Monadic aspects of the ideal lattice functor on the category of distributive lattices

TL;DR

This paper explores the monadic structure of the ideal lattice functor on distributive lattices, extending existing results on idempotent approximations and establishing new equivalences with coherent frames.

Contribution

It extends B. Jacobs' results on lax idempotent monads using Fakir construction and proves the equivalence between distributive lattices and coherent frames.

Findings

Successive iterations of the ideal functor do not produce new categories.

The Fakir construction provides a new proof of the equivalence between distributive lattices and coherent frames.

When the first inductive step is the identity monad, the category is equivalent to free algebras.

Abstract

It is known that the construction of the frame of ideals from a distributive lattice induces a monad whose algebras are precisely the frames and frame homomorphisms. Using the Fakir construction of an idempotent approximation of a monad, we extend B. Jacobs' results on lax idempotent monads and show that the sequence of monads and comonads generated by successive iterations of this ideal functor on its algebras and coalgebras do not strictly lead to a new category. We further extend this result and provide a new proof of the equivalence between distributive lattices and coherent frames by showing that when the first inductive step in the Fakir construction is the identity monad, then the ambient category is equivalent to the free algebras.