Sub-APS of groups

This article gives a basic definition in the following area: APS theory
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Definition

Let $(G,\Phi )$ be an APS of groups. A sub-APS $H$ of $G$ is, for every $n$, a subgroup $H_{n}$ of $G_{n}$ such that $\Phi _{m,n}(g,h)\in H_{m+n}$ whenever $g\in H_{m},h\in H_{n}$. Thus, $H$ can be viewed as an APS of groups, in its own right.