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Total preorder

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 ab or ba.

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 ab for all a and b in X.