Does concatenating two empty strings result in an empty string?

I have a question on the multiplication of sets of words as it is defined in Cohen's Intro To Computer Theory. He gives the definition: if S and T are sets of strings of letters, the product of the sets, ST = {All combinations of string from S concatenated with strings from T}.

The definition is easy enough to understand. But after giving a few standard examples, he gives an example along the lines of: S = {$\alpha$ a b}, T = {$\alpha$ a}

ST = {$\alpha$ a a aa b ab}

(I"ve used $\alpha$ to denote the empty string)

Basically, why is $\alpha \alpha$ = $\alpha$ in the product set and $\alpha$a = a ?

• Okay. It makes sense now. The last line in your answer really drives it home. So $\alpha+\alpha$ = ' ' + ' ' = ' ' = $\alpha$. Thanks. – Sadio Apr 20 '16 at 14:25