External semidirect product of semigroup and group

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:

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