Modal Operators for Coequations

We present the dual to Birkhoff's variety theorem in terms of predicates over the carrier of a cofree coalgebra. 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 that they commute, and we prove the invariance theorem, which is the formal dual of the completeness theorem.