Metric Spaces Over Partially Ordered Rings

Abstract

In this paper, we present a systematic study of a new class of generalized metric spaces, called POR-metric spaces, in which the distance function takes its values in a partially ordered ring satisfying specific algebraic compatibility conditions that ensure coherence between the algebraic structure and the order relation. This work falls within current directions in the generalization of metric spaces, which seek to move beyond the traditional numerical framework of distance by replacing real-valued distances with ordered or algebraic structures, thereby combining additional algebraic information into the topological structure.

The significance of this approach lies in enabling a deeper interaction between topological concepts and algebraic structures, as well as providing greater flexibility in notions of convergence, continuity, and topology generation. In this context, we establish the fundamental properties of POR-metric spaces, and we define and study POR-balls, open and closed sets, interior, exterior, and boundary points, as well as certain separation axioms, highlighting similarities and differences with the classical metric case. Overall, POR-spaces constitute a coherent extension of existing concepts and provide a flexible and extensible framework for further research in topology, fixed point theory, and abstract analysis.