  We calculate the one-loop corrections to the soft anomalous dimension
matrices for the production of a top-antitop quark pair in association with a
jet at hadron colliders. This is a step forward towards implementing a
procedure for the resummation of soft-gluon emission logarithms for the
$t\bar{t}j + X$ hadroproduction process, that will enable the improvement of
the accuracy of the $t\bar{t} j + X$ cross sections, beyond the current degree
of knowledge. The latter, so far, has reached the next-to-leading order (NLO)
level, complemented by the accuracy of the Shower Monte Carlo approaches used
in matching the NLO computations to parton showers (PS) and by merging with
matrix elements with a different number of light jets.
