Questions of Canonicity
We review progress on the long standing question of whether
every \textit{canonical} modal logic must be characterized by an
\textit{elementary} class of Kripke frames, or equivalently, whether
every
canonical variety of Boolean algebras with operators must be
generating by the complex algebras of an elementary class of relational
structures. We verify that this does hold for certain families
$\RA_n$,
$\S\nr_\b\CA_\a$ and
$\S\Ra\CA_\a$ of varieties related to relation algebras and
cylindric
algebras. Canonical extensions of structures are shown to be free
objects
in certain categories of structures with topology, and to be associated
with a monad on the category of sets that generalizes Manes' Theorem to
relational structures.