Partial orders and complementation

Given a bounded lattice we can form a symmetric binary relation of that relates elements to one another if they are complements of one another. Here are some examples of complementation relations produced from bounded lattices:

(= (complementation-relation (weak-order [#{0} #{1} #{2}]))
   #relation{
     :vertices #{0 1 2}
     :edges #{[0 2] [2 0]}})

(= (complementation-relation (weak-order [#{0} #{1 2} #{3}]))
   #relation{
     :vertices #{0 1 2 3}
     :edges #{[0 3] [1 2] [2 1] [3 0]}})

We know that complemented lattices are those bounded lattices that have functional complementation relations and uniquely complemented lattices are those bounded lattices that have complementation relations that are involutions.

(= (complemented-lattice? order)
   (unary-operation? (complementation-relation order)))
(= (uniquely-complemented-lattice? order)
   (involution? (complementation-relation order)))

It is worth noting that the complementation relation itself can be partially ordered by the complementary pairs ordering function to get a partially ordered relation.