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This paper establishes a new combinatorial framework for the study of coarsemedian spaces, bridging the worlds of asymptotic geometry, algebra andcombinatorics. We introduce a simple and entirely algebraic notion of coarsemedian algebra which simultaneously generalises the concepts of boundedgeometry coarse median spaces and classical discrete median algebras. We studythe coarse median universe from the perspective of intervals, with a particularfocus on cardinality as a proxy for distance. In particular we prove that themetric on a quasi-geodesic coarse median space of bounded geometry can beconstructed up to quasi-isometry using only the coarse median operator. Finallywe develop a concept of rank for coarse median algebras in terms of thegeometry of intervals and show that the notion of finite rank coarse medianalgebra provides a natural higher dimensional analogue of Gromov's concept of$\delta$-hyperbolicity.

Coarse median algebras: The intrinsic geometry of coarse median spaces and their intervals