1. Introduction

The aim of the present work is to study a fundamental algebraic generalisation of classical modules, namely semimodules, which play an essential role in automata theory, tropical mathematics and theoretical computer science. Distributivity lies at the heart of this structural programme. An

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The passage from modules to semimodules is not entirely formal. Semimodules generally lack additive inverses, so several arguments that are routine for modules — those involving homomorphic images, inverse images, or additive decompositions — need extra hypotheses once that inverse structure is gone

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Level cuts raise a related difficulty. Fuzzy addition is usually defined through a supremum over additive decompositions, and an ordinary, non-strict level cut need not commute with this operation unless the supremum happens to be attained. The use of cut-based representations has also played an important role in recent studies of semiring-valued fuzzy structures

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These two observations shape the paper's main contributions. First, we introduce fuzzy distributive semimodules over commutative semirings and characterize them through their strict level semimodules. Second, we determine how fuzzy distributivity behaves under inverse images of injective homomorphisms and under isomorphisms, in each case under hypotheses suited to semimodules rather than modules. Third, we examine direct sums and identify the structural assumptions needed to control their subsemimodules. Recent work on semimodules has likewise shown that direct-sum and direct-summand properties often require additional structural conditions

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The rest of the paper is organized as follows. Section 2 recalls the algebraic and fuzzy preliminaries needed later and establishes the strict-level characterization of fuzzy distributive semimodules. Section 3 turns to their behaviour under homomorphisms and isomorphisms. Section 4 takes up direct sums and states the assumptions under which distributivity is preserved. Section 5 relates chained and level-uniserial fuzzy semimodules and discusses where the converse implication breaks down. Section 6 collects examples and counterexamples that mark out the scope of the results, and Section 7 closes the paper.

2. Algebraic and Fuzzy Preliminaries

To ground the algebraic formalism utilized herein, we briefly review semirings, semimodules, and their fuzzy extensions. A semiring is defined as an additively commutative monoid with a zero identity, paired with a multiplicative monoid where multiplication distributes over addition and the additive identity acts as a multiplicative annihilator

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Transitioning to fuzzy topologies, a fuzzy set maps a non-empty set into the real unit interval to assign membership degrees [4]. A fuzzy subset of a semimodule qualifies as a fuzzy semimodule under three conditions: the additive identity holds absolute unity in membership, the membership of any sum is bounded below by the minimum membership of its addends, and scalar multiplication does not strictly decrease an element's membership [5]. A fuzzy subsemimodule independently satisfies these axioms.

3. Fuzzy Distributive Semimodules

A fuzzy distributive semimodule extends the standard distributive law to fuzzy systems. Let

For a fuzzy subset

for its strict level cut. When

The next proposition records how these two operations interact with strict level cuts.

For the sum, take

so some decomposition

Conversely, if

For the last claim, let

the same inequality holds trivially whenever at least one of

and again the inequality is trivial when

that is,

for every

A fuzzy subset is determined by its strict level cuts: if two fuzzy subsets disagreed in membership at some point, any threshold lying between the two values would separate their strict level cuts. Consequently,

1. F is a fuzzy distributive semimodule;

2. Fₜ is distributive for every t ∈ [0,1).

Proof. By hypothesis, Fₜ is distributive for every t ≥ t₀. For t < t₀we have Fₜ = Fₜ₀, which is itself distributive, so in fact Fₜis distributive for every t ∈ [0,1). 3.3. Theorem then gives the conclusion.

4. Homomorphisms of Fuzzy Distributive Semimodules

This section analyses the algebraic invariance of fuzzy distributive semimodules under homomorphic mappings. Using the structural properties of strict level semimodules, we establish preservation under injective inverse images and under isomorphisms.

Similarly, for

so

Proof. For every t ∈ [0,1), (f(F))ₜ= f(Fₜ). 3.2. Theorem shows Fₜ is distributive, and since f is an isomorphism, f(Fₜ) is isomorphic to Fₜand hence distributive as well. Every strict level semimodule of f(F) is therefore distributive, and 3.3. Theorem shows that f(F) is fuzzy distributive.

5. Direct Sums of Fuzzy Distributive Semimodules

This section examines fuzzy distributivity for finite direct sums. Preservation is obtained under an explicit componentwise decomposition condition, while the converse requires no Artinian or Noetherian hypothesis.

For fuzzy semimodules

The following theorem gives a preservation result for direct sums under a componentwise decomposition assumption.

and computing componentwise yields

Proof. For every t ∈ [0,1), Fₜ = (F₁)ₜ ⊕ (F₂)ₜ, and 4.2. Theorem shows this semimodule is distributive. Since (F₁)ₜ ⊕ {0} and {0} ⊕ (F₂)ₜ are subsemimodules of Fₜ, they too are distributive, and they are naturally isomorphic to (F₁)ₜand (F₂)ₜ, respectively. As this holds for every t ∈ [0,1), 3.3. Theorem shows that F₁ and F₂ are both fuzzy distributive.

6. Relations with Chained and Uniserial Fuzzy Semimodules

This section elucidates the algebraic interplay between chained and distributive fuzzy semimodules. We establish that chained fuzzy semimodules are distributive and show by counterexample that the converse can fail even when every strict level semimodule is uniserial.

A fuzzy semimodule

The case

The converse of 6.3. Theorem fails: level-wise uniseriality alone does not force a fuzzy semimodule to be chained.

Take

so

Let

Consider now two fuzzy subsemimodules

Both satisfy the fuzzy subsemimodule conditions, yet

so neither A ≤ B nor B ≤ A. Hence F is not chained, even though it is fuzzy distributive and level-uniserial. This example shows that these two properties together still fall short of chainedness.

7. Examples and Counterexamples

To substantiate the established theoretical framework, this section provides concrete algebraic models that clarify the delineation between fuzzy distributivity and chained structures.

Let

for every

Let

Let

The subsemimodules of

their lattice is distributive but not linearly ordered, so

The fuzzy subsemimodules corresponding to the two coordinate subsemimodules, however, are incomparable, so

Let

for every

Taken together, these examples mark out both the reach and the limits of the results established above. Fuzzy distributivity does not imply chainedness, not even under the level-uniserial condition introduced in Section 5, and the direct-sum examples explain why transferring distributivity from individual components to their direct sum calls for the additional structural assumptions imposed in Section 4.

8. Conclusion

This paper developed a framework for fuzzy distributive semimodules over commutative semirings. Working with strict level cuts avoids the attainment issue that arises when fuzzy addition is defined through a supremum, and it leads to a precise characterization: a fuzzy semimodule is distributive exactly when each of its strict level semimodules is distributive.

We also reconsidered how this property behaves under semimodule mappings, in a form that respects the absence of additive inverses. In particular, we obtained preservation results for injective inverse images and for isomorphisms without appealing to the unrestricted arguments available for modules. For direct sums, distributivity is preserved under an explicit componentwise decomposition condition, and distributivity of a direct sum implies distributivity of its components without any Artinian or Noetherian assumption.

The relation to chained structures came out clearly as well: every chained fuzzy semimodule is fuzzy distributive, but the converse fails in general, and the counterexamples show that requiring all strict level semimodules to be uniserial still does not recover chainedness at the fuzzy level.

Taken together, these results show that several statements familiar from module theory need extra care once additive inverses are no longer available. The framework developed here may also serve as a starting point for further work on fuzzy automata, weighted algebraic systems, and decision models, though concrete applications in these directions will require problem-specific analysis left for future work. Natural extensions include hesitant fuzzy and interval-valued semimodules, provided their distributive operations and level structures are treated on their own terms.

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