ALGEBRAIC STRUCTURES
Algebraic Structure (Set + Operation + Rules) A non empty set S is called an algebraic structure w.r.t binary operation (*) if it follows following axiom (rules):
1) 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.
2) 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.
(N, +) is a semi group.
(N, .) is a semi group.
(N, – ) is not a semi group……. WHY? (3-5 = -2, and -2 does not
belong to N)
3)IDENTITY ELEMENT :
The element of a set of numbers that when combined with another number in a particular operation leaves that number unchanged.
For example, 0 is the identity element under addition for the real numbers, since if a is any real number, a + 0 = 0 + a = a.
Similarly, 1 is the identity element under multiplication for the real numbers, since a × 1 = 1 × a = a.
4) 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.
5) 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.
Example: Z- set of –ve and +ve integers (Z,+) ( What is 5-1? Ans: -5)
6) 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
Note : (Z,+) is a example of Abelian Group
LETS COMPARE ALL
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Ex. Let (Z, *) be an algebraic structure, where Z is the set of integers and the operation * is defined by n * m = maximum of (n, m). Show that (Z, *) is a semi group and not a monoid.
Solution:
Let a , b and c are any three integers.
Closure property: Now, a * b = maximum of (a, b) ∈ Z for all a, b ∈ Z
Associativity : (a * b) * c = maximum of {a, b, c} = a * (b * c)
So, (Z, *) is a semi group.
Identity : There is no integer x such that a * x = maximum of (a, x) = a for all a ∈ Z
As Identity element does not exist. Hence, (Z, *) is not a monoid.
Ex. Show that the set of all positive rational numbers forms an abelian group under the composition * defined by a * b = (ab)/2 .
Solution:
Let A = set of all positive rational numbers.
Let a,b,c be any three elements of A.
1. Closure property: We know that, Product of two positive rational numbers is again a rational number. i.e., a *b ∈ A for all a,b ∈ A .
2. Associativity: (a*b)*c = (ab/2) * c = (abc) / 4 a*(b*c) = a * (bc/2) = (abc) / 4
3. Identity : Let e be the identity element.
We have a*e = (a e)/2 …(1) , By the definition of * Try to find out an element ‘e’ such that a*e = a …..(2) ,
E.g. (5X2)/2 =2 e = 2 and 2 ∈ A
Identity element exists, and ‘2’ is the identity element in A.
4. Inverse: Let a ∈ A let us suppose b is inverse of a
Now, a * b = (a b)/2 ….(1)
Again, a * b = e = 2 …..(2) (By definition of inverse)
From (1) and (2), it follows that
(a b)/2 = 2 🡪 b = (4 / a) ∈ A So (A ,*) is a group.
5. Commutativity: a * b = (ab/2) = (ba/2) = b * a
Hence, (A,*) is an abelian group.