**ALGEBRAIC STRUCTURE**

Gà a non-empty set.

G with one or more binary operations is known as algebraic structures.

**For examples:**

1)

**, where ‘*’ is an binary operation on Set/Group ‘G’. Than (G,*) is an algebraic group.***(G, *)*
2)

**, where ‘+’ is an binary operation on Set/Group ‘N’,set of natural numbers.***(N, +)*
3)

**, where ‘+’ is an binary operation on Set/Group ‘I’, set of integer numbers.***(I, + )*
4)

**, where ‘-‘ is an binary operation on Set/Group ‘I’, set of integer numbers.***(I, - )*
5)

**, where ‘ + ‘ and ‘ * ‘ are two binary operations on Set/Group ‘R’, set of real numbers.***(R, +, *)*
6)

*(R, +, .)*
7)

**etc.***(I, +, .)***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).