The polyhedral geometry of Wajsberg hoops

Abstract

AbstractWe show that the algebraic category of finitely presented Wajsberg hoops is equivalent to a non-full subcategory of finitely presented MV-algebras. We use this connection to show how methods and techniques developed to study MV-algebras can be adapted to study Wajsberg hoops, as well. In particular, we show that finitely presented Wajsberg hoops are dually equivalent to a subcategory of rational polyhedra with $\mathbb {Z}$-maps. We use the duality to provide a geometrical characterization of finitely generated projective and exact Wajsberg hoops. As applications, we study logical properties of the $0$-free fragment of Łukasiewicz logic, seen as a substructural logic. We show that, while deducibility in the fragment is equivalent to deducibility among 0-free formulas in Łukasiewicz logic, the same is not true for the admissibility of rules: there are rules written in the $0$-free language that are admissible in Wajsberg hoops but not in MV-algebras. Moreover, we show that the unification type of Wajsberg hoops is nullary, while the exact unification type is unitary, therefore showing decidability of admissible rules in the fragment.