Twisted forms of commutative monoid structures on affine spaces

In this paper, we study affine commutative algebraic monoid structures on affine spaces over an arbitrary field of characteristic zero. We obtain full classification of such structures on ${\mathbb{A}}_K^2$ and ${\mathbb{A}}_K^3$ and describe some generalizations on ${\mathbb{A}}_K^n$ for any dimension $n$.