For some alphabet $\sum = \{a,b\}$ what is the concatenation $\sum\sum$?

Question

Does this mean that:

$\sum^2= \{a,b\}\{a,b\}= \{a,aa,ab,ba,bb\}$

What I am trying to ascertain is if every concatenation on a alphabet include $\lambda$ a.k.a the empty string, if so then:

$\sum^2= \{\lambda,a,b\}\{\lambda,a,b\} =\{a,b,aa,ab,ba,bb\}$

I seem to be getting confused as to if it is implied that every alphabet includes $\lambda$ or not. Any insight would be highly appreciated.

Answer 1

The empty string $\lambda$ is a string over the alphabet, not a member of the alphabet.

Think of any programming language: If the characters in a string in the language consist of (for example) Unicode symbols, the alphabet corresponds to the set of Unicode symbols. The empty string is a valid string in the language, but there is no Unicode symbol corresponding to the empty string.

So $\Sigma \Sigma = \{aa, ab, ba, bb\}$.