. . Definitions The basic arithmetic operations on are addition ( + ) , subtraction ( - ), multiplication ( × ) , and division ( ÷ ). Eminent mathematicians of the latter part of th century and in th century like Abel, Cayley, Cauchy, and others, tried to generalize the properties satisfied by these usual arithmetic operations.
To this end they developed new abstract algebraic structures through the axiomatic approach . This new branch of algebra dealing with these abstract algebraic structures is known as abstract algebra . To begin with, consider a simple example involving the basic usual arithmetic operations addition and multiplication of any two natural numbers. ∈ ; m n × ∈ , ∀ m n { , , ,...} Each of the above two operations yields the following observations: ( ) At a time exactly two elements of are processed.
( ) The resulting element (outcome) is also an element of . Any such operation defined on a nonempty set is called a binary operation or a binary composition on the set in abstract algebra. Definition . Any operation * defined on a non-empty set S is called a binary operation on S if the following conditions are satisfied: (i) The operation * must be defined for each and every ordered pair ( , ) a b ∈ S S × (ii) It assigns a unique element a b ∗ of S to every ordered pair ( , ) a b ∈ S S × In other words, any binary operation * on S is a rule that assigns to each ordered pair of elements of S a unique element of S .
Also * can be regarded as a function ( mapping ) with input in the Cartesian product S S × and the output in S . ∗ × : S S S ; ∗ ∗∈ a b a b S , where a b * is an unique element.