Unique Factorisation Monoids

We examine the properties of commutative unique factorisation monoids and with these in mind, we propose a definition for a class of noncommutative unique factorisation monoids. Examples include graph monoids, and cancellative monoids in which all right ideals are projective and such that any principal left ideal is contained in only finitely many ideals. The structure of the monoids in the latter class was recently studied by M V Lawson who showed that such a monoid is a Zappa-Szep product of a group and a free monoid. We show that a unique factorisation monoid in the class we consider is a Zappa-Szep product of a group and a graph monoid.