Classification of noncommutative monoid structures on normal affine surfaces

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