a normal subgroup $N$ of a group $G$ is amenable and $G/N$ is amenable, then $G$ is amenable, and second, Spielberg’s result on amenability of groupoids [@Spielberg2014 Proposition 9.3].

In Theorem \[(G\*,P\*) is left ql\] we give conditions under which an HNN extension of a quasi-lattice ordered group is quasi-lattice ordered. This result allows us to construct many new examples of quasi-lattice ordered groups. Finally, we use Theorem \[p-controlled map implies amenable\] to prove that an HNN extension of an amenable quasi-lattice ordered group is amenable (Theorem \[HNN extensions are amenable as expected\]).

Preliminaries
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Let $P$ be a subsemigroup of a discrete group $G$ such that $P\cap P^{-1}=\{e\}$. There is a partial order on $G$ defined by $$x\leq y\Leftrightarrow x^{-1}y\in P\Leftrightarrow y\in xP.$$ The order is left-invariant in the sense that $x\leq y$ implies $zx\leq zy$ for all $z \in G$. A partially ordered group $(G, P)$ is *quasi-lattice ordered* if every finite subset of $G$ with a common upper bound in $P$ has a least common upper bound in $P$ [@Nica1992 Definition 2.1]. By [@CrispLaca2002 Lemma 7], $(G,P)$ is quasi-lattice ordered if and only if: $$\label{Crisp Laca}
\ \hfill
\parbox{0.9\textwidth}{
 if $x\in PP^{-1}$, then there exist a pair  