# Looking for a half-remembered reference on 'magnitude algebras'

I've been trying and failing to find a paper/article/blog post (I think it was a paper) on a particular algebraic structure. The paper describes a structure consisting of something like a constant $$0$$, an associative binary operation $$+$$ for which $$0$$ is the unit, and an $$\omega$$-ary operation $$\Sigma$$ that agrees with repeated addition if cofinitely many of the arguments are $$0$$. There were some other requirements I believe, because this is not enough for the following result. One of the points the paper made was that the set $$[0,\infty]$$ (the 'set of magnitudes') with its usual addition was the free object on one point for this algebraic structure, so this gave an elementary construction of the real numbers. This is also why the structure was called something like a 'magnitude algebra'.

I feel like this must be enough information to dig up the paper, but I have had no luck after a lot of trying, so I can only conclude I'm misremembering something. Does anyone know the text I'm looking for?

• Also, please let me know if this would be more at home on the Math SE! I figured that since I'm looking for a bit of research for my own research it qualifies as 'research level', but I may have misjudged. Jun 30 at 20:00
• How about this: cs.bham.ac.uk/~mhe/papers/lics2001-revised.pdf ("A universal characterisation of the closed Euclidean interval", by Escardo and Simpson). It's certainly in the same spirit, but the technical details are a bit different from what you said. Jun 30 at 20:28

@Neil Strickland gave me all I needed in his comment (thank you!). Escardó and Simpson's paper is not what I was looking for, but they do cite it: A Universal Characterization of $$[0,\infty]$$, by Denis Higgs.

For those curious but not curious enough to find a text of the paper: a magnitude module is a variety of algebras with a constant $$0$$, a unary operation $$h$$, and an $$\omega$$-ary operation $$\Sigma$$ satisfying the identities

• $$\Sigma(\Sigma(x_{00},x_{01},\dots),\Sigma(x_{10},x_{11},\dots),\dots) = \Sigma(\Sigma(x_{00},x_{10},\dots),\Sigma(x_{01},x_{11},\dots),\dots)$$.
• $$\Sigma(0,\dots,0,x,0,\dots) = x$$.
• $$h (\Sigma(x_0,x_1,\dots)) = \Sigma(h(x_0),h(x_1),\dots)$$.
• $$\Sigma(h(x),h^2(x),h^3(x),\dots) = x$$.

Because $$[0,\infty]$$ is the free magnitude module on one point, every magnitude module $$M$$ admits a scalar multiplication-like action $$[0,\infty] \times M \to M$$, hence the name 'magnitude module'.

• You can accept your own answer if you're happy with it. Jun 30 at 23:11
• @Sam Hopkins I need to wait 2 days before I'm able to :) Jul 1 at 7:47
• I'm exactly the person you describe: 'curious but not curious enough to find a text of the paper'. Hence my question here: what does this $h$ correspond to in the real world? 'Half' as in 'multiply by 1/2'? Jul 1 at 9:55
• @Vincent Exactly, yes. If we define $x + y := \Sigma(x,y,0,\dots)$ then we can prove from the above axioms that $h(x + x) = x$. Jul 1 at 13:57
• Super, thank you! Jul 1 at 14:02