 Automatic structure for a group

An automatic structure for a group consists of a finite generating alphabet $A$, a surjective representative regular language $L\subseteq A^{\ast }$, and finite-state automata accepting the padded convolutions of words $(u,v)$ in $L$ satisfying $\overline{u}x=\overline{v}$ for each $x\in A\cup \{\epsilon \}$. Symmetry of $A$ and uniqueness of representatives are additional conditions, not part of this definition.