On the sets of lengths of Puiseux monoids generated by multiple geometric sequences
Commun. Korean Math. Soc. 2020 Vol. 35, No. 4, 1057-1073 https://doi.org/10.4134/CKMS.c200017 Published online September 7, 2020 Printed October 31, 2020
Harold Polo University of Florida
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.
Keywords : Puiseux monoids, factorization theory, factorization invariants, set of lengths, union of sets of lengths, set of distances, delta set
