The term orderings which are compatible with composition II

Polynomial composition is the operation of replacing the variables of a polynomial with other polynomials. Let> be a term ordering, if for all terms p and q, p>q implies that p lt>(Θ)>q lt>(Θ), then we say that composition by Θ is compatible with the term ordering >. In (J. Symb. Comput. 25 (1998) 643), Hong proposed the question of how to test if a term ordering is compatible with composition. In this paper, by using elementary rational row operations for matrices, we obtain a decision procedure for Hong’s question.