A book I am reading gives the following definitions:
A collection $\{L_i:i=0,1,2,...\}$ of halfgroupoids $L_i$ is called an extension chain if $L_{i+1}$ is an extension of $L_i$ for each $i$. If $G$ is a subhalfgroupoid of the halfgroupoid $H$, the maximal extension chain of $G$ in $H$ is defined as follows: $G_0 = G$ and, for each nonnegative integer $i$, (a) $G_i$ is a subhalfgroupoid of $H$; (b) $G_{i+1}$ is an extension of $G_i$; (c) if $x, y \in G_i$ and if $xy=z$ in $H$, then $xy=z$ in $G_{i+1}$. The halfgroupoid $K=\bigcup_{i=0}^\infty G_i$ may be characterized as follows: (i) G is a subhalfgroupoid of K; (ii) K is a closed subhalfgroupoid of H; (iii) if $G$ is a subhalfgroupoid of the closed subhalfgroupoid $L$ of $H$, then $K$ is a subhalfgroupoid of $L$. We call $K$ the subhalfgroupoid of $H$ generated by G. In particular, $G$ generates $H$ if $K=H$.
I am confused about (iii) of the chacterizing properties of $K$. If we generate the halfgroupoid $K$ from the maximal extension chain of $G$ in $H$, shouldn't any closed subhalfgroupoid $L$ of $H$ that has $G$ as a subhalfgroupoid be itself a subhalfgroupoid of $K$, rather than the other way around?
