# Varieties of positive modal algebras and structural completeness

## TL;DR

This paper investigates the structure and properties of positive modal algebras, particularly positive S4-algebras, revealing their non-local finiteness, finite free algebras, and characterizing structurally complete varieties.

## Contribution

It establishes the non-local finiteness of positive S4-algebras, describes the lattice of their varieties, and characterizes structurally complete varieties of positive K4-algebras.

## Key findings

- Positive S4-algebras are not locally finite.
- The free one-generated positive S4-algebra is finite.
- Characterization of structurally complete varieties of positive K4-algebras.

## Abstract

Positive modal algebras are the positive-subreducts of modal algebras. We prove that the variety of positive S4-algebras is not locally finite. On the other hand, the free one-generated positive S4-algebra is shown to be finite. Moreover, we describe the bottom part of the lattice of varieties of positive S4-algebras. Building on this, we characterize (passively, hereditarily) structurally complete varieties of positive K4-algebras.