This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of \(\delta \)-hyperbolicity.

本文建立了一个新的组合框架，用于研究粗略的中位数空间，将渐近几何，代数和组合数学的世界联系起来。我们引入了一个简单且完全为代数的粗中位数代数概念，该概念同时概括了有界几何粗中位数空间和经典离散中位数代数的概念。我们从时间间隔的角度研究粗略的中值宇宙，特别关注基数作为距离的代理。特别地，我们证明了仅使用粗糙中值算子就可以构造有界几何体的准大地测量粗糙中值空间上的度量，直到准等距。\（\ delta \）-双曲性。