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Inverse Element

Let \(A\) be a set , and \(\ast:A \times A \rightarrow A\) a binary operation on \(A\). For any element \(a\) in \(A\), we define an inverse element of \(a\) to be an element \(b \in A\) such that:

\[ a \ast b=b \ast a=e \]

where \(e\) is defined to be an identity element of \(A\).

(sidenote: We remark that when \(A\) is a group , then the inverse elements are unique. )