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Using the Fakir construction of an idempotent approximation of a monad, we extend B. Jacobs\u2019 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 category of free algebras.<\/jats:p>","DOI":"10.1007\/s10485-025-09811-5","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T14:03:54Z","timestamp":1752156234000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Monadic Aspects of the Ideal Lattice Functor on the Category of Distributive Lattices"],"prefix":"10.1007","volume":"33","author":[{"given":"Ando","family":"Razafindrakoto","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,10]]},"reference":[{"key":"9811_CR1","doi-asserted-by":"crossref","unstructured":"Banaschewski, B., Niefield, S.B.: Projective and supercoherent frames. 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