Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and state-transition systems.

An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal ” sets of formulas that have natural syntactic closure properties. The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination ” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if if the deducibility relation is generated by countably many inference rules. A counter-example to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable. 1

Abstract: A class of Kripke models for intuitionistic propositional logic is ‘axiomatic’ if it is the class of all models of some set of formulas (axioms). This paper discusses various structural characterisations of axiomatic classes in terms of closure under certain constructions, including images of bisimulations, disjoint unions, ultrapowers and ‘prime extensions’. The prime extension of a model is a new model whose points are the prime filters of the lattice of upwardly-closed subsets of the original model. We also construct and analyse a ‘definable ’ extension whose points are prime filters of definable sets. A structural explanation is given of why a class that is closed under images of bisimulations and invariant under prime/definable extensions must be invariant under arbitrary ultrapowers. This uses iterated ultrapowers and saturation.