Flexible loop

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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 Diassociative loop, Extra loop, Flexible G-loop, Moufang loop, Universally flexible loop|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 Diassociative loop|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 Universally flexible loop|FULL LIST, MORE INFO