in [@Spielberg2012 Theorem 2.11] that $({\operatorname{BS}}(c,d), {\operatorname{BS}}(c,d)^+)$ is quasi-lattice ordered for all $c,d> 0$; he also proved in [@Spielberg2012 Lemma 2.12] that $({\operatorname{BS}}(c,-d),{\operatorname{BS}}(c,-d)^+)$ is not quasi-lattice ordered unless $c=1$.

To work with an HNN extension we use a normal form for its elements from [@LyndonSchupp1977 Theorem 2.1]. We choose $X$ to be a complete set of left coset representatives for $G/A$ such that $xA\neq x'A$ for $x\neq x'\in X$. Similarly, choose a complete set $Y$ of left coset representatives for $G/B$. Then a (right) *normal form relative to $X$ and $Y$* of $g\in G$ is a product $$g=g_0t^{\epsilon_1}g_1t^{\epsilon_2}\ldots g_{n-1}t^{\epsilon_n}g_n$$ where:

1.  $g_n$ is an arbitrary element of $G$.

2.  If $\epsilon_i=1$, then $g_{i-1}$ is an element of $X$

3.  If $\epsilon_i=-1$, then $g_{i-1}$ is an element of $Y$.

By [@LyndonSchupp1977 Theorem 2.1], for every choice of complete left coset representatives $X$ and $Y$, each $g\in G^*$ has a unique normal form.

Our goal is to generalise the properties of the Baumslag-Solitar group with $c, d>0$ to construct quasi-lattice ordered HNN extensions of other quasi-lattice ordered groups.

Let $(G,P)$ be a quasi-lattice ordered group. Let $G^*$ be the HNN extension of $G$ with respect to subgroups $A$ and $B$ with an 