0P5E Remark 3.2.13. The same method as the one used in the proof of Proposition 3.2.9 shows that 𝔖^n+\hat{{\mathfrak{S}}}_{n}^{+} is the free (𝔖n,𝔖n)({\mathfrak{S}}_{n},{\mathfrak{S}}_{n})-monoid on a generator cc with relations c⋅sr=sr+1⋅cc\cdot s_{r}=s_{r+1}\cdot c for r∈{1,…,n−1}r\in\{1,\ldots,n-1\} and c2⋅sn−1=s1⋅c2c^{2}\cdot s_{n-1}=s_{1}\cdot c^{2}.