  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.
