## Abstract

The fundamental relation \(\theta ^{*}\) on *M* can be defined as the smallest equivalence relation such that the quotient \(M/\theta ^{*}\) be a module over the corresponding fundamental ring such that \(M/\theta ^{*}\) as a group is not abelian. Moreover, the fundamental ring is not commutative with respect to both sum and product. Now, we would like the fundamental module as a group to be abelian and the fundamental ring to be commutative with respect to both sum and product. Also, we assign fundamental functor between the category of fuzzy hypermodules and the category of abelian groups to convey its features and related commutative diagram. Finally, we find necessary and sufficient conditions such that \(\theta \) is transitive.

## Keywords

Fuzzy hypermodule Fuzzy hyperring Fuzzy strongly regular equivalence relation Fundamental functor## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This paper does not contain any studies with human participants or animals performed by any of the authors.

### Informed consent

Informed consent was obtained from all individual participants included in the study.

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