Loop

A loop is a quasigroup with an identity element. That is, a set $L$ with a binary operation such that:

1. Identity: There exists $e \in L$ with $ea = ae = a$ for all $a \in L$
2. Divisibility: For all $a, b \in L$, the equations $ax = b$ and $ya = b$ have unique solutions

A loop that is also associative is a group . The smallest non-associative loop has 5 elements.

Examples:

• Unit octonions under multiplication (non-associative!)
• Moufang loops
• All groups are loops