Minimal-family characterization of the ideals \mathcal{I}_{h,n}

Let $\mathcal{I}_{h,n}$ be a family of polynomial ideals in the variables $X_1,\ldots,X_n$, and let $e_i(\mathbf{X}_n)$ denote the elementary symmetric polynomials. The family is required to satisfy the three closure properties introduced earlier, and to contain all polynomials listed in Proposition~. Minimal-family characterization. The family $\mathcal{I}_{h,n}$ is the minimal family satisfying

$$\mathcal{I}_{0,n}=\langle X_i \mid i=1,\ldots,n\rangle,$$ $$\mathcal{I}_{n-1,n}=\mathcal{I}_{n,n}=\langle e_i(\mathbf{X}_n)\mid i\geq 1\rangle,$$

with $0\leq h\leq n$, all three closure properties, and inclusion of every polynomial listed in Proposition~. This characterization is part of the ideal framework used to study polynomial identities for monotone trapezoids with prescribed bottom row; its resolution status is not specified in the supplied text.