This paper develops structural results for distributive hesitant fuzzy ℛ-modules and separates the genuinely distributive consequences from the functorial identities that hold for arbitrary homomorphisms. The author’s contribution consists in establishing precise compatibility conditions under which classical submodule operations lift to the hesitant fuzzy setting. We prove that distributivity is invariant under ℛ-module isomorphisms and is preserved by hesitant fuzzy direct sums under the condition ann(ℳ₁) + ann(ℳ₂) = ℛ; a counterexample shows that this hypothesis cannot simply be omitted. The homomorphism results are reformulated with exact assumptions: images, inverse images and sums are treated separately from the preservation of intersections, for which a kernel-saturation condition is required. A new direct theorem, proved without level subsets, shows that a surjective hesitant fuzzy homomorphic image of a distributive hesitant fuzzy ℛ-module is distributive. We also prove directly that all chained hesitant fuzzy ℛ-modules have the distributive property, while the converse fails, and establish closure of distributive hesitant fuzzy submodules under nonempty intersections.
1. Introduction
The fuzzy concept of a Distributive Module was introduced as a natural extension of the classical Distributive Module by adding fuzziness to the submodules involved in the operation of distributive laws using fuzzy intersection and sum operations
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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
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On the other hand, in
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The present paper continues the investigation initiated in
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The scientific novelty and the author’s contribution are expressed in the following results:
• an explicit isomorphism-invariance theorem for distributive hesitant fuzzy ℛ-modules, based on compatibility of hesitant fuzzy sums and intersections (Lemma 2.1 and Proposition 2.2);
• a detailed direct-sum criterion under ann(ℳ₁) + ann(ℳ₂) = ℛ, including the missing annihilator argument at every level and a counterexample showing that the hypothesis cannot simply be removed (Theorem 2.4 and Example 2.5);
• a corrected homomorphism transfer proposition that states the standard image, inverse-image and sum identities without an unnecessary distributivity hypothesis, and gives the exact kernel-saturation condition for preservation of intersections (Proposition 2.6);
• a new level-free theorem proving that surjective hesitant fuzzy homomorphic images preserve distributivity (Theorem 2.9);
• direct structural results showing that the chain condition is strictly stronger than distributivity and that nonempty intersections of distributive hesitant fuzzy submodules remain distributive (Theorem 2.10, Remark 2.11 and Proposition 2.12).
Thus, the contribution lies in the precise lifting mechanism, the formulation of sharp additional assumptions, the direct proofs inside the hesitant fuzzy submodule lattice and the counterexamples delimiting the obtained statements.
The present subsection gathers the elementary notions, the terminology and the auxiliary facts on which the later sections depend. Taken together, they supply the theoretical basis for the arguments that follow and delimit the framework within which the study is carried out.
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The notions collected above provide the platform on which distributive structures inside hesitant fuzzy ℛ-modules will be analysed. A single notational convention is kept throughout the text: fuzzy modules are written 𝒳 and 𝒴; hesitant fuzzy ℛ-modules are written ℱ, ℱ₁, ℱ₂ and 𝒦; fuzzy or hesitant fuzzy submodules are written 𝒢, ℋ, ℐ, indices being attached where needed; and 𝒢𝑗Missing Mark : sub stands for the level subset of 𝒢 taken at a level 𝑗 ∈ (0, 1]. Readers are directed to
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2. Structural Properties of Distributive Hesitant Fuzzy ℛ-Modules
New results on distributive hesitant fuzzy ℛ-modules are obtained in this section by transferring classical fuzzy distributive structures into the hesitant fuzzy setting. Throughout what follows ℛ stands for a commutative ring with identity, while ℱ stands for a hesitant fuzzy ℛ-module defined over the ℛ-module ℳ.
Since equality of hesitant fuzzy submodules may be tested level by level, both claims follow.
Choose hesitant fuzzy submodules 𝒢, ℋ, ℐ of ℱ.
Because
Accordingly:
Hence the relation [LATEX_FORMULA]\mathcal{G} \cap(\mathscr{H}+\mathscr{J})=(\mathcal{G} \cap \mathscr{H})+(\mathcal{G} \cap \mathscr{J})[/LATEX_FORMULA] is equivalent to:
Therefore ℱ is distributive exactly when
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Distributivity of ℱ₁ and of ℱ₂ combined with Remark 1. 1.6 shows that (ℱ₁)ⱼ and (ℱ₂)ⱼ are distributive ℛ-modules. In addition, any element annihilating ℳᵢ annihilates each of its submodules, so that
The sum ann((ℱ₁)ⱼ) + ann((ℱ₂)ⱼ) therefore contains ann(ℳ₁) + ann(ℳ₂) = ℛ, from which
Invoking Lemma 2.3 for the pair (ℱ₁)ⱼ, (ℱ₂)ⱼ and appealing to (2.1), one concludes that (ℱ₁ ⊕ ℱ₂)ⱼ is distributive at every j ∈ (0,1]. To complete the picture, pick arbitrary hesitant fuzzy submodules 𝒜, ℬ and 𝒞 of ℱ₁ ⊕ ℱ₂. Then
[LATEX_FORMULA]\begin{aligned}
{[\mathcal{A} \cap(\mathscr{B}+\mathcal{C})]_{\mathrm{j}} } & =\mathcal{A}_{\mathrm{j}} \cap\left(\mathscr{B}_{\mathrm{j}}+\mathcal{C}_{\mathrm{j}}\right) \\
=\left(\mathcal{A}_{\mathrm{j}} \cap \mathscr{B}_{\mathrm{j}}\right)+\left(\mathcal{A}_{\mathrm{j}} \cap \mathcal{C}_{\mathrm{j}}\right) & =[(\mathcal{A} \cap \mathscr{B})+(\mathcal{A} \cap \mathcal{C})]_{\mathrm{j}} .
\end{aligned}[/LATEX_FORMULA]
As the identity is valid at each j, one obtains 𝒜 ∩ (ℬ + 𝒞) = (𝒜 ∩ ℬ) + (𝒜 ∩ 𝒞), and consequently ℱ₁ ⊕ ℱ₂ is distributive.
Work with ℛ = ℤ and ℳ = ℤ ⊕ ℤ, and set ℱ(x) = {1} at every x ∈ ℳ. Given a crisp submodule A ≤ ℳ, write  for the characteristic hesitant fuzzy submodule specified by Â(x) = {1} when x ∈ A and by Â(x) = {0} in all remaining cases. Now put
One has H + I = ℳ, which gives Ĝ ∩ (Ĥ + Î) = Ĝ ≠ 0. The intersections G ∩ H = 0 and G ∩ I = 0, however, force (Ĝ ∩ Ĥ) + (Ĝ ∩ Î) = 0, so distributivity of ℱ fails. Since ann(ℤ) + ann(ℤ) = 0 ≠ ℤ in this situation, no conflict with Theorem 2.4 arises; the example rather makes clear why the annihilator assumption is indispensable for the sufficient criterion stated there.
1. Whenever 𝒢 is a hesitant fuzzy submodule of ℱ, the image 𝑓 (𝒢) is a hesitant fuzzy submodule of 𝑓 (ℱ), and therefore also of 𝒦 provided 𝑓 (ℱ) ⊆ 𝒦.
2. Whenever ℋ is a hesitant fuzzy submodule of 𝒦, the preimage 𝑓⁻¹ (ℋ) is a hesitant fuzzy submodule of ℱ.
3. Given hesitant fuzzy submodules 𝒢₁ and 𝒢₂ of ℱ, one has 𝑓 (𝒢₁ + 𝒢₂) = 𝑓 (𝒢₁) + 𝑓 (𝒢₂).
4. Given hesitant fuzzy submodules ℋ₁ and ℋ₂ of 𝒦, one has 𝑓⁻¹ (ℋ₁ ∩ ℋ₂) = 𝑓⁻¹ (ℋ₁) ∩ 𝑓⁻¹ (ℋ₂); injectivity need not be assumed here.
5. For hesitant fuzzy submodules 𝒢₁ and 𝒢₂ of ℱ one has
Equality is attained provided 𝒢₁ and 𝒢₂ are 𝑓 -saturated, that is, provided ker 𝑓 ⊆ (𝒢₁)ⱼ ∩ (𝒢₂)ⱼ at every j ∈ (0,1]. Injectivity of 𝑓 is a particular case of this situation.
6. Surjectivity of 𝑓 implies 𝑓⁻¹ (ℋ₁ + ℋ₂) = 𝑓⁻¹ (ℋ₁) + 𝑓⁻¹ (ℋ₂) for every pair of hesitant fuzzy submodules ℋ₁, ℋ₂ of 𝒦.
Turning to (3), linearity supplies at each level
so that the levelwise equality criterion delivers the claimed identity of hesitant fuzzy submodules. Item (4) is a consequence of the purely set-theoretic rule 𝑓⁻¹ (A ∩ B) = 𝑓⁻¹ (A) ∩ 𝑓⁻¹ (B) applied at all levels at once.
As regards (5), the inclusion never fails. Suppose in addition that ker 𝑓 ⊆ (𝒢₁)ⱼ ∩ (𝒢₂)ⱼ and pick y ∈ 𝑓 ((𝒢₁)ⱼ) ∩ 𝑓 ((𝒢₂)ⱼ). Elements x₁ ∈ (𝒢₁)ⱼ and x₂ ∈ (𝒢₂)ⱼ may then be found with y = 𝑓 (x₁) = 𝑓 (x₂). The difference x₁ − x₂ lies in ker 𝑓 ⊆ (𝒢₁)ⱼ ∩ (𝒢₂)ⱼ, whence x₁ = x₂ + (x₁ − x₂) ∈ (𝒢₂)ⱼ and consequently x₁ ∈ (𝒢₁)ⱼ ∩ (𝒢₂)ⱼ. This places y in 𝑓 ((𝒢₁)ⱼ ∩ (𝒢₂)ⱼ) and establishes equality at each level.
It remains to treat (6). Let 𝑓 be surjective and take x ∈ 𝑓⁻¹ ((ℋ₁)ⱼ + (ℋ₂)ⱼ), so that 𝑓 (x) = y₁ + y₂ for suitable yᵢ ∈ (ℋᵢ)ⱼ. Surjectivity permits a choice of xᵢ ∈ ℳ with 𝑓 (xᵢ) = yᵢ, and then k = x − x₁ − x₂ ∈ ker 𝑓. Since 0 lies in both level submodules, the inclusion ker 𝑓 ⊆ 𝑓⁻¹ ((ℋᵢ)ⱼ) holds, so the decomposition x = (x₁ + k) + x₂ exhibits x as an element of 𝑓⁻¹ ((ℋ₁)ⱼ) + 𝑓⁻¹ ((ℋ₂)ⱼ). The opposite inclusion is clear, and (6) is proved.
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The distributivity of ℱ turns the right-hand side into
The preimages of 𝒜 ∩ (ℬ + 𝒞) and of (𝒜 ∩ ℬ) + (𝒜 ∩ 𝒞) therefore coincide. Surjectivity of f allows one to pass from equality of preimages to equality of the corresponding hesitant fuzzy submodules of 𝒦, so 𝒦 is distributive. The argument has been conducted entirely within the operations on hesitant fuzzy submodules, no level subsets being involved.
Theorem 2.10. Distributivity is enjoyed by every chained hesitant fuzzy ℛ-module.
Distributivity of ℱ follows. The reasoning is direct and makes no appeal to level subsets.
Regard ℳ as an ℛ-module and let ℱ be a hesitant fuzzy ℛ-module defined over it. Fix an arbitrary level j with j ∈ (0, 1] and suppose that the family [LATEX_FORMULA]\left\{\left(\mathcal{G}_{i}\right)_{j}\right\}_{i=1}^{n}[/LATEX_FORMULA] consists exclusively of distributive ℛ-submodules of ℱ𝑗. Suppose further that the intersection [LATEX_FORMULA]\bigcap_{i=1}^{n}\left(\mathcal{G}_{i}\right)_{j}[/LATEX_FORMULA] retains distributivity for all j ∈ (0, 1]. Under these hypotheses the full intersection [LATEX_FORMULA]\bigcap_{i=1}^{n} \mathcal{G}_{i}[/LATEX_FORMULA] is a hesitant fuzzy ℛ-submodule of ℱ which is itself strictly distributive.
Proof. For the direct implication, let us initially presume that ℱ operates as a hesitant fuzzy ℛ-module that is simultaneously distributive and chained. Given the isomorphic nature of the mapping , we can deduce from Proposition 2.2 that 𝒦 inherently becomes a distributive hesitant fuzzy ℛ-module. The subsequent step requires us to establish that 𝒦 is chained. To achieve this, we pick arbitrary hesitant fuzzy submodules . Owing to being an isomorphism, its pre-images, specifically and qualify as hesitant fuzzy submodules located in ℱ. Acknowledging the chained nature of ℱ, it becomes mandatory that one of the following inclusions holds true: either [LATEX_FORMULA]f^{-1}(\mathcal{A}) \subseteq f^{-1}(\mathcal{B})[/LATEX_FORMULA] or [LATEX_FORMULA]f^{-1}(\mathcal{B}) \subseteq f^{-1}(\mathcal{A}) .[/LATEX_FORMULA] When we apply the bijective mapping across these inclusions, it translates directly to either 𝒜⊆ or ⊆𝒜. This demonstrates that any two hesitant fuzzy submodules belonging to 𝒦 are comparable, thereby confirming that 𝒦 represents a chained hesitant fuzzy ℛ-module. For the reverse scenario, let's assume that 𝒦 is the module identified as both distributive and chained. Recognizing that the inverse function equally serves as an ℛ-module isomorphism, we can utilize the identical logical framework on [LATEX_FORMULA]f^{-1}[/LATEX_FORMULA]. This parallel reasoning validates that ℱ must be distributive and chained as well. Ultimately, this confirms the biconditional statement: ℱ Is distributive and chained ⇔ 𝒦 is distributive and chained.
3. Conclusion
The structural content of distributive hesitant fuzzy ℛ-modules has been clarified here by drawing a line between homomorphism identities of a standard kind and statements whose proofs really consume distributivity. Invariance under isomorphism was obtained from an explicit compatibility of hesitant fuzzy sums with hesitant fuzzy intersections. The direct-sum theorem was supplied with the annihilator argument at every level, while the accompanying example makes clear why its hypothesis cannot simply be discarded. Proposition 2.6 was restated with sharp assumptions: intersections survive passage to preimages with no injectivity required, whereas their survival under images rests on a kernel-saturation condition. The main new contribution is Theorem 2.9, which shows directly, and without recourse to level subsets, that a surjective hesitant fuzzy homomorphic image of a distributive module stays distributive. Level-free proofs were likewise supplied for the implication chained ⇒ distributive and for closure under nonempty intersections. Since the reverse implication fails, the chain condition is strictly stronger than distributivity. Taken together, these results place further study of quotients, decompositions and lattice-theoretic properties of hesitant fuzzy modules on firmer ground.
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