A binary operation defined by ∗ × : S S S ; ∗ ∗∈ a b a b S demands that the output a b ∗ must always lie the given set S and not in the complement of it. Then we say that ‘ ∗ is closed on S ’ or ‘ S is closed with respect to ∗ ’. This property is known as the closure property . Definition .
Any non-empty set on which one or more binary operations are defined is called an algebraic structure . Another way of defining a binary operation ∗ on S is as follows: ∀ ∗ a b S a b is unique and a b S ∗∈ . Note It follows that every binary operation satisfies the closure property. Note The operation ∗ is just a symbol which may be + × − , , , ÷, matrix addition, matrix multiplication, etc.
depending on the set on which it is defined. For instance, though + and × are binary on , - is not binary operation on . To verify this, consider ( , ) ∈ × . ∗ = − = −∉ a b .
Hence - is not binary operation on . So is to be extended to in order that - becomes binary operation on . Thus is closed with respect to + × , , and . Thus ( , + × − is an algebraic structure.
Observations The binary operation depends on the set on which it is defined. (a) The operation – which is not binary operation on but it is binary on . The set is extended to include zero and negative integers. We call the included set .
(b) The operation ÷ on is not binary operation on . For instance, for ( , ) ∈ × , ÷