Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras
Authors: Vincenzo Marra and Luca Spada
Abstract: We prove that the unification type of Lukasiewicz (infinite-valued propositional) logic and of it sequivalent algebraic semantics, the variety of MV-algebras, is nullary. The proof rests upon Ghilardi’s algebraic char- acterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra — a fundamental result that, albeit known to specialists, seems to appear in print here for the first time.
