On a problem of Nazarova and Roiter

In the present paper we introduce the notion of representations of a bush which is a generalization of matrix problems (self reproducing systems) introduced by Nazarova and Roiter. We show that the problem of classifying representations of clannish algebras come down to such generalized matrix problems. Based on the classification of Crawley-Boevcy, we provide a description of indecomposable representations of bushes over any field. The proof is based on a categorical formulation of the matrix reduction of Nazarova. and Roiter.