Example of a closure operator
Think of the preorder of arithmatic expressions such as \(12, 4+2+4, 9*(2+3)\), where the \(\leq\) operators denotes whether an expression can be simplified to another.
A computer program \(j\) that evaluates expressions is a monotonic function on the preorder to itself (if you can reduce x to y, then \(j(x)\) should be able to be rewritten as \(j(y)\).
The requirements of closure operator say that \(j\) should be a simplification, and that trying to reduce an expression which has already been reduced will do nothing further.