Enlargements of Polynomial Coalgebras
We continue a programme of study of
the model theory of coalgebras of
polynomial functors on the category of sets. Each such coalgebra $\a$ is
shown to have an ``enlargement'' to a new coalgebra $E\a$ whose states
are
certain ultrafilters on the state-set of $\a$.
This construction is used to give a new characterization, in terms of
structural closure properties, of classes of coalgebras that are
defined by
``observable'' formulas, these being Boolean combinations of equations
between terms that take observable values.
It is shown that the $E$-construction can be replaced by a modification
that
restricts to ultrafilters whose members are definable in $\a$. Both
constructions are examined from the category-theoretic perspective, and
shown
to generate monads on the category of coalgebras concerned.