  A generalized Chan-Paton construction is presented which is analogous to the
tensor product of vector bundles. To this end open string theories are
considered where the space of states decomposes into sectors whose product is
described by a semigroup. The cyclicity properties of the open string theory
are used to prove that the relevant semigroups are direct unions of Brandt
semigroups. The known classification of Brandt semigroups then implies that all
such theories have the structure of a theory with Dirichlet-branes. We also
describe the structure of an arbitrary orientifold group, and show that the
truncation to the invariant subspace defines a consistent open string theory.
Finally, we analyze the possible orientifold projections of a theory with
several kinds of branes.
