the same for both paths. Since $b_1 - b_2$ and $\alpha -b_2$ have the same (negative) sign, we can homotope $\gamma' \sigma \gamma$ to a path with the same real part (for both coordinates), and with imaginary part the same as $\sigma$, up to possibly diminishing the chosen factor $\eps$ in the definition of the detours. Choosing then an homotopy in $\R_{\leq b_2}^2$ between the real parts of these two paths, provides an homotopy between them in $Y$.

Parabolic submonoids
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We apply the results of section \[sectembed\] on submonoids to the monoid $N = M(e,e,r)$, with generators $S = \{ t_0,\dots,t_{e-1}, s_3,\dots,s_r \}$. Let $\mathcal{C} = \{ t_0 , \dots ,t_{e-1} \}$. For $S_0 \subset
S$, let $M(S_0)$ be the monoid generated by $S_0$ with the defining relations of $M(e,e,r)$ which involve only elements of $S_0$. We get a natural morphism $\varphi : M(S_0) \to M(S) = M(e,e,r)$. We ask for the following extra assumption on $S_0$ : $$S_0 \cap \mathcal{C} \in \{ \emptyset, \mathcal{C}, \{ t_i \} \} \mbox{ for some } i \in \{0,1,\dots,e-1 \} .$$ In other terms, $S_0$ contains none, all or exactly one of the $t_i$’s. Note that all the corresponding monoids are known 