Let us consider the set of all complexes of a group , which in nothing but power set of . Let it be denoted by . Now, we define three binary composition in . The two compositions namely union and intersection of sets are familiar ones. How we define the multiplication of complexes.
Multiplication is Complexes
Let and be two complexes of a group whose composition has been denoted multiplicatively, then the product of and denoted by is defined as
In other words is the set of all possible products of elements of with those of . It is evident that .
Thus the product of two complexes is also a complex of the group.
Multiplication of Complexes is Associative
Let and be three complexes of a group whose composition is denoted multiplicatively. Then
Because , multiplication in being associative
Inverse of Complexes in a Group
Let be any complex of and let us define
Then is the complex of consisting of the inverse of the elements of . This is called the inverse of complex .