19.62 (2018)

Let $\mathfrak{A} = \{\mathfrak{A}_r \mid r \in \Phi\}$ be an elementary carpet of type $\Phi$ of rank $l \geqslant 2$ (see 7.28). For $p \in \Phi$, define a set of additive subgroups $\mathfrak{B}_p = \sum c_{ij,rs}\mathfrak{A}_r^i\mathfrak{A}_s^j$, where the sum is taken over all natural numbers $i, j$ and roots $r, s \in \Phi$ such that $ir + js = p$. It is known that the set $\mathfrak{B} = \{\mathfrak{B}_p \mid p \in \Phi\}$ is a carpet called the derived carpet of $\mathfrak{A}$. It is also known that for $\Phi = A_l$ the set $\mathfrak{B}$ is a closed (admissible) carpet, which means that its carpet subgroup does not contain new root elements. Is every derived carpet of type $\Phi$ over a commutative ring closed (admissible)?