In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element.
Is the empty string an element of the underlying set of the free monoid?
In the free monoid the elements of the underlying set are finite sequences of letters from an alphabet: $ \{ab,aab,a,b...\} $ I'm just wondering how exactly is the empty sequence represented in this underlying set?
Take a look at Wikipedia Kleene star:
If V is a set of symbols or characters, then $V*$ is the set of all strings over symbols in $V$, including the empty string $\epsilon$.