Let $X$ be a set with an order on it (say a partial order or total order). Suppose also that $X$ is a metric space with metric $d$. Then does $x \leq y \leq z$ imply $d(x,z) = d(x,y) + d(y,z)$?
If this is not necessarily true, what other assumptions might make it true? Does it depend on whether we assume it is a partial order versus assuming it is a total order? Or perhaps some other kind of order?
My motivation for asking this question comes from thinking of, say, the real numbers (with the usual ordering and metric), where it is true. In general it seems that whenever we have something that resembles a “line”, then this should be true.
You have to assume some connection between the order and the metric, otherwise the ordering of the elements does not imply anything for the metric. Consider for example the real numbers with the usual order on it and the discrete metric, i.e. $d(x,y)=1$ if $x\neq y$ and $d(x,y)=0$ if $x=y$ then clearly for any $x<y<z$ we have $1=d(x,z)<d(x,y)+d(y,z)=1+1=2$. Although i do not know what exactly would imply such a behaviour as described by you.