Saturated sub-APS

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This article gives a basic definition in the following area: APS theory
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This article gives a property of a sub-APS in an APS of groups, where the condition is purely set-theoretic in terms of the concatenation maps

Definition

A sub-APS H of an APS (G,\Phi) is termed a saturated sub-APS if for any (m,n), the inverse image via \Phi_{m,n} of H_{m+n} is precisely H_m \times H_n.

For groups

For an APS G of groups with a sub-APS H, the following are equivalent:

Further, the following are equivalent: