 An element \(x\) of \(X\) is a left identity for \(f\) when \(f(x, y) = y\) for all \(y\) in \(X\), and a right identity when \(f(y, x) = y\) for all \(y\) in \(X\). An identity for \(f\) is an element that is both a left identity and a right identity for \(f\). An operation \(f\) can have only one identity, because when \(x\) and \(y\) are identities they are both equal to \(f(x,y)\).