Finite regular language partition of free monoid $L^*$?

Question

If $L^*$ is free monoid ($L$ finite), is there exist a finite partition $L_1, L_2, \ldots, L_r \subset L$ which all $L_i$ is regular language?

My conjecture is not, but I don't know how to start.

Answer 1

For $a \in L$, let $M = L - \{a\}$.

• $M^*$ is a regular language.
• The language $N$ of all strings that include at least one $a$ is a regular language: it is recognized by a finite automaton with 2 states, start state $S_0$ being "no $a$ so far" and accept state $S_1$ being "an $a$ has been encountered", with $a$ the transition from $S_0$ to $S_1$.

Then $M^*,N$ is a partition of $L^*$.

EDIT: as requested, here is a diagram for $N$ finite state machine.