Padding-saturated sub-IAPS

This article defines a group sub-IAPS property: a property that can be evaluated to true/false given an IAPS of groups and a sub-IAPS.

Definition

Let $(G, Φ)$ be an IAPS of groups and $H$ be a sub-IAPS. We say that $H$ is padding-saturated in $G$ if both these conditions hold for all positive integers $m$ and $n$ :

The intersection of $H_{m + n}$ with the image of the restriction of the concatenation map $Φ_{m, n} : G_{m} \times G_{n} \to G_{m + n}$ to $1 \times G_{n}$ is precisely $1 \times H_{n}$ .
he intersection of $H_{m + n}$ with the image of the restriction of the concatenation map $Φ_{m, n} : G_{m} \times G_{n} \to G_{m + n}$ to $G_{m} \times 1$ is precisely $H_{m} \times 1$ .

In other words, we cannot use padding on either the left or the right to take elements outside $H$ inside a later member of $H$ .

Examples

Inside the GL IAPS, the special linear groups form a padding-saturated sub-IAPS but not a saturated sub-IAPS. This means, concretely, that block concatenation with the identity matrix cannot take stuff outside the special linear group into the special linear group; however, block concatenation in general is capable of concatenating two elements outside their respective special linear groups into an element in the special linear group (for instance, two 1-dimensional matrices with value -1 concatenate to a determinant 1 matrix). In fact, this is an example of a padding-saturated sub-IAPS that is also a contrasaturated sub-IAPS.
Inside the IAPS of symmetric groups, the sub-IAPS of alternating groups is padding-saturated but not saturated. In fact, this is an example of a padding-saturated sub-IAPS that is also a contrasaturated sub-IAPS.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
saturated sub-IAPS	concatenating things outside the sub-IAPS cannot get inside the sub-IAPS	(obvious)	sub-IAPS of alternating groups inside IAPS of symmetric groups, special linear groups inside GL IAPS