广义算术序列生成的数值幺半群的 Delta 集

中文摘要 AI 辅助

设 \\[ S_h=\langle a,ah+d,ah+2d,\ldots,ah+kd\rangle, \qquad \gcd(a,d)=1,\quad 1<k<a,\quad h> 1, \\] 是由广义算术序列生成的数值幺半群。本文确定了其 Delta 集 \\[\Delta(S_h),\\] 解决了 Christopher O'Neill 和 Roberto Pelayo 在 2014 年提出的一个开放问题。记 $e=h-1$，并令 $E(e,d)$ 表示对 $e$ 和 $d$ 应用减法欧几里得算法所得值的集合，我们证明 \\[ \Delta(S_h) = E(h-1,d)\cup \{M-r(h-1):M-r(h-1)>0,\\ r\geq0\}, \\] 其中 \\[ M=\left\lceil\frac{a}{k}\right\rceil(h-1)+d \\] 是 $\Delta(S_h)$ 的最大元素。特别地，$d$ 和 $h-1$ 总是属于 $\Delta(S_h)$。我们进一步建立了一个约化定理，表明 $\Delta(S_h)$ 是满足 $\gcd(d,h-1)=1$ 的原始幺半群的 Delta 集的倍数，并推出最小 Delta 为 $\gcd(d,h-1)$。我们研究了这些幺半群的 Delta 集与链式次数之间的关系，得到了它们最大值之间的显式联系，并提出了若干关于链式次数集的猜想。我们还发现了一种通过 Delta 集生成金属斐波那契序列的有趣方法。这些结果为广义算术数值幺半群的分解不变量提供了结构描述，并提出了若干进一步研究的方向。

英文摘要

Let \[ S_h=\langle a,ah+d,ah+2d,\ldots,ah+kd\rangle, \qquad \gcd(a,d)=1,\quad 1<k<a,\quad h> 1, \] be a numerical monoid generated by a generalized arithmetic sequence. In this paper, we determine the delta set \[Δ(S_h),\] resolving an open problem posed by Christopher O'Neill and Roberto Pelayo in 2014 \cite{ONeill}. Writing $e=h-1$ and letting $E(e,d)$ denote the set of values arising from the subtractive Euclidean algorithm applied to $e$ and $d$, we prove that \[ Δ(S_h) = E(h-1,d)\cup \{M-r(h-1):M-r(h-1)>0,\ r\geq0\}, \] where \[ M=\left\lceil\frac{a}{k}\right\rceil(h-1)+d \] is the maximal element of $Δ(S_h)$. In particular, $d$ and $h-1$ always belong to $Δ(S_h)$. We further establish a reduction theorem showing that $Δ(S_h)$ is a multiple of the delta set of a primitive monoid satisfying $\gcd(d,h-1)=1$, and deduce that the minimum delta is $\gcd(d,h-1)$. We investigate the relationship between delta sets and catenary degrees for these monoids, obtaining an explicit connection between their maximal values and formulating several conjectures concerning the catenary degree set. We also find an interesting way to generate metallic fibonacci sequences through Delta Sets. These results provide a structural description of factorization invariants for generalized arithmetic numerical monoids and suggest several directions for further study.

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