Let
F
\mathcal {F}
be a free semigroup and let
A
A
be an automorphism group of
F
\mathcal {F}
. A description is given of the space of real functions
φ
\varphi
on semigroup
F
\mathcal {F}
satisfying the following conditions: 1) the set
{
φ
(
x
y
)
−
φ
(
x
)
−
φ
(
y
)
;
x
,
y
∈
F
}
\{\varphi (xy)-\varphi (x)-\varphi (y);\,\, x,y\in \mathcal {F} \}
is bounded; 2)
φ
(
x
n
)
=
n
φ
(
x
)
\varphi (x^{n}) = n\varphi (x)
for any
x
∈
F
x\in \mathcal {F}
and
n
∈
N
n\in N
; 3)
φ
(
x
τ
)
=
φ
(
x
)
∀
x
∈
F
\varphi (x^{\tau }) = \varphi (x) \quad \forall x\in \mathcal {F}
, and
∀
τ
∈
A
\forall \tau \in A
.