Corecursive Algebras, Corecursive Monads and Bloom Monads

TL;DR

This paper introduces the concept of corecursive algebras and monads, showing how free corecursive algebras are constructed and characterizing Bloom algebras as their Eilenberg-Moore algebras, advancing the theory of iterative and corecursive structures.

Contribution

It establishes the structure of free corecursive algebras and monads, generalizing iterative monads and characterizing Bloom algebras, thus extending the theoretical framework of corecursion.

Findings

Free corecursive algebras are coproducts of terminal coalgebras and free algebras.

The free corecursive monad is characterized as the free corecursive monad.

Bloom algebras are the Eilenberg-Moore algebras for the free corecursive monad.

Abstract

An algebra is called corecursive if from every coalgebra a unique coalgebra-to-algebra homomorphism exists into it. We prove that free corecursive algebras are obtained as coproducts of the terminal coalgebra (considered as an algebra) and free algebras. The monad of free corecursive algebras is proved to be the free corecursive monad, where the concept of corecursive monad is a generalization of Elgot's iterative monads, analogous to corecursive algebras generalizing completely iterative algebras. We also characterize the Eilenberg-Moore algebras for the free corecursive monad and call them Bloom algebras.