# Moss' logic for ordered coalgebras

## TL;DR

This paper introduces a finitary coalgebraic logic for ordered coalgebras using a single modality, establishing its semantics, properties, and a complete proof system, extending Moss's logic to a finitary setting.

## Contribution

It develops a finitary version of Moss' coalgebraic logic for ordered coalgebras, including semantics, a proof system, and the Hennessy-Milner property, under specific functor preservation conditions.

## Key findings

- The logic has the Hennessy-Milner property for similarity.
- A complete sequent proof system is constructed.
- Semantics are given via relation lifting with functor preservation assumptions.

## Abstract

We present a finitary version of Moss' coalgebraic logic for $T$-coalgebras, where $T$ is a locally monotone endofunctor of the category of posets and monotone maps. The logic uses a single cover modality whose arity is given by the least finitary subfunctor of the dual of the coalgebra functor $T_\omega^\partial$, and the semantics of the modality is given by relation lifting. For the semantics to work, $T$ is required to preserve exact squares. For the finitary setting to work, $T_\omega^\partial$ is required to preserve finite intersections. We develop a notion of a base for subobjects of $T_\omega X$. This in particular allows us to talk about the finite poset of subformulas for a given formula. The notion of a base is introduced generally for a category equipped with a suitable factorisation system. We prove that the resulting logic has the Hennessy-Milner property for the notion of similarity based on the notion of relation lifting. We define a sequent proof system for the logic, and prove its completeness.