J-equivalence for associative algebras

We study the combinatorics of an analogue of Green’s \({\mathcal {J}}\) $\mathcal{J}$ -relation (a.k.a. the two-sided relation) for the bicategory of finite-dimensional bimodules over finite-dimensional associative algebras over a fixed field. In particular, we provide a number of exameples for both \({\mathcal {J}}\) $\mathcal{J}$ -equivalent and inequivalent but \({\mathcal {J}}\) $\mathcal{J}$ -comparabe algebras.