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Total preorderA 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. 
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Total preorder".