_{1}, t_{2}} + k < 1 . \end{matrix}$

This completes the proof. □

The following example shows that, if 0 ≤ r < k < 1, then an (∝, ⋐_r)-fuzzy subsemihypermodule of M may not be an (∝, ⋐_k)-fuzzy subsemihypermodule of M.

Example 4.8. Consider the hypermodule M and the fuzzy subset μ defined in Example 3.4. Set r = 0.14 and k = 0.2. By Example 3.4, it can be seen that μ is an (∝, ⋐_0.14)-fuzzy subsemihypermodule, but is not an (∝, ⋐_0.2)-fuzzy subsemihypermodule.

Now, we state a new characterization for (∝, ⋐_k)-fuzzy subsemihypermodules.

Let S be a subset of the hypermodule M. Consider a fuzzy subset μ_S in M, where, for all x ∈ M, we define $μ_{S} (x) = {\begin{array}{l} 0, & x \in S \\ 1, & x \in S \end{array}$

Theorem 4.9. A nonempty subset S of M is a subsemihypermodule if and only if the fuzzy subset μ_S in M is an (∝, ⋐_k)-fuzzy subsemihypermodule of M.

Proof. Let S be a subsemihypermodule of M. Then $(\bar{μ_{S}})_{t}$ is clearly a subsemihypermodule of M, for all $t \in (\frac{1 - k}{2}, 1]$ . Hence, μ_S is an (∝, ⋐_k)-fuzzy subsemihypermodule of M, by Theorem 3.3.

Conversely, suppose that μ_S is an (∝, ⋐_k)-fuzzy subsemihypermodule of M. Let x, y ∈ S. Then, for all z ∈ x ⊕ y, we have $\begin{matrix} μ_{S} (z) & \leq max {μ_{S} (x), μ_{S} (y), \frac{1 - k}{2}} \\ = max {0, \frac{1 - k}{2}} = \frac{1 - k}{2} < 1, \end{matrix}$ for all k ∈ [0, 1). Hence, μ_S (z) =0 and so z ∈ S. Therefore, x ⊕ y ⊆ S. Similarly, for all x ∈ S and r ∈ R, we can show that r ⊙ x ⊆ S. Hence, S is a subsemihypermodule of M. □

Theorem 4.10. Let S be a subsemihypermodule of M. Then, for every $t \in (\frac{1 - k}{2}, 1]$ , there exists an(∝, ⋐_k)-fuzzy subsemihypermodule μ of M such that $\bar{μ_{t}} = S$ .

Proof. Let μ be a fuzzy subset of the hypermodule M, defined by $μ (x) = {\begin{array}{l} 0, & x \in S \\ 1, & x \in S \end{array}$ where $t \in (\frac{1 - k}{2}, 1]$ . Then, $\begin{matrix} \bar{μ_{t}} & = & {x \in M ∣ μ (x) < t} \\ = & {x \in M ∣ μ (x) = 0} = S . \end{matrix}$

Assume that assertion (i) in Theorem 3.2 is not valid. Then, similar to the proof of Theorem 3.2, we have $μ (z) \geq t > max {μ (a), μ (b), \frac{1 - k}{2}}$ , for a, b ∈ M, t ∈ (0, 1) and z ∈ a ⊕ b. Hence, μ (a) < t and μ (b) < t. It implies that $a, b \in \bar{μ_{t}} = S$ . Since S is a subsemihypermodule of M, then z ∈ a ⊕ b ⊆ S and so μ (z) =0 < t, for all t ∈ (0, 1), which is a contradiction. Therefore, $μ (z) \leq max {μ (x), μ (y), \frac{1 - k}{2}}$ , for all z ∈ x ⊕ y and x, y ∈ M. Similarly, assertion (ii) in Theorem 3.2 is valid, too. Hence, μ is an (∝, ⋐_k)-fuzzy subsemihypermodule of M. □

In almost all (fuzzy) algebraic (hyper)structures, the intersection of a family of (fuzzy) (hyper) substructures of a certain (fuzzy) (hyper)structure is a (fuzzy) (hyper) substructure of the same type, while the union is not. In the case of (∝, ⋐_k)-fuzzy subsemihypermodules, we show that this property is conversely valid.

Theorem 4.11. Let {μ_i ∣ i ∈ I} be a family of(∝, ⋐_k)-fuzzy subsemihypermodules of M. Then μ = ⋃ _i∈I μ_i is an (∝, ⋐_k)-fuzzy subsemihypermodule of M.

Proof. Let x, y ∈ M and t₁, t₂ ∈ (0, 1] such that (x) _{t
₁} ∝ μ and (y) _{t
₂} ∝ μ. Suppose that (z) _{max(t₁,t₂)} ∈ ∧ q_k μ, for some z ∈ x ⊕ y. Then μ (z) ≥ max {t₁, t₂} and μ (z) + max {t₁, t₂} >1 - k, which implies that $μ (z) \geq \frac{1 - k}{2}$ (∗).

Now, let Γ₁ = {i ∈ I ∣ (z) _{max(t₁,t₂)} ∝ μ_i} and $Γ_{2} = {i \in I ∣ (z)_{max (t_{1}, t_{2})} \bar{q_{k}} μ_{i}} \cap {j \in I ∣ (z)_{max (t_{1}, t_{2})} \bar{\propto} μ_{j}} .$ Then I = Γ₁ ∪ Γ₂ and Γ₁∩ Γ₂ = ∅. If Γ₂ =∅, then (z) _{max(t₁,t₂)} ∝ μ_i, for all i ∈ I, that is, μ (z) < max {t₁, t₂}, for all i ∈ I, which is a contradiction. Hence Γ₂≠ ∅, and thereby, for every i ∈ Γ₂, we have μ_i (z) ≥ max {t₁, t₂} and μ_i (z) + max {t₁, t₂} ≤1 - k. It follows that $max {t_{1}, t_{2}} \leq \frac{1 - k}{2}$ . Now, since (x) _{t
₁} ∝ μ it follows that $μ_{i} (x) \leq μ (x) < t_{1} \leq max {t_{1}, t_{2}} \leq \frac{1 - k}{2},$ for all i ∈ I. Similarly $μ_{i} (y) < \frac{1 - k}{2}$ , for all i ∈ I. Then, suppose that $t = μ_{i} (z) \geq \frac{1 - k}{2}$ .

Take now $t > r > \frac{1 - k}{2}$ . Hence, (x) _r ∝ μ_i and (y) _r ∝ μ_i, which by hypothesis must imply that (z) _r⋐_k μ_i, but we have μ_i (z) = t > r and $μ_{i} (z) + r > \frac{1 - k}{2} + \frac{1 - k}{2} = 1 - k$ , meaning that ( $(z)_{r} \bar{\propto} μ_{i}$ and (z) _rq_k μ_i) μ_i is not an (∝, ⋐_k)-fuzzy subsemihypermodule of M, which is a contradiction. Hence $μ_{i} (z) < \frac{1 - k}{2}$ , for all i ∈ I, and so $μ (z) < \frac{1 - k}{2}$ , that contradicts relation (∗). Therefore (z) _{max(t₁,t₂)}⋐_k μ, for all z ∈ x ⊕ y. Similarly, for z ∈ r ⊙ x and r ∈ R, we have (z) _{t
₁}⋐_k μ. Consequently μ is an (∝, ⋐_k)-fuzzy subsemihypermodule of M. □

The following example shows that there exists k ∈ [0, 1) such that the intersection of two (∝, ⋐_k)-fuzzy subsemihypermodules may be not an (∝, ⋐_k)-fuzzy subsemihypermodule of M.

Example 4.12. Let M = {0, a, b, c}, R = {0, 1} and define the hyperoperations ⊕ and ⊙ as x ⊕ 0 = {x}, $x \oplus y = {\begin{array}{l} {z}, & x \neq y \\ {0}, & x = y \end{array}, r ⊙ x = {\begin{array}{l} {x}, & r \neq 0 \\ {0}, & r = 0 \end{array}$ for all r ∈ R and x ≠ y ≠ z ≠ 0 ∈ M. Also, set $\begin{matrix} ⊎ & 0 & 1 \\ 0 & {0} & {1} \\ 1 & {1} & {0} \end{matrix} \begin{matrix} \circ & 0 & 1 \\ 0 & {0} & {0} \\ 1 & {0} & {1} \end{matrix}$

Then (M, ⊕ , ⊙) is a hypermodule over (R, ⊎ , ∘). Define two fuzzy subsets μ and ν on M as follows: $\begin{matrix} μ (0) & = 0.37, μ (a) = 0.3, μ (b) = μ (c) = 0.42, \\ ν (0) & = 0.33, ν (b) = 0.4, ν (a) = ν (c) = 0.42, \end{matrix}$ noting that both of them are (∝, ⋐_0.2)-fuzzy subsemihypermodules of M, accordingly with Theorem 3.3, since $\begin{matrix} \bar{μ_{t}} & = & {\begin{matrix} {0, a}, & t \in (0.4, 0.42] \\ M, & t \in (0.42, 1] \end{matrix}, \\ \bar{ν_{t}} & = & {\begin{matrix} {0, b}, & t \in (0.4, 0.42] \\ M, & t \in (0.42, 1] \end{matrix} \end{matrix}$

Now, we have (μ ∩ ν) (0) =0.33, (μ ∩ ν) (a) =0.3, (μ ∩ ν) (b) =0.4 and (μ ∩ ν) (c) =0.42. Hence, $\bar{{(μ \cap ν)}_{t}} = {\begin{array}{l} {0, a, b}, & t \in (0.4, 0.42] \\ M, & t \in (0.42, 1] \end{array}$

Since {0, a, b} is not a subsemihypermodule of M, it follows that μ ∩ ν is not an (∝, ⋐_0.2)-fuzzy subsemihypermodule of M, by Theorem 3.3.

5 Conclusions

In this note, we have defined and studied a new type of fuzzy subsemihypermodules, which was created using the inverse concepts of “belongs to” jointly with “quasi coincidence”. One of our aims has been to make a context for comparing the concepts and results regarding this type of fuzzy subsemihypermodules based on the “max” operator with the similar results from [2] and [22], where the “min” operator was involved. Now it is clear that our results can be applied also to other algebraic hyperstructures, that is why we intend to continue and extend this study in the following contexts:

interval-valued form of (∝, ⋐_k)-fuzzy subsemihypermodules;

generalization of this concept to (m, n)-hyperrings and (m, n)-hypermodules, similarly to the works in [3 –5];

adapt and study this concept in the framework Γ-hyperrings and Γ-hypermodules.

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A new type of fuzzy subsemihypermodules

Abstract

Keywords

1 Introduction

2 Preliminaries

3 (∝, ⋐ k )-fuzzy subsemihypermodules

4 Properties of (∝, ⋐ k )-fuzzy subsemihypermodules

5 Conclusions

References

3 (∝, ⋐_k)-fuzzy subsemihypermodules

4 Properties of (∝, ⋐_k)-fuzzy subsemihypermodules