Writing a set of all possible (symmetric) products condensely? [closed]

Question

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?

Answer 1

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 $\mathbb N_0:=\{0,1,\dots\}$, $a_i^0:=1$, $b_j^0:=1$.