Q:

Regular expression arithmetics

What are the rules of regular expression arithmetics ?
For example: Let $\Sigma=\{0,1\}$ 
$1. 1+01=(\epsilon+0)1$.
$2. (\epsilon+00)^*=(00)^*$

A:

Off the top of my head you have at least the following:

$\epsilon\alpha=\alpha=\alpha\epsilon$.  
$\alpha+\beta=\beta+\alpha$.  
$\alpha+\alpha=\alpha$.  
$\alpha\beta+\alpha\gamma=\alpha(\beta+\gamma)$ and $\beta\alpha+\gamma\alpha=(\beta+\gamma)\alpha$.  
$(\epsilon+\alpha)^*=\epsilon+\alpha^*=\alpha^*$.  
$\epsilon^*=\epsilon$.

If your formalization includes $\varnothing$ (no string) distinct from $\epsilon$ (empty string), you have $\varnothing+\alpha=\alpha$.

