For a finite subset M ⊂ [ x 1 ,…, x d ] of monomials, we describe how to constructively obtain a monomial ideal I ⊆ R = K [ x 1 , … , x d ] $I\subseteq R = K[x_{1},\ldots ,x_{d}]$ such that the set of monomials in Soc( I ) ∖ I is precisely M , or such that M ¯ ⊆ R / I $\overline {M}\subseteq R/I$ is a K -basis for the the socle of R / I . For a given M we obtain a natural class of monomials ideals I with this property. This is done by using solely the lattice structure of the monoid [ x 1 ,…, x d ]. We then present some duality results by using anti-isomorphisms between upsets and downsets of the lattice ( ℤ d , ≼ ) $({\mathbb {Z}}^{d},\preceq )$ . Finally, we define and analyze zero-dimensional monomial ideals of R of type k , where type 1 are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in ℤ d ${\mathbb {Z}}^{d}$ .

中文翻译：

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论单项式理想及其社会

对于单项式的有限子集 M ⊂ [ x 1 ,…, xd ]，我们描述了如何建设性地获得单项式理想 I ⊆ R = K [ x 1 , … , xd ] $I\subseteq R = K[x_{1 },\ldots ,x_{d}]$ 使得 Soc( I ) ∖ I 中的单项式集合恰好是 M ，或者使得 M ¯ ⊆ R / I $\overline {M}\subseteq R/I$ 是R / I 的系数的 K 基。对于给定的 M，我们获得了具有这个性质的单项式理想 I 的自然类。这是通过仅使用幺半群 [ x 1 ,..., xd ] 的晶格结构来完成的。然后，我们通过使用晶格 ( ℤ d , ≼ ) $({\mathbb {Z}}^{d},\preceq )$ 的倒置和倒置之间的反同构来呈现一些对偶结果。最后，我们定义并分析了类型 k 的 R 的零维单项式理想，其中类型 1 正是 Artinian Gorenstein 理想，