Regular expression arithmetics

Question

What are the rules of regular expression arithmetics ?

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

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

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

@martini - "What are the rules of regular expression arithmetics ?" in $\mathbb{R}$ for exmaple you have $ab+ac=a(b+c)$ and in the examples I gave there are cirtin rules for regular expression arithmetics as well...I ask about what those rules are (although I do understand that in $\mathbb{R}$ this was an axiom and in our case it can be proven)
What regular expression syntax are you using? In the RE syntaxes I know + is either an ordinary character, or means "one or more times the regular expression preceding it" (i.e. like *, but excluding the empty string). Here, it seems to mean something different (maybe "or")?
@celtschk: Belgi is clearly using it for OR; I’m more familiar with both $\mid$ and $\lor$ in this context, but I’ve also seen $+$ used.

Brian M. Scott · Accepted Answer · 2012-08-12 09:08:23Z

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 $\varnothing+\alpha=\alpha$.

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