A loop is a pair (L, ·) where L is a set and (a, b) 7→ a · b is a binary operation on L relative to which there is a two-sided identity element and with the property that for each a ∈ L, the left and right translation maps R(a) and L(a), defined by

xR(a) = xa, xL(a) = ax,

are bijections. For a, b, c ∈ L, the commutator (a, b) and associator (a, b, c) are, respectively, the unique elements satisfying

ab = ba(a, b), ab · c = [a · bc](a, b, c).