# Saturated sub-APS

## Contents

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:

• $H$ 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: