Connection between monads and posets? I understand this is broad, but could any elucidate and/or direct me on which structures, areas, and objects to study to get a deep understanding of the relationship between monads and partially ordered sets, especially with regard to relations? Could you provide some formal definitions for these connections, expound upon closure operations over partially ordered sets in the context of monads?

I think you are referring to the fact that a closure operator is a monad on a poset. < Note that every poset can be viewed as a category with at most one arrow between each pair of objects. Your question would be better stated as "What is the connection between monads and closure operators?"