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In this manuscript we show that the metric mean dimension of a free semigroupaction satisfies three variational principles: (a) the first one is based on adefinition of Shapira's entropy, introduced in \cite{SH} for a singles dynamicsand extended for a semigroup action in this note; (b) the second one treatsabout a definition of Katok's entropy for a free semigroup action introduced in\cite{CRV-IV}; (c) lastly we consider the local entropy function for a freesemigroup action and show that the metric mean dimension satisfies avariational principle in terms of such function. Our results are inspired inthe ones obtained by \cite{LT2019}, \cite{VV}, \cite{GS1} and \cite{RX}.

Some variational principles for the metric mean dimension of a semigroup action