The partial isometry homology groups Hn dened in Power  and a related chain complex homology CH are calculated for various triangular operator algebras, including the disc algebra. These invariants are closely connected with K-theory. Simplicial homotopy reductions are used to identify both Hn and CHn for the lexicographic products A(G)?A with A(G) a digraph algebra and A a triangular subalgebra of the Cuntz algebra Om. Specifically Hn(A(G)?A) =Hn((G))ZK0(C(A)) and CHn(A(G) ? A) is the simplicial homology group Hn((G);K0(C(A))) with coecients in K0(C(A)).