Order Theory

Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that".
Order theory focus on transitive binary relations, we start our definitions from a basic one -- PreOrder.
A preorder is a thin category -- hom_set(A, B).size <= 1.
From PreOrder, we can add antisymmetric to get Order:
Note that, there will be no cycle in a partial order, and we can implement topological sort for a partial order.
TODO We can implement sort algorithms for TotalOrder:
  • quick sort
  • bubble sort
  • merge sort
From PreOrder again, we can get equivalence relation by adding symmetric.