Study on Anti Q-Pythagorean Fuzzy Ideals in Semiring

Abstract

In this paper, we introduce the notion of anti Q-Pythagorean fuzzy left ideal, anti Q-Pythagorean fuzzy right ideal, anti Q-Pythagorean fuzzy ideal and bi-ideal in semiring some interesting properties, results are discussed in this paper.

Introduction

The study builds upon foundational work in algebraic structures and fuzzy set theory. Initially, Γ-semirings were developed as a generalization of rings, semirings, γ-rings, ternary semirings, and semigroups by researchers like Nobusawa and Murali Krishna Rao. These structures involve operations defined using an external set Γ and have been used to extend results from classical algebra.

On the fuzzy side, Zadeh introduced fuzzy sets, Atanassov expanded on this with intuitionistic fuzzy sets, and Yager further generalized the concept with Pythagorean fuzzy sets, where the sum of squares of membership and non-membership degrees is ≤ 1. Pythagorean fuzzy sets have shown strong potential in modeling uncertainty in practical applications and are more expressive than intuitionistic fuzzy sets.

This paper focuses on defining and analyzing anti Q-Pythagorean fuzzy ideals in semirings. These are fuzzy subsets defined with respect to a set Q that satisfy specific algebraic properties related to fuzzy addition and multiplication:

Key Definitions:

• Γ-Semiring: An algebraic structure defined over two commutative semigroups with a ternary operation satisfying certain distributive and associative-like properties.

• Pythagorean Fuzzy Set: A fuzzy set where the square of the membership and non-membership degrees sums to at most 1.

• Anti Q-Pythagorean Fuzzy Subsemiring / Ideals: A fuzzy subset of a semiring that satisfies upper/lower bounds on membership and non-membership values under addition and multiplication.

Main Results:

• Intersection Theorem: The intersection of any family of anti Q-Pythagorean fuzzy right (or left) ideals is also an anti Q-Pythagorean fuzzy right (or left) ideal.

• Union Theorem: The union of any family of anti Q-Pythagorean fuzzy right (or left) ideals is also an anti Q-Pythagorean fuzzy right (or left) ideal.

These theorems confirm that the class of anti Q-Pythagorean fuzzy ideals is closed under intersection and union, supporting their robustness in algebraic modeling.

References

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Copyright

Copyright © 2025 J. Jayaraj, R. Raghu. 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.