Relation between group action in abstract algebra and semi-automaton

Question

If $G$ is a set acting on a set $A$, can it define a semi-automaton? How is this concept of abstract algebra related to semi-automata?

Answer 1

Your notation is a bit misleading, since $A$ usually denotes an alphabet. So let me switch the role of $G$ and $A$. Suppose that $A$ acts on $G$, that is, there is a map $(g,a) \mapsto g \cdot a$ from $G \times A$ to $G$. This map can be extended, to an action on $G$ of every word $u$ of $A^*$. This can be done by induction on the length of $u$. Let $g \in G$. For the empty word $1$, just set $g \cdot 1 = g$. Now if $u$ is a word and $a$ is a letter, set $g \cdot (ua) = (g \cdot u) \cdot a.\ $ You define in this way a semi-automaton $(G, A)$ where $G$ is the set of states and $A$ is the alphabet.