Cluster structures from 2-Calabi–Yau categories with loops

We generalise the notion of cluster structures from the work of Buan–Iyama–Reiten–Scott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Hom-finite 2-Calabi–Yau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2-cycles in the quivers of their endomorphism rings. We apply this result to the cluster category of a tube, and show that this category forms a good model for the combinatorics of a type B cluster algebra.