Abstract
This paper continues the investigation, started in [21], of infinitary propositional logics from the perspective of their algebraic completeness and lter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every nitary logic, (completely) intersectionprime theories form a basis of the closure system of all theories.
In this article we consider the open problem of whether these properties can be transferred to lattices of lters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of innitary logics that we separate with natural examples.
As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruiful new notion of natural expansion, and increase the understanding of semilinear logics.