  We study several notions of topological pressure and capacities for
multi-potentials $\Phi \in \mathcal C(X;\mathbb R)^m$, with respect to finitely
generated continuous semigroups $G$ on a compact metric space $X$. We introduce
the amalgamated pressure, the condensed pressure, the trajectory pressure, the
exhaustive pressure, and the respective capacities on non-compact sets $Y$, for
multi-potentials $\Phi$. This is done by using Carath\'eodory-Pesin structures.
Several properties of these types of pressure, and relations between them are
explored. The inverse limit of the semigroup and its relations to the above
pressures are studied. These notions can be used to classify semigroup actions.
We introduce a notion of measure-theoretic amalgamated entropy, and prove a
Partial Variational Principle for the amalgamated pressure. Local amalgamated
entropies and local exhaustive entropies are introduced for probability
measures on X, and we show they provide estimates for the amalgamated and the
exhaustive entropies of non-compact sets. Also we find a bound for the local
exhaustive entropy for marginal measures on $X$. We apply the amalgamated
pressure of the unstable multi-potential to estimate the dimensions of slices
in $G$-invariant saddle-type sets.
