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We examine the semicrossed products of a semigroup action by$*$-endomorphisms on a C*-algebra, or more generally of an action on anarbitrary operator algebra by completely contractive endomorphisms. The choiceof allowable representations affects the corresponding universal algebra. Weseek quite general conditions which will allow us to show that the C*-envelopeof the semicrossed product is (a full corner of) a crossed product of anauxiliary C*-algebra by a group action.  Our analysis concerns a case-by-case dilation theory on covariant pairs. Inthe process we determine the C*-envelope for various semicrossed products of(possibly nonselfadjoint) operator algebras by spanning cones andlattice-ordered abelian semigroups.  In particular, we show that the C*-envelope of the semicrossed product ofC*-dynamical systems by doubly commuting representations of $\mathbb{Z}^n_+$(by generally non-injective endomorphisms) is the full corner of a C*-crossedproduct. In consequence we connect the ideal structure of C*-covers toproperties of the actions. In particular, when the system is classical, we showthat the C*-envelope is simple if and only if the action is injective andminimal.  The dilation methods that we use may be applied to non-abelian semigroups. Weidentify the C*-envelope for actions of the free semigroup $\mathbb{F}_+^n$ byautomorphisms in a concrete way, and for injective systems in a more abstractmanner. We also deal with C*-dynamical systems over Ore semigroups when theappropriate covariance relation is considered.

Semicrossed Products of Operator Algebras by Semigroups