  In this manuscript we show that the metric mean dimension of a free semigroup
action satisfies three variational principles: (a) the first one is based on a
definition of Shapira's entropy, introduced in \cite{SH} for a singles dynamics
and extended for a semigroup action in this note; (b) the second one treats
about 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 free
semigroup action and show that the metric mean dimension satisfies a
variational principle in terms of such function. Our results are inspired in
the ones obtained by \cite{LT2019}, \cite{VV}, \cite{GS1} and \cite{RX}.
