Structural Properties of Distributive Hesitant Fuzzy ℛ-Modules

The fuzzy concept of a Distributive Module was introduced as a natural extension of classical Distributive Module by adding fuzziness to the submodules involved in the operation of distributive laws using fuzzy intersection and sum operations

Various fundamental properties of hesitant fuzzy ℛ-modules have been studied, including homomorphisms, submodule configurations and lattice-theoretic properties. In particular, chained hesitant fuzzy ℛ-modules have been studied due to their strong structural properties, in which the set of hesitant fuzzy submodules is totally ordered with inclusion

On the other hand, in

The aim of this paper is to study distributive hesitant fuzzy ℛ-modules by means of level submodule techniques, without introducing any new definitions. The principal goal is to develop new structural theorems relating distributivity and the chained property. The preservation of distributivity under isomorphisms, homomorphisms and direct sums with appropriate annihilator conditions is studied in particular. Furthermore, several level-based characterizations are derived in order to elucidate the relationship between distributive and chained hesitant fuzzy ℛ-modules.

The main new contributions of the paper are the following:

• the invariance of distributivity of hesitant fuzzy ℛ-modules under ℛ-module isomorphisms (Proposition 2.2);

• a criterion for the distributivity of hesitant fuzzy direct sums under the annihilator condition ann(ℳ₁) + ann(ℳ₂) = ℛ, together with an example showing that this condition cannot be omitted (Theorem 2.4, Example 2.5);

• a level-wise description of the behaviour of images and inverse images of hesitant fuzzy submodules of distributive hesitant fuzzy ℛ-modules under homomorphisms (Proposition 2.6);

• a proof that every chained hesitant fuzzy ℛ-module is distributive, whereas the converse fails in general (Theorem 2.9, Remark 2.10). Although the corresponding classical statements are well known, the results listed above are obtained entirely within the hesitant fuzzy framework, where the level-set technique is used to transfer the arguments between the classical and the hesitant fuzzy settings.

1.1. Preliminaries and Basic Definitions

In this subsection some of the basic definitions, concepts and preliminary results needed for the subsequent development of this paper are presented. The materials contain the theoretical underpinning for the arguments and framework of the study.

Definition 1.1.1.

Definition 1.1.2.

Definition 1.1.3.

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Definition 1.1.4.

Remark 1.1.5.

Definition 1.1.6.

Remark 1.1.7.

Definition 1.1.8.

Definition 1.1.9.

These foundational concepts establish the groundwork for examining distributive structures in hesitant fuzzy ℛ-modules. Throughout the paper a unified system of notation is adopted: 𝒳 and 𝒴 denote fuzzy modules; ℱ, ℱ₁, ℱ₂ and 𝒦 denote hesitant fuzzy ℛ-modules;

This section develops novel findings on distributive hesitant fuzzy ℛ-modules by extending classical fuzzy distributive structures to the hesitant fuzzy framework. The mathematical notation ℛ denotes a commutative ring possessing an identity element while ℱ represents a hesitant fuzzy ℛ-module that operates on the ℛ-module ℳ.

Lemma 2.1. Let

Proof. For every 𝑗 ∈ (0, 1] we have

Proposition 2.2. Distributivity of hesitant fuzzy ℛ-modules remains invariant under ℛ-module isomorphisms.

Proof. Suppose

Let

Since 𝑔 constitutes an ℛ-module isomorphism, it follows from Lemma 1 that 𝑔 preserves sums and intersections of hesitant fuzzy submodules.

Thus, we obtain:

This demonstrates that the equality

Consequently, ℱ exhibits distributivity if and only if 𝑔(ℱ) does as well.

Lemma 2.3. [1] Let ℳ₁ , ℳ₂ be ℛ-modules satisfying the condition that the sum of the annihilators of ℳ₁ and ℳ₂ equals the entire ring ℛ. The direct sum ℳ₁ ⊕ ℳ₂ forms a distributive ℛ module when both ℳ₁ and ℳ₂ function as distributive ℛ-modules.

Theorem 2.4. Let ℱ₁ and ℱ₂ be distributive hesitant fuzzy ℛ-modules on ℛ-modules ℳ₁ and ℳ₂, respectively. If ann(ℳ₁) + ann(ℳ₂) = ℛ, then the hesitant fuzzy direct sum ℱ₁ ⊕ ℱ₂ constitutes a distributive hesitant fuzzy ℛ-module on ℳ₁ ⊕ ℳ₂.

Proof. For any level 𝑗 ∈ (0, 1], we have:

Given that ℱ₁ and ℱ₂ are distributive hesitant fuzzy modules, (ℱ₁)𝑗 and (ℱ₂)𝑗 are distributive ℛ-modules for all 𝑗. By the classical result concerning direct sums of distributive modules under the annihilator condition (as per Lemma 2.3), (ℱ₁)𝑗 ⊕ (ℱ₂)𝑗 exhibits distributivity. Hence, (ℱ₁ ⊕ ℱ₂)𝑗 is distributive for all 𝑗, and consequently ℱ₁ ⊕ ℱ₂ qualifies as a distributive hesitant fuzzy ℛ-module.

Example 2.5. Take ℛ = ℤ, with ℳ = ℤ ⊕ ℤ. Define ℱ(𝑥) = {1} for all 𝑥 ∈ ℳ. Consider the hesitant fuzzy submodules corresponding to

This example demonstrates that assigning complete hesitant membership to every element fails to guarantee fulfillment of the distributive property for hesitant fuzzy ℛ-modules. Note also that here ann(ℤ) + ann(ℤ) = 0 ≠ ℤ, so the example does not contradict Theorem 1; on the contrary, it shows that the annihilator condition of Theorem 2.4 cannot be omitted.

Proposition 2.6. Let ℱ and 𝒦 be hesitant fuzzy ℛ-modules on ℛ-modules ℳ and

1. For every hesitant fuzzy submodule

2. For every hesitant fuzzy submodule ℋ of 𝒦, the inverse image 𝑓⁻¹(ℋ) forms a hesitant fuzzy submodule of the distributive hesitant fuzzy ℛ-module ℱ.

3. If

4. If 𝑓 is a monomorphism and ℋ₁, ℋ₂ are hesitant fuzzy submodules of 𝒦, then

5. For any hesitant fuzzy submodules

Proof. We establish the assertions using the level-wise characterization of hesitant fuzzy submodules. Let 𝑗 ∈ (0, 1]. Since

Similarly, for any hesitant fuzzy submodule ℋ of 𝒦, the level subset (𝑓⁻¹(ℋ))𝑗 = 𝑓⁻¹(ℋ𝑗) forms a submodule of ℱ𝑗, demonstrating that 𝑓⁻¹(ℋ) is a hesitant fuzzy submodule of ℱ. This proves (2).

Assume now that

If 𝑓 is injective, then for any hesitant fuzzy submodules ℋ₁, ℋ₂ of 𝒦, we have for all 𝑗 ∈ (0, 1], 𝑓⁻¹((ℋ₁)𝑗 ∩ (ℋ₂)𝑗) = 𝑓⁻¹((ℋ₁)𝑗) ∩ 𝑓⁻¹((ℋ₂)𝑗), which implies 𝑓⁻¹(ℋ₁ ∩ ℋ₂) = 𝑓⁻¹(ℋ₁) ∩ 𝑓⁻¹(ℋ₂), confirming (4).

Finally, for hesitant fuzzy submodules

which yields

Remark 2.7. The ℱ-invariance condition proves essential to ensure preservation of intersections under the image mapping (see

Remark 2.8. Although the statements of Proposition 2.6 resemble the classical properties of images and inverse images of submodules under module homomorphisms, they cannot be deduced directly from the known results of the literature: a hesitant fuzzy submodule is determined by the whole family of its level subsets simultaneously, so the classical statements have to be verified level-wise for every 𝑗 ∈ (0, 1] and then lifted to the hesitant fuzzy setting. Proposition 2.6 makes this transfer explicit and, in items (3) and (4), specifies the additional assumptions (ℱ-invariance and injectivity of 𝑓) under which intersections are preserved in the hesitant fuzzy environment.

Theorem 2.9. Every chained hesitant fuzzy ℛ-module is a distributive hesitant fuzzy ℛ-module.

Proof. Let ℱ be a chained hesitant fuzzy ℛ-module on ℳ and let

Remark 2.10. The converse of Theorem 2.9 does not hold in general; consequently, distributivity is independent of chain conditions. Indeed, take ℛ = ℤ and ℳ = ℤ, and define ℱ(𝑥) = {1} for all 𝑥 ∈ ℤ. For every 𝑗 ∈ (0, 1] the level subset ℱ𝑗 = ℤ is a distributive ℤ-module, since the lattice operations on the submodules 𝑛ℤ of ℤ are given by greatest common divisors and least common multiples, which distribute over each other. Hence ℱ is a distributive hesitant fuzzy ℤ-module. Nevertheless, ℱ is not chained, because the hesitant fuzzy submodules corresponding to 2ℤ and 3ℤ are incomparable with respect to inclusion. Thus a distributive hesitant fuzzy ℛ-module need not satisfy any chain-type condition.

In this paper, distributive hesitant fuzzy ℛ-modules were studied through a structural and level-based approach. The obtained results extend several classical properties of distributive modules to the hesitant fuzzy setting and provide a clearer understanding of the interaction between distributivity and hesitant fuzzy submodule structures. It was shown that distributivity is preserved under ℛ-module isomorphisms and under hesitant fuzzy direct sums satisfying suitable annihilator conditions. Moreover, the behaviour of distributivity with respect to homomorphisms was analysed by level submodule methods. It was also proved that every chained hesitant fuzzy ℛ-module is distributive (Theorem 2.9), whereas the converse is not true in general (Remark 2.10); hence distributivity is independent of chain conditions in the internal structure of hesitant fuzzy modules. In general, the above results are helpful for the development of hesitant fuzzy module theory and can be used as a starting point for studying further the more general fuzzy algebraic structures and corresponding module-theoretic properties.