Left Bol loop
From Groupprops
This article defines a property that can be evaluated for a loop.
View other properties of loops
Contents
Definition
Definition with symbols
An algebra loop with binary operation is said to be a left Bol loop if it satisfies the following identity for all :
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Left Bruck loop | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Power-associative loop | powers are well-defined | |FULL LIST, MORE INFO | ||
Left-inverse property loop | left inverses (in a strong sense) are well-defined | |FULL LIST, MORE INFO | ||
Left alternative loop | |FULL LIST, MORE INFO |
Property obtained by the opposite operation
The dual notion to that of left Bol loop is that of a right Bol loop. The theory runs exactly the same.
Facts
Inverses
A left Bol loop has the property that the subloop generated by any element is a subgroup. Thus, we can define the inverse of an element in the left bol loop as its inverse in that subgroup. Also, it is clear that this is both the left and right inverse in the whole algebra loop.
Further, we can talk of the order of an element in a left Bol loop as the order of the subgroup generated by it.
References
- Infinite simple Bol loops by Hubert Kiechle and Michael K. Kinyon
- Finite Bruck loops by MIchael Aschbacher, Michael K. Kinyon, and J. D. Phillips