automatic group

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

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

• $L$ can be checked by a finite automaton (http://planetmath.org/DeterministicFiniteAutomaton)

• The language of all convolutions of $x,y$ where $f(x)=f(y)$ can be checked by a

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

$(A,L)$ 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.