Neutrosophic cubic N-fuzzy Ideals with Tention Semigroup

Abstract

The aim of this paper is to define neutrosophic cubic ?-ternion subsemigroup, neutrosophic(neut.) cubic ?- left ideal(LI), (resp. lateral ideal(LI), right ideal(RI)) of ternion semigroup(SG) with suitable example, to define characteristic neut. cubic ?-structure(?-S) of ternion SG. Additionally intersection of two neut. cubic ?-LI(resp. LI, RI) is also a neut. cubic ?-LI(resp. LI, RI). We find intersection between two neut. cubic ?-LI(resp. LI, RI) in ternion SG is neut. cubic ?-LI(resp. LI, RI) in ternion SG. Further if we have an neut. cubic ?-LI(resp. LI, RI) then its pre-image is also neut. cubic ?-LI(resp. LI, RI) of ternion SG. In this study a new algebraic approach has been developed in neut. cubic ?-ideals in ternion SG. In future this neut. cubic ?-fuzzy ideal concept can be used in semiring, ternion semiring etc.

Introduction

Background and Motivation

Several researchers have contributed to the study of semigroups (SG), ternary and quaternary structures, and fuzzy/neutrosophic concepts:

• Chinram et al. (2023): Introduced covered left ideals in ternary SG.

• Nongmanee & Leeratanavalee (2022): Studied quaternary rectangular bands and representations in ternary SG.

• Zadeh (1975): Introduced interval-valued fuzzy sets.

• Atanassov: Developed intuitionistic fuzzy sets.

• Smarandache: Proposed neutrosophic sets.

• Lehmer (1932): Explored ternary analogues of abelian groups.

• Khan et al.: Studied neutrosophic N-subsemigroups.

• Kumaran & Selvaraj (2023): Focused on interval-valued neutrosophic ℵ-fuzzy ideals.

• Other contributions explored cubic and lateral ideals in ternary or near-ring settings.

Objectives of the Paper

This work introduces and investigates the neutrosophic cubic ℵ-fuzzy ideals in ternion semigroups (SG). Key concepts studied include:

• Left, Right, and Two-sided (LI, RI, and ideals)

• Characterization using membership (T), indeterminacy (I), and non-membership (F)

• Level sets, characteristic functions

• Operations such as direct product, intersection, and homomorphisms

Methodology

• Defines new structures such as neutrosophic cubic ℵ-ternion SSGs and characterizes their behavior using interval and scalar fuzzy values.

• Studies algebraic operations (product and intersection) over these structures and proves closure under them.

• Uses formal definitions and examples to demonstrate key properties and non-properties.

Key Definitions and Concepts

• Neutrosophic Cubic ℵ-S: A structure defined with interval-valued and scalar-valued functions T,I,FT, I, FT,I,F over a semigroup SSS.

• Neutrosophic Cubic ℵ-LI/RI/Ideal: Specialized structures satisfying inequality relations among the T, I, F values under ternion operations.

• Product and Intersection Operations: Definitions for combining ℵ-S structures.

• Homomorphism Preservation: Shows that preimages under SG homomorphisms preserve the ℵ-LI/RI structure.

• Characteristic Functions (χ): Defined to describe subset-based neutrosophic ideals.

Results and Examples

• Examples 3.4 & 3.5: Show concrete ternary SGs with defined neutrosophic cubic ℵ-S structures and demonstrate when these do or do not satisfy the ideal conditions.

• Theorems 3.7–3.12: Prove properties such as:

  • Closure under intersection (Theorem 3.7, Corollary 3.8).

  • Stability under neutrosophic product for LI/RI (Theorems 3.9, 3.10).

  • Homomorphism preimages of neutrosophic ℵ-ideals remain ℵ-ideals (Theorem 3.12).

Conclusion

This paper explores the notion of a SSG within the framework of neut. cubic ?-ternion SG, investigating its inherent properties. The study encompasses neut. cubic ?-fuzzy ideals within a ternion SG, scrutinizing their algebraic features. Through illustrative examples, we establish that the combination of two neut. cubic ?-fuzzy ideals within a ternion SG results in another neut. cubic ?-fuzzy ideal within the same ternion SG. Additionally, we present the notion of the direct product for neut. cubic ?-fuzzy ideals in a ternion SG, emphasizing its nature as a neut. cubic ?-fuzzy ideal. Furthermore, we can extend this neut. cubic fuzzy ternion SG to a neut. cubic fuzzy ternion semi-ring etc.

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Copyright

Copyright © 2025 S. Karunanithi, G. Shobana, S. Selvaraj. 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.