# metric spaces as algebraic systems

Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, for all positive rational $\alpha$: $(x,y)\in R_\alpha$ iff ${\mathrm{dist}}(x,y)<\alpha$. Is this the first paper where this algebraic system was defined?

Update 1: I need it because in my paper, I want to call these algebraic systems KSTT-systems. They satisfy axioms 0) $(x,x)\in R_\alpha$ for every $\alpha$, 1) $(x,y)\in R_\alpha$ iff $(y,x)\in R_\alpha$, 2) $(x,y)\in R_\alpha, (y,z)\in R_\beta\to (x,z)\in R_{\alpha+\beta}$, 3) $(x,y)\in R_\alpha\to (x,y)\in R_\beta$ for every $\beta\ge \alpha$. The original metric space $X$ can be elementary defined inside $A(X)$, and for every KSTT-system $A$, and an element $o\in A$, one can canonically (elementary) define a pointed metric space. This can be used to show that (modulo Continuum Hypothesis), for every metric space $X$ and every asymptotic cone $C$ of $X$, $C$ is isometric to any ultralimit of $C$. Thus, the only question remains: whether it is appropriate to call these KSTT-systems or somebody has introduced them earlier.

Update 2: I guess I was not clear enough. I need to know who was the first to consider metric spaces as algebraic systems with countable set of relations (as above). I am not interested in equivalent categories.

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Note that the fourth author is Simon Thomas, who is here at MathOverflow. –  Joel David Hamkins Oct 31 '10 at 12:38
Joel: I did ask Simon Thomas, of course. –  Mark Sapir Oct 31 '10 at 12:41
I think what you have described is either a nearness relation or some encoding of a metrizable uniformity. –  Michael Blackmon Feb 5 '11 at 20:03
@Michael: I do not know what "metrizable uniformity" is. Is it an algebraic system? The whole and only point of the construction is to view metric spaces as algebraic systems with countable signature (so that one can use model theory to treat metric spaces). –  Mark Sapir Feb 6 '11 at 2:00

Hi, I just want to add that your definition is very similar to Bourbaki's "uniformity" and "entourages."

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There is some similarity. But the main difference (besides the fact that the axioms are different) is that Bourbaki do not consider a uniformity as an algebraic system. And the set of entourages is allowed to be arbitrary large, i.e., say, uncountable. –  Mark Sapir Oct 17 '10 at 6:34

you KSTT-System is like a axioms for a monoidal symmetric category (order). Do a look in : http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html

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I cannot find it there. Are these categories algebraic structures? –  Mark Sapir Oct 31 '10 at 12:40