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 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.
Example:
(Set of integers,*) is Monoid as 1 is an integer which is also an 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 that lies in which category one must always check axioms one by one starting from closure property and so on.
Here are 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 |
Note:
Every abelian group is a group, monoid, semigroup, and algebraic structure.
Here is a Table with different nonempty 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 |