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 Følner sequence (Fn)n∈N such that Fn is a monotile of Fn+1 for every n∈N). We study in details the class of locally monotileable groups, also in relation with already existing notions of monotileability for groups, introduced by B. Weiss and developed further by other authors recently.