the corresponding HNN extension of $G$ and let $P^*$ be the subsemigroup of $G^*$ generated by $P\cup\{t\}$. Let $$L_A:=\{p\in P:  q\in pA\cap P\Rightarrow p\leq q\}.$$ The following two statements are equivalent:

1.  Every left coset $gA\in G/A$ such that $gA\cap P\neq \emptyset$ has a minimal coset representative $p\in P$;

2.  There exists a complete set $X$ of left coset representatives such that $L_A\subseteq X$ and every $\alpha\in P^*$ has normal form $$\label{normal form}
          \alpha=p_0tp_1t\ldots p_{n-1}tp_n  \text{\ where $p_i\in L_A$ for all $0\leq i< n$, $p_n\in P$.}$$

Assume (1). Choose a complete set $X$ of coset representatives which contains $L_A$. Let $\alpha\in P^*$. If $\theta(\alpha)=0$ then $\alpha\in P$, and $\alpha$ has form trivially.

We proceed by induction on $\theta(\alpha)\geq 1$. Suppose that $\theta(\alpha)=1$. We may write $\alpha=q_0tq_1$ for some $q_0,q_1\in P$. Then $q_0A\cap P\neq \emptyset$, and there exists $p_0\in L_A$ such that $p_0A=q_0A$ and $p_0\leq q_0$. Thus$p_0^{-1}q_0\in P\cap A$. Hence $q_0 =p_0a$ for some $a\in A\cap P$. Thus $\alpha$ has normal form $$\alpha=p_0atq_1=p_0t\phi(a)q_1.$$ Since $\phi(A\cap P)=B\cap P$ we have $\phi(a)\in P$ and so $X$ satisfies (2).

Suppose that all $\alpha$ with $1\leq \theta(\alpha)\leq k$ have normal form . Consider $\alpha$ with $\theta(\alpha)=k+1$. We write $$\alpha=q_0tq_1t\ldots tq_ktq_{k+1}.$$ By assumption, we can write 