An ideal in a polynomial ring generated by monomials. Its exponent vectors form an upward-closed subset of under the componentwise order. The Dickson lemma makes the minimal exponent set finite. This description permits Maclagan's theorem to compare ideals by reverse inclusion independently of the coefficient field.
Upward-closed subsets of , including the empty subset, form a well-quasi-ordering by reverse inclusion. For , tails correspond to the well-order . Inductively slice in the last coordinate. The slices increase and eventually stabilize: their union has a finite minimal basis by the Dickson lemma, all of which has appeared in one finite slice. Encode an upset by the word of its pre-stabilization slices and its final slice. The Higman lemma and finite product closure of well-quasi-orderings compare these encodings in the reverse-inclusion slice order. A subsequence map has , so containment of the matched later slice implies containment of the slice at ; the final-slice comparison handles the tail. The entire earlier upset therefore contains the later upset.
