The addition theorem for locally monotileable monoid actions

We prove an instance of the so-called Addition Theorem for the algebraic entropy of actions of cancellative right amenable monoids $S$ on discrete abelian groups $A$ by endomorphisms, under the hypothesis that $S$ is locally monotileable (that is, $S$ admits a right F\o lner sequence $(F_n)_{n\in\mathbb N}$ such that $F_n$ is a monotile of $F_{n+1}$ for every $n\in\mathbb N$). We study in details the class of locally monotileable groups, also in relation with already existing notions of monotileability for groups, introduced by Weiss and developed further by other authors recently.