Correspondence Theory for Generalized Modal Algebras

TL;DR

This paper develops a new correspondence theory for generalized modal algebras and spaces, introducing a novel approach to valuations that extends beyond traditional clopen valuations, with implications for duality theories.

Contribution

It presents a systematic study of correspondence theory for generalized modal algebras, notably extending the topological Ackermann lemma to valuations in DK(X).

Findings

Introduces valuations in DK(X) that are closed but not necessarily open.

Extends the topological Ackermann lemma to this new valuation setting.

Differentiates from existing Stone/Priestley duality frameworks.

Abstract

In the present paper, we give a systematic study of the correspondence theory of generalized modal algebras and generalized modal spaces. The special feature of the present paper is that in the proof of the (right-handed) topological Ackermann lemma, the admissible valuations are not the clopen valuations anymore, but values in the set DK(X) which are only closed and satisfy additional properties, not necessarily open. This situation is significantly different from existing settings using Stone/Priestley-like dualities, where all admissible valuations are clopen valuations.