Saturated sub-APS

This article gives a basic definition in the following area: APS theory
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This article defines a group sub-IAPS property: a property that can be evaluated to true/false given an APS of groups and a sub-APS.

Definition

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

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.

When the APS is an IAPS of groups, the saturated sub-APS becomes a saturated sub-IAPS.

Examples

• In the power APS of a group, the power APS of any subgroup is saturated.
• In the GL IAPS (the IAPS of general linear groups over a unital ring), the GL IAPS for any unital subring is saturated. So are the unitriangular sub-IAPS and Borel sub-IAPS.
• The IAPS of symmetric groups is saturated in the GL IAPS.