**6. Conclusions**

When data for real world complex situations come from *m* factors (*m* ≥ <sup>2</sup>), then *m*-PFS is used to deal such problems. The structure of semigroups is investigated using the idea of *m*-PFS in this research paper. Shabir et al. [27] used LA-semigroups as the basis for their algebraic structure, which we converted into semigroups. Most importantly, we proved some results related to fuzzy ideals in semigroups in terms of *m*-PFIs in semigroups. This paper presents a significant number of *m*-PFS theory applications. We also studied the characterization of regular and intra-regular semigroups by *m*-PFIs (left) (resp. *m*-PFIs right) and *m*-PFBI.

Our future plans are to study the *m*-PFIs in terms of semirings, ternary semigroups, ternary semirings, near rings and hyperstructures.

**Author Contributions:** Conceptualization: S.B. and M.S.; Methodology: S.S.; Software: S.B.; Validation: A.N.A.-K.; Formal Analysis: S.S.; Investigation: S.S.; Resources: A.N.A.-K.; Data Curation, M.S.; Writing—Original Draft Preparation: S.S.; Writing—Review and Editing: S.B.; Visualization: S.S.; Supervision: S.B.; Project Administration: S.S.; funding acquisition: A.N.A.-K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** We didn't use any data for this research work.

**Conflicts of Interest:** The authors declare that they have no conflict of interest.
