Normal subloop

This article defines a property that can be evaluated for a subloop of a loop| View other such properties

ANALOGY: This is an analogue in algebra loops of the subgroup property:
View other analogues of normality | View other analogues in algebra loops of subgroup properties

Definition

Definition with symbols

A subloop $N$ of an algebra loop $L$ is said to be normal if, for any $a,b\in L$, the following holds:

$(a*b)*N=a*(b*N)=a*(N*b)$

Note that the equality of the firsts two is not guaranteed because we do not assume the algebra loop to be associative.

Facts

Quotient by a normal subloop

Given a loop, and a normal subloop, we can define a corresponding quotient loop. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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Left multiplication group corresponding to a subloop

The following are true:

• Given a normal subloop, the left multiplication group corresponding to that subloop, is a normal subgroup of the left multiplication group corresponding to the whole algebra loop. Notice that for this, we crucially need the equality of all three parts: $a*(b*N)=(a*b)*N=a*(N*b)$.
• Further, the left multiplication group of the quotient loop equals the quotient of the left multiplication group of the whole loop, by that of the subgroup.