2024/04/30 by Ando Razafindrakoto, Razafindrakoto, Ando · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Category Theory (math.CT) #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2404.19642
openalex publication_date 2024/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.