A name for a mathematical structure of geometric type

Question

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:

1. a metric space $(X,d)$ in which $xy\leqslant uv$ is defined as $d(x,y)\le d(u,v)$;
2. an ordered group $(X,+,\le)$ in which $xy\leqslant uv$ is defined as $|x-y|\le |u-v|$;
3. 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''.

Answer 1

After a long search, I have finally found an existing well-known geometric tool that does exactly what is required: it compares distances without expressing them in real numbers. This measuring instrument is called a rope and has been used in the human civilization since the ancient times.

For example, at this picture taken from the tomb of Menna ($\approx$ 1350 BC) we can see so called harpedonaptai, who professionally used such a rope:

So, I suggest the following terminology:

Definition 1. A rope on a set $X$ is a relation ${\leqslant}\subseteq X^2\times X^2$ satisfying three axioms:

(TR) $\forall a,b,u,v,x,y\in X\;(ab\leqslant uv\leqslant xy\to ab\leqslant xy)$;

(ES) $\forall x,y\in X\; (xy\leqslant yx)$;

(ZD) $\forall x,y\in X\;(xx\leqslant yy)$;

whose notations are abbreviations of Transitivity, End Symmetry, and Zero Distance.

In this definition for two elements $x,y\in X$ the ordered pair $(x,y)$ is denoted by $xy$.

Definition 2. A rope $\leqslant$ on a set $X$ is called

$\bullet$ linear if $\forall a,b,x,y\in X\; (ab\leqslant xy\;\vee\; xy\leqslant ab)$;

$\bullet$ alternating if $\forall x,y,z\in X\;(xy\leqslant zz\;\vee\; zz\leqslant xy)$;

$\bullet$ non-negative if $\forall x,y,z\in X\;(xx\leqslant yz)$;

$\bullet$ positive if $\forall x,y\in X\; (x=y\leftrightarrow \forall a,b\in X\;xy\leqslant ab)$.

Example 1. For every metric space $(X,d)$ the relation $${\leqslant}:=\{(xy,ab)\in X^2\times X^2:d(x,y)\le d(a,b)\}$$ is a linear positive rope on $X$.

Example 2. For every linear space $X$ over an ordered field $F$ with linear order $\le_F$ and every quadratic form $q:X\to F$, the relation $${\leqslant}:=\{(ab,xy)\in X^2\times X^2:q(a-b)\le_F q(x-y)\}$$ is a linear alternating rope on $X$. This rope is positive if and only if the quadratic form $q$ is positive definite.

Besides the linear alternating rope, the quadratic form $q$ also induces a non-negative rope $${\leqslant}:=\{(ab,xy)\in X^2\times X^2:|q(a-b)|\le_F|q(x-y)|\;\wedge\;0\le_F q(a-b)\cdot q(x-y)\},$$ which is not linear, in general.

Therefore, the $n$-Euclidean space $\mathbb R^n$ carries a canonical linear positive rope, and the Minkowski space-time $\mathbb R^{1,3}$ carries a linear alternating rope and also a non-linear non-negative rope, but both these ropes on $\mathbb R^{1,3}$ are not positive.

Definition 3. A rope structure is a pair $(X,\leqslant)$ consisting of a set $X$ and a rope $\leqslant$ on $X$.

Definition 4. Two rope structures $(X,\leqslant_X)$ and $(Y,\leqslant_Y)$ are isomorphic if there exists a bijective function $F:X\to Y$ such that $\forall x,y,u,v\in X \big(xy\leqslant_X uv\leftrightarrow F(x)F(y)\leqslant_Y F(u)F(v)\big)$.

Classical Euclidean Geometry actually studies the rope structure of the Euclidean spaces. The rope structure of the Euclidean space was characterized (up to isomorphism) by Tarski and his students (Szmielew and Gupta). To formulate this characterization we should introduce the congruence and betweenness relations for rope structures.

Let $(X,\leqslant)$ be a rope structure. Given points $a,b,x,y\in X$ we write $ab\equiv xy$ if $ab\leqslant xy$ and $xy\leqslant ab$.

For points $a,b,c\in X$ the sets $$\overline{c{\equiv}ab}:=\{x\in X:cx\equiv ab\}\quad\mbox{and}$$and $$\overline{c{\leqslant}ab}:=\{x\in X:(cc\leqslant cx\leqslant ab)\vee(ab\leqslant cx\leqslant cc)\}$$are called the sphere and the ball of radius $ab$ centered at $c$.

For a subset $A\subseteq X$, the sets $$[A]_\equiv:=\{x\in X:\{x\}=\bigcap_{a\in A}\overline{a{\equiv}ax}\}\quad\mbox{and}\quad [A]_{\leqslant}:=\{x\in X:\{x\}=\bigcap_{a\in A}\overline{a{\leqslant}ax}\}$$ are called the affine and convex hulls of $A$, respectively.

Definition 5. The cardinal $$GPS(X,\leqslant):=\min\{|A|:A\subseteq X=[A]_\equiv\}$$is called the GPS-complexity of the rope structure $(X,\leqslant)$.

Example. The GPS-complexity of the rope structure of the $n$-dimensional Euclidean space equals $n+1$.

For two points $a,b\in X$ the convex hull $[\{a,b\}]_{\leqslant}$ of the doubleton $\{a,b\}$ is denoted by $[ab]_{\leqslant}$ and is called the segment with end-points $a,b$.

Given points $a,x,b\in X$, we write $\mathbf Baxb$ if $x\in [ab]_{\leqslant}$ and say that $x$ is betweeen $a$ and $b$.

Definition 5. A Tarski rope space is a rope structure $(X,\leqslant)$ whose rope is positive and satisfies three axioms:

(SC) $\forall a,b,x,y\in X\;\exists z\;(\mathbf Bxyz\wedge yz\equiv ab)$;

(PA) $\forall a,p,b,q,c\;(\mathbf Bapb\wedge\mathbf Bbqc\to \exists x\in X\;(\mathbf Baxq\wedge\mathbf Bqxc))$;

(FS) $\forall \hat x,\hat y,\hat z,\hat v,\check x,\check y,\check z,\check v\;(\hat x\ne\hat y\wedge\mathbf B\hat x\hat y\hat z\wedge\mathbf B\check x\check y\check z\wedge \hat x\hat v\equiv\check x\check v\wedge \hat y\hat v\equiv \check y\check v)\to \hat z\hat v\equiv\check z\check v)$, called the axioms of Segment Construction, the Pasch Axiom, and Five-Segments Axiom.

The following characterization theorem should be attributed to Tarski. It can be proved by the methods developed in the known book of Schwabhäuser, Szmielew, and Tarski:

Theorem. A rope structure $(X,\leqslant)$ is isomorphic to the rope structure of the Euclidean space $\mathbb R^n$ of dimension $n\ge 2$ if and only if $(X,\leqslant)$ is a Tarski rope space satisfying the axioms:

(BA) $\forall x,y,z\in X\;(\mathbf Bxyz\vee\mathbf Bxzy\vee\mathbf Byxz\vee\exists c (cx\equiv cy\equiv cz))$;

(CA) $\forall A,B\subseteq X\;\forall o\;\big((\forall a\in A\;\forall b\in B\;\;\mathbf B oab)\to (\exists x\;\forall a\in A\;\forall b\in B\; \mathbf Baxb)\big)$;

(Dn) $GPS(X,\leqslant)=n+1$.

The axioms (BA),(CA), and (Dn) are called the Bolyai Axiom, the Continuity Axiom, and the Dimension Axiom.

Therefore, the geometry of the $n$-dimensional Euclidean space can be uniquely described as the geometry whose unique undefined notions are "point" and "rope", which is a binary relation $\leqslant$ between pairs of points. The rope $\leqslant$ determines the congruence and betweenness relations $\equiv$ and $\mathbf B$, which should satisfy 9 axioms:

(TR) $\forall a,b,u,v,x,y\; (ab\leqslant uv\leqslant xy\to ab\leqslant xy)$

(ES) $\forall x,y\;(xy\leqslant yx)$

(PD) $\forall x,y (x=y\leftrightarrow \forall a,b (xy\leqslant ab))$;

(SC) $\forall a,b,x,y\exists z\;(\mathbf Bxyz\wedge yz\equiv ab)$;

(PA) $\forall a,p,b,q,c\;(\mathbf Bapb\wedge\mathbf Bbqc\to \exists x\;(\mathbf Baxq\wedge\mathbf Bqxc))$;

(FS) $\forall \hat x,\hat y,\hat z,\hat v,\check x,\check y,\check z,\check v\;(\hat x\ne\hat y\wedge\mathbf B\hat x\hat y\hat z\wedge\mathbf B\check x\check y\check z\wedge \hat x\hat v\equiv\check x\check v\wedge \hat y\hat v\equiv \check y\check v)\to \hat z\hat v\equiv\check z\check v)$,

(BA) $\forall x,y,z\in X\;(\mathbf Bxyz\vee\mathbf Bxzy\vee\mathbf Byxz\vee\exists c (cx\equiv cy\equiv cz))$;

(CA) $\forall A,B\subseteq X\;\forall o\;\big((\forall a\in A\;\forall b\in B\;\;\mathbf B oab)\to (\exists x\;\forall a\in A\;\forall b\in B\; \mathbf Baxb)\big)$;

(Dn) $\min\{|A|:A\subseteq X=[A]_\equiv\}=n+1$.