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Let F be a free semigroup and let A be an automorphism group of F. A description is given of the space of real functions y on semigroup F satisfying the following conditions: 1) the set {y (xy) p(x) p(y); x, y E F} is bounded; 2) Op(xn) = nyp(x) for any x E F and n E N; 3) p(xr) = p(x) Vx c F, and VT E A.

Pseudocharacters on free semigroups invariant with respect to their automorphism groups