On length densities

For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x ∈ M {x\in M} , denoted LD ⁡ ( x ) {\operatorname{LD}(x)} , and the entire monoid M, denoted LD ⁡ ( M ) {\operatorname{LD}(M)} . This invariant is related to three widely studied invariants in the theory of nonunit factorizations, L ⁢ ( x ) {L(x)} , ℓ ⁢ ( x ) {\ell(x)} , and ρ ⁢ ( x ) {\rho(x)} . We consider some general properties of LD ⁡ ( x ) {\operatorname{LD}(x)} and LD ⁡ ( M ) {\operatorname{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 ) {\operatorname{LD}(M)} is rational and there is a nonunit element x ∈ M {x\in M} with LD ⁡ ( M ) = LD ⁡ ( x ) {\operatorname{LD}(M)=\operatorname{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)} , ℓ ⁢ ( x ) {\ell(x)} , and ρ ⁢ ( x ) {\rho(x)} (denoted L ¯ ⁢ ( x ) {\overline{L}(x)} , ℓ ¯ ⁢ ( x ) {\overline{\ell}(x)} , and ρ ¯ ⁢ ( x ) {\overline{\rho}(x)} ) always exist, we show the somewhat surprising result that LD ¯ ⁢ ( x ) = lim n → ∞ ⁡ LD ⁡ ( x n ) {\overline{\operatorname{LD}}(x)=\lim_{n\rightarrow\infty}\operatorname{LD}(x^% {n})} may not exist. We also give some finiteness conditions on M that force the existence of LD ¯ ⁢ ( x ) {\overline{\operatorname{LD}}(x)} .