Структурные соотношения между дистрибутивными и цепочечными нечеткими модулями с колеблющейся неопределенностью

The theory of hesitant fuzzy sets has been emerging as a crucial extension of classical fuzzy sets, since it allows to model uncertainty due to several potential membership values. Since the inception it has found a lot of interest in various mathematical and applied fields, especially in algebraic structures where uncertainty and vagueness abound naturally. In this class of structures, hesitant fuzzy modules are considered as a flexible structure to research the generalized module properties under the context of the hesitant membership environment.

Distributive and chained modules are basic classes of modules in module theory which have important roles to play in the understanding of lattice structure of submodules and decomposition behavior. The above ideas extend to the uncertain fuzzy setting, providing new opportunities for analysis of algebraic systems that are influenced by uncertainty. There are still not enough studies in the literature about the structural relationship between distributive HFMs and chained HFMs.

This paper is dedicated to establishing new relationships between these two families of modules with hesitations, by level subset techniques and homomorphic characterizations. A few sufficient conditions for the distributivity are deduced, especially with the chain structure of level modules. Further, the preservation of distributivity under hesitant fuzzy epimorphisms is also explored. Based on equivalence results of chained hesitant fuzzy modules with their level submodules, the paper further develops the results.

For purposes of illustration, concrete examples to illustrate the theoretical results based on direct sums of integer modules are created. The above examples show that distributivity strictly weaker than chained property in hesitant fuzzy setting, particularly by direct sum construction. Thus, the results of this study come to a better understanding of the structure of the hesitant fuzzy modules and can serve as a basis for further study in hesitant fuzzy algebra and its applications and these results are an extension of the research work reported in the sources

2.1. Theorem 1

Let

Proof:

Let 𝒢, ℋ, and ℐ be hesitant fuzzy submodules of 𝑓(ℱ). The definition of inverse image enables construction of hesitant fuzzy submodules through the sets 𝑓⁻¹(𝒢), 𝑓⁻¹(ℋ), and 𝑓⁻¹(ℐ), all residing within the framework of ℱ. The distributive hesitant fuzzy ℛ-module structure of ℱ permits establishment of the equality:

The properties of inverse images under fuzzy homomorphisms enable derivation of:

Applying 𝑓 to both sides combined with the surjective property of 𝑓 and the fact that every hesitant fuzzy submodule of ℱ remains ℱ-invariant leads to the deduction that:

Consequently, the expression 𝒢 ∩ (ℋ + ℐ) = (𝒢 ∩ ℋ) + (𝒢 ∩ ℐ) holds true. Thus 𝑓(ℱ) demonstrates adherence to the distributive property, thereby establishing itself as a distributive hesitant fuzzy ℛ-module.

2.2. Theorem 2

Every chained hesitant fuzzy ℛ-module qualifies as a distributive hesitant fuzzy ℛ-module.

Proof:

Let ℱ be a chained hesitant fuzzy ℛ-module and let 𝒢, ℋ, ℐ be hesitant fuzzy submodules of ℱ. Since ℱ is chained, the family {𝒢, ℋ, ℐ} is totally ordered under inclusion. Without loss of generality, assume 𝒢⊆ ℋ ⊆ ℐ. Then we obtain:

On the other hand:

Thus,

2.3. Theorem 3

Let ℱ be a hesitant fuzzy ℛ-module. If for every 𝑗 ∈ (0, 1], the level module ℱ𝑗 constitutes a chained ℛ-module, then ℱ forms a distributive hesitant fuzzy ℛ-module.

Proof:

Take 𝑗 ∈ (0, 1]. By hypothesis, ℱ𝑗 is a chained ℛ-module. Hence, for any submodules 𝒢𝑗, ℋ𝑗, ℐ𝑗 of ℱ𝑗, we have:

Thus, ℱ𝑗 is a distributive ℛ-module for all 𝑗 ∈ (0, 1]. By the level characterization of distributive hesitant fuzzy modules, it follows that ℱ qualifies as a distributive hesitant fuzzy ℛ-module.

2.3.1 Proposition 1

Let ℱ be a distributive H.F. ℛ-module. If for every 𝑗 ∈ (0, 1], the lattice of submodules of ℱ𝑗 is linearly ordered, then ℱ constitutes a chained hesitant fuzzy ℛ-module.

Proof:

Let 𝒢 and ℋ be hH.F. submodules of ℱ. Then for each 𝑗 ∈ (0, 1], the level subsets 𝒢𝑗 and ℋ𝑗 are submodules of ℱ𝑗. Since the lattice of submodules of ℱ𝑗 is linearly ordered, either 𝒢𝑗 ⊆ ℋ𝑗 or ℋ𝑗 ⊆ 𝒢𝑗. This holds for all 𝑗 ∈ (0, 1]. Hence, by the characterization of hesitant fuzzy inclusion via level subsets, either 𝒢 ⊆ ℋ or ℋ ⊆ 𝒢. Therefore, ℱ qualifies as a chained H.F. ℛ-module.

2.4. Theorem 4

Let ℱ denote a hesitant fuzzy module over the ring ℛ. The subsequent statements will demonstrate their equivalence to one another:

1. ℱ is a chained hesitant fuzzy ℛ-module.

2. For every 𝑗 ∈ (0, 1], ℱ𝑗 is a chained ℛ-module.

Proof:

(1) ⇒ (2): Assume ℱ is chained. Let 𝒢𝑗, ℋ𝑗 be submodules of ℱ𝑗. Then there exist hesitant fuzzy submodules 𝒢, ℋ such that 𝒢𝑗 and ℋ𝑗 are their level subsets. Since ℱ is chained, either 𝒢 ⊆ ℋ or ℋ ⊆ 𝒢, hence 𝒢𝑗 ⊆ ℋ𝑗 or ℋ𝑗 ⊆ 𝒢𝑗. Thus ℱ𝑗 is chained.

(2) ⇒ (1): Assume each ℱ𝑗 is chained. Let 𝒢, ℋ be hesitant fuzzy submodules of ℱ. Then for all 𝑗∈ (0, 1], either 𝒢𝑗⊆ ℋ𝑗 or ℋ𝑗⊆𝒢𝑗. Hence, 𝒢⊆ ℋ or ℋ ⊆𝒢. Therefore, ℱ is chained.

The preceding results demonstrate that the chained property is stronger than distributivity in the hesitant fuzzy setting, while distributivity alone does not guarantee the chained condition unless additional structure (such as linear ordering at each level) is imposed.

2.5. Theorem 5

Let ℱ be a hesitant f. ℛ-module on an ℛ-module ℳ. Assume that for every level 𝛼 ∈ (0, 1], the level submodule ℱ𝛼 constitutes a chained ℛ-module. Then ℱ is a distributive H.F ℛ-module.

Proof:

Assume that for every 𝛼∈ (0, 1], the level submodule ℱ𝛼 is chained. Since every chained module exhibits distributivity, it follows that for all 𝛼∈ (0, 1]:

Let 𝒢, ℋ, ℐ be hesitant fuzzy submodules of ℱ. Then:

Given that ℱ𝛼 is distributive, we obtain:

Thus:

2.6. Corollary 1

If ℱ is a distributive hesitant fuzzy ℛ-module and for every 𝛼 ∈ (0, 1], ℱ𝛼 is Noetherian, then ℱ𝛼 is chained for all 𝛼 ∈ (0, 1].

Proof:

Assume ℱ is distributive and each ℱ𝛼 is Noetherian. Then each ℱ𝛼 constitutes a distributive ℛ-module. A classical result implies that every Noetherian distributive module decomposes into chained components. Since level submodules inherit minimal structure from the hesitant fuzzy setting, each ℱ𝛼 must be chained.

2.7. Example 1

Let ℛ = ℤ, and consider the ℤ-modules ℳ₁ = ℤ and ℳ₂ = ℤ. Define ℳ = ℳ₁ ⊕ ℳ₂. Let ℱ₁ and ℱ₂ be hesitant fuzzy ℛ-modules on ℳ₁ and ℳ₂, respectively, defined as follows:

Define the hesitant fuzzy direct sum ℱ = ℱ₁ ⊕ ℱ₂ on ℳ by:

Now we analyze the level subsets. For 𝛼 ∈ (0, 1], we have: ℱ𝛼 = (ℱ₁)𝛼 ⊕ (ℱ₂)𝛼. Observe that:

Now note:

· ℤ, 2ℤ, and 3ℤ are chained ℤ-modules.

· ann(

By the classical result concerning direct sums of distributive modules, each level ℱ𝛼 constitutes a distributive module. Therefore, ℱ qualifies as a distributive hesitant fuzzy ℛ-module. However, note that ℱ𝛼 is generally not chained (since direct sums of chained modules are not necessarily chained).

This illustration demonstrates that distributivity of hesitant fuzzy modules is strictly weaker than the chained property under direct sum constructions. Thus, the direct sum structure preserves distributivity but not the chain condition, highlighting the depth of the main theorem.

The preceding example shows that while the chain condition guarantees distributivity, the converse fails under direct sum constructions. This reveals a structural gap between hesitant fuzzy chained and distributive modules.

2.8. Example 2

Let ℛ = ℤ and consider the ℛ-modules:

Define two hesitant fuzzy ℛ-modules ℱ₁ and ℱ₂ on ℳ₁ and ℳ₂, respectively, as follows:

Define the hesitant fuzzy direct sum ℱ = ℱ₁ ⊕ ℱ₂ on ℳ by:

Step 1: Level structure. For each 𝛼 ∈ (0, 1], we have ℱ𝛼 = (ℱ₁)𝛼 ⊕ (ℱ₂)𝛼. By direct computation:

Step 2: Distributivity. Each of the modules ℤ, 2ℤ, and 3ℤ is a distributive ℤ-module. Moreover:

Hence, by the classical result on direct sums of distributive modules, each level module ℱ𝛼 is distributive. Therefore, ℱ is a distributive hesitant fuzzy ℛ-module.

Step 3: Failure of the chain condition. Consider the following submodules of ℳ: 𝒢 = 2ℤ ⊕ 0, ℋ = 0 ⊕ 3ℤ. Then: 𝒢⊈ ℋ and ℋ ⊈𝒢. Thus, ℳ is not a chained ℛ-module. Consequently, for suitable hesitant fuzzy submodules corresponding to 𝒢 and ℋ, the inclusion relation fails, and hence ℱ is not a chained H.F ℛ-module.

In this paper, several fundamental properties of distributive and chained hesitant fuzzy modules were investigated through the study of level submodules and hesitant fuzzy homomorphisms. The obtained results established that every chained hesitant fuzzy module is distributive, while distributivity alone is generally insufficient to guarantee the chain condition without additional structural assumptions. Moreover, distributivity was shown to be fully characterized by the distributive behavior of the associated level modules.

The study also demonstrated that distributive hesitant fuzzy modules remain stable under suitable epimorphic images and that the structure of level submodules plays a decisive role in determining both distributive and chained properties. Furthermore, explicit direct sum examples clarified the structural gap between distributive and chained hesitant fuzzy modules by showing that distributivity may be preserved even when the chain condition fails.

The results presented in this paper contribute to enhancing the theoretical understanding of hesitant fuzzy module structures and provide more links between classical module theory and hesitant fuzzy algebra. The obtained results can be used as a starting point for future studies on decomposition theory, homological properties and generalized fuzzy algebraic systems.