The Delta Set of a Numerical Monoid Generated by a Generalized Arithmetic Sequence

Abstract

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 $\Delta (S_h)$. In particular, $d$ and $h-1$ always belong to $\Delta (S_h)$. We further establish a reduction theorem showing that $\Delta (S_h)$ 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.

arXiv comment24 Pages