ql\] Let $(G,P)$ be a quasi-lattice ordered group with subgroups $A$ and $B$. Suppose that:

1.  There is an isomorphism $\phi:A\to B$ such that $\phi(A\cap P)=B\cap P$;

2.  Every left coset $gA\in G/A$ such that $gA\cap P\neq \emptyset$ has a minimal coset representative $p\in P$: $q\in gA\cap P \Rightarrow p\leq q$;

3.  For every $x,y\in B$, $x\vee y<\infty\Rightarrow x\vee y\in B$.

Let $G^*=\langle G,t\mid t^{-1}at=\phi(a),a\in A\rangle$ be the HNN extension of $G$ and let $P^*$ be the subsemigroup of $G^*$ generated by $\{P,t\}$. Then $(G^*,P^*)$ is quasi-lattice ordered.

Before we can prove Theorem \[(G\*,P\*) is left ql\], we need to prove two lemmas. The first shows that elements of $P^*$ are guaranteed to have normal forms made up of elements of $P$ and $t$ if and only if condition (2) of Theorem \[(G\*,P\*) is left ql\] holds. The second is a technical lemma which we will use several times in Theorem \[(G\*,P\*) is left ql\] and in later proofs.

\[right lower bound gives unique normal form in P\] Suppose that $(G,P)$ is a quasi-lattice ordered group with subgroups $A$ and $B$. Suppose that $\phi:A\to B$ is a group isomorphism such that $\phi(A\cap P)=B\cap P$. Let $G^*=\langle G,t\mid t^{-1}at=\phi(a),a\in A\rangle$ be 