> 
> Let $M$ be some finite set. Do we then have that
>  $M^{\mathbb{Z}}=M^{\mathbb{Z}\_{\leq 0}}\times M^{\mathbb{Z}\_{>0}}$?
> 
> 
> 

Since $M^{\mathbb{Z}}=\prod\_{i\in\mathbb{Z}}M\_i$ with $M\_i=M$ for all $i\in\mathbb{Z}$, it should be possible to split this product, ie.
$$
M^{\mathbb{Z}}=\prod\_{i\in\mathbb{Z}}M\_i=\prod\_{i\in\mathbb{Z}\_{\leq 0}}M\_i\times\prod\_{i\in\mathbb{Z}\_{>0}}M\_i=M^{\mathbb{Z}\_{\leq 0}}\times M^{\mathbb{Z}\_{>0}}.
$$

Or am I completely wrong and think too naively?