Projective limit construction of a semigroup 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?
 A: 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.
A: Dear Phoenix87, I don't know where to write the answer to your latest question, so I put it here, though I'm afraid this will be interpreted as another answer to the original question. Anyway, two books where profinite monoids and semigroups are developed are: "Finite semigroups and universal algebra" by J. Almeida (in this book the term "profinite" is not used, yet such topics are treated) and "The q-theory of finite semigroups" by J. Rhodes and B. Steinberg.
Maybe I should make a further remark to your example which perhaps supports  your understanding.
As I mentioned already, $S_0$ is the free proACom object on a single generator (lets call it $x_1$); more generally, $S_n$ is the free proACom object on a finite space consisting of $2^n$ elements, lets call this $X_n$ (the set consisting of $x_1,x_2,\dots, x_{2^n}$). Now the homomorphism $\phi_n:S_{n+1}\to  S_n$ is completely determined by the values it takes on the free generators $X_{n+1}$ of $S_{n+1}$, these values are all in $X_n$ (free generators of $S_n$), according to your definition this mapping is just 
$$x_1,x_2\mapsto x_1, x_3,x_4\mapsto x_2, x_5,x_6\mapsto x_3 \dots.$$ 
So you have an inverse system of finite discrete spaces 
$$\dots \to X_{n+1}\to X_n\to X_{n-1}\to \dots\to X_0.$$
From the universal property of the projective limit it follows that the free object on the projective limit of the $X_n$s is just the projective limit of the free objects on the $X_n$s (you may interchange forming the projective limit and the free object). But the projective limit of the $X_n$s is just the Cantor set.  
