As part of the bordered Floer homology package, Lipshitz, Ozsvath and D. Thurston have associated to a parametrized oriented surface a certain differential graded algebra. I will describe a decomposition theorem for this algebra, corresponding to cutting the surface along a circle. Moreover, I will discuss a decomposition theorem for bordered modules associated to nice diagrams, corresponding to cutting a 3-manifold with boundary along a surface transverse to the boundary. This is joint work with Christopher Douglas.