Editing Free monoid (section)

===Equidivisibility===
A free monoid is '''equidivisible''': if the equation ''mn'' = ''pq'' holds, then there exists an ''s'' such that either ''m'' = ''ps'', ''sn'' = ''q'' (example see image) or ''ms'' = ''p'', ''n'' = ''sq''.<ref name=Sak26>Sakarovitch (2009) p.26</ref> This result is also known as [[Levi's lemma]].<ref name="LucaVarricchio2011">{{cite book|author1=Aldo de Luca|author2=Stefano Varricchio|title=Finiteness and Regularity in Semigroups and Formal Languages|year=1999|publisher=Springer Berlin Heidelberg|isbn=978-3-642-64150-3|page=2}}</ref>

A monoid is free if and only if it is [[Graded monoid|graded]] (in the strong sense that only the identity has gradation 0) and equidivisible.<ref name=Sak26/>