Dense completeness theorem for protoalgebraic logics

The paper [3] started a new approach to Abstract Algebraic Logic in which, instead of the usual equivalence-based classification of logical systems leading to the well-known Leibniz hierarchy of protoalgebraic logics (see [5]), we proposed an alternative setting based on implication connectives. By studying the properties of weak p-implications (understood as generalized connectives defined by sets of formulae ⇒ in two variables and, possibly, with parameters), we obtained a refinement of the Leibniz hierarchy. Moreover, it was shown that they define an order relation in the semantical counterpart of these logics, i.e. in their reduced matrix models. This yielded a natural definition of semilinear weak pimplications as those that endow the logic with a complete semantics of linearly ordered matrix models. In particular examples studied in the literature of manyvalued logics such completeness is often refined to particular kinds of linearly ordered models, typically those where the order is dense (see e.g. [2]). The aim of this contribution is to provide a general characterization of the logics that admit such completeness theorem with respect to their densely ordered linear matrix models. Let L be a logic with a weak p-implication⇒ (i.e. an arbitrary protoalgebraic logic) in a propositional language L. Assume that the cardinality of both the set of variables and the set of all formulae is the same and denote it as κ. Let A be an L-algebra. A filter F ∈ FiL(A) is ⇒-dense if for every a, b ∈ A: