Computing the image of a regular language by the transitive closure of a relation is a central question in Regular Model Checking. In a recent paper Bouajjani, Muscholl and Touili proved that the class of APC regular languages is closed under all semi-commutation relations R. Moreover a recursive algorithm on the regular expression is given to compute the image of an APC language by the transitive closure of R. This paper provides a new approach, based on automata, of the same question. Our approach produces a simpler and more efficient algorithm which furthermore works for a larger class of regular languages closed under union, intersection, semi-commutation relations and conjugacy. The existence of this new class, PolC, answered the open question proposed in Bouajjani and al.'s paper.