Automatic structure

Definition

An automatic structure on a finitely generated group $G$ with respect to a finite generating set $S$ is defined as the following collection of finite state automata:

1. word acceptor: This accepts, for every element of $G$, at least one word using letters from $S\cup S^{-1}$ that evaluates to that element.
2. multiplier, one for each $s\in S$: This accepts a pair of words $(w_{1},w_{2})$ iff both $w_{1},w_{2}$ are accepted by the word acceptor and $w_{1}s=w_{2}$.