A contravariant duality is constructed between the category of coalgebras
of a
given signature, and a category of Boolean algebras with operators,
including
modal operators corresponding to state transitions in coalgebras, and
distinguished elements abstracting the sets of states defined by observable
equations.
This duality is used to give a new proof that a class of coalgebras
is definable
by Boolean combinations of observable equations if it is closed under
disjoint
unions, domains and images of coalgebraic morphisms, and ultrafilter
enlargements. The proof reduces the problem to a direct application
of
Birkhoff's variety theorem characterising equational classes of algebras.