  Let $\sS$ be a countable, abelian semigroup of continuous surjections on a
compact metric space $X$. Corresponding to this dynamical system we associate
two operator algebras, the tensor algebra, and the semicrossed product. There
is a unique smallest C$^*$-algebra into which an operator algebra is completely
isometrically embedded, which is the C$^*$-envelope. We provide two distinct
characterizations of the C$^*$-envelope of the tensor algebra; one developed in
a general setting by Katsura, and the other using tools of projective and
inductive limits, which gives the C$^*$-envelope as a crossed product
C$^*$-algebra. We also study two natural classes of representations, the left
regular representations and the orbit representations. The first is Shilov, and
the second has a Shilov resolution.
