Classification of two-dimensional algebraic projective semigroups

In this article, we address the classification of smooth projective algebraic surfaces over complex numbers admitting algebraic semigroup structures. We give a full description of those surfaces $S$, which has at least one non-trivial algebraic semigroup structure, when the Kodaira dimension of $S$ is $ -\infty$ and $ 0$. For the case "$ \kappa (S)=1$", we give a description of one special type of elliptic surfaces which admit non-trivial algebraic semigroup laws. \\ For a given surface $S$, it is an interesting problem to describe all algebraic semigroup structures on it and determine the dimension of this moduli. In this article, we solve this problem for case "$ \kappa (S)\ge 0$".