Geodesic
The \v Cerný conjecture for aperiodic automata
Discrete mathematics & theoretical computer science, Tome 9 (2007) no. 2.

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A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n-1)2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n-1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true. 
@article{DMTCS_2007_9_2_a7,
     author = {Trahtman, A. N.},
     title = {The \v {Cern\'y} conjecture for aperiodic automata},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {9},
     number = {2},
     year = {2007},
     doi = {10.46298/dmtcs.395},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.395/}
}
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Trahtman, A. N. The \v Cerný conjecture for aperiodic automata. Discrete mathematics & theoretical computer science, Tome 9 (2007) no. 2. doi : 10.46298/dmtcs.395. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.395/

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