J. Symbolic Computation (1999) 11, 1-000

We prove that any simpli cation order over arbitrary terms is an extension of an order by weight, by considering a related monadic term algebra called the spine. We show that any total ground-stable simpli cation order on the spine lifts to an order on the full term algebra. Conversely, under certain restrictions, a simpli cation ordering on the term algebra de nes a weight function on the spine, which in turn can be lifted to a weight order on the original ground terms which contains the original order. We investigate the Knuth Bendix and polynomial orders in this light. We provide a general framework for ordering terms by counting embedded patterns, which gives rise to many new orderings. We examine the recursive path order in this context.