  The one-variable fragment of any first-order logic may be considered as a
modal logic, where the universal and existential quantifiers are replaced by a
box and diamond modality, respectively. In several cases, axiomatizations of
algebraic semantics for these logics have been obtained: most notably, for the
modal counterparts S5 and MIPC of the one-variable fragments of first-order
classical logic and intuitionistic logic, respectively. Outside the setting of
first-order intermediate logics, however, a general approach is lacking. This
paper provides the basis for such an approach in the setting of first-order
lattice-valued logics, where formulas are interpreted in algebraic structures
with a lattice reduct. In particular, axiomatizations are obtained for modal
counterparts of one-variable fragments of a broad family of these logics by
generalizing a functional representation theorem of Bezhanishvili and Harding
for monadic Heyting algebras. An alternative proof-theoretic proof is also
provided for one-variable fragments of first-order substructural logics that
have a cut-free sequent calculus and admit a certain bounded interpolation
property.
