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 "$κ(S)=1$", 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 "$κ(S)\ge 0$".

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