Neutrosophic primal structure with closure and proximity operators

Mesfer H. Alqahtani; Mesfer H. Alqahtani

doi:10.3934/math.2026028

AIMS Mathematics

2026, Volume 11, Issue 1: 644-660. doi: 10.3934/math.2026028

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Neutrosophic primal structure with closure and proximity operators

Mesfer H. Alqahtani ^,

Department of Mathematics, University College of Umluj, University of Tabuk, Tabuk 48322, Saudi Arabia

Received: 05 August 2025 Revised: 23 October 2025 Accepted: 10 November 2025 Published: 08 January 2026
MSC : 03E72, 54A40

This paper aimed to introduce novel neutrosophic primal and neutrosophic proximity operators, derived from a new abstract framework called "neutrosophic primal topology". We began by examining the core properties of neutrosophic primal operators. Next, we defined a neutrosophic primal closure operator derived from the neutrosophic primal operator and explored the relationships between them. Based on this neutrosophic primal closure operator, we constructed a neutrosophic topology and identified the conditions under which the image of a neutrosophic primal remained a neutrosophic primal. In the next stage, we defined the neutrosophic point-primal proximity operator and explored a range of fundamental properties characterizing neutrosophic primal proximity topological spaces derived from this operator. We also introduced the concept of neutrosophic proximal closed sets and demonstrated that the collection of complements of neutrosophic primal closed sets constituted a neutrosophic topology. Finally, we defined a neutrosophic operator on a neutrosophic primal proximity topological space that satisfies the neutrosophic Kuratowski closure axioms and used it to construct a neutrosophic topology. All results established in this study were thoroughly supported and clarified through illustrative examples.
- neutrosophic primal space,
- neutrosophic primal proximity space,
- neutrosophic primal operator,
- neutrosophic primal proximity topology
Citation: Mesfer H. Alqahtani. Neutrosophic primal structure with closure and proximity operators[J]. AIMS Mathematics, 2026, 11(1): 644-660. doi: 10.3934/math.2026028

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Abstract

This paper aimed to introduce novel neutrosophic primal and neutrosophic proximity operators, derived from a new abstract framework called "neutrosophic primal topology". We began by examining the core properties of neutrosophic primal operators. Next, we defined a neutrosophic primal closure operator derived from the neutrosophic primal operator and explored the relationships between them. Based on this neutrosophic primal closure operator, we constructed a neutrosophic topology and identified the conditions under which the image of a neutrosophic primal remained a neutrosophic primal. In the next stage, we defined the neutrosophic point-primal proximity operator and explored a range of fundamental properties characterizing neutrosophic primal proximity topological spaces derived from this operator. We also introduced the concept of neutrosophic proximal closed sets and demonstrated that the collection of complements of neutrosophic primal closed sets constituted a neutrosophic topology. Finally, we defined a neutrosophic operator on a neutrosophic primal proximity topological space that satisfies the neutrosophic Kuratowski closure axioms and used it to construct a neutrosophic topology. All results established in this study were thoroughly supported and clarified through illustrative examples.

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