  We investigate the structure of the generalized Weierstrass semigroups at
several points on a curve defined over a finite field. We present a description
of these semigroups that enables us to deduce properties concerned with the
arithmetical structure of divisors supported on the specified points and their
corresponding Riemann-Roch spaces. This characterization allows us to show that
the Poincar\'e series associated with generalized Weierstrass semigroups carry
essential information to describe entirely their respective semigroups.
