Bol loop

In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in .
A loop, "L", is said to be a left Bol loop if it satisfies the identity
while "L" is said to be a right Bol loop if it satisfies
These identities can be seen as weakened forms of associativity.
A loop is both left Bol and right Bol if and only if it is a Moufang loop. Different authors use the term "Bol loop" to refer to either a left Bol or a right Bol loop.
Bruck loops.
A Bol loop satisfying the "automorphic inverse property," ("ab")−1 = "a"−1 "b"−1 for all "a,b" in "L", is known as a (left or right) Bruck loop or K-loop (named for the American mathematician Richard Bruck). The example in the following section is a Bruck loop.
Bruck loops have applications in special relativity; see Ungar (2002). Left Bruck loops are equivalent to Ungar's (2002) gyrocommutative gyrogroups, even though the two structures are defined differently.
Example.
Let "L" denote the set of "n x n" positive definite, Hermitian matrices over the complex numbers. It is generally not true that the matrix product "AB" of matrices "A", "B" in "L" is Hermitian, let alone positive definite. However, there exists a unique "P" in "L" and a unique unitary matrix "U" such that "AB = PU"; this is the polar decomposition of "AB". Define a binary operation * on "L" by "A" * "B" = "P". Then ("L", *) is a left Bruck loop. An explicit formula for * is given by "A" * "B" = ("A B"2 "A")1/2, where the superscript 1/2 indicates the unique positive definite Hermitian square root.
Bol algebra.
A (left) Bol algebra is a vector space equipped with a binary operation formula_3 and a ternary operation formula_4 that satisfies the following identities: 
and
and
and
If is a left or right alternative algebra then it has an associated Bol algebra , where formula_9 is the commutator and formula_10 is the Jordan associator.