On length densities

TL;DR

This paper introduces the concept of length density in commutative cancellative monoids, explores its properties through various examples, and investigates conditions under which certain limits exist or are rational.

Contribution

It defines length density for monoids, relates it to existing invariants, and studies its properties, including examples with irrational values and conditions for rationality and limit existence.

Findings

Length density relates to known factorization invariants.

Finitely generated monoids have rational length density.

The limit of length density for powers may not always exist.

Abstract

For a commutative cancellative monoid $M$, we introduce the notion of the length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three widely studied invariants in the theory of non-unit factorizations, $L(x)$, $\ell(x)$, and $\rho(x)$. We consider some general properties of $\mathrm{LD}(x)$ and $\mathrm{LD}(M)$ 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 $\mathrm{LD}(M)$ is rational and there is a nonunit element $x\in M$ with $\mathrm{LD}(M)=\mathrm{LD}(x)$ (such a monoid is said to have accepted length density). While it is well-known that the much studied asymptotic versions of $L(x)$, $\ell (x)$ and $\rho (x)$ (denoted $\overline{L}(x)$, $\overline{\ell}(x)$, and $\overline{\rho} (x)$) always exist, we show the somewhat surprising result that $\overline{\mathrm{LD}}(x) = \lim_{n\rightarrow \infty} \mathrm{LD}(x^n)$ may not exist. We also give some finiteness conditions on $M$ that force the existence of $\overline{\mathrm{LD}}(x)$.