2017/03/13 by Gotti, Felix, O'Neill, Christopher
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1703.04207
Let M be an atomic monoid and let x be a non-unit element of M. The elasticity of x, denoted by ρ(x), is the ratio of its largest factorization length to its shortest factorization length, and it measures how far is x from having a unique factorization. The elasticity ρ(M) of M is the supremum of the elasticities of all non-unit elements of M. The monoid M has accepted elasticity if ρ(M) = ρ(m) for some m ∈ M. In this paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of ℚ≥ 0). First, we characterize the Puiseux monoids M having finite elasticity and find a formula for ρ(M). Then we classify the Puiseux monoids having accepted elasticity in terms of their sets of atoms. When M is a primary Puiseux monoid, we describe the topology of the set of elasticities of M, including a characterization of when M is a bounded factorization monoid. Lastly, we give an example of a Puiseux monoid that is bifurcus (that is, every nonzero element has a factorization of length at most 2).