A VARIANT OF THE LINDENBAUM LEMMA FOR INFINITARY CLOSURE SYSTEMS

Abstract The Lindenbaum lemma saying that completely meet-irreducible closed sets form a basis of any finitary closure system is an easy-to-prove yet crucial result transcending algebraic logic. While the finitarity restriction is crucial for its usual proof, it is not necessary: there are indeed works proving it (or its variant for a larger class of finitely meet-irreducible closed sets) for non-finitary closure systems arising from particular infinitary logics (i.e., substitution-invariant consequence relations). There is also a general result proving it for a wide class of logics with strong p-disjunction and a countable Hilbert-style axiomatization. Identifying the essential properties of strong p-disjunctions we prove a variant of the Lindenbaum lemma for closure systems which are 1) defined over countable sets, 2) countably axiomatized, and 3) frames (in the order-theoretic sense) but not necessarily substitution-invariant.