I-simple IAPS

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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.