19.61 (2018)

Let $\mathfrak{A} = \{\mathfrak{A}_r \mid r \in \Phi\}$ be an elementary carpet of type $\Phi$ over a commutative ring $K$ (see 7.28), and let $\Phi(\mathfrak{A}) = \langle xr(\mathfrak{A}_r) \mid r \in \Phi\rangle$ be its carpet subgroup. Define the closure of the carpet $\mathfrak{A}$ to be the set of additive subgroups $\overline{\mathfrak{A}} = \{\overline{\mathfrak{A}}_r \mid r \in \Phi\}$, where $\overline{\mathfrak{A}}_r = \{t \in K \mid xr(t) \in \Phi(\mathfrak{A})\}$. Is the closure $\overline{\mathfrak{A}}$ of a carpet $\mathfrak{A}$ always a carpet?