AI 中文总结
研究分次素理想的多种推广，包括分次n-吸收等理想及子群。通过将主单项式理想与指数向量识别，为分次n-吸收主单项式理想建立组合模型，对应单纯形格点，实现哈斯图为凯莱图1-骨架，给出分次n-吸收的几何和组合解释。
AI 中文摘要
分次n-吸收理想通过将吸收性质扩展到(n + 1)个齐次元素的乘积来推广分次素理想。我们研究了分次素理想的几种推广，包括分次n-吸收、分次弱n-吸收、分次强n-吸收和分次n-吸收准素理想，以及相关的分次n-吸收子群。我们的主要结果为具有标准分次的多项式环中的分次n-吸收主单项式理想建立了一个组合模型。通过将主单项式理想与\(\mathbb{N}^{m}\)中的指数向量进行识别，我们表明分次n-吸收主单项式理想恰好对应于单纯形\(\{\alpha \in \mathbb{N}^{m}: \vert \alpha \vert \leq n\}\)中的格点。因此，主单项式理想的哈斯图被实现为\(\mathbb{N}^{m}\)的凯莱图的1-骨架，给出了分次n-吸收的几何和组合解释。
英文摘要
Graded $n$-absorbing ideals generalize graded prime ideals by extending absorption properties to products of $(n+1)$ homogeneous elements. We study several generalizations of graded prime ideals, including graded $n$-absorbing, graded weakly $n$-absorbing, graded strongly $n$-absorbing, and graded $n$-absorbing primary ideals, as well as related graded $n$-absorbing subgroups. Our primary result establishes a combinatorial model for graded $n$-absorbing principal monomial ideals in polynomial rings with the standard grading. By identifying principal monomial ideals with exponential vectors in $\mathbb{N}^{m}$, we show that the graded $n$-absorbing principal monomial ideals correspond precisely to lattice points in the simplex $\{α\in \mathbb{N}^{m}: \vert α\vert \leq n\}.$ Consequently, the Hasse diagram of principal monomial ideals is realized as the 1-skeleton of the Cayley graph of $\mathbb{N}^{m}$, yielding a geometric and combinatorial interpretation of graded $n$-absorption.
Comments21 pages, 4 figures
