  We introduce hybrid algebras as algebraic semantics for hybrid languages with
nominals and, possibly, the satisfaction operator. We establish a duality
between hybrid algebras and the descriptive two-sorted general frames of Ten
Cate. We show that all axiomatic extensions of the basic hybrid logics, with or
without the satisfaction operator, are complete with respect to their classes
of hybrid algebras. Moreover, we show that by adding the usual non-orthodox
rules to these logics, they become complete with respect to their classes of
permeated hybrid algebras, corresponding to strongly descriptive two-sorted
general frames.
