In the mathematical area of order theory, a total preorder over a set X is a preorder ≤ over X that is total; that is, for all a and b in X, it holds that a ≤ b or b ≤ a.
The difference between a total preorder and a total order is that a total preorder (since it is only a preorder) is not required to be antisymmetric.
An example is (X, ≤) where X = {1, 2} and a ≤ b for all a and b in X.