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

Part of: Semigroups

Published online by Cambridge University Press:  09 September 2014

SCOTT T. CHAPMAN ,
MARLY CORRALES ,
ANDREW MILLER ,
CHRIS MILLER  and
DHIR PATEL
SCOTT T. CHAPMAN*
Affiliation:
Sam Houston State University, Department of Mathematics, Box 2206, Huntsville, TX 77341, USA email scott.chapman@shsu.edu
MARLY CORRALES
Affiliation:
University of Southern California, Department of Mathematics, 3620 S. Vermont Ave., KAP 104, Los Angeles, CA 90089-2532, USA email marly.corrales@usc.edu
ANDREW MILLER
Affiliation:
Amherst College, Department of Mathematics, Box 2239 P.O. 5000, Amherst, MA 01002-5000, USA email admiller15@amherst.edu
CHRIS MILLER
Affiliation:
The University of Wisconsin at Madison, Department of Mathematics, 480 Lincoln Dr., Madison, WI 53706-1325, USA email crmiller2@wisc.edu
DHIR PATEL
Affiliation:
Rutgers University, Department of Mathematics, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA email dp553@eden.rutgers.edu
*
✉scott.chapman@shsu.edu
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Abstract

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Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ 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.

Keywords

catenary degreetame degreenumerical monoid

MSC classification

Secondary: 20M13: Arithmetic theory of monoids 20M14: Commutative semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 3 , December 2014 , pp. 289 - 300
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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