  For every univariate formula $\chi$ we introduce a lattices of intermediate
theories: the lattice of $\chi$-logics. The key idea to define chi-logics is to
interpret atomic propositions as fixpoints of the formula $\chi^2$, which can
be characterised syntactically using Ruitenburg's theorem. We develop an
algebraic duality between the lattice of $\chi$-logics and a special class of
varieties of Heyting algebras. This approach allows us to build five distinct
lattices corresponding to the possible fixpoints of univariate formulas|among
which the lattice of negative variants of intermediate logics. We describe
these lattices in more detail.
