By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ satisfying all Tarski's axioms except for the Euclid Axiom, the Axiom of Continuity, and the Dimension Axioms. In other words, Tarski spaces should satisfy seven Tarski's axioms: (1) Reflexivity of Congruence, (2) Identity of Congruence, (3) Transitivity of Congruence, (4) Identity of Betweenness, (5) Five Segments, (6) Segment Construction, (7) Inner Pasch Axiom.
A Tarski space is called a Tarski plane if it has affine dimension 2.
For a Tarski space $(X,\mathsf B,\equiv)$, the quotient set $X^2_{\equiv}$ of $X^2$ by the equivalence relation $\equiv$ is called the distance monoid of $X$. It carries a natural commutative associative operation of addition of distances: $a+b=c$ iff there exist points $x,y,z\in X$ such that $\mathsf Bxyz$ and $xy\in a$, $yz\in b$ and $xz\in c$.
As shown in the book of Schwabhäuser, Szmielew, and Tarski, a substantial portion of familiar (absolute) geometry holds in Tarski spaces of dimension $\ge 2$. In particular, one can construct right triangles with given sides.
This allows us to define a hypotenuse operation $\triangledown$ on $X^2_\equiv$ letting $a\triangledown b=c$ iff there exists a right triangle with sides $a,b$ and hypotenuse $c$.
In the Euclidean geometry the hypotenuse operation is defined by the Pithagoras Theorem: $a\triangledown b=\sqrt{a^2+b^2}$, which implies that this operation is associative: $$(a\triangledown b)\triangledown c=\sqrt{a^2+b^2+c^2}=a\triangledown (b\triangledown c)$$
In the hyperbolic geometry, $a\triangledown b=\cosh^{-1}(\cosh(a)\cosh(b))$ and hence $$(a\triangledown b)\triangledown c=\cosh^{-1}(\cosh(a)\cosh(b)\cosh(c))=a\triangledown (b\triangledown c),$$ so again the hypotenuse operation is associative.
For every Tarski space $X$ of dimension $\ge 2$ the hypotenuse operation on $X^2_\equiv$ is commutative and has the zero distance $0=\{xx:x\in X\}$ as the neutral element. Moreover, if the Tarski space $X$ has dimension $\ge 3$, then the hypotenuse operation on $X^2_\equiv$ is associative (by a simple stereometry argument). So, $X^2_\equiv$ has at least two natural monoid operations with the same zero.
Problem. Let $X$ be a Tarski plane. Is the hypotenuse operation on $X^2_\equiv$ always associative?
Remark. The answer to this problem is affirmative if the Tarski plane satisfies the Axiom of Continuity because in this case the Tarski plane is isomorphic either to the Euclidean plane or to the hyperbolic plane, where, as we know, the hypotenuse operations are associative. Also the answer to the Problem is affirmative for Tarski planes that embed into 3-dimensional Tarski spaces, which makes the Problem relevant to this MO-question.
