Group action on a variety

Remarks

On points, with $g \cdot x = \sigma(g, x)$, the clauses say $e \cdot x = x$ and $(gh) \cdot x = g \cdot (h \cdot x)$. So for each $k$-algebra $R$ one gets a group action of $G(R)$ on $X(R)$.

Example: $\mathrm{GL}_n$ acts on $\mathbb{A}^n$ by $\sigma(g, v) = g v$. The coordinates of $g v$ are polynomials in the entries of $g$ and $v$, the identity matrix fixes every $v$, and $(gh) v = g (h v)$.

Non-example: the morphism $\sigma \colon \mathbb{G}_m \times \mathbb{A}^1 \to \mathbb{A}^1$, $\sigma(t, x) = x + t - 1$, satisfies clause 1. It fails clause 2, since $\sigma(st, x) = x + st - 1$ while $\sigma(s, \sigma(t, x)) = x + s + t - 2$.