The modal theory of linear orders

TL;DR

This paper explores the modal theory of linear orders, establishing modality elimination, definability of scatteredness, and computing modal validities across different order embeddings.

Contribution

It introduces modality elimination results, shows scatteredness is modally definable via condensations, and calculates exact modal validities for key cases.

Findings

Proves modality elimination for embeddings and monotone maps.

Shows scatteredness is modally definable through condensations.

Computes exact propositional modal validities in main cases.

Abstract

I study the modal theory of linear orders under embeddings, monotone maps, condensations, and end-extensions. I prove modality elimination for embeddings and monotone maps, show that condensations make scatteredness modally definable, and compute exact propositional modal validities in the main cases.