  A reduction $\varphi$ of an ordered group $(G,P)$ to another ordered group is
an order homomorphism which maps each interval $[1,p]$ bijectively onto $[1,
\varphi(p)]$. We show that if $(G,P)$ is weakly quasi-lattice ordered and
reduces to an amenable ordered group, then there is a gauge-invariant
uniqueness theorem for $P$-graph algebras. We also consider the class of
ordered groups which reduce to an amenable ordered group, and show this class
contains all amenable ordered groups and is closed under direct products, free
products, and hereditary subgroups.
