 
Summary: 1. Binary operations.
Suppose X is a set
Definition 1.1. We say is a binary operation on X if
: X × X X.
We say such a binary operation is commutative or Abelian if
(x, y) = (y, x) whenever x, y X.
Definition 1.2. We say e X is an identity element (for the binary opera
tion on X) if
(x, e) = x and (e, x) = x whenever x X.
If e1 and e2 are identity elements for we have
e1 = (e1, e2) = e2.
Thus an identity element for a binary operation, if it exists, is unique and we may
speak of the identity element for the binary operation.
Definition 1.3. We say the binary operation on X is associative if
((x1, x2), x3) = (x1, (x2, x3)) whenever x1, x2, x3 X.
Suppose is associative. For each positive integer n 2 and each j {1, . . . , n
1} we define the map
j,n : Xn
Xn1
on (x1, . . . , xn) Xn
