Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A WEAKER NOTION OF THE FINITE FACTORIZATION PROPERTY

  • Received : 2023.07.19
  • Accepted : 2024.02.26
  • Published : 2024.04.30
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Abstract

An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years.

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Acknowledgement

We are grateful to our mentors, Prof. Jim Coykendall and Dr. Felix Gotti, for proposing this project and for their guidance during the preparation of this project. While working on this paper, we were part of PRIMES-USA, a year-long math research program hosted by MIT. We would like to express our collective gratitude to the MIT PRIMES program for arranging such an engaging research experience.

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