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

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

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

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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.
Title  automatic group 

Canonical name  AutomaticGroup 
Date of creation  20130322 14:16:54 
Last modified on  20130322 14:16:54 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  7 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 20F10 
Related topic  AutomaticPresentation 
Defines  automatic semigroup 
Defines  automatic structure 