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?



Looking for a half-remembered reference on 'magnitude algebras'
@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'.


