2020/08/15 by Chapman, Scott T., O'Neill, Christopher, Ponomarenko, Vadim
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2008.06725
For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x∈ M, denoted LD(x), and the entire monoid M, denoted LD(M). This invariant is related to three widely studied invariants in the theory of non-unit factorizations, L(x), ℓ(x), and ρ(x). We consider some general properties of LD(x) and LD(M) and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD(M) is rational and there is a nonunit element x∈ M with LD(M)=LD(x) (such a monoid is said to have accepted length density). While it is well-known that the much studied asymptotic versions of L(x), ℓ (x) and ρ(x) (denoted L(x), ℓ(x), and ρ (x)) always exist, we show the somewhat surprising result that LD(x) = limn→ ∞ LD(xn) may not exist. We also give some finiteness conditions on M that force the existence of LD(x).