Regular expression arithmetics

What are the rules of regular expression arithmetics ?

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

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

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

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What exactly is you question? –  martini Aug 12 '12 at 8:49
@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) –  Belgi Aug 12 '12 at 8:53
Perhaps look into Kleene algebras. –  Arthur Fischer Aug 12 '12 at 8:58
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 Aug 12 '12 at 9:14
@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 Aug 12 '12 at 9:16

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 $\varnothing+\alpha=\alpha$.