There is a very classical correspondence between projective planes and division algebras: given a plane, each choice of three distinct points (zero, one and infinity) on each line determines addition, subtraction, multiplication and division on points of this line with the infinity removed, and division algebras corresponding to all such choices are isomorphic. Furthermore associativity of multiplication corresponds to the Desargues axiom/theorem/property and commutativity to the Pappus axiom.
Is there a similar way to introduce any kind of algebraic structure on a line in the hyperbolic plane? It seems that accordingly rigged versions of most of the ordinary constructions are available in this case too (like, finding a line through a point parallel to two given lines). Besides, in the Beltrami-Klein model one has remnants of "ordinary" parallelism too.
Can one do something with that?