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.

中文翻译：

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适合半群动作的代数熵

我们通过自同态在离散阿贝尔群 $A$ 上为可取消右服从半群 $S$ 的作用引入代数熵的两个概念；这些扩展了阿贝尔群的自同态的经典代数熵，对应于 $S=\mathbb N$ 的情况。我们研究了代数熵的基本​​性质并在几个例子中计算它，特别注意 S 是一个服从群的情况。对于取消右服从幺半群对扭转阿贝尔群的作用，我们证明了所谓的加法定理。在同一设置中，我们看到桥接定理通过庞特里亚金对偶性将代数熵与对偶作用的拓扑熵连接起来，