Mikołaj Bojańczyk

**Idempotent semigroup. **Call a semigroup *idempotent* if every element is idempotent.

0. Show that if a semigroup is idempotent and -trivial, then it is commutative.

1. Prove that if a semigroup is idempotent, then every -class is closed under multiplication (i.e. it is a sub-semigroup)

2. Prove that for every there are finitely many idempotent semigroups with generators.

**Strongly connected automata. **Let be a regular language with the following properties:

- the minimal DFA is strongly connected (i.e. every state is reachable from every other state);
- the minimal DFA of the reverse of is strongly connected;
- the syntactic monoid of is aperiodic.

Prove that is trivial, i.e. empty or full.

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