Antichains of monomial ideals are finite (Q2706571)

The main result of the paper is the following: Let \({\mathbf I}\) be an infinite collection of monomial ideals in a polynomial ring. Then there are two ideals \(I,J\in{\mathbf I}\) with \(I\subset J\) (theorem 1.1). With other words, polynomial rings do not contain infinite antichains of monomial ideals. This statement is also true for dual order ideals of \({\mathbb N}^n\) and for the generalized Young's lattice (theorem 1.2 and 1.3). The main result of the paper implies the following two known facts: NEWLINENEWLINENEWLINE-- any polynomial ideal admits only finitely many distinct initial ideals, NEWLINENEWLINENEWLINE-- there are only finitely many monomial ideals having the same Hilbert series (corollary 2.1 and 2.2). NEWLINENEWLINENEWLINEThe paper finishes with some new, but very technical applications to SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) and convex polytopes.