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Bernd Losert (May 12 2023 at 11:51):Is there any research or results regarding F-algebras where F : Pos -> Pos (and Pos is the category of partially ordered sets)?
Morgan Rogers (he/him) (May 12 2023 at 18:11):For algebras for a monad or an endofunctor?
Bernd Losert (May 13 2023 at 08:51):Both, but I'll take what I can get.
Morgan Rogers (he/him) (May 13 2023 at 13:34):I'll answer for Eilenberg-Moore categories then.
In the first instance, there are plenty of general results for categories of algebras which come from studying [[monadic functors]]: because the category of posets is complete, this will be true for any category of algebras (by Proposition 3.1 here ).
Of course, monads on Set are far more intensely studied, but some of the inheritance results will still hold for Pos. For instance, it might be worth examining Vitale's characterization of categories of algebras for monads over Set and seeing which analogous properties of Pos can be lifted. On the other hand, beware that Pos lacks some of the regularity properties that Set has!
Evan Washington (May 13 2023 at 20:00):You may be interested in this paper https://arxiv.org/abs/2011.14796, which studies finitary monads on Pos. Just as equational algebraic theories present finitary monads on Set, "inequational" algebraic theories present finitary monads on Pos.
Bernd Losert (May 14 2023 at 10:44):Thanks for the pointers. I will investigate.