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The one-variable fragment of any first-order logic may be considered as amodal logic, where the universal and existential quantifiers are replaced by abox and diamond modality, respectively. In several cases, axiomatizations ofalgebraic semantics for these logics have been obtained: most notably, for themodal counterparts S5 and MIPC of the one-variable fragments of first-orderclassical logic and intuitionistic logic, respectively. Outside the setting offirst-order intermediate logics, however, a general approach is lacking. Thispaper provides the basis for such an approach in the setting of first-orderlattice-valued logics, where formulas are interpreted in algebraic structureswith a lattice reduct. In particular, axiomatizations are obtained for modalcounterparts of one-variable fragments of a broad family of these logics bygeneralizing a functional representation theorem of Bezhanishvili and Hardingfor monadic Heyting algebras. An alternative proof-theoretic proof is alsoprovided for one-variable fragments of first-order substructural logics thathave a cut-free sequent calculus and admit a certain bounded interpolationproperty.

Algebraic semantics for one-variable lattice-valued logics