Perfect basis theory for quantum Borcherds-Bozec algebras

In this paper, we develop the perfect basis theory for quantum Borcherds-Bozec algebras $U_q(\mathfrak{g})$ and their irreducible highest weight modules $V(\lambda )$. We show that the lower perfect graph (resp. upper perfect graph) of every lower perfect basis (resp. upper perfect basis) of $U_q^{-}(\mathfrak{g})$ (resp. $V(\lambda )$) is isomorphic to the crystal $B(\mathrm{\infty })$ (resp. $B(\lambda )$).