Set Representation of Dynamic De Morgan algebras

TL;DR

This paper introduces tense operators in De Morgan algebras to model tense logic with negation satisfying double negation law, and explores frame representations for dynamic De Morgan algebras.

Contribution

It extends De Morgan algebras with tense operators and characterizes frame representations for dynamic De Morgan algebras.

Findings

Defined tense operators G and H in De Morgan algebras.

Provided a construction for frames corresponding to dynamic De Morgan algebras.

Solved the problem of representing tense operators via frames.

Abstract

By a De Morgan algebra is meant a bounded poset equipped with an antitone involution considered as negation. Such an algebra can be considered as an algebraic axiomatization of a propositional logic satisfying the double negation law. Our aim is to introduce the so-called tense operators in every De Morgan algebra for to get an algebraic counterpart of a tense logic with negation satisfying the double negation law which need not be Boolean. Following the standard construction of tense operators $G$ and $H$ by a frame we solve the following question: if a dynamic De Morgan algebra is given, how to find a frame such that its tense operators $G$ and $H$ can be reached by this construction.