  We prove that a map onto a nilpotent group $Q$ has finitely generated kernel
if and only if the preimage of the positive cone is coarsely connected as a
subset of the Cayley graph for every full archimedean partial order on $Q$. In
case $Q$ is abelian, we recover the classical theorem that $N$ is finitely
generated if and only if $S(G,N) \subseteq \Sigma^1(G)$. Furthermore, we
provide a way to construct all such orders on nilpotent groups. A key step is
to translate the classical setting based on characters into a language of
orders on $G$.
