Algebraic Structure
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. 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. Monoid A non-empty set S, (S,*) is called a monoid 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 ...