What's a semidirect product of semigroups?

Question

I see many references to the notion in the title on the internet, but I can't find a definition. Could you please give one? A short introduction to the theory of such products (especially one relating them to the group case) would be well-received, but I'll accept a bare definition if that's all I can get.

Answer 1

Here is the definition of the semidirect product of two semigroups.

Let $S$ and $T$ be semigroups. Let us write the product in $S$ additively to provide a more transparent notation, but it is not meant to suggest that $S$ is commutative. A left action of $T$ on $S$ is a map $(t,s) \mapsto t \cdot s$ from $T^1\times S$ to $S$ such that, for all $s, s_1, s_2 \in S$ and $t, t_1, t_2 \in T$,

1. $t_1 \cdot (t_2 \cdot s) = (t_1t_2) \cdot s$
2. $t \cdot (s_1+s_2) = t \cdot s_1 + t \cdot s_2$
3. $1 \cdot s = s$

If $S$ is a monoid with identity $0$, the action is unitary if it satisfies, for all $t\in T$,

4. $t \cdot 0=0$

The semidirect product of $S$ and $T$ (with respect to the given action) is the semigroup $S * T$ defined on $S\times T$ by the multiplication $$ (s,t)(s',t') = (s + t \cdot s', tt') $$