Neutrosophic Quotient Spaces and the First Isomorphism Theorem for Topological Groups

Abstract

In this paper, the concept of Neutrosophic Topological Groups (NTGs) is introduced and studied. By combining Smarandache’s Neutrosophic logic and topological groups theory [1], the NTGs are defined. Since classical set theories and classical logic are not enough to deal with indeterminate data, neutrosophic sets have been used for this work. In the neutrosophic sets, each element has a truth-membership function, indeterminacy-membership function and false-membership function. The continuity of multiplication and inversion operations for the NTGs are defined. Moreover, the NTGs quotient spaces which are induced by normal subgroups are constructed. The main target of this study is to prove the continuity and openness of the projection map. Therefore, this paper ends with formulating and proving a Neutrosophic version of the First Isomorphism Theorem for topological groups.

Introduction

The text studies Neutrosophic Topological Groups (NTGs), a mathematical framework that combines abstract algebra, topology, and neutrosophic logic to handle systems involving truth, indeterminacy, and falsehood.

Neutrosophic logic, introduced by Florentin Smarandache, extends fuzzy and intuitionistic fuzzy logic by treating indeterminacy as an independent component. This makes it suitable for mathematical systems containing incomplete, inconsistent, or uncertain information.

Background

The development of uncertainty-based mathematics progressed from:

• Fuzzy sets introduced by Zadeh, which represent degrees of membership.
• Fuzzy topological spaces, which extended fuzzy concepts to topology.
• Intuitionistic fuzzy sets introduced by Atanassov, which added non-membership.
• Neutrosophic sets, which introduced an independent indeterminacy component.

A neutrosophic set A is described using three independent functions:

• Truth membership TA(x)
• Indeterminacy membership IA(x)
• Falsehood membership FA(x)

This allows mathematical objects to represent information that may be true, false, or uncertain simultaneously.

Neutrosophic Topological Groups

A Neutrosophic Topological Group (NTG) is a group equipped with a neutrosophic topology such that the two fundamental group operations are neutrosophically continuous:

1. Multiplication: m(x,y)=xy
2. Inversion: x→x−1

The paper focuses on understanding how algebraic operations interact with neutrosophic open sets.

Translation Invariance

One of the main results is the Translation Invariance Theorem.

For any element a in a neutrosophic topological group, the mappings

• Left translation: La(x)=ax
• Right translation: Ra(x)=xa

are both neutrosophic homeomorphisms.

The proof establishes that translations are:

• Bijective.
• Neutrosophically continuous.
• Open mappings.

For the left translation, the inverse is La−1. The membership functions of translated sets are shown to transform consistently:

TLa(U)(y)=TU(a−1y)

with equivalent relationships for indeterminacy and falsehood. Therefore, translations preserve the essential neutrosophic topological structure of the group.

Neutrosophic Quotient Spaces

The paper also investigates quotient topologies created using normal subgroups.

If H is a normal subgroup of a neutrosophic topological group G, the canonical projection

p:G→G/H,p(x)=xH

is shown to be a neutrosophic open mapping.

The quotient topology is defined so that a neutrosophic set in G/H is open when its inverse image under the projection is open in G. The proof examines the truth, indeterminacy, and falsehood membership functions of these projected sets.

Research Gap and Significance

The text identifies a gap in existing research: although fuzzy, intuitionistic fuzzy, and neutrosophic structures have been studied separately in various algebraic and topological contexts, there has been comparatively less systematic work specifically on Neutrosophic Topological Groups.

The research therefore aims to establish a stronger theoretical foundation by studying:

• Neutrosophic topological groups.
• Translation mappings.
• Neutrosophic homeomorphisms.
• Quotient topological spaces.
• Canonical projection mappings.
• Continuous neutrosophic homomorphisms.
• Extensions of classical algebraic results, including possible isomorphism theorems, to the neutrosophic setting.

Conclusion

In this study, we were able to generalize the classical and fundamental notions of topological groups to the very flexible neutrosophic environment. We proved that left, and right group translations are still homeomorphisms in case of Neutrosophic Topological Groups. We were able to define the so-called neutrosophic quotient topology and proved that the corresponding canonical projection is an open map. This allowed us to formulate and prove the Neutrosophic First Isomorphism Theorem. Our results create the fertile ground for future research as the researchers can follow our lead, and try to establish the basic neutrosophic separation axioms and the various forms of compactness, and investigate their behavior in different quotient spaces.

References

[1] F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability. American Research Press, 1999. [2] L. A. Zadeh, “Fuzzy Sets,” Information and Control, pp. 338–353, 1965. [3] C.-L. Chang, “Fuzzy Topological Spaces,” J. Math. Anal. Appl., pp. 182–190, 1968. [4] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 20, no. 1, pp. 87–96, 1986. [5] A. A. Salama and S. A. Alblowi, “Neutrosophic set and neutrosophic topological spaces,” IOSR Journal of Mathematics, vol. 3, no. 4, pp. 31–35, 2012. [6] T. Husain, Introduction to Topological Groups. Philadelphia, 1981. [7] L. S. Pontryagin, Topological Groups. New York, 1966. [8] M. Ali, F. Smarandache, M. Shabir, and L. Vladareanu, “Generalization of neutrosophic rings and neutrosophic fields,” Neutrosophic Sets and Systems, vol. 5, pp. 9–14, 2014. [9] A. A. A. Agboola, A. D. Akinola, and O. Y. Oyebola, “Neutrosophic rings I,” International Journal of Mathematical Combinatorics, vol. 4, pp. 1–14, 2015. [10] T. Bera and N. K. Mahapatra, “Introduction to neutrosophic soft topological space,” Opsearch, vol. 54, no. 4, pp. 841–867, 2017.

Copyright

Copyright © 2026 Shadab Ali Ahmad, Dr. Md. Mushtaque Khan. 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.