Flexible loop

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

Definition

Symbol-free definition

A loop is termed a flexible loop if its underlying magma is a flexible magma.

Definition with symbols

A loop $(L,*)$ is termed a flexible loop if it satisfies the following identity for all $x,y\in L$:

$\!x*(y*x)=(x*y)*x$

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Commutative loop	any two elements commute	commutative implies flexible	flexible not implies commutative	|FULL LIST, MORE INFO
Group	any three elements associate			|FULL LIST, MORE INFO
Left Bol loop	$(x*y)*(x*z)=(x*(y*x))*z$			|FULL LIST, MORE INFO
Right Bol loop				|FULL LIST, MORE INFO
Moufang loop	both left and right Bol			|FULL LIST, MORE INFO
Universally flexible loop	all loops isotopic to it are flexible			|FULL LIST, MORE INFO
Flexible G-loop	flexible, isomorphic to all isotopic loops			|FULL LIST, MORE INFO