论文标题
有限弹性的表征
The Characterization of Finite Elasticities
论文作者
论文摘要
我们的激励目标是在Krull域中分解$ h $，其中有限生成的集体组$ g $。弹性$ρ（h）$是$ k $原子的任何重新分离的原子数量。弹性与零和序列$ b（g_0）$组合的组合的弹性相同，其中$ g_0 \ subseteq g $是具有高度一prime的类。我们表征了有限生成的类组的任何Krull域何时有限弹性。我们的结果对于更通用的转移Krull Monoids（在有限生成的Abelian Group $ G $的子集$ G_0 $上）是有效的。我们显示有一个最小的$ s \ leq（d+1）m $，其中$ d $是免费等级，$ m $是扭力指数，因此$ρ_s（h）<\ infty $含义$ρ_K（h）<\ infty $ at $ k \ e k \ geq feqq \ egeq 1 $ $。这样可以确保$ρ（h）<\ infty $，并且仅当$ρ_{（d+1）m}（h）<\ infty $。 Our characterization is in terms of a simple combinatorial obstruction to infinite elasticity: there existing a subset $G_0^\diamond\subseteq G_0$ and bound $N$ such that there are no nontrivial zero-sum sequences with terms from $G_0^\diamond$, and every minimal zero-sum sequence has at most $N$ terms from $G_0\setminus g_0^\钻石$。我们对$ g_0^\ diamond $的明确说明在凸的几何形状上，$ g_0 $ modulo torsion子组$ g_t \ leq g $，并且显示有限的弹性与此明确定义的子集的元素相同的元素不存在正面线性的组合，均等$ 0 $ $ g_ $ g_ $ g_ $ g_ $ g_ $ g_。我们使用结果表明有限的弹性意味着一组距离$δ（h）$，capenary级$ \ Mathsf C（H）$（对于Krull Monoids）和弱化的驯化度（对于Krull Monoids）也是有限的，并且对于Unions的结构定理。我们在转移Krull单体方面进行分解的结果是通过在凸几何形状中开发出广泛的正碱基的广泛理论来实现的。
Our motivating goal is factorization in Krull Domains $H$ with finitely generated class group $G$. The elasticity $ρ(H)$ is the maximal number of atoms in any re-factorization of a product of $k$ atoms. The elasticities are the same as those of a combinatorial monoid of zero-sum sequences $B(G_0)$, where $G_0\subseteq G$ are the classes with height one primes. We characterize when finite elasticity holds for any Krull Domain with finitely generated class group. Our results are valid for the more general class of Transfer Krull Monoids (over a subset $G_0$ of a finitely generated abelian group $G$). We show there is a minimal $s\leq (d+1)m$, where $d$ is the torsion free rank and $m$ is the torsion exponent, such that $ρ_s(H)<\infty$ implies $ρ_k(H)<\infty$ for all $k\geq 1$. This ensures $ρ(H)<\infty$ if and only if $ρ_{(d+1)m}(H)<\infty$. Our characterization is in terms of a simple combinatorial obstruction to infinite elasticity: there existing a subset $G_0^\diamond\subseteq G_0$ and bound $N$ such that there are no nontrivial zero-sum sequences with terms from $G_0^\diamond$, and every minimal zero-sum sequence has at most $N$ terms from $G_0\setminus G_0^\diamond$. We give an explicit description of $G_0^\diamond$ in terms of the Convex Geometry of $G_0$ modulo the torsion subgroup $G_T\leq G$, and show finite elasticity is equivalent to there being no positive linear combination of the elements of this explicitly defined subset equal to $0$ modulo $G_T$. We use our results to show finite elasticity implies the set of distances $Δ(H)$, the catenary degree $\mathsf c(H)$ (for Krull Monoids) and a weakened tame degree (for Krull Monoids) are all also finite, and that the Structure Theorem for Unions holds. Our results for factorization in Transfer Krull Monoids are accomplished by developing an extensive theory in Convex Geometry generalizing positive bases.
