  The universal C*-algebras of discrete product systems generalize the
Toeplitz- Cuntz algebras and the Toeplitz algebras of discrete semigroups. We
consider a semigroup P which is quasi-lattice ordered in the sense of Nica,
and, for a product system p:E\to P, we study those representations of E, called
covariant, which respect the lattice structure of P. We identify a class of
product systems, which we call compactly aligned, for which there is a purely
C*-algebraic characterization of covariance, and study the algebra
C*_{cov}(P,E) which is universal for covariant representations of E. Our main
theorem is a characterization of the faithful representations of C*_{cov}(P,E)
when P is the positive cone of a free product of totally-ordered amenable
groups.
