We study some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study here consists of all atomic monoids of the form S r := r n | n ∈ N 0 , where r is a positive rational. As the atomic monoids S r are nicely generated, we are able to give detailed descriptions of many of their factorization invariants. One distinguishing characteristic of S r is that all its sets of lengths are arithmetic sequences of the same distance, namely |a − b|, where a, b ∈ N are such that r = a/b and gcd(a, b) = 1. We prove this, and then use it to study the elasticity and tameness of S r .

In this paper, we study the atomic structure of Puiseux monoids generated by monotone sequences. To understand this atomicity, it is often useful to know whether the monoid is bounded, in the sense that it has a bounded generating set. We provide necessary and sufficient conditions for atomicity and boundedness to be transferred from a monotone Puiseux monoid to all its submonoids. Finally, we present two special subfamilies of monotone Puiseux monoids and fully classify their atomic structure.

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If M is an atomic monoid and x is a nonzero non-unit element of M , then the set of lengths L(x) of x is the set of all possible lengths of factorizations of x, where the length of a factorization is the number of irreducible factors (counting repetitions). In a recent paper, F. Gotti and C. O'Neil studied the sets of elastici-Here we take this study a step further and explore the local k-elasticities of the same class of monoids. We find conditions under which Puiseux monoids have all their local elasticities finite as well as conditions under which they have infinite local k-elasticities for sufficiently large k. Finally, we focus our study of the k-elasticities on the class of primary Puiseux monoids, proving that they have finite local k-elasticities if either they are boundedly generated and do not have any stable atoms or if they do not contain 0 as a limit point.

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