Anindya - you gave a very nice answer to Puzzle 46 in #31 and #32. Congratulations!
There's another sort of answer where we look at a poset \\(A\\) in which every \\(S\\)-tuple of elements has a join (for some fixed set \\(S\\).) The join then defines a function
$$ \bigvee: A^S \to A $$
This is a monotone function that's the left adjoint of the "generalized diagonal"
$$ \Delta : A \to A^S $$