Abstract
Lindenbaum lemma and its strengthening known as Pair Ex- tension Theorem are among the crucial results in algebraic logic. They are however limited to logics with Hilbert style axiomatization with ni- tary rules only.
In this contribution we prove variants of these results for a wide class of innitary logics, we relate them to dierent forms of cut rule, and demonstrate their application for a variety of such logics studied in the literature.