Let $G$ be a finitely generated group. Let $A$ be a finite generating set for $G$ closed under inverses.

$G$ is an automatic group if there is a language $L\subseteq A^*$ and a surjective map $f:L\rightarrow G$ such that

• $L$ can be checked by a finite automaton
• The language of all convolutions of $x,y$ where $f(x)=f(y)$ can be checked by a finite automaton
• For each $a\in A$ , the language of all convolutions of $x,y$ where $f(x).a=f(y)$ can be checked by a finite automaton

$(A, L)$ is said to be an automatic structure for $G$ .

Note that by taking a finitely generated semigroup $S$ , and some finite generating set $A$ , these conditions define an automatic semigroup.