  In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine
commutative algebraic monoids. In this work, we extend this classification to
noncommutative monoid structures on normal affine surfaces. We prove that
two-dimensional algebraic monoids are toric. We also show how to find all
monoid structures on a normal toric surface. Every such structure is induced by
a comultiplication formula involving Demazure roots. We also give descriptions
of opposite monoids, quotient monoids, and boundary divisors.
