Simple refinements, derived cones, and total extensions in partially ordered integral domains

Maximality in partially ordered rings was traditionally considered in the context of Fields. The extension of such relations to integral domains then raises the question of their maximality via constructive methods, particularly their cone refinements and localizations. In this note, we describe the structure of partial orders on integral domains under simple refinements, derived cones, and field-of-fractions mappings. Using binary linear combinations, Zorn’s lemma, and cross-multiplication, we analyze the interaction of a partial order with its refinements. We characterize simple cone refinements, determining sufficient conditions for their maximality. For an integral domain, we prove that the derived set D(P) belongs to the positive cone and that any maximal refinement contains it as a subset and is total. We establish that any partial order on an integral domain has a unique extension to its field of fractions, which gives a bijective correspondence between their total orders. Our findings generalize analogous results for ordered fields, providing new insight into maximality and localization in ordered domains.