Left Bol loop

Definition with symbols
An algebra loop $$L$$ with binary operation $$*$$ is said to be a left Bol loop if it satisfies the following identity for all $$x, y, z \in L$$:

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

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.

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.