A name for a mathematical structure of geometric type I am looking for (maybe existing) name for a mathematical structure $(X,\leqslant)$ consisting of a set $X$ and a transitive relation ${\leqslant}\subseteq X^2\times X^2$ such that $xx\leqslant yz\leqslant zy$ and $(xy\leqslant zz\;\Rightarrow\; x=y)$ for every $x,y,z\in X$. Here for elements $x,y\in X$ I denote by $xy$ the ordered pair $(x,y)$.
There are at least three important examples of such mathematical structure:

*

*a metric space $(X,d)$ in which $xy\leqslant uv$ is defined as $d(x,y)\le d(u,v)$;

*an ordered group $(X,+,\le)$ in which $xy\leqslant uv$ is defined as $|x-y|\le |u-v|$;

*the Euclidean or hyperbolic plane in which $xy\leqslant ab$ means that $xy$ is congruent to $ac$ for some $c\in[a,b]$.


Question. What would be a good name for such a structure $(X,\leqslant)$?

I thought about protometric spaces but this name is already occupied for something different.
Maybe to call it a compass space? Because the standard compass can be used for comparing distances (this is exactly what this mathematical structure describes). What do you think? Google shows nothing mathematical for the search ``compass space''.
 A: After a long search, I have finally found an existing geometric tool that does exactly what is required: it compares distances without expressing them in real numbers. This measuring instrument is called a divider. So, the corresponding mathematical structure $(X,\leqslant)$ should be called a divider space.
Dividers and compasses are similar geometric tools (by their design) but serve to different purposes. A compass is used for drawing circles with a given center and a given point on the circle. In ruler-and-compass constructions, a compass cannot be used for transferring distances. The latter function is reserved for dividers, which by their construction cannot draw circles (because dividers have two sharp needles at the ends of both legs).
Nothing else but a divider is used in the famous masonic symbol:

I have got interested in dividers studying the Tarski axioms of Euclidean geometry. In contrast to the standard Euclid axioms that formalize the straightedge-and-compass geometric constructions, the Tarski axioms (without the Continuity Axiom) formalize straightedge-and-divider geometric constructions, which are more restrictive compared to the standard straightedge-and-compass constructions. For example, using only a straightedge and a divider it is possible to construct a right triangle with two given catheta but it is impossible to construct a right triangle by a cathetus and hypotenuse.
The Hilbert axioms system also differs from the standard Euclid axiom system because it actually formalizes the use of a straightedge, a divider, and a protractor (but not a compass). But a protractor adds nothing new comparing to a straightedge and a divider, which can copy angles. So, constructible distances in Tarski's and Hilbert's Geometry are the same: they form a Pithagorean field. In contrast, the straightedge-and-compass geometric constructions produce a Euclidean field (which is closed under taking square roots). To produce such a field one should add to Tarski (or Hilbert) axioms the Segment-Circle Axiom postulating the existence of a point of the intersection of a circle and a segment with endpoints at different sides of the circle.
