Communications of the
Korean Mathematical Society CKMS

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

MSC numbers: Primary 20M13; Secondary 06F05, 20M14

Supported by: While work-ing on this manuscript, the author was supported by the University of Florida Mathematics Department Fellowship.