If you know what a group is, then you can define a loop to be a set with a binary operation that satisfies all the group properties except possibly associativity. Specifically,

A LOOP is a nonempty set, call it G, together with an operation, call it *, that satisfies the following four properties:

1. For all elements a and b of G, a*b is an element of G.
   
   
2. For all elements a and b of G, there is a unique element x in G for which a*x = b.
   
   
3. For all elements a and b of G, there is a unique element y in G for which y*a = b.
   
   
4. G has an identity element, call it e: e*x = x*e = x for all elements x in G.

Chris