Close as dups:
If uv is a unit in a monoid M, can we know that u and v are both units? http://math.stackexchange.com/q/103797
what is a set of normal forms for this group http://math.stackexchange.com/q/103787
If uv is a unit in a monoid M, can we know that u and v are both units? http://math.stackexchange.com/q/103797
what is a set of normal forms for this group http://math.stackexchange.com/q/103787