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Abstract
For a numerical monoid
⟨ n 1 , … , n k ⟩
minimally generated by
n 1 , … , n k
∈
ℕ ,
with
n 1
<
⋯
< n k ,
the longest and shortest factorization lengths of an element
x , denoted
as
L ( x ) and
ℓ ( x ) , respectively, follow
the identities
L ( x
+ n 1 )
=
L ( x )
+ 1 and
ℓ ( x
+ n k )
=
ℓ ( x )
+ 1 for sufficiently
large elements
x .
We characterize when these identities hold for all elements of numerical monoids of
embedding dimension three.
Keywords
numerical monoid, factorization length, Betti element
Mathematical Subject Classification
Primary: 11D75, 20M13, 20M14
Milestones
Received: 15 June 2022
Revised: 24 September 2022
Accepted: 10 October 2022
Published: 31 October 2023
Communicated by Scott T. Chapman
© 2023 MSP (Mathematical Sciences
Publishers).