Superamalgamation for modal lattices via non-distributive dualities

TL;DR

This paper proves that the variety of modal lattices possesses the superamalgamation property, leading to the Craig interpolation property for certain modal logics, using duality theory based on modal L-spaces.

Contribution

It establishes superamalgamation for modal lattices and extends the result to various weak positive modal logics using duality methods.

Findings

Modal lattices have the superamalgamation property.

Weak positive modal logic has the Craig interpolation property.

Duality for modal lattices via modal L-spaces is employed.

Abstract

We show that the variety of modal lattices has the superamalgamation property. As a consequence, we obtain that the weak positive modal logic has the Craig interpolation property. Our proof employs the recent duality for modal lattices based on modal L-spaces. Moreover, we extend this result to a number of other weak positive modal logics axiomatized by modal axioms corresponding to universal Horn sentences.