* on empty string

Question

we know that {a,b}* = { ε, ab, abab, ab, a...}. and ∅* = {ε}. what about ε*. My thinking is that since * is regular operation, not string operation, which means we can't apply * on empty string. please point out if anything I missed or please simply confirm my thought.

Answer 1

If $L$ is any language of $A^*$, $L^*$ is the submonoid of $A^*$ generated by $L$. Thus if $L = \{1\}$, where $1$ is the empty word, then $L^° = L$ since $1$ is the identity of $A^*$. More generally, the submonoid generated by the identity of a monoid only contains the identity.