Semi Group

A non-empty set S, (S,*) is called a semigroup if it follows the following axiom: 

• Closure:(a*b) belongs to S for all a, b ? S.  
• Associativity: a*(b*c) = (a*b)*c ? a, b ,c belongs to S.

Note: A semi-group is always an algebraic structure. 

Example: (Set of integers, +), and (Matrix ,*) are examples of semigroup.  

Algebraic Structure

A non empty set S is called an algebraic structure w.r.t binary operation (*) if it follows the following axioms: 

Closure:(a*b) belongs to S for all a,b ? S. 

Example: 

S = {1,-1} is algebraic structure under * 
As 1*1 = 1, 1*-1 = -1, -1*-1 = 1 all results belong to S. 

But the above is not an algebraic structure under + as 1+(-1) = 0 not belongs to S.