# Algebraic entropy for amenable semigroup actions

## TL;DR

This paper extends the concept of algebraic entropy to actions of cancellative right amenable semigroups on abelian groups, establishing fundamental properties, computing examples, and linking it to topological entropy via duality.

## Contribution

It introduces two new notions of algebraic entropy for semigroup actions, proves the Addition Theorem, and connects algebraic and topological entropy through Pontryagin duality.

## Key findings

- Defined algebraic entropy for semigroup actions
- Proved the Addition Theorem for torsion abelian groups
- Established the Bridge Theorem linking algebraic and topological entropy

## Abstract

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case $S=\mathbb N$. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of the Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.