# On free completely iterative algebras

## TL;DR

This paper investigates the structure of free algebras for finitary set functors, showing how they can be extended to free completely iterative algebras with a canonical partial order, and providing methods to solve recursive equations.

## Contribution

It establishes a canonical partial order on free algebras and characterizes free completely iterative algebras as their conservative completions for bicontinuous functors.

## Key findings

- Free algebras carry a canonical partial order.
- For bicontinuous functors, the free completely iterative algebra is the conservative completion of the free algebra.
- Recursive equations have solutions as joins of omega-chains of approximate solutions.

## Abstract

For every finitary set functor F we demonstrate that free algebras carry a canonical partial order. In case F is bicontinuous, we prove that the cpo obtained as the conservative completion of the free algebra is the free completely iterative algebra. Moreover, the algebra structure of the latter is the unique continuous extension of the algebra structure of the free algebra. For general finitary functors the free algebra and the free completely iterative algebra are proved to be posets sharing the same conservative completion. And for every recursive equation e in the free completely iterative algebra we present an omega-chain of approximate solutions in the free algebra whose join is the solution of e.