An Introduction to the Algebraic Structure of Type-1 Neutrosophic-Set
Theory
Adel Mohammed Al-Odhari1.∗
1Faculty of Education, Humanities and Applied Sciences ( khawlan) and Department of Foundations of
Sciences, Faculty of Engineering, Sana’a University. Box:13509, Sana’a, Yemen
Email:a.aleidhri@su.edu.ye
Abstract
This article presents a focused investigation of type-1 neutrosophic sets, derived from classical sets by introduc-
ing an indeterminacy component, I. type-1 neutrosophic sets generalize classical set theory by incorporating
four-valued logic, which was generated by Boolean logic in our work. This work will appear in the future.
As we know, a neutrosophic set is based on a many-valued logic defined by three independent membership
functions: truth, indeterminacy, and falsehood. This work systematically re-examines and consolidates foun-
dational research conducted between 2024 and 2025, isolating type-1 structures from the broader frameworks
of type-2 and type-3 neutrosophic sets for clearer axiomatic and theoretical development. We establish core
concepts, terminology, operations, and properties specific to type-1 neutrosophic sets, constructing and ana-
lyzing the type-1 neutrosophic Cartesian product. In addition, we introduce and investigate the properties of
type-1 neutrosophic ordered pairs and their corresponding products. This foundation formally defines type-1
neutrosophic relations and neutrosophic partially ordered relations, establishing their core properties. Fur-
thermore, the article explores type-1 neutrosophic functions, detailing their various types, including injective,
surjective, and bijective functions and their respective properties. A significant focus is placed on invertible
neutrosophic functions, where we examine the conditions for invertibility and prove key related theorems.By
focusing exclusively on type-1, we aim to create a more dynamic and effective foundation for application
across diverse neutrosophic fields, including neutrosophic algebra, number theory, and logic. This focused
approach is intended to open new research pathways within the neutrosophic sciences.
Keywords: Type-1 Neutrosophic Set and their properties; Operations on Type-1 Neutrosophic Set and Their
Properties; Cartesian Product of Type-1; Neutrosophic Relations of Type-1; Neutrosophic Functions of Type-
1; Invertible Netrosophic of Type-1