Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S\_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. Introduce the transition maps $\{\phi\_n:S\_{n+1}\to S\_n\}\_{n\in\mathbb N}$ defined by


$$\phi\_{n-1}(n\_1,n\_2,n\_3,n\_4,\ldots,n\_{2^n-1},n\_{2^n}) := (n\_1+n\_2,n\_3+n\_4,\ldots,n\_{2^n-1}+n\_{2^n}),$$


i.e. summing all consecutive pairs of integers.


Now consider the sequence $(S\_n,\phi\_n)\_{n\in\mathbb{N}}$. Does the projective limit


$$S=\lim\_{\longleftarrow}S\_n\subset\prod\_{k\in\mathbb N}S\_n$$


have a name as an Abelian semigroup, namely is such $S$ an object in a well-studied family of semigroups so that it can be actually identified among them?



Projective limit construction of a semigroup
Let ACom be the class of all finite aperiodic commutative monoids (aperiodic means each subgroup is trivial, i.e. the monoid satisfies some identity of the form x^n=x^{n+1}); let pro-ACom be the class of all projective limits of monoids in ACom. The class pro-ACom has free objects on each compact totally disconnected space, in particular on each finite set (considered as a discrete space). Now \tilde\mathbb{N}=S\_0 is the free pro-ACom object on a singleton (namely {1}) --- in this context S\_0 has to be considered as one-point compactification of the discrete \mathbb{N}\_0=S\_0\setminus{\infty}. The monoid S in question is the free pro-ACom object on the Cantor set. Inside S the Cantor set of generators can be identified as those elements whose projection to S\_0 is 1.


