  We say that a nonselfadjoint operator algebra is partly free if it contains a
free semigroup algebra. Motivation for such algebras occurs in the setting of
what we call free semigroupoid algebras. These are the weak operator topology
closed algebras generated by the left regular representations of semigroupoids
associated with finite or countable directed graphs. We expand our analysis of
partly free algebras from previous work and obtain a graph-theoretic
characterization of when a free semigroupoid algebra with countable graph is
partly free. This analysis carries over to norm closed quiver algebras. We also
discuss new examples for the countable graph case.
