Inverse property loop
This article defines a property that can be evaluated for a loop.
View other properties of loops
Contents
Definition
A loop is termed an inverse property loop or inverse loop or IP-loop if it satisfies the following equivalent conditions:
- Existence of left and right inverses: There exist bijective maps such that .
- Existence of two-sided inverses: There exists a bijective map such that for all .
Equivalence of definitions
Further information: equivalence of definitions of inverse property loop
Note that for a quasigroup, the existence of both left and right inverses does not guarantee the existence of two-sided inverses.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Group | |FULL LIST, MORE INFO | |||
Automorphic inverse property loop | |FULL LIST, MORE INFO | |||
Moufang loop | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Left-inverse property loop | ||||
Right-inverse property loop |