Monadic Aspects of the Ideal Lattice Functor on the Category of Distributive Lattices
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Date
2025
Authors
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Journal ISSN
Volume Title
Publisher
Springer Science and Business Media B.V.
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 category of free algebras
Description
Keywords
Monad, Algebras, Distributive lattices, Frames, Continuous lattices
Citation
Razafindrakoto, A. (2025) Monadic Aspects of the Ideal Lattice Functor on the Category of Distributive Lattices. Applied categorical structures. [Online] 33 (4), .