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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?

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I'm not certain that all of the conditions you mention in your "definition" are satisfied in this setting, but I think you should look at $\Gamma$-Ultrametric spaces, c.f. notes by Ackerman, where $\Gamma$ is a complete lattice with minimal element (denoted 0 for convenience) and the map $d_M: M \times M \rightarrow \Gamma$ is a $\Gamma$-ultrametric.

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Many examples can be obtained by the following scheme: Let $x$ assign to each pair $(p,q)$ in $S$ either the fixed element $z_X$ if $p=q$, or, if $p\neq q$, some entity that encodes the distance $d(p,q)$ along with some other information about $(p,q)$. Let $X$ be the set of all such encodings and $z_X$. Define $r_d$ to be the function which, given two encodings, decodes them, extracts the metric values (discarding the additional information), and produces the ratio of those distances. For example, if $S$ is the plane, $x(p,q)$ could be the vector $p-q$, where the additional information encoded is the direction.

Because of the great variety of options for the additional information (for various particular metric spaces $S$) as well as the great variety of options for the encoding scheme, I would doubt that this general concept has acquired a name. Note also, that the additional information could be chosen in perverse ways and still satisfy your definition. For example, in the case of the plane, I could take $x(p,q)$ to be the vector $p-q$ as above when this points into the upper half-plane but just $d(p,q)$ otherwise. So it seems to me that your definition is too broad to give a natural concept.

It also seems to me that such "encodings of distances plus other information" are the only examples satisfying your definition. Or do you have essentially different ones?

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Thanks, Jeremy and Andreas, for your comments. In the notes by Ackerman ( already the first paragraph has the clue:

[...] the distance function takes values in a fixed complete lattice (and not necessarily in the reals)
. What I had not imagined when asking my question was that people would speak of a "distance function" in such a case at all, and not at least for instance of a "generalized distance function"; considering that (which redirects to [[Metric space]]) and even specificly require that any distance function should take values in the reals.
(Or could you perhaps suggest another online reference for terminology than Wikipedia?)

So a generalization of the notion "distance function" (perhaps even the essential generalization) is apparently achieved by generalizing its codomain from the reals, $R$, to

[... the triple] $(\Gamma, \leq, 0)$ [which is] a complete lattice with minimal element 0
(cmp. Definition 1.1 of Ackerman's paper).

There's an additional point to my question, though, which I now can try to express more specificly -- perhaps:

What exactly is required to induce a partial order "$\leq$" in a given set $X$ or $\Gamma$?

I suppose: some particular relation (or collection of relations) to some other, already given partially ordered set(s); i.e. something that may well be called an "encoding" (although I don't know if that's a precise, technical term, of course ...)

p.s. There are only 600 characters available for a comment?!? And it can't even be formatted sensibly? ... Sorry -- then I rather reply by posting an "Answer" than trying to "comment".

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