On the local k-elasticities of Puiseux monoids

Abstract

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.

Factorization invariants of Puiseux monoids generated by geometric sequences

“…[4,7,15,43]. In particular, the unions of sets of lengths and the local elasticities of Puiseux monoids have been considered in [34]. By [20,Section 1.4], the elasticity of an atomic monoid can be expressed in terms of its local elasticities as follows…”

Approximating length-based invariants in atomic Puiseux monoids

“…Example 3.2. In [18] the author proved that there exists a Puiseux monoid without 0 as a limit point that has no finite local elasticities. With this purpose, she pieces together a Puiseux monoid M by creating a strictly increasing sequence of finite subsets of positive rationals (A i ) i≥1 satisfying the following three conditions:…”

“…Since A(N i ) ⊆ A(M) for each i ∈ N, it follows that ρ 2 (M) = ∞. For details see [18,Proposition 3.6].…”

“…[18, Proposition 3.1] Let M be an atomic Puiseux monoid. If M contains a stable atom a ∈ A(M) then ρ k (M) is infinite for all sufficiently large k.Proof.…”

For Which Puiseux Monoids Are Their Monoid Rings Over Fields Ap?

“…Note that the additive submonoids of Q + have received a lot of attention by the researchers in the areas of commutative semigroup theory and factorization theory during the last few years (see, for example, [4,5,9,12,22,23,25,26,32]) and they even got a special name, Puiseux monoids, so we will be using that name from now on. Of course, Puiseux monoids were also used before; the paper [27] is one of the well-known instances.…”