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- by Roshan Maharjan 3 months ago

Algebraic Structure:

A set with one or more binary operations gives rise to what is commonly known as a algebraic structure. In particular ,the set of integers under the addition operation '+' is an algebraic structure .It is commonly denoted by (Z,+). In the same way ,the set of rational numbers Q under the usual multiplication operation '*',and denoted by (Q,*),is another algebraic structure .A more complicated algebraic structure is the set of real numbers R together with the two usual operations: addition '+' and multiplication '*' . Such an algebraic structure is denoted by (R,+,*).Algebraic structures with one or more binary operations are given special names depending upon additional properties involved.

An algebraic structure consisting of a set G under an operation * on G, and denoted by(G,*), may enjoy one or more of the following characteristics :

given a,b,c,..... as the elements of the set G, the algebraic structure (G,*) may be

- Closed if a*b belongs to G for each a,b belonging to G.
- Commutative if a*b =b*a belongs to G, for each a,b belonging to G.
- Associative if (a*b)*c =a*(b*c), for each a,b,c belonging to G.
- Existence of identity element: For each 'a' belonging to G, if there exists an element 'e' belonging to G, such that a*e=a=e=e*a then 'e' is called the identity element.
- Existence of the inverse element : For each 'a' belonging to G, if there exists an element a' belonging to G such that a*a'=e=a'*a then a' is called the inverse of the element a.

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