Simple loop

This article defines a property that can be evaluated for a loop.
View other properties of loops

ANALOGY: This is an analogue in algebra loop of a property encountered in group. Specifically, it is a algebra loop property analogous to the group property: simple group
View other analogues of simple group | View other analogues in algebra loops of group properties (OR, View as a tabulated list)

Definition

Symbol-free definition

An algebra loop is said to be simple if it has no proper nontrivial normal subloop. In other words, the only normal subloops are the whole loop and the trivial element.

Definition with symbols

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Analogy

The definition of simple loop generalizes, to algebra loops, the definition of simple group for groups.

Recall that a group is said to be simple if it has no proper nontrivial normal subgroup. The analogue of normal subgroup in algebra loops is normal subloop, and this explains the definition of simple loop.

Relation with other properties

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Facts

Relation with the left multiplication group

If the left multiplication group of an algebra loop is a simple group, then the algebra loop is simple. This follows from the fact that the left multiplication group corresponding to any normal subloop of an algebra group, is a normal subgroup of the left multiplicat group.