2
$\begingroup$

I'm investigating the origin of the following notion:

Let $S=(S, +, <, 0)$ be a partially ordered semigroup with minimum $0$ (such that $<$ is invariant by the action of $+$ on both sides). A $S$-metric on a space $X$ is a function $d: X\times X\to S$ that respect the usual laws of metrics, replacing $\mathbb{R}^+$ with $S$.

Question 1: Do you know who first introduced the notion of $S$-metric space with values in a partial order $S$?

Question 2: Do you know any results (possibly of topological nature) on the subject, or any survey that treats the argument?

To give a bit of explanation:

I've found that there is a lot of confusion in literature about who first introduced the notion and what are the known result about it. It seems that similar notions have been rediscovered several times in several different field, and it seems to me that most of the times people working in one field did not cite any paper of people who worked/introduced the concept in another field.

In this question there is a similar discussion about the case of totally ordered semigroups. For this case the situation seems a bit more clear to me: much has been said on this class, and there are several results on the topological nature of spaces of this class.

Here instead I wanted to ask about $S$-metric spaces when $S$ is only a partial order (not necessarily total).

I've read that is notion has been used before, for example by S. Braunfeld in his paper "The lattice of definable equivalence relations in homogeneous n-dimensional permutation structure". In another source (MR3402850 A.-P. Grecianu, A. V. Kvaschuk, A. G. Myasnikov, D. E. Serbin, Groups acting on hyperbolic Λ-metric spaces) I've read that the notion has been attributed to a paper by Morgan and Shalen of 1984. However, I'm convinced that the notion is older. I recall reading a paper about this subject that I think was dated in the 60s (and it was in french, if I recall correctly). Unfortunately, I can not find the reference anymore, so I might just be wrong.

Anyway, I would expect that someone had studied the possible analogies/differences between this metrics and metrics over totally ordered semigroups. In particular, I would expect to find in literature some metrization theorems for $S$-metrizable spaces of the same fashion of the one proved for the case of total orders. However, I could find really little about them.

Improve this question
$\endgroup$
2
  • $\begingroup$ Lawvere pointed out long ago that metric spaces amount to categories enriched over a monoidal category of real numbers. From that observation, free association leads me to think you might want your posets to be quantales. A small point in favor of this notion is that "quantale valued metric spaces" produces some hits on Google. $\endgroup$
    – Andreas Blass
    Commented Nov 22, 2021 at 1:47
  • $\begingroup$ Thank you, that seems a good direction to look at. But I would be surprised if using quantales instead of other structures makes the difference. In the case of totally ordered semigroups, most of the structures one consider are not quantales (the order is not complete), and results obtained are usually independent from the choice of $S$, as long as it satisfies certain basic conditions (e.g. being a continuous semigroup, as defined in the other MO answer I linked). $\endgroup$
    – Cla
    Commented Nov 22, 2021 at 10:18

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .