数值半群的Delta集的实现问题
The Realization Problem for Delta Sets of Numerical Semigroups
摘要：
数值半群$S$的δ集，记作$Δ(S)$，是一个因子分解不变量，它度量了$S$中元素长度集合的复杂性。我们研究了下列问题：哪些有限集作为数值半群$S$的δ集出现？已知$\min Δ(S) = \gcd Δ(S)$是必要条件。对于任何二元集$\{d,td\}$，我们产生一个半群$S$与此δ集。然后证明了对于$t\ge 2$，集合$\{d,td\}$作为嵌入维数为3的某个数值半群的δ集出现当且仅当$t=2$ .
英文摘要：
The delta set of a numerical semigroup $S$, denoted $Δ(S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min Δ(S) = \gcd Δ(S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup $S$ with this delta set. We then show that for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension three if and only if $t=2$.
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