Compactly-aligned discrete product systems, and generalizations of O-infinity

The universal C*-algebras of discrete product systems generalize the Toepli tz-Cuntz algebras and the Toeplitz algebras of discrete semigroups. We cons ider a semigroup P which is quasi-lattice ordered in the sense of Nica, and , for a product system p : E --> P, we study those representations of E, ca lled covariant, which respect the lattice structure of P. We identify a cla ss 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 m ain 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 am enable groups.