A regular language is star-free if and only if it is accepted by an automaton with a finite and aperiodic transition monoid.
This result of algebraic automata theory is due to Marcel-Paul Schützenberger.
[1] In particular, the minimum automaton of a star-free language is always counter-free (however, a star-free language may also be recognized by other automata that are not aperiodic).
(This is automatically true when y is the empty string, but becomes a nontrivial condition when y is non-empty.)
An aperiodic automaton satisfies the Černý conjecture.