# I-simple IAPS

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

*This article defines a property that can be evaluated for an IAPS of groups*

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

View other analogues of simple group | View other analogues in IAPSs of group properties (OR, View as a tabulated list)

## Definition

### Symbol-free definition

An IAPS of groups is termed **i-simple** if it has no proper nontrivial saturated sub-IAPS. Equivalently, it is a simple object in the category of IAPSes with IAPS homomorphisms.

## Definition

### Stronger properties

## Facts

### I-simple Abelian IAPSes

The only i-simple Abelian IAPSes are the power APSes over simple Abelian groups. The proof is direct.

### I-simple non-Abelian IAPSes

I-simple non-Abelian IAPSes satisfy a similar principle as simple non-Abelian groups. Namely, any sub-IAPS-defining function on such a thing must either give a contrasaturated sub-IAPS, or the trivial sub-IAPS. This principle shows, in particular, that any i-simple IAPS is i-perfect.