Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE SETS OF LENGTHS OF PUISEUX MONOIDS GENERATED BY MULTIPLE GEOMETRIC SEQUENCES

  • Polo, Harold (Department of Mathematics University of Florida)
  • Received : 2020.01.18
  • Accepted : 2020.07.20
  • Published : 2020.10.31
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Abstract

In this paper, we study some of the factorization aspects of rational multicyclic monoids, that is, additive submonoids of the nonnegative rational numbers generated by multiple geometric sequences. In particular, we provide a complete description of the rational multicyclic monoids M that are hereditarily atomic (i.e., every submonoid of M is atomic). Additionally, we show that the sets of lengths of certain rational multicyclic monoids are finite unions of multidimensional arithmetic progressions, while their unions satisfy the Structure Theorem for Unions of Sets of Lengths. Finally, we realize arithmetic progressions as the sets of distances of some additive submonoids of the nonnegative rational numbers.

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Acknowledgement

The author wants to thank Felix Gotti for his guidance throughout the different stages of this manuscript and for many useful conversations about factorization theory. The author extends his thanks to anonymous referees whose feedback improve the final version of this paper. While working on this manuscript, the author was supported by the University of Florida Mathematics Department Fellowship.

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