  To each discrete product system E of finite-dimensional Hilbert spaces we
associate a C*-algebra O_E. When E is the n-dimensional product system over N,
O_E is the Cuntz algebra O_n, and the irrational rotation algebras appear as
O_E for certain one-dimensional product systems over N^2. We give conditions
which ensure that O_E is simple, purely infinite, and nuclear. Our main
examples are the lexicographic product systems, for which we obtain slightly
stronger results.
