On Positive Cone and Partial Order in a Generalized Algebraic System

One of the extensions of a nearring and a gamma ring is the concept of a gamma nearring, which allows for a more general multiplication operation. In this paper, we aim to establish the concept of a partial order in a Γ-nearring, thereby extending the notion of partial order observed in a nearring. We introduce several key concepts such as partial order, positive cone, convex ideal, and others, within the context of a Γ-nearring. Additionally, we provide proofs for various classical results pertaining to these notions. Moreover, we investigate different types of prime ideals within a lattice-ordered Γ-nearring and examine their properties. By exploring the characteristics and behavior of these prime ideals, we enhance our understanding of lattice-ordered Γ-nearrings and their structural properties.