Algebraic Structures
G -> a non-empty set.
G with one or more binary operations is known as algebraic structures.
For examples
1) (G, ) , where ‘’ is an binary operation on Set/Group ‘G’. Than (G,*) is an algebraic group.
2) (N, +), where ‘+’ is an binary operation on Set/Group ‘N’,set of natural numbers.
3) (I, + ), where ‘+’ is an binary operation on Set/Group ‘I’, set of integer numbers.
4) (I, – ), where ‘-‘ is an binary operation on Set/Group ‘I’, set of integer numbers.
5) (R, +, *), where ‘ + ‘ and ‘ * ‘ are two binary operations on Set/Group ‘R’, set of real numbers.
6) (R, +, .)
7) (I, +, .) etc.
Properties of an Algebraic Structure
1) Associative and Commutative Laws
(a * b)* c = a * (b * c)
(a * b ) = (b * a)
2) Identity element and Inverses
a * e = e * a = a, where e à identity element
Left identity element,
e * a = a.
Right identity element,
a * e = a.
If an binary operation ‘ * ‘ is not having an identity element,
Than,
inverse of an element ‘a’ in set is ‘b’.
a * b = b * a = e
3) Cancellation Laws
Left cancellation law:
a * b = a * c, implies b = c ( ‘a’ of both sides get cancelled).
Right cancellation law:
b * a = c * a, implies b = c (‘a’ of both sides get cancelled).