Fermat’z Fuzzy Ideals over Fuzzy Sub Lattice with Respect to (S,T)-Norms

Abstract

This research delves into the exploration of fermat’z fuzzy sub lattices and fermat’z fuzzy ideals within the context of lattice theory. Through a rigorous analysis of structural theorem concerning these concepts derived fermat’z fuzzy sets. We uncover significant parallels with classical theory. Additionally, we investigate the behavior of fermat’z fuzzy ideals under lattice homomorphism. Our finding shed light on the applicability and utility of fermat’z fuzzy theory in lattice-based structures, offering insights into their properties and relationships.

Introduction

The study explores Fermat’z fuzzy sets as an extension of classical and fuzzy set theories, building on foundational work in fuzzy, intuitionistic fuzzy, neutrosophic, and Pythagorean fuzzy sets to address uncertainty and imprecision in mathematical structures. In particular, it focuses on Fermat’z fuzzy sublattices and fuzzy ideals within lattices, defining membership and non-membership functions J(x)J(x)J(x) and K(x)K(x)K(x) satisfying 0≤Jn(x)+Kn(x)≤10 \le J^n(x) + K^n(x) \le 10≤Jn(x)+Kn(x)≤1. Key operations (union, intersection, complement, product) and relations are formalized, along with definitions for Fermat’z fuzzy ideals and prime ideals.

The work establishes correspondences between fuzzy sets and their level subsets, proving that a Fermat’z fuzzy sublattice (or ideal/prime ideal) is equivalent to each of its non-empty (γ1,γ2)(\gamma_1, \gamma_2)(γ1?,γ2?)-level subsets being a sublattice (or ideal/prime ideal). Additionally, the intersection of Fermat’z fuzzy sublattices (or ideals) retains the same structure, reinforcing their stability under lattice operations. The study provides rigorous mathematical formulations bridging abstract fuzzy concepts with practical lattice theory applications.

Conclusion

To summarize up, our study investigate the complex characteristics of fermat’z fuzzy ideals and sub lattices. We defined operations for these fuzzy ideals and proved their preservation in distributive lattices. We also performed a thorough examination of fermat’z fuzzy ideals homomorphic images and pre-images, which resulted in the creation of invariant Fermat’s fuzzy sets. This work culminated in establishing a correspondence theorem that connects the f-invariant fermat’z fuzzy ideals of a lattice to its homomorphic image.

References

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Copyright

Copyright © 2025 R. Nagarajan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.