Regular expression arithmetics

Question

What are the rules of regular expression arithmetics ?

For example: Let $\Sigma=\{0,1\}$

$1. 1+01=(\epsilon+0)1$.

$2. (\epsilon+00)^*=(00)^*$

Answer 1

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

1. $\epsilon\alpha=\alpha=\alpha\epsilon$.
2. $\alpha+\beta=\beta+\alpha$.
3. $\alpha+\alpha=\alpha$.
4. $\alpha\beta+\alpha\gamma=\alpha(\beta+\gamma)$ and $\beta\alpha+\gamma\alpha=(\beta+\gamma)\alpha$.
5. $(\epsilon+\alpha)^*=\epsilon+\alpha^*=\alpha^*$.
6. $\epsilon^*=\epsilon$.

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