What is the coalgebraic analogue of Birkhoff's variety theorem?

Abstract

Logical definability is investigated for certain classes of coalgebras related to state-transition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special “observationally rich” filters on the state set of A. The ultrafilter enlargement is the subcoalgebra A ∗ of A + whose states are ultrafilters. Boolean combinations of equations between terms of observable (or output) type are identified as a natural class of formulas for specifying properties of coalgebras. These observable formulas are permitted to have a single-state variable, and form a language in which modalities describing the effects of state transitions are implicitly present. A ∗ and A + validate the same observable formulas. It is shown that a class of coalgebras is definable by observable formulas iff the class is closed under disjoint unions, images of bisimulations, and (ultra)filter enlargements. (Closure under images of bisimulations is equivalent to closure under images and domains of coalgebraic morphisms.) Moreover, every set of observable formulas has the same models as some set of conditional equations. Examples are constructed to show that the use of enlargements is essential in these characterisations, and that there are classes of coalgebras definable by conditional observable equations, but not by equations alone. The main conclusion of the paper is that to structurally characterise classes of coalgebras that are logically definable by modal languages requires a new construction, of “Stone space” type, in addition to the coalgebraic duals of the three constructions (homomorphisms, subalgebras, direct products) that occur in Birkhoff's original variety theorem for algebras.