The lattice of strengthenings of a strongly finite consequence operation

First, we prove that the lattice of all structural strengthenings of a given strongly finite consequence operation is both atomic and coatomic, it has finitely many atoms and coatoms, each coatom is strongly finite but atoms are not of this kind — we settle this by constructing a suitable counterexample. Second, we deal with the notions of hereditary: algebraicness, strong finitisticity and finite approximability of a strongly finite consequence operation. Third, we formulate some conditions which tell us when the lattice of all structural strengthenings of a given strongly finite consequence operation is finite, and subsequently we give some applications of them.This paper was read at the Third Autumn School on Strongly Finite Sentential Calculi organized by the Section of Logic, Polish Academy of Sciences, Institute of Philosophy and Sociology, in Ustronie (Poland), November 1979.