Generalized Weierstrass Semigroups and their Poincaré Series

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.