Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 27 / 10 / c

Absorption says $xy\le x$; commutativity also gives $xy\le y$. Iterating this inequality shows that a monomial with larger exponents is smaller in the natural order. This is the link from exponent vectors to the reverse-inclusion well-quasi-ordering of monomial ideals in solution.