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 (Bol 1937).

A loop, L, is said to be a left Bol loop if it satisfies the identity

${\displaystyle a(b(ac))=(a(ba))c}$, for every a,b,c in L,

while L is said to be a right Bol loop if it satisfies

${\displaystyle ((ca)b)a=c((ab)a)}$, for every a,b,c in L.

These identities can be seen as weakened forms of associativity, or a strengthened form of (left or right) alternativity.

A loop is both left Bol and right Bol if and only if it is a Moufang loop. Alternatively, a right or left Bol loop is Moufang if and only if it satisfies the flexible identity a(ba) = (ab)a . Different authors use the term "Bol loop" to refer to either a left Bol or a right Bol loop.