P-simple IAPS

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This term is related to: APS theory
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ANALOGY: This is an analogue in IAPS of a property encountered in group. Specifically, it is a IAPS property analogous to the group property: simple group
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This article defines a property that can be evaluated for an IAPS of groups

Definition

Symbol-free definition

An IAPS of groups is termed p-simple if it has no strongly proper nontrivial normal sub-IAPS.

Definition with symbols

An IAPS of groups $(G,\Phi )$ is termed p-simple if there is no sub-IAPS $H$ of $G$ satisfying all these conditions:

$H_{n}\triangleleft G_{n}$ for every $n$
There are infinitely many indices $n$ for which $H_{n}$ is properly contained in $G_{n}$
$H_{n}$ is nontrivial for at least some value of $n$