Superamalgamation for modal lattices via non-distributive dualities

TL;DR

It is shown that the variety of modal lattices has the superamalgamation property, and obtained that the weak positive modal logic has the Craig interpolation property, and extended to a number of other weak positive modal logics axiomatized by modal axioms corresponding to universal Horn sentences.

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.

Figures (7)

• Figure 1: Amalgamation of lattices
• Figure 2: Craig interpolation and superamalgamation on free algebras
• Figure 3: An epimorphism which is not surjective
• Figure 4: The second L-morphism condition
• Figure 5: Summary of dualities, dual adjunctions and inclusions

Theorems & Definitions (71)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8
• Proposition 2.9
• Theorem 2.10: Algebraic Completeness Theorem