Direct factor of a loop

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 loop of a property encountered in group. Specifically, it is a subloop property analogous to the subgroup property: direct factor
View other analogues of direct factor | View other analogues in loops of subgroup properties (OR, View as a tabulated list)

Definition

A direct factor of an algebra loop $L$ is a subloop $S$ such that there exists a subloop $T$ of $S$ satisfying:

• Every element of $L$ can be written uniquely in the form $s*t,s\in S,t\in T$.
• For any $s_{1},s_{2}\in S,t_{1},t_{2}\in T$, we have $\!(s_{1}*t_{1})*(s_{2}*t_{2})=(s_{1}*s_{2})*(t_{1}*t_{2})$. In particular, by an idea analogous to the Eckmann-Hilton principle, every element of $S$ commutes with every element of $T$.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
central factor of a loop	Product with another subloop is whole loop, they commute in the strong sense indicated here		(proof for groups suffices)	|FULL LIST, MORE INFO
normal subloop				|FULL LIST, MORE INFO
Lagrange-like subloop				|FULL LIST, MORE INFO
retract of a loop	image of the whole loop under a retraction, i.e., an endomorphism whose fixed point set equals its image			|FULL LIST, MORE INFO