isomorphism $\phi:A\to B$. Let $P^*$ be the subsemigroup of $G^*$ generated by $P$ and $t$. In general, $(G^*,P^*)$ is not a quasi-lattice ordered group. For example, if $c>1$, then $({\operatorname{BS}}(c,-d),{\operatorname{BS}}(c,-d)^+)$ is not quasi-lattice ordered by [@Spielberg2012 Lemma 2.12]. We need some conditions on our subgroups $A$ and $B$ and on the isomorphism $\phi$ which ensure that $(G^*,P^*)$ is quasi-lattice ordered.

There are two reasons why $({\operatorname{BS}}(c,d),{\operatorname{BS}}(c,d)^+)$ is easy to work with. The first is that there are natural choices of coset representatives: $\{0,\ldots,d-1\}$ for $A=d{\mathbb{Z}}$ and $\{0,\ldots,c-1\}$ for $B=c{\mathbb{Z}}$. The second is that the subgroup isomorphism $\phi:md\mapsto mc$ takes positive elements to positive elements. In particular, using this choice of representatives, every element $\omega\in{\operatorname{BS}}(c,d)^+$ has a unique normal form $$\omega=x^{m_0}tx^{m_1}t\ldots x^{m_{n-1}}tx^{m_n}$$ where $0\leq m_i<d$ for $i<n$ and $m_n\in {\mathbb{N}}$. This choice of coset representatives is associated to the division algorithm on ${\mathbb{N}}$: for every $n\in {\mathbb{N}}$ we can uniquely write $n=md+r$ for some $m\in {\mathbb{N}}$ and $0\leq r\leq d-1$.

In general, for $G^*$ we would like a natural choice of coset representatives for $G/A$ and $G/B$ so that every element of $P^*$ has a unique normal form that is a sequence of elements in $P$ and $t$.

\[(G\*,P\*) is left 