Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let $x\in X$. To avoid missing any important conditions, let us for simplicity also assume that $M=\mathbb C^{n\times n}$ is the monoid of square matrices. Is $M.x$ an affine variety? 



Are orbits of an affine algebraic monoid affine?
The orbits of a monoid aren't even varieties. Let $M$ be $\mathbb{A}^2$ with the monoid structure $(x\_1, y\_1) \cdot (x\_2, y\_2) = (x\_1 x\_2, y\_1 y\_2)$. Let $M$ act on $\mathbb{A}^2$ by $(x,y) \cdot (t,u) = (xt, xyu)$. Then $M \cdot (1,1)$ is the image of the map $(x,y) \mapsto (x,xy)$, which isn't a variety at all. Note that the unit group is $\mathbb{G}\_m^2$, which is reductive.


