Mean weak length

Abstract

We introduce a weak version of the classical length function, termed the weak length function, defined on subsets of $R$-modules relative to the ambient $R$-modules over a unital ring $R$. We further consider the concept of mean weak length for each $R\Gamma$-module relative to a bigger $R\Gamma$-module associated with an amenable group $\Gamma$. Under an appropriate upgrading condition together with certain mild assumptions, we establish that the mean weak length satisfies an addition formula with respect to short exact sequences. This result has three applications. 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 Virili using different approaches. Third, we recover a particular case of amenable extension for Sylvester rank functions in the spirit of Jiang-Li.

We modify the definition of weak length in a more reasonable way, and the main results are adjusted accordingly