Using associativity we can define a very basic structure.
Definition
A semi-group is a structure (S,*) consisting of a set S and a binary associative operation *, called multiplication.
More advanced structures, as we will see later, on usually consist of a semi-group with some additional operations.
When considering a semi-group (S,*), we often speak of the semi-group S if it is clear what the associative multiplication * is.
Definition
e*a = a*e = a
A unit e in a semi-group
S is an element e of S with the property that, for
all a 
S,