In this paper, a new general decomposition theory inspired from modular graph decomposition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are still efficient. This theory not only unifies the usual modular decomposition generalisations such as modular decomposition of directed graphs or decomposition of 2-structures, but also star cutsets and bimodular decomposition. Our general framework provides a decomposition algorithm which improves the best known algorithms for bimodular decomposition.