The polyhedral geometry of Wajsberg hoops

TL;DR

This paper establishes a duality between finitely presented Wajsberg hoops and rational polyhedra, providing geometric insights into their structure and logical properties, including decidability of admissible rules in the positive fragment of Lukasiewicz logic.

Contribution

It introduces a duality framework linking Wajsberg hoops to rational polyhedra and characterizes projective and exact hoops geometrically, advancing understanding of their logical and algebraic properties.

Findings

Duality between finitely presented Wajsberg hoops and rational polyhedra.

Decidability of admissible rules in the positive fragment of Lukasiewicz logic.

Nullary unification type and unitary exact unification type of Wajsberg hoops.

Abstract

We show that the category of finitely presented Wajsberg hoops with homomorphisms is dually equivalent to a particular subcategory of rational polyhedra with 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 positive fragment of Lukasiewicz logic. We show that, while deducibility in the fragment is equivalent to deducibility among positive formulas in Lukasiewicz logic, the same is not true for admissibility of rules. 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.