This page ( https://www.pls-lab.org/en/Egli-Milner_order ) on the Egli-Milner order is technically correct but it would be nice to see some of the interesting facts about it too. For instance, it's a preorder with a corresponding equivalence relation, and the largest subset equivalent to a given set is its convex hull (in the order-theoretic sense). The lower order has a similar property, except the largest subset equivalent to a given set is its downward closure; the upclosure is the corresponding set for the upper order. In fact the lower order is the pullback of the subset relation through the downclosure operator, which gives a straightforward proof that the lower order really is a preorder (similarly for the upper order). There are connections with modal logic too!

#math #ComputerScience #OrderTheory