THE CATENARY AND TAME DEGREES ON A NUMERICAL MONOID ARE EVENTUALLY PERIODIC

Abstract

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ be a commutative cancellative monoid. For $m$ a nonunit in $M$, the catenary degree of $m$, denoted $c(m)$, and the tame degree of $m$, denoted $t(m)$, are combinatorial constants that describe the relationships between differing irreducible factorizations of $m$. These constants have been studied carefully in the literature for various kinds of monoids, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid $S$ that the sequences $\{c(s)\}_{s\in S}$ and $\{t(s)\}_{s\in S}$ are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence $\{\Delta (s)\}_{s\in S}$ of delta sets of $S$ also satisfies a similar periodicity condition.