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REALISABLE SETS OF CATENARY DEGREES OF NUMERICAL MONOIDS

Part of: Semigroups

Published online by Cambridge University Press:  04 December 2017

CHRISTOPHER O’NEILL*
Affiliation:
Mathematics Department, University of California, Davis, One Shields Ave, Davis, CA 95616, USA email [email protected]
ROBERTO PELAYO
Affiliation:
Mathematics Department, University of Hawai‘i at Hilo, Hilo, HI 96720, USA email [email protected]
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Abstract

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The catenary degree is an invariant that measures the distance between factorisations of elements within an atomic monoid. In this paper, we classify which finite subsets of $\mathbb{Z}_{\geq 0}$ occur as the set of catenary degrees of a numerical monoid (that is, a co-finite, additive submonoid of $\mathbb{Z}_{\geq 0}$). In particular, we show that, with one exception, every finite subset of $\mathbb{Z}_{\geq 0}$ that can possibly occur as the set of catenary degrees of some atomic monoid is actually achieved by a numerical monoid.

Type
Research Article
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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