The algebraic structure of a line in a (Tarski) plane By a Tarski plane (resp. plane) I understand a mathematical structure $(X,B,\equiv)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and the 4-ary congruence relation ${\equiv}\subseteq X^2\times X^2$ satisfying the Tarski's axioms minus the Euclides parallel postulate (in which the Axiom schema of continuity is replaced by two axioms of segment-circle continuity and circle-circle continuity).
Since axioms of a Tarski plane include the axioms of a plane, each Tarski plane is a plane. In a plane one can produce standard geometric constructions with a compass and ruler.
If a plane satisfies the Euclid parallel postulate,
then it is called a Euclidean plane.
For distinct points $o,e\in X$ of a plane $(X,B,\equiv)$, the set $$L(o,e)=\{ x\in X:Bxoe\vee Boxe\vee Boex \}$$is called the line containing the points $o,e$.
The line $L(o,e)$ carries a unique structure of a linearly ordered commutative group $(L(o,e),+,\le)$ such that
$\bullet$ $o$ is zero of $L(o,e)$ and $o<e$,
$\bullet$ for any $x,y\in L(o,e)$ we have $x\le y$ iff $Bxyo\vee 
 Bxoy\vee Boxy$,
$\bullet$ for any $x,y,z\in L(o,e)$ with $o\le x$ and $o\le y$ we have $z=x+y$ if and only if $Boxz$ and $xz\equiv oy$.
If the plane is Euclidean (and Tarski), then the line $L(o,e)$ carries a structure of a (real closed) ordered field such that $e$ is the unit of $L(o,e)$ and for any element $y\in L(o,e)$ with $o\le y$ there exists $x\in L(o,e)$ such that $x^2=y$.

Question. What is an algeraic structure of a line $L(o,e)$ in a (Tarski) plane? It should be something more general than the structure of a (real-closed) ordered field, closed under taking square roots.

Or more precisely:

Problem. Characterize ordered groups, which are isomorphic to the lines in (Tarski) planes.

 A: This question is addressed by W. Schwabhäuser on p. 156 of his paper Metamathematical methods in foundations of geometry. Logic, Methodology and Philosophy of Science (Proc. 1964 Internat. Congr.) North-Holland, Amsterdam, 1965, pp. 152–165. If the Tarski plane (as you have defined it) is hyperbolic (i.e. not Euclidean), then by a result of W. Szmielew (A new analytic approach to hyperbolic geometry. Fund. Math. 50 (1961/62), pp. 129–158) the plane is isomorphic to a Klein space over a Euclidean ordered field if and only if the plane satisfies Hilbert's hyperbolic axiom of parallels. If it does not satisfy Hilbert's axiom then the algebraic characterization is substantially more complicated and can be obtained from Pejas's classification of Hilbert planes, as discussed in W. Pejas, Die Modelle des Hilbertschen Axiomensystems der absoluten Geometrie, Math. Ann. 143 (1961), 212–235 and in F. Bachmann Zur Parallelenfrage Abh. Math. Sem. Univ. Hamburg 27 (1964), 173–192. Unfortunately, I no longer have these papers and it's been so long since I've studied the matter, I can't be of help with the details.
