Commutative loop
(Redirected from Abelian loop)
This article defines a property that can be evaluated for a loop.
View other properties of loops
ANALOGY: This is an analogue in loop of a property encountered in group. Specifically, it is a loop property analogous to the group property: abelian group
View other analogues of abelian group | View other analogues in loops of group properties (OR, View as a tabulated list)
Definition
A loop is termed a commutative loop or abelian loop if its binary operation is a commutative binary operation, i.e., it is a commutative magma under its multiplication.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Commutative Moufang loop | ||||
| Abelian group |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Commutative magma | ||||
| Flexible loop |