Mathematics | Algebraic Structure
Algebraic Structure
A non empty set S is called an algebraic structure w.r.t binary operation (*) if it follows following axioms:
- Closure:(a*b) belongs to S for all a ,b ∈ S.
Ex : S = {1,-1} is algebraic structure under *
As 1*1 = 1, 1*-1 = -1, -1*-1 = 1 all results belongs to S.
But above is not 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.
Ex : (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 to S.
- Identity Element: There exists e ∈ S such that a*e = e*a = a ∀ a ∈ S
Note: A monoid is always a semi-group and algebraic structure.
Ex : (Set of integers,*) is Monoid as 1 is an integer which is also identity element .
(Set of natural numbers, +) is not Monoid as there doesn’t exist any identity element. But this is Semigroup.
But (Set of whole numbers, +) is Monoid with 0 as identity element.
Group
A non-empty set G, (G,*) is called a group if it follows the following axiom:
- Closure:(a*b) belongs to G for all a, b ∈ G.
- Associativity: a*(b*c) = (a*b)*c ∀ a, b, c belongs to G.
- Identity Element: There exists e ∈ G such that a*e = e*a = a ∀ a ∈ G
- Inverses:∀ a ∈ G there exists a-1 ∈ G such that a*a-1 = a-1*a = e
Note:
- A group is always a monoid, semigroup, and algebraic structure.
- (Z,+) and Matrix multiplication is example of group.
Abelian Group or Commutative group
A non-empty set S, (S,*) is called a Abelian group 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.
- Identity Element: There exists e ∈ S such that a*e = e*a = a ∀ a ∈ S
- Inverses:∀ a ∈ S there exists a-1 ∈ S such that a*a-1 = a-1*a = e
- Commutative: a*b = b*a for all a, b ∈ S
For finding a set lies in which category one must always check axioms one by one starting from closure property and so on.
Here are the some important results-
Must Satisfy Properties | |
Algebraic Structure | Closure |
Semi Group | Closure, Associative |
Monoid | Closure, Associative, Identity |
Group | Closure, Associative, Identity, Inverse |
Abelian Group | Closure, Associative, Identity, Inverse, Commutative |
Here a Table with different non empty set and operation
N=Set of Natural Number
Z=Set of Integer
R=Set of Real Number
E=Set of Even Number
O=Set of Odd Number
M=Set of Matrix
+,-,×,÷ are the operations.
Set, Operation | Algebraic Structure | Semi Group | Monoid | Group | Abelian Group |
N,+ | Y | Y | X | X | X |
N,- | X | X | X | X | X |
N,× | Y | Y | Y | X | X |
N,÷ | X | X | X | X | X |
Z,+ | Y | Y | Y | Y | Y |
Z,- | Y | X | X | X | X |
Z,× | Y | Y | Y | X | X |
Z,÷ | X | X | X | X | X |
R,+ | Y | Y | Y | Y | Y |
R,- | Y | X | X | X | X |
R,× | Y | Y | Y | X | X |
R,÷ | X | X | X | X | X |
E,+ | Y | Y | Y | Y | Y |
E,× | Y | Y | X | X | X |
O,+ | X | X | X | X | X |
O,× | Y | Y | Y | X | X |
M,+ | Y | Y | Y | Y | Y |
M,× | Y | Y | Y | X | X |
This article is contributed by Abhishek Kumar.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.