Mean weak length

TL;DR

This paper introduces a weak length function for modules over rings and demonstrates its additivity, providing algebraic proofs for properties like algebraic entropy's additivity, previously shown through topological methods.

Contribution

It defines a weak length function and proves its additivity under certain conditions, offering new algebraic proofs for entropy and mean length properties.

Findings

Weak length function is additive for short exact sequences.

Provides algebraic proof of algebraic entropy's additivity.

Offers an alternative proof for mean length's additivity.

Abstract

We introduce a weak version of the classical length function, termed the weak length function, defined on subsets of $R$-modules over a unital ring $R$, and further consider the concept of mean weak length for $R\Gamma$-modules associated with an amenable group $\Gamma$. Under an appropriate upgrading condition together with certain mild assumptions, we establish that the mean weak length function is additive with respect to short exact sequences. This result has two consequences. First, we provide a purely algebraic proof of the additivity of algebraic entropy, which is a property originally established via topological entropy methods. Second, within our unified framework, we give an alternative and conceptual proof of the additivity of mean length, previously obtained by Li-Liang and Virilli using different approaches.