Nice stuff!

[Juan wrote](https://forum.azimuthproject.org/discussion/comment/16784/#Comment_16784):

> I don't know if the other two functions, `join'` and `duplicate'`, give any interesting properties to \\(f \circ g\\) and \\(g \circ f\\).

You seem to have proved all the interesting properties I know. Mathematicians would say that whenever we have a left adjoint functor \\(f\\) and a right adjoint functor \\(g\\), the composite \\(f \circ g\\) is a monad and the composite \\(g \circ f\\) is a comonad. We are considering the special case where the categories involved are preorders. In this case \\(f \circ g\\) is always an **idempotent monad**, meaning

$$ f(g(f(g(q))) \cong f(g(q)) $$

for all \\(q\\), and similarly \\(g \circ f\\) is always an **idempotent comonad**, meaning

$$ g(f(g(f(p))) \cong g(f(p)) $$

for all \\(p\\). In a preorder these isomorphisms become equations.

An idempotent monad on a preorder is called a **[closure operator](https://en.wikipedia.org/wiki/Closure_operator#Closure_operators_on_partially_ordered_sets)** because the classic example is taking the closure of a subset of a topological space. Similarly, an idempotent comonad on a preorder is called **co-closure operator** and the classic example is taking the interior of a subset of a topological space.

[Juan wrote](https://forum.azimuthproject.org/discussion/comment/16784/#Comment_16784):

> I don't know if the other two functions, `join'` and `duplicate'`, give any interesting properties to \\(f \circ g\\) and \\(g \circ f\\).

You seem to have proved all the interesting properties I know. Mathematicians would say that whenever we have a left adjoint functor \\(f\\) and a right adjoint functor \\(g\\), the composite \\(f \circ g\\) is a monad and the composite \\(g \circ f\\) is a comonad. We are considering the special case where the categories involved are preorders. In this case \\(f \circ g\\) is always an **idempotent monad**, meaning

$$ f(g(f(g(q))) \cong f(g(q)) $$

for all \\(q\\), and similarly \\(g \circ f\\) is always an **idempotent comonad**, meaning

$$ g(f(g(f(p))) \cong g(f(p)) $$

for all \\(p\\). In a preorder these isomorphisms become equations.

An idempotent monad on a preorder is called a **[closure operator](https://en.wikipedia.org/wiki/Closure_operator#Closure_operators_on_partially_ordered_sets)** because the classic example is taking the closure of a subset of a topological space. Similarly, an idempotent comonad on a preorder is called **co-closure operator** and the classic example is taking the interior of a subset of a topological space.