The Half-String Oscillator (HS) approach to String Field Theory is com
pleted. The ghost sector of the theory is formulated using the fermion
ic representation of the ghosts. The Fock space of the HS functionals
is constructed. Integration, multiplication, and the interaction verte
x are constructed explicitly. In the HS language integration and multi
plication are viewed as a trace and a matrix multiplication. The proof
that the formalism is both reparametrization and BRST invariant is gi
ven.