Metric mean dimension of free semigroup actions for non-compact sets

In this paper, we introduce the notions of upper metric mean dimension, $u$-upper metric mean dimension, $l$-upper metric mean dimension of free semigroup actions for non-compact sets via Carath\'{e}odory-Pesin structure. Firstly, the lower and upper estimations of the upper metric mean dimension of free semigroup actions are obtained by local metric mean dimensions. Secondly, one proves a variational principle that relates the $u$-upper metric mean dimension of free semigroup actions for non-compact sets with the corresponding skew product transformation. Furthermore, using the variational principle above, $\varphi$-irregular set acting on free semigroup actions shows full upper metric mean dimension in the system with the gluing orbit property. Our analysis generalizes the results obtained by Carvalho et al. \cite{MR4348410}, Lima and Varandas \cite{MR4308163}.