This is a review of those aspects of the theory of varieties of
Boolean algebras
with operators (BAO's) that emphasise connections with modal logic
and
structural properties that are related to natural properties of logical
systems.
It begins with a survey of the duality that exists between BAO's and
relational
structures, focusing on the notions of bounded morphisms, inner substructures,
disjoint and
bounded unions, and canonical extensions of structures that originate
in the study of
validity-preserving operations on Kripke frames. This duality is then
applied to {\em
polymodal\/} propositional logics having finitary intensional connectives
that
generalise the Box and Diamond connectives of unary modal logic. Issues
discussed include validity in canonical structures, completeness and
incompleteness under
the relational semantics, and characterisations of logics by elementary
classes of
structures and by finite structures.
It turns out that a logic is {\em strongly\/} complete for the relational
semantics iff
the variety of algebras it defines is {\em complex\/}, which means
that every algebra in
the variety is embeddable into a full powerset algebra that is also
in the
variety. A hitherto unpublished formulation and proof of this is given
(Theorem 5.6.1) that
applies to {\em quasi\/}-varieties. This is followed by an algebraic
demonstration that
the temporal logic of Dedekind complete linear orderings defines a
complex variety,
adapting Gabbay's model-theoretic proof that this logic is strongly
complete.