Projective relative unification through duality

Abstract Unification problems can be formulated and investigated in an algebraic setting, by identifying substitutions to modal algebra homomorphisms. This opens the door to applications of the notorious duality between Heyting or modal algebras and descriptive frames. Through substantial use of this correspondence, we give a necessary and sufficient condition for formulas to be projective. A close inspection of this characterization will motivate a generalization of standard unification, which we dub relative unification. Applying this result to a number of different logics, we then obtain new proofs of their projective—or non-projective—character. Aside from reproving known results, we show that the projective extensions of $\textbf{K5}$ are exactly the extensions of $\textbf{K45}$. This resolves the open question of whether $\textbf{K5}$ is projective.