I am interested in an set-theoretic operation (which is potentially related to the Power Set $\mathcal{P} (X)$).
Let's say, $X=\{a,b\}$, then I want an operation $\mathcal{Q_n} (X)$, with the following property:
$$ \mathcal{Q_0}(X)=\{\{\}\} \\ \mathcal{Q_1}(X)=\mathcal{Q_0}(X) \cup \{\{a\},\{b\}\} \\ \mathcal{Q_2}(X)=\mathcal{Q_1}(X) \cup\{\{a,a\},\{a,b\},\{b,b\}\} \\ \mathcal{Q_3}(X)=\mathcal{Q_2}(X) \cup\{\{a,a,a\},\{a,a,b\},\{a,b,b\},\{b,b,b\}\} \\ \mathcal{Q_4}(X)=\mathcal{Q_3}(X) \cup\{\{a,a,a,a\},\{a,a,a,b\},\{a,a,b,b\},\{a,b,b,b\},\{b,b,b,b\}\} \\ ... $$
That means, for every order $n$, it contains every combintation of elements from $X$ up to $n$ elements. In particular, it allows for multiple appearances of the same element (which is in contrast to the Power Set).
Question: What is the name of the operation $\mathcal{Q_n}(X)$?