Synchronization of some DFA

A word w is called synchronizing (recurrent, reset, directable) word of deterministic finite automaton (DFA) if w brings all states of the automaton to an unique state. Cerny conjectured in 1964 that every n-state synchronizable automaton possesses a synchronizing word of length at most (n - 1)(2). The problem is still open. It will be proved that the minimal length of synchronizing word is not greater than (n - 1)(2) /2 for every n-state (n > 2) synchronizable DFA with transition monoid having only trivial subgroups (such automata are called aperiodic). This important class of DFA accepting precisely star-free languages was involved and studied by Schutzenberger. So for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true. Some properties of an arbitrary synchronizable DFA and its transition semigroup were established. http://www.cs.biu.ac.il/similar to trakht/syn.html.