Modal Operators for Coequations

We present the dual to Birkhoff’s variety theorem in terms of predicates over the carrier of a cofree coalgebra (i.e., in terms of “coequations”). We then discuss the dual to Birkhoff’s completeness theorem, showing how closure under deductive rules dualizes to yield two modal operators acting on coequations. We discuss the properties of these operators and show, given a familiar condition on the endofunctor, the two operators commute, and we prove as main result the invariance theorem, which is the formal dual of Birkhoff’s completeness theorem.