Simple loop

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.

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 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.