  In this paper we develop a theory of monomial preorders, which differ from
the classical notion of monomial orders in that they allow ties between
monomials. Since for monomial preorders, the leading ideal is less degenerate
than for monomial orders, our results can be used to study problems where
monomial orders fail to give a solution. Some of our results are new even in
the classical case of monomial orders and in the special case in which the
leading ideal defines the tangent cone.
