  In this paper, we construct, for a certain class of semigroup dynamical
systems, two operator algebras that are universal with respect to their
corresponding covariance conditions: one being self-adjoint, and another being
non-self-adjoint. We prove that the $C^*$-envelope of the non-self-adjoint
operator algebra is precisely the self-adjoint one. This result leads to a
number of new examples of operator algebras and their $C^*$-envelopes, with
many from number fields and commutative rings. We further establish the
functoriality of these operator algebras along with their applications.
