I'm trying to understand the statement:
The regular languages over $A$ are the homomorphic pre-images in $A^∗$ of subsets of finite monoids.
which appears in the Wikipedia article on free monoids: http://en.wikipedia.org/wiki/Free_monoid. ($A^*$ is the free monoid over $A$.)
Can anyone explain what the statement means? I gather there's a homomorphism from A* to the set of subsets of some finite monoid (?). But what is the finite monoid, and what is the homomorphism?
I'd like to understand if this is actually a deep statement about regular languages or not :)
Edit: Let me write here what I gather from Thomas Andrews' answer. We let $A$ be some alphabet and let $A^*$ be the free monoid over $A$. So $A^*$ is just the strings made of letters from $A$.
We think of each element of monoid $M$ as being some state of a finite state machine. So if $a$ and $b$ are some strings in $A^*$, then $\phi(a)$ and $\phi(b)$ are some states in the FSM. (They are the states you would get to if you started on the start state and entered $a$ or $b$.)
$\phi$ then makes sense as a homomorphism: $\phi(a + b) = \phi(a) \star \phi(b)$ means "the state the string $a + b$ gets you to in the FSM is the same as first running string $a$ and then string $b$."
I don't think the monoid operation in $M$ (i.e. $\star$) makes sense though...