Saturated sub-APS

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:


 * $$H$$ is a saturated sub-APS of $$G$$.
 * The left congruence induced by $$H$$ is a saturated APS relation.
 * The coset space APS of $$H$$ in $$G$$ is an IAPS (of sets)

Further, the following are equivalent:


 * $$H$$ is a saturated normal sub-APS of $$G$$.
 * The congruence induced by $$H$$ is a saturated APS congruence.
 * The quotient APS is an IAPS of groups.