Abstract
An action X x A + → X: (x,w) → x*w of the free semigroup A + over the alphabet A is synchronised by a word w ∈ A + if x*w = y*w for all x, y ∈ X. Cerny conjectured that if there is such a w and |X| = n, then there is one of length ≤ (n - 1) 2 . The conjecture remains open, except for particular classes of actions. An action is circular if there is c ∈ A acting as a circular permutation, and biaised if there is b ∈ A with only one x ∈ X such that |x* b -1 |≥ 2. We confirm here Cerny conjecture for biaised circular actions.
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