External semidirect product of semigroup and group

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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')