# 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 of an APS is termed a **saturated sub-APS** if for any , the inverse image via of is precisely .

## For groups

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

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

Further, the following are equivalent:

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