  Duality relations for simple exclusion processes with general open boundaries
are discussed. It is shown that a combination of spin operators and bosonic
operators enables us to have an unified discussion for the duality relations
with the open boundaries. As for the symmetric simple exclusion process (SSEP),
more general results than previous studies are obtained; it is clarified that
not only the absorbing sites, but also additional sites, called copying sites,
are needed for the boundaries in the dual process for the SSEP. The role of the
copying sites is to conserve information about the particle states on the
boundary sites. The similar discussions are applied to the asymmetric simple
exclusion process (ASEP), in which the q-analogues are employed, and it is
clarified that the ASEP with open boundaries has a complicated dual process on
the boundaries.
