Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s\_1,...s\_k\in S$ such that the sets $s\_i\cdot W$ are pairwise disjoint and cover $S$. For instance, tiles for the semigroup $(\mathbb N,+)$ are given by the multiples of some fixed natural number. Here is my question: is there any explicit example of tile for the semigroup $(\mathbb N,\cdot)$?


A literature remark: I do not know if the notion of "tile" is already defined/used somewhere. I know the existence of "syndetic sets" that are pretty similar but different. I hope you like the name "tile"!


Thanks in advance, Valerio


If someone of you wants to know more specifically my problem:
Let $(S,\cdot)$ be a countable amenable semigroup and let $W\subset S$ be a subset. Let me denote by $\chi\_W$ the characteristic function of $W$ and define the following numbers
$$
W^-=\inf [f(\chi\_W),f \text{ bi-invariant mean}]
$$
$$
W^+=\sup[f(\chi\_W), f \text{ bi-invariant mean}]
$$
I am first of all interested in the case $W^-=W^+$ (this is why I'm interested in tiles). That would be the case when one can define some notion of intrinsic probability of the set $W$. Then it would be nice to find some description for the number $W^+-W^-$ and hopefully finding how the numbers $f(\chi\_W)$ are distributed into the interval $[W^-,W^+]$.



How can we pave the multiplicative semigroup $(\mathbb N,\cdot)$?
Prime factorization provides an isomorphism between the semigroup $(\mathbb N,\cdot)$ and an infinite direct sum of copies of $(\mathbb N,+)$.
So you can reduce your problem to the case that you already know how to solve.


*Warning:* my first $\mathbb N$ does not contain the element zero, whereas my second $\mathbb N$ does!


Concretely, here is an example of a tile of $(\mathbb N,\cdot)$: the set of all $n\in\mathbb N$ whose $p$-valuation is a multiple of $k$, where $p$ is a prime number, and $k$ is arbitrary.


