The partial isometry homology groups H_n defined in Power [17] 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 H_n and CH_n for the lexicographic products A(G) ★ A with A(G) a digraph algebra and A a triangular subalgebra of the Cuntz algebra O_m. Specifically H_n (A(G) ★ A) = H_n (Δ (G)) ⊗_Z K₀ (C*(A)) and CH_n (A(G) ★ A) is the simplicial homology group H_n (Δ (G) ; K₀ (C*(A))) with coefficients in K₀ (C*(A)).