2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79536We classify all total orders having a certain convex property on the positive root system of an arbitrary untwisted affine Lie algebra ${\frak g}$. Such total orders are called convex orders and are used to construct convex bases of Poincaré-Birkhoff-Witt type of the upper triangular subalgebra $U_q^+$ of the quantized enveloping algebra $U_q({\frak g})$.30pages, AMS-LaTeXQuantum AlgebraRepresentation Theorytext