Let the pair $( S, d \, )$ be a metric space, i.e.

$d\!: S^2 \rightarrow R$, where for any three distinct elements $k$, $p$, $q$ $\in S$:

$d[ \, p, q \, ] = d[ \, q, p \, ] > 0$,

$d[ \, p, q \, ] + d[ \, q, k \, ] \geq d[ \, p, k \, ]$, and

$d[ \, p, p \, ] = 0$.

My question concerns a (any) set $X$, and

a (any) map $x\!: S^2 \rightarrow X$

which satisfies that there exists an element $z_{X}$ $\in X$ such that

for each element $p$ $\in S$: $ x[ \, p, p \, ] = z_{X} $ and

for any two distinct elements $p$, $q$ $\in S$: $ x[ \, p, q \, ] \ne z_{X} $, and

a map $r_d\!: (X \verb|\| z_{X})^2 \rightarrow R$ which is defined such that

for any two not necessarily distinct elements $a$, $b$ $\in (X \verb|\| z_{X})$

and for any four not necessarily all distinct elements $j$, $k$, $p$, $q$ $\in S$ such that

$ x[ \, j, k \, ] = a $ and $ x[ \, p, q \, ] = b $

the values of map $r_d$ are defined as
$ r_d[ \, a, b \, ] := d[ \, j, k \, ] / d[ \, p, q \, ] $.

Note that the pair $( S, x \, )$ is not necessarily a metric space since set $X$ is not necessarily the set of real numbers, $R$; nor necessarily some subset of $R$.

I'd like to know:

How do you call such a set $X$, or such maps $x$ or $r_d$, in general, please?