We consider the submonoids $M_k$ of the Thompson group $F$ that are generated by the first $k+1$ generators of the infinite presentation \[F = \big\langle x_0, x_1, x_2, \ldots \mid \ x_jx_i=x_ix_{j+1}\ \text{for }j>i\big\rangle.\]

The standard normal form for $F$ breaks down for these monoids but we give a new normal form that works in $F$ and every $M_k$ and allows us to analyze their constructible right ideal structure. We show that there exist embeddings $M_k \hookrightarrow M_{k+1}$ for which the associated Toeplitz algebras are functorial, and then we study the directed system of Toeplitz algebras $\mathcal T_\lambda(M_k) \hookrightarrow \mathcal T_\lambda(M_{k+1})$. Using recent results of Sehnem and mine we characterize faithful representations and show that the boundary quotients are purely infinite simple.

This is joint work with A. an Huef, B. Nucinkis, I. Raeburn and C. Sehnem.