External semidirect product of semigroup and group

Definition

Definition with the left action convention

Suppose $S$ is a semigroup and $H$ is a group along with a homomorphism $\rho :H\to \operatorname {Aut} (S)$ (the automorphism group of $S$). The external semidirect product of $S$ by $H$ with respect to $\rho$, denoted $S\rtimes H$, is defined as the following semigroup $T$. As a set $T=S\times H$, with the multiplication given by:

$\!(a,b)(a',b')=(a(\rho (b)(a')),bb')$

If we denote $\rho (b)(a')$ by $b\cdot a'$, this can be rewritten as:

$\!(a,b)(a',b')=(a(b\cdot a'),bb')$