Abstract
We prove that each proper ideal in the lattice of axiomatic, resp. standard strengthenings of the intuitionistic propositional logic is of cardinality 2ℵ0. But, each proper ideal in the lattice of structural strengthenings of the intuitionistic propositional logic is of cardinality 22ℵ0. As a corollary we have that each of these three lattices has no atoms.
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