In abstract interpretation, the notion of forward-completeness with respect to a logic guarantees that an abstract and a concrete structure satisfy exactly the same formulae. We apply completeness to study models of positive modal logic, which are logics that lack negation and implication. These models, which were discovered independently by re- searchers in abstract interpretation, modal logic and model checking, are Kripke structures with an order on states and their lattice-theoretic duals. We show that forward-complete abstractions are models of certain modal logics. The Kripke structures corresponding to these models satisfy a saturation condition ensuring that modal operators have the same properties as best abstract transformers. We synthesize a new notion of simulation for these models using the constructive characterization of complete abstractions.