Under the cross product, $\mathbb{R}^3$ satisfies closure (cross product of two vectors in $\mathbb{R}^3$ is in $\mathbb{R}^3$) and associativity (the cross product is NOT associative in general — actually it is not associative). It lacks an identity element. So it is a groupoid (closed binary operation) at best. The cross product is not associative: $(\mathbf{a}\times\mathbf{b})\times\mathbf{c} \neq \mathbf{a}\times(\mathbf{b}\times\mathbf{c})$ in general. Per key: semi-group.