By a Tarski space I understand a mathematical structure $(X,B,E)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and a 4-ary equidistance relation $E\subseteq X^2\times X^2$ that satisfy all Tarski's axioms except for the Axiom Schema of Continuity, Euclid parallel Axiom and Upper Dimension axiom.
By a deep result of Gupta, any segment in a Tarski space has a midpoint. This allows us to mimic the standard vector addition in Tarski spaces.
Namely, given a Tarski space $X$, fix any point $o\in X$ and call it the origin of the space. Given two points $x,y\in X$, find the midpoint $m$ of the segment $[x,y]$ and let $x\oplus y$ be the unique point of $X$ such that $m$ is the midpoint of the segment $[o,x\oplus y]$.
It is easy to see that $X$ endowed with the binary operation $\oplus$ is a commutative loop. This loop is a group if and only if the Tarski space satisfies the Euclid parallel axiom. The homogeneity of a Tarski space (via central symmetries) implies that the isomorphism type of the loop $(X,\oplus)$ does not depend on the choice of the origin. So, it is a kind of "canonical" algebraic structure associated with the Tarski space whose special case is the standard vector space associated with a Euclidean space.
I am interested in this associated loop structure for the hyperbolic plane (or more generally, hyperbolic space). Since the addition $\oplus$ is very natural, I suggest that someone has already studied this algebraic structure.
The closest references, which I have found are the gyrogroups of Ungar and geometric loops of Karzel, but those loops seem to be non-commutative.
So, the
Question. What is known about algebraic properties of the commutative loop structure of Tarski spaces, in particular, of hyperbolic planes and hyperbolic spaces of higher dimension?
