I have a set of elements $\{a\_1, a\_2, a\_3...\}$ and $\{b\_1, b\_2, b\_3...\}$ and I want to condensely formally write the set of all possible products of these elements, where the ordering does not matter, e.g. $a\_i b\_j = b\_j a\_i$, i.e. product is commutative.
The set is thus: $\{1, a\_1, b\_1,a\_1^2, a\_1 b\_1, ... ,a\_1^3 b\_1^2 a\_2^6 b\_7^1,...\}$. Preferably I would write ordered.


One option, I think, is $$ \left\{\prod\_{i\in\mathbb{N}} a\_i^{m\_i} b\_i^{n\_i} \right\}\_{\vec m, \vec n \in \mathbb{N}^\infty}.$$
I'm not sure this is correct, nor clear. Does it change if I need only finite products?



Writing a set of all possible (symmetric) products condensely?
The set of all finite commutative products can be written as 
$$ \Big\{\prod\_{i=1}^k a\_i^{m\_i} b\_i^{n\_i}\colon k\in\mathbb N\_0,\vec m\in \mathbb N\_0^k, \vec n \in \mathbb N\_0^k\Big\},$$
where $\mathbb N\_0:=\{0,1,\dots\}$, $a\_i^0:=1$, $b\_j^0:=1$. 


