  In this paper, we introduce the notions of topological pressure and
measure-theoretic entropy for a free semigroup action. Suppose that a free
semigroup acts on a compact metric space by continuous self-maps. To this
action, we assign a skew-product transformation whose fiber topological
pressure is taken to be the topological pressure of the initial action. Some
properties of these two notions are given, and then we give two main results.
One is the relationship between the topological pressure of the skew-product
transformation and the topological pressure of the free semigroup action, the
other is the partial variational principle about the topological pressure.
Moreover, we apply this partial variational principle to study the
measure-theoretic entropy and the topological entropy of finite affine
transformations on a metrizable group.
